Concepts

The canonical list of every concept the curriculum names. Each maps to at most one unit. Browse 307 concepts; 306 have shipped units.

• spin-geometry.clifford.clifford-algebraopen unit 03.09.02 →

  Clifford algebra

  Graded algebra structure; relation to exterior algebra and Grassmann algebra; real vs complex Clifford algebras.

  requires: linalg.bilinear-form, linalg.vector-space, algebra.tensor-algebra, algebra.quotient-algebra

• spin-geometry.clifford-chessboardopen unit 03.09.11 →

  Clifford algebra classification — the 8×8 chessboard

  ABS classification of ${\mathrm{Cl}}_{p,q}$ as matrix algebras over $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$. Bridging identity ${\mathrm{Cl}}_{r+1,s+1}\cong {\mathrm{Cl}}_{r,s}\otimes {\mathrm{Cl}}_{1,1}$. Real eight-fold periodicity ${\mathrm{Cl}}_{p,q+8}\cong {\mathrm{Cl}}_{p,q}\otimes M_{16}(\mathbb{R})$; complex two-fold periodicity. Module quotient ${\hat{\mathfrak{M}}}_n\cong K{O}^{-n}(\mathrm{pt})$.

  requires: spin-geometry.clifford.clifford-algebra, algebra.tensor-algebra, algebra.quotient-algebra

• spin-geometry.kr-theoryopen unit 03.09.12 →

  KR-theory and the $(1,1)$-periodicity theorem

  Atiyah's bigraded $K{R}^{p,q}$ unifying $K$, $KO$, and $KSp$. The $(1,1)$-periodicity $K{R}^{p,q+1}\cong K{R}^{p+1,q}$ is the $K$-theoretic incarnation of the Clifford bridging identity. Bott periodicity in eight steps via the volume element of ${\mathrm{Cl}}_{0,8}$.

  requires: spin-geometry.clifford-chessboard, k-theory.bott-periodicity, topology.classifying-space

• spin-geometry.trialityopen unit 03.09.13 →

  Triality on $\mathrm{Spin}(8)$ and exceptional Lie group constructions

  Triality as the outer automorphism of $\mathrm{Spin}(8)$ permuting the vector representation $V$ and the two half-spinor representations ${\mathfrak{S}}^{\pm }$. Spinor squaring builds the octonions and constructs $G_2\subset \mathrm{Spin}(7)\subset \mathrm{Spin}(8)$, $F_4$, $E_6,E_7,E_8$ via the Freudenthal magic square.

  requires: spin-geometry.spin-group, spin-geometry.clifford-chessboard, lie.classification.cartan-weyl

• spin-geometry.dirac.dirac-operatoropen unit 03.09.08 →

  Dirac operator on a spin manifold

  First-order elliptic operator on the spinor bundle, $D=c\circ {\nabla }^S$. Principal symbol is Clifford multiplication, hence elliptic. Role in Lichnerowicz formula, positive scalar curvature, and Atiyah-Singer index theorem.

  requires: spin-geometry.clifford.clifford-algebra, spin-geometry.structure.spin-structure, spin-geometry.spinor-bundle, diffgeo.connection.connection, diffgeo.elliptic-operators, functional-analysis.fredholm.operators

• foundations.real-numbersopen unit 00.01.01 →

  Real numbers, integers, rationals

  Number-system hierarchy $\mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}$. Field axioms (closure, associativity, commutativity, distributivity, identities, inverses) and order axioms for $\mathbb{Q}$ and $\mathbb{R}$. Completeness axiom (least upper bound) and Archimedean property. Two parallel constructions of $\mathbb{R}$ from $\mathbb{Q}$: Dedekind cuts and Cauchy-sequence completion modulo null sequences. Uniqueness of complete ordered fields up to unique order-isomorphism. Irrationality of $\sqrt{2}$ (classical contradiction proof). Density of $\mathbb{Q}$ in $\mathbb{R}$ via Archimedean property. Cardinality jump $|\mathbb{Z}|=|\mathbb{Q}|={\aleph }_0$ versus $|\mathbb{R}|=2^{{\aleph }_0}$ via Cantor's diagonal argument. Foundational floor for the algebra strand (linalg.field), the analysis strand (metric-space completion), and the complex-analysis strand.

  requires: none

• foundations.absolute-valueopen unit 00.01.02 →

  Absolute value and the triangle inequality

  Definition $|x|=x$ if $x\ge 0$ else $-x$, the four signature properties (positivity, multiplicativity, triangle inequality, reverse triangle), and the case-on-sign proof of the triangle inequality. The triangle inequality is the load-bearing axiom — the absolute-value function on $\mathbb{R}$ is the prototype of a norm, and the metric $d(x,y)=|x-y|$ is the prototype of a metric. Master-tier scope adds the ${\ell }^p$ norms via Minkowski's inequality, the discrete metric as a non-norm, the $p$-adic absolute value with its ultrametric strengthening, and Ostrowski's theorem classifying all non-trivial absolute values on $\mathbb{Q}$ as either Euclidean or $p$-adic. Foundational floor for sequence convergence (analysis.sequence-limit), metric-space axioms (topology.metric-space), normed vector spaces (functional-analysis.banach-spaces), and number-theoretic completions (number-theory.p-adic-numbers).

  requires: foundations.real-numbers

• foundations.polynomialsopen unit 00.01.03 →

  Polynomials and rational expressions

  A polynomial over a field $k$ is a finite formal sum $p(x)={\sum }_{i=0}^n{a}_i{x}^i$ with $a_i\in k$ and $a_n\ne 0$ (the leading coefficient); $\mathrm{deg}p=n$. The set $k[x]$ is a commutative ring under term-by-term addition and convolution multiplication. For $k$ a field, $k[x]$ is a Euclidean domain via the polynomial division algorithm: for any $p,q\in k[x]$ with $q\ne 0$ there exist unique $a,r\in k[x]$ with $p=aq+r$ and $\mathrm{deg}r<\mathrm{deg}q$. Factor theorem: $r$ is a root of $p$ iff $(x-r)\mid p(x)$. A degree-$n$ polynomial has at most $n$ roots in any field. Rational expression: ratio $p/q$ with $q\ne 0$, organised into the field of fractions $k(x)$ when one quotients by the equivalence $p_1/q_1\sim p_2/q_2⟺p_1{q}_2=p_2{q}_1$. Master-tier scope: $k[x]$ Euclidean $\Rightarrow$ PID $\Rightarrow$ UFD, with the chain breaking for $k[x_1,\dots ,x_n]$ ($n\ge 2$) which is UFD (Gauss) but not PID; fundamental theorem of algebra (every non-constant $p\in \mathbb{C}[x]$ has a root, Gauss 1799 thesis); real-coefficient polynomials factor into real linear and irreducible real quadratic factors via conjugate-pair roots; algebraic closure $\bar{k}$ and the algebraic numbers $\bar{\mathbb{Q}}\subset \mathbb{C}$; partial-fraction decomposition over $k(x)$ as the dual to division. Originators: Diophantus Arithmetica (~250 CE); al-Khwārizmī al-jabr (~825 CE); Cardano-Tartaglia 1545 Ars Magna (cubic formula); Ferrari 1545 (quartic formula); Abel 1824 / Ruffini 1799 (no general quintic formula in radicals); Galois 1832 (the symmetry theory behind solvability); Gauss 1799 (fundamental theorem of algebra). Foundational floor for the algebra strand (linalg.field, polynomial-ring algebra), complex analysis (complex-analysis.fundamental-theorem-of-algebra), and integration of rational functions (partial fractions).

  requires: foundations.real-numbers

• foundations.linear-equations-lineopen unit 00.03.01 →

  Linear equations and the line

  A linear equation in $n$ variables over a field $k$ is an equation $a_1{x}_1+\cdots +a_n{x}_n=b$ with coefficients $a_i,b\in k$ and the $a_i$ not all zero; the solution set in $k^n$ is a hyperplane (codimension-$1$ affine subspace). For $n=2$ over $\mathbb{R}$, the solution set of $ax+by=c$ is a line in ${\mathbb{R}}^2$, equivalently the affine $1$-flat $\{p_0+tv:t\in \mathbb{R}\}$ where $p_0$ is any solution and $v$ is a direction vector orthogonal to the normal $(a,b)$. Slope-intercept form $y=mx+b^{\prime }$ exists for non-vertical lines, with $m$ the slope and $b^{\prime }$ the y-intercept; two non-vertical lines are parallel iff their slopes agree and perpendicular iff $m_1{m}_2=-1$. Classification of two-line intersections: the system $\{a_{1}x+b_{1}y=c_1,a_{2}x+b_{2}y=c_2\}$ has a unique solution iff the determinant $D=a_1{b}_2-a_2{b}_1\ne 0$ (Cramer's rule: $x=(c_1{b}_2-c_2{b}_1)/D$, $y=(a_1{c}_2-a_2{c}_1)/D$); if $D=0$, the lines are parallel (no solution) or coincident (every common point), distinguished by whether the triples $(a_i,b_i,c_i)$ are proportional. Master-tier scope: linear-versus-affine distinction (linear subspace through origin, affine subspace = translate); hyperplanes and flats in ${\mathbb{R}}^n$; Frobenius / Kronecker-Capelli theorem ($Ax=b$ consistent iff $\mathrm{rank}(A)=\mathrm{rank}([A\mid b])$); Cramer's rule for $n\times n$ systems via the adjugate-inverse formula; projective line ${\mathbb{P}}^1(\mathbb{R})=\mathbb{R}\cup \{\infty \}$ with $\mathrm{PGL}(2,\mathbb{R})$ acting by Möbius transformations, sharply $3$-transitive on triples of distinct points; linear codes over ${\mathbb{F}}_q$; convex polytopes as bounded intersections of half-spaces (Minkowski-Weyl duality). Originators: Euclid Elements Book I (~300 BCE) for the geometric line; Descartes 1637 La Géométrie and Fermat ~1636 letters (algebra-geometry correspondence); Cramer 1750 (determinant rule); Cauchy 1812 (systematic determinant theory); Klein 1872 Erlangen Programm (projective hierarchy). Foundational floor for the determinant in linear-algebra.determinant, linear maps and rank-nullity in linear-algebra.linear-transformation-rank-nullity, multivariable Jacobian determinants in multivariable-calculus.jacobian, Möbius transformations and the Riemann sphere in complex-analysis.mobius-transformations, and the affine-versus-linear distinction throughout differential geometry.

  requires: foundations.real-numbers

• foundations.quadratic-formulaopen unit 00.03.02 →

  Quadratic equations and the quadratic formula

  A quadratic equation over a field $k$ of characteristic not two is $a{x}^2+bx+c=0$ with $a,b,c\in k$ and $a\ne 0$. Completing the square rewrites the equation as $(x+b/(2a){)}^2=(b^2-4ac)/(4a^2)$, and taking square roots gives the quadratic formula $x=(-b\pm \sqrt{\Delta })/(2a)$, where $\Delta =b^2-4ac$ is the discriminant. Over the real numbers the discriminant classifies the solution count: $\Delta >0$ gives two distinct real roots, $\Delta =0$ a single repeated real root (the vertex of the parabola $y=a{x}^2+bx+c$ touches the $x$-axis), and $\Delta <0$ no real roots (two complex-conjugate solutions $x=(-b\pm i\sqrt{-\Delta })/(2a)$). The Vieta formulas express the elementary symmetric functions of the two roots in terms of the coefficients: $x_1+x_2=-b/a$ and $x_1{x}_2=c/a$. Master-tier scope: the discriminant of a degree-$n$ polynomial ${\Delta }_n=a_n^{2n-2}{\prod }_{i<j}(r_i-r_j{)}^2$ as a symmetric function of the roots, expressible in the coefficients by Vieta; Cardano's 1545 cubic and Ferrari's 1540 quartic formulas; Abel-Ruffini 1824 and Galois 1832 (no general formula in radicals for the quintic and beyond, originating Galois theory); the binary quadratic form $Q(x,y)=a{x}^2+bxy+c{y}^2$ and the conic-section discriminant $b^2-4ac$ classifying hyperbola / parabola / ellipse; the general quadratic form on ${\mathbb{R}}^n$ as $Q(v)=v^{\top }Av$, Sylvester's law of inertia on the signature $(p,q)$ under change of basis; Gauss's 1801 quadratic reciprocity for Legendre symbols; the fundamental theorem of algebra (Gauss 1799) extending complete factorisation from quadratics to all polynomials over $\mathbb{C}$. Originators: Babylonian scribes ~2000 BCE (geometric completion-of-the-square on specific quadratics); al-Khwārizmī ~825 CE Kitāb al-jabr wa-l-muqābala (systematic Arabic method, with the word algebra deriving from al-jabr); Cardano 1545 Ars Magna (cubic formula and modern algebraic form); Viète 1591 In artem analyticen isagoge (letter notation for coefficients, sum-and-product of roots); Gauss 1799 doctoral thesis (fundamental theorem of algebra); Gauss 1801 Disquisitiones (quadratic reciprocity). Foundational floor for conic-section classification (foundations.conic-sections), bilinear and quadratic forms (linalg.bilinear-quadratic-form), the algebra-strand discriminant of a polynomial, the Galois-theoretic solvability story (algebra.galois-theory), and the algebraic-closure picture (complex-analysis.fundamental-theorem-of-algebra).

  requires: foundations.polynomials

• foundations.inequalitiesopen unit 00.04.01 →

  Inequalities (linear and quadratic)

  An inequality in one real variable replaces the equals sign in an equation by one of $<,\le ,>,\ge$. The solution set is a region of the number line rather than a finite set of points. The basic manipulation rules match those for equations except that multiplying or dividing both sides by a negative number reverses the direction of the inequality, since the order on $\mathbb{R}$ is incompatible with multiplication by negatives in the opposite sense from addition. A linear inequality $ax+b\le c$ has solution set $\{x:x\le (c-b)/a\}$ when $a>0$ (and $\{x:x\ge (c-b)/a\}$ when $a<0$), a half-line in either case. A quadratic inequality $a{x}^2+bx+c\le 0$ is solved by sign analysis: factor as $a(x-r_1)(x-r_2)$ when the discriminant $\Delta =b^2-4ac$ is non-negative, then read the sign of the product in each of the three regions cut off by the roots. The solution set is a closed interval when $a>0$ and $\Delta >0$, the complement of an open interval when $a<0$ and $\Delta >0$, a single point when $\Delta =0$, and either the entire line or the empty set when $\Delta <0$. Named load-bearing inequalities at this tier: the triangle inequality $|x+y|\le |x|+|y|$ on $\mathbb{R}$, the arithmetic-mean-geometric-mean inequality $(x_1+\cdots +x_n)/n\ge (x_1\cdots x_n{)}^{1/n}$ for non-negative reals with equality iff all $x_i$ coincide, and the Cauchy-Schwarz inequality $|u\cdot v|\le \Vert u\Vert \Vert v\Vert$ for vectors in a real or complex inner-product space with equality iff the vectors are linearly dependent. The standard proof of Cauchy-Schwarz observes that the quadratic $t\mapsto \Vert u-tv{\Vert }^2$ is non-negative everywhere, hence its discriminant in $t$ is non-positive — a direct invocation of the discriminant trichotomy from foundations.quadratic-formula. Master-tier scope: Hölder's inequality $\sum |x_i{y}_i|\le (\sum |x_i{|}^p{)}^{1/p}(\sum |y_i{|}^q{)}^{1/q}$ for conjugate exponents $1/p+1/q=1$ (with Cauchy-Schwarz as the case $p=q=2$); Minkowski's inequality $(\sum |x_i+y_i{|}^p{)}^{1/p}\le (\sum |x_i{|}^p{)}^{1/p}+(\sum |y_i{|}^p{)}^{1/p}$ supplying the triangle inequality on ${\ell }^p$ and $L^p$; Jensen's inequality $\varphi (\mathbb{E}[X])\le \mathbb{E}[\varphi (X)]$ for $\varphi$ convex (generalising AM-GM via $\varphi =-\log$); the isoperimetric inequality on ${\mathbb{R}}^2$ (and Lévy-Gromov on manifolds); polynomial inequalities and the Tarski-Seidenberg theorem (1948) on the decidability of the first-order theory of $\mathbb{R}$ via quantifier elimination over real-closed fields; semi-algebraic sets as a category. Originators: Cauchy 1821 Cours d'analyse (the original finite-sum form of the inequality, in the language of bilinear sums); Schwarz 1885 (the inner-product generalisation used in his minimal-surfaces work); Hölder 1889 (the eponymous inequality, generalising Cauchy-Schwarz to conjugate exponents); Minkowski 1896 (the triangle inequality in ${\ell }^p$); Jensen 1906 (the convex-function inequality and its probabilistic formulation); Tarski 1948 (the decision procedure for the elementary theory of $\mathbb{R}$). Foundational floor for the metric-space triangle inequality (metric-spaces.metric-space), the $L^p$-norm theory (functional-analysis.lp-spaces), inner-product geometry (functional-analysis.hilbert-space — the Cauchy-Schwarz inequality is the reason the angle $\cos \theta =⟨u,v⟩/(\Vert u\Vert \Vert v\Vert )$ takes values in $[-1,1]$), and concentration inequalities in probability (probability.concentration).

  requires: foundations.quadratic-formula

• linalg.fieldopen unit 01.01.01 →

  Field

  Commutative division rings with $0\ne 1$, field homomorphisms, characteristic, prime fields, finite fields in elementary examples, and field extensions as the scalar context for vector spaces and algebras.

  requires: set-theory.function

• linalg.vector-spaceopen unit 01.01.03 →

  Vector space over a field

  The single most-reused concept in the curriculum. Definition over arbitrary field, not just $\mathbb{R}$ or $\mathbb{C}$. Counterexamples: function spaces, polynomial rings as $K$-vector spaces.

  requires: linalg.field, linalg.set-and-function

• linear-algebra.subspace-basis-dimensionopen unit 01.01.04 →

  Subspace, basis, dimension

  The structural triple every linear-algebra computation rests on. Subspace = subset closed under the two operations; basis = linearly independent spanning set; dimension = common cardinality of any basis. Steinitz replacement theorem (1913) is the load-bearing result. Generalises to rank over a PID and to module-theoretic invariants for general modules; in the categorical view, finite-dimensional $K$-vector spaces form a category equivalent to $\mathbb{N}$ with morphisms given by matrices, and $K_0(K{\text{-Vect}}^{\mathrm{fd}})=\mathbb{Z}$.

  requires: linalg.vector-space

• linear-algebra.linear-transformation-rank-nullityopen unit 01.01.05 →

  Linear transformation: kernel, image, rank-nullity

  The fundamental dimension-counting theorem for linear maps. Linear map $T:V\to W$ is the morphism in the category of vector spaces; kernel $\ker T$ and image $\mathrm{im}T$ are its canonical sub-objects. Rank-nullity $\dim V=\dim (\ker T)+\dim (\mathrm{im}T)$ packages dimension as a conserved quantity through any linear map. Categorical sharpening: in $K\text{-Vect}$ every short exact sequence splits, so $V\cong \ker T\oplus \mathrm{im}T$ — the splitting fails for general modules (e.g. $0\to \mathbb{Z}\stackrel{2}{\to }\mathbb{Z}\to \mathbb{Z}/2\to 0$). Functional-analytic generalisation: the index $\mathrm{ind}(T)=\dim \ker T-\dim \mathrm{coker}T$ for Fredholm operators (Atiyah-Singer). Pairs with the first isomorphism theorem $V/\ker T\cong \mathrm{im}T$.

  requires: linear-algebra.subspace-basis-dimension

• linear-algebra.determinantopen unit 01.01.07 →

  Determinant: axiomatic + expansion + properties

  The scalar invariant of a square matrix. Three equivalent definitions: axiomatic (the unique multilinear-alternating-normalised function on rows / columns), Leibniz permutation sum $\det A={\sum }_{\sigma \in S_n}\mathrm{sgn}(\sigma ){\prod }_i{a}_{i,\sigma (i)}$, and recursive Laplace / cofactor expansion. Geometric content: signed volume of the parallelepiped spanned by the columns; $\det A=0$ iff the columns are linearly dependent. Multiplicativity $\det (AB)=\det A\cdot \det B$ from the axiomatic characterisation. Builds toward (i) the change-of-variables Jacobian $|\det J|$ in multivariable calculus, (ii) the characteristic polynomial ${\chi }_A(t)=\det (tI-A)$ for eigenvalues, (iii) the top exterior power ${\Lambda }^{n}A:{\Lambda }^{n}V\to {\Lambda }^{n}V$ acting as multiplication by $\det A$, and (iv) the determinant line bundle $\det V={\Lambda }^{\mathrm{rank}V}V$ in geometry. Originators: Seki Takakazu (1683) and Leibniz (1693); Cauchy (1812) unified the modern theory; Cayley (1858) the matrix notation; Vandermonde (1771) the special case; Artin (1942) the modern axiomatic presentation; Bourbaki the multilinear-algebra framing.

  requires: linear-algebra.linear-transformation-rank-nullity

• linear-algebra.eigenvalue-eigenvectoropen unit 01.01.08 →

  Eigenvalue, eigenvector, characteristic polynomial

  The spectral structure of a linear operator on a finite-dimensional vector space. Eigenvalue equation $Tv=\lambda v$ with $v\ne 0$; eigenspace $E_{\lambda }=\ker (T-\lambda I)$; characteristic polynomial ${\chi }_T(t)=\det (tI-T)$ with eigenvalues as roots; algebraic multiplicity vs geometric multiplicity. Key result: eigenvectors for distinct eigenvalues are linearly independent, so an operator with $n=\dim V$ distinct eigenvalues is diagonalisable. Cayley-Hamilton: every operator satisfies its own characteristic polynomial, ${\chi }_T(T)=0$; the spectral theorem (finite-dim) refines this for self-adjoint operators on an inner-product space, giving an orthonormal eigenbasis with real eigenvalues; Jordan canonical form classifies operators on a finite-dim $\mathbb{C}$-vector space up to similarity by Jordan-block data, with the Segre characteristic recording block sizes. Generalises to Banach-space spectral theory (point / continuous / residual spectrum), the resolvent $(T-\lambda I{)}^{-1}$ as an analytic function off the spectrum, spectral measures for self-adjoint operators on Hilbert space (Stone-von Neumann), and Frobenius eigenvalues for $\ell$-adic Galois representations. Originators: Cauchy (1829) for symmetric matrices; Cayley (1858) and Frobenius (1878) for Cayley-Hamilton; Jordan (1870) for Jordan canonical form; Hilbert (1904-) for the infinite-dim spectral theorem; von Neumann (1929) for self-adjoint spectral measures.

  requires: linear-algebra.determinant

• linear-algebra.jordan-canonical-formopen unit 01.01.11 →

  Jordan canonical form and minimal polynomial

  The complete similarity classification of linear operators on a finite-dimensional $\mathbb{C}$-vector space. Jordan block $J_k(\lambda )$: $k\times k$ matrix with $\lambda$ on the diagonal and $1$s on the superdiagonal. Minimal polynomial $m_T(t)$: monic generator of the ideal $\{p\in K[t]:p(T)=0\}$; $m_T\mid {\chi }_T$ by Cayley-Hamilton, and $T$ is diagonalisable iff $m_T$ is square-free. Primary decomposition: $V={⨁}_i{G}_{{\lambda }_i}$ with $G_{{\lambda }_i}=\ker (T-{\lambda }_{i}I{)}^{m_i}$ the generalised eigenspaces. Existence + uniqueness theorem: over an algebraically closed field every operator is similar to a direct sum of Jordan blocks, unique up to block reordering. Segre characteristic: multiset of block sizes at each eigenvalue, encoding the partition of $\dim G_{\lambda }$ by Jordan-chain lengths. Builds toward (i) rational canonical form over arbitrary fields via companion matrices of invariant factors, (ii) Smith normal form for matrices over a PID, (iii) the structure theorem for finitely generated modules over a PID via $V\cong {⨁}_{i}K[t]/(p_i^{e_i})$, (iv) holomorphic functional calculus $f(J_k(\lambda ))={\sum }_{j<k}\frac{f^{(j)}(\lambda )}{j!}{N}^j$ used to compute $e^{tA}$, $\sin A$, $\cos A$, (v) GIT quotient ${\mathrm{GL}}_n//{\mathrm{GL}}_n$ parametrising conjugacy classes. Originators: Weierstrass (1858, elementary divisor theory); Smith (1861, PID version); Jordan (1870, Traité des substitutions); Frobenius (1879, rational canonical form); Lang (1965, modern module-theoretic packaging in Algebra Ch. III).

  requires: linear-algebra.eigenvalue-eigenvector

• linear-algebra.singular-value-decompositionopen unit 01.01.12 →

  Singular value decomposition (finite-dim)

  The factorisation $A=U\Sigma V^{\ast }$ for an arbitrary $m\times n$ matrix $A$ over $\mathbb{R}$ or $\mathbb{C}$: $U$ unitary $m\times m$, $V$ unitary $n\times n$, $\Sigma$ an $m\times n$ diagonal matrix with non-negative entries ${\sigma }_1\ge {\sigma }_2\ge \cdots \ge {\sigma }_r>0$ called the singular values of $A$, with $r=\mathrm{rank}A$. The singular values are the non-negative square roots of the eigenvalues of $A^{\ast }A$ (equivalently of $A{A}^{\ast }$); the right singular vectors (columns of $V$) are an orthonormal eigenbasis of $A^{\ast }A$, the left singular vectors (columns of $U$) are an orthonormal eigenbasis of $A{A}^{\ast }$, and the two bases are paired by $A{v}_i={\sigma }_i{u}_i$ for ${\sigma }_i>0$. Existence proof: spectral theorem on $A^{\ast }A$, then define $u_i=A{v}_i/{\sigma }_i$ and verify orthonormality. Uniqueness: the singular values are uniquely determined; $U$ and $V$ are unique up to a unitary block on each constant-singular-value subspace (and a phase rotation on simple singular values in the complex case). Companion structures: (i) Moore-Penrose pseudoinverse $A^{+}=V{\Sigma }^{+}{U}^{\ast }$ with ${\Sigma }^{+}$ the transpose of $\Sigma$ with non-zero entries reciprocated; gives the minimum-norm least-squares solution to $Ax=b$. (ii) Polar decomposition $A=QP$ with $Q=U{V}^{\ast }$ unitary and $P=V\Sigma V^{\ast }$ Hermitian positive semidefinite — the matrix analogue of the polar form $z=r{e}^{i\theta }$. (iii) Eckart-Young theorem: for any unitarily invariant norm, the best rank-$k$ approximation to $A$ is $A_k={\sum }_{i=1}^k{\sigma }_i{u}_i{v}_i^{\ast }$; in the operator norm $\Vert A-A_k\Vert ={\sigma }_{k+1}$, in the Frobenius norm $\Vert A-A_k{\Vert }_F=({\sigma }_{k+1}^2+\cdots +{\sigma }_r^2{)}^{1/2}$. (iv) Schmidt decomposition for compact operators on Hilbert space: $A={\sum }_n{\sigma }_n{u}_n{v}_n^{\ast }$ with ${\sigma }_n\to 0$, foundation for trace-class and Hilbert-Schmidt operators. (v) GL action: the bi-unitary action of $U(m)\times U(n)$ on $m\times n$ complex matrices has orbits parametrised by the tuple $({\sigma }_1,\dots ,{\sigma }_r)$ of singular values — SVD is the orbit-decomposition. Applications: principal component analysis (right singular vectors = principal directions, ${\sigma }_i^2/N$ = variances); least-squares regression via the pseudoinverse; low-rank approximation in image compression, latent semantic indexing, recommender systems; condition number $\kappa (A)={\sigma }_1/{\sigma }_r$ for numerical sensitivity; numerical rank via small-singular-value thresholding. Originators: Beltrami (1873) and Jordan (1874), independently for square matrices; Sylvester (1889) for the rectangular case; Schmidt (1907) for the integral-operator / Hilbert-space generalisation; Weyl (1912) for the modern unified treatment; Eckart-Young (1936) for the low-rank approximation theorem; Mirsky (1960) extended Eckart-Young to all unitarily invariant norms; Golub-Kahan (1965) gave the numerical SVD algorithm.

  requires: linear-algebra.eigenvalue-eigenvector

• set-theory.functionopen unit 00.02.05 →

  Function

  Functions as total single-valued relations, graphs, image, composition, identity maps, inverse criterion for bijections, and categorical monomorphism / epimorphism behavior in Set. Foundational prerequisite for vector spaces, groups, maps, and actions.

  requires: none

• linalg.bilinear-formopen unit 01.01.15 →

  Bilinear form / quadratic form

  Symmetric / antisymmetric / hermitian sub-cases. Polarization identity. Gram matrix. Signature for real symmetric forms (Sylvester's law of inertia).

  requires: linalg.vector-space, linalg.dual-space

• algebra.tensor-productopen unit 03.01.01 →

  Tensor product of vector spaces

  Universal bilinear map $V\times W\to V{\otimes }_{K}W$, pure tensors, basis $e_i\otimes f_j$, functoriality, uniqueness by representing property, and the bridge to tensor algebra, tensor powers, vector-bundle operations, and Clifford algebra.

  requires: linalg.field, linalg.vector-space

• algebra.associative-algebraopen unit 03.01.02 →

  Associative algebra

  Unital associative $K$-algebras, bilinear multiplication, central scalar action, algebra homomorphisms, kernels as two-sided ideals, commutator Lie algebra, matrix and polynomial examples, and the ambient structure for tensor and quotient algebras.

  requires: linalg.field, linalg.vector-space

• algebra.idealopen unit 03.01.03 →

  Ideal in an algebra

  Left, right, and two-sided ideals in associative algebras; kernels of algebra homomorphisms; intersections and preimages; two-sided ideals as the data needed for quotient algebra multiplication; polynomial examples and relation to Clifford algebra.

  requires: algebra.associative-algebra

• algebra.tensor-algebraopen unit 03.01.04 →

  Tensor algebra of a vector space

  $T(V)=⨁V^{\otimes n}$. Universal property among unital associative algebras under $V$. Distinguish from symmetric / exterior / Clifford algebras as quotients of $T(V)$.

  requires: linalg.vector-space, linalg.tensor-product

• algebra.quotient-algebraopen unit 03.01.05 →

  Quotient algebra by a two-sided ideal

  Universal property: factoring through the quotient is equivalent to killing the ideal. Foundational for: Clifford algebras (quotient of $T(V)$), exterior algebras, polynomial rings modulo relations.

  requires: algebra.associative-algebra, algebra.two-sided-ideal

• algebra.groupopen unit 01.02.01 →

  Group

  Algebraic groups in the elementary sense: identity, inverse, associativity, homomorphisms, subgroups, kernels, quotient-ready normal subgroups, and the foundation for group actions and Lie groups.

  requires: set-theory.set-and-function

• algebra.group-actionopen unit 03.03.02 →

  Group action

  Left and right group actions, orbit-stabilizer theorem, torsors, equivariance, homogeneous spaces, and the principal-bundle fiber action.

  requires: algebra.group, set-theory.set-and-function

• lie-groups.orthogonal-groupopen unit 03.03.03 →

  Orthogonal group

  Orthogonal group of a real inner-product space, matrix equation $A^{T}A=I$, determinant components, special orthogonal group, Lie algebra of skew-symmetric matrices, and role in frame bundles and spin geometry.

  requires: algebra.group, linalg.bilinear-form, lie-groups.lie-group

• spin-geometry.structure.spin-structureopen unit 03.09.04 →

  Spin structure on an oriented Riemannian manifold

  Lift of the orthonormal frame bundle through Spin(n) → SO(n). Existence obstructed by w_2; classification (when nonempty) by H^1(M; Z/2). Pin± analogues for non-orientable case.

  requires: spin-geometry.clifford.clifford-algebra, spin-geometry.spin-group, bundle.principal-bundle, bundle.frame-bundle.orthonormal, topology.cover.double-cover, char-classes.stiefel-whitney

• topology.cover.double-coveropen unit 03.05.05 →

  Double cover

  Two-sheeted covering maps, fibers with two points, deck involutions, relation to principal $\mathbb{Z}/2$-bundles, and the role of $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ as the double cover behind spin structures.

  requires: topology.topological-space, topology.covering-space

• topology.continuous-mapopen unit 02.01.02 →

  Continuous map

  Continuity between topological spaces, preimage characterization, composition, homeomorphisms, metric-space epsilon-delta comparison, and use in pullbacks and classifying maps.

  requires: topology.topological-space

• bundle.frame-bundle.orthonormalopen unit 03.05.03 →

  Orthogonal frame bundle

  Orthonormal frames of a Riemannian vector bundle, principal $\mathrm{O}(n)$-bundle structure, oriented $\mathrm{SO}(n)$ reduction, tautological frame action, and use as the bundle lifted by a spin structure.

  requires: bundle.principal-bundle, bundle.vector-bundle, diffgeo.smooth-manifold, lie-groups.orthogonal-group

• diffgeo.stokes-theoremopen unit 03.04.05 →

  Stokes' theorem

  ${\int }_{M}d\omega ={\int }_{\partial M}\omega$ on oriented compact manifolds with boundary. Unifies fundamental theorem of calculus, Green's, classical Stokes, divergence theorem. Extends to chains (de Rham theorem) and supports Poincaré duality.

  requires: diffgeo.differential-forms, topology.integration-on-manifolds, diffgeo.exterior-derivative, diffgeo.smooth-manifold

• diffgeo.exterior-derivativeopen unit 03.04.04 →

  Exterior derivative

  Unique linear operator $d:{\Omega }^k\to {\Omega }^{k+1}$ characterised by action on functions, linearity, graded Leibniz, $d^2=0$. Local formula. Naturality (commutes with pullback). Cartan magic formula. Poincaré lemma. Maurer-Cartan equation on Lie groups. Covariant exterior derivative on bundle-valued forms.

  requires: diffgeo.differential-forms, diffgeo.smooth-manifold

• topology.homotopyopen unit 03.12.01 →

  Homotopy and homotopy group

  Homotopy as continuous deformation; homotopy equivalence; fundamental group ${\pi }_1$ via loop concatenation; higher homotopy groups ${\pi }_n$ for $n\ge 2$ (abelian by Eckmann-Hilton). Functoriality, homotopy invariance, Seifert-Van Kampen, Hurewicz, universal cover, long exact sequence of a fibration. Eilenberg-MacLane spaces.

  requires: topology.topological-space, topology.continuous-map

• topology.fundamental-groupopen unit 03.12.00 →

  Fundamental group

  ${\pi }_1(X,x_0)$ as homotopy classes of based loops $\gamma :[0,1]\to X$, $\gamma (0)=\gamma (1)=x_0$, under concatenation. Identity = constant loop; inverse = reversed loop. Functoriality $f_{\ast }:{\pi }_1(X,x_0)\to {\pi }_1(Y,y_0)$, $(g\circ f{)}_{\ast }=g_{\ast }\circ f_{\ast }$. Homotopy invariance: $f\simeq g$ implies $f_{\ast }=g_{\ast }$. Basepoint independence up to inner automorphism via path conjugation ${\beta }_{\eta }$. Standard examples: ${\pi }_1({\mathbb{R}}^n)=0$, ${\pi }_1(S^1)=\mathbb{Z}$ (winding number via universal cover), ${\pi }_1(S^n)=0$ for $n\ge 2$, ${\pi }_1({\mathbb{T}}^n)={\mathbb{Z}}^n$, ${\pi }_1({\mathbb{RP}}^n)=\mathbb{Z}/2$, ${\pi }_1({\Sigma }_g)$ surface group, ${\pi }_1$(figure-eight) $=F_2$. Loop space $\Omega (X,x_0)$ with compact-open topology; ${\pi }_0(\Omega X)={\pi }_1(X)$ and ${\pi }_n(X)={\pi }_{n-1}(\Omega X)$. Galois correspondence between subgroups and connected covers (topology.covering-space). ${\pi }_1(X,x_0)={\mathrm{Aut}}_{{\pi }_1(X)}(x_0)$ in the fundamental groupoid (topology.fundamental-groupoid). Originator Poincaré 1895.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.fundamental-groupoidopen unit 03.12.08 →

  Fundamental groupoid

  Small category ${\pi }_1(X)$ on points of $X$ with morphisms = path-homotopy classes; partial composition (concatenation when endpoints match); inverses by reversed paths; identity = constant path. Functor ${\pi }_1:\mathbf{Top}\to \mathbf{Groupoid}$. Equivalent to one-object $\mathrm{B}{\pi }_1(X,x_0)$ when $X$ is path-connected. Brown's groupoid Seifert-van Kampen: ${\pi }_1(X,A)$ as a pushout, no path-connectedness hypothesis on $U\cap V$. Galois correspondence as equivalence $\mathbf{Cov}(X)\simeq {\mathbf{Set}}^{{\pi }_1(X)}$.

  requires: topology.homotopy, topology.topological-space, topology.continuous-map

• topology.singular-homologyopen unit 03.12.11 →

  Singular homology

  Singular $n$-simplex $\sigma :{\Delta }^n\to X$. Free abelian chain group $C_n(X)$. Boundary $\partial =\sum (-1{)}^i{d}^i$, ${\partial }^2=0$. Homology $H_n(X)=\ker {\partial }_n/\mathrm{im}{\partial }_{n+1}$. Functoriality, homotopy invariance via prism/chain-homotopy. Reduced homology, augmentation. Coefficients $H_n(X;G)$. Mayer-Vietoris. Long exact sequence of a pair. Computations $H_n(\mathrm{pt})$, $H_n(S^k)$. Originator: Eilenberg 1944 Singular homology theory (Ann. Math. 45); precursors Vietoris 1927, Lefschetz 1933.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.simplicial-homologyopen unit 03.12.12 →

  Simplicial and $\Delta$-complex homology

  $\Delta$-complex / semi-simplicial structure on $X$ via characteristic maps with face conditions. Simplicial chain complex $C_n^{\Delta }(X)$ — finite-dim when $X$ is finite. Simplicial homology $H_n^{\Delta }(X)$. Comparison theorem (Hatcher 2.27): $H_n^{\Delta }(X)\cong H_n(X)$ via the natural chain map. Standard computations: $S^n$, ${\mathbb{RP}}^n$, ${\mathbb{CP}}^n$, genus-$g$ surfaces, lens spaces. Originator: Poincaré 1895 (simplicial chain complex); Eilenberg-Zilber 1950 / Eilenberg-Steenrod 1952 modern $\Delta$-framing.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.cellular-homologyopen unit 03.12.13 →

  Cellular homology and cellular approximation

  For a CW complex $X$, $C_n^{\mathrm{cell}}(X)=H_n(X^{(n)},X^{(n-1)})$. Cellular boundary via long-exact-sequence connecting map; explicit degree formula $d_n(e_{\alpha }^n)=\sum \mathrm{deg}({\varphi }_{\alpha \beta })e_{\beta }^{n-1}$. Cellular = singular: $H_n^{\mathrm{cell}}(X)\cong H_n(X)$ (Hatcher 2.35). Computational power: ${\mathbb{CP}}^n$ has zero boundaries (no consecutive cells); ${\mathbb{RP}}^n$ has $\partial =1+(-1{)}^k$. Cellular approximation theorem: every continuous CW-map is homotopic to a cellular map (Hatcher 4.8). Originator: J.H.C. Whitehead 1949 Combinatorial homotopy II.

  requires: topology.cw-complex, topology.singular-homology

• topology.excisionopen unit 03.12.14 →

  Excision theorem

  For $Z\subseteq A\subseteq X$ with $\overline{Z}\subseteq \mathrm{int}(A)$, the inclusion induces $H_n(X\setminus Z,A\setminus Z)\cong H_n(X,A)$. Equivalent open-cover form: $X=\mathrm{int}(A)\cup \mathrm{int}(B)\Rightarrow H_n(B,A\cap B)\cong H_n(X,A)$. Proof via barycentric subdivision + Lebesgue-number argument. Consequences: Mayer-Vietoris derivation; $H_n(D^k,S^{k-1})\cong {\tilde{H}}_{n-1}(S^{k-1})$; cellular boundary formula. Originator: Eilenberg-Steenrod 1952 (axiomatic elevation); precursor in Eilenberg 1944.

  requires: topology.singular-homology, topology.cellular-homology

• topology.eilenberg-steenrodopen unit 03.12.15 →

  Eilenberg-Steenrod axioms

  Axioms for ordinary homology: homotopy, long exact sequence of a pair, excision, additivity, naturality, dimension ($h_n(\mathrm{pt})=0$ for $n\ne 0$), and weak-equivalence. Uniqueness theorem: any two theories satisfying dimension and agreeing on $\mathrm{pt}$ are naturally isomorphic on CW pairs. Generalised cohomology theories drop dimension axiom (K-theory, cobordism, stable homotopy). Brown representability via spectra. Singular = simplicial = cellular = Čech all satisfy axioms; uniqueness explains agreement.

  requires: topology.singular-homology, topology.excision

• topology.poincare-dualityopen unit 03.12.16 →

  Poincaré duality

  For closed oriented $n$-manifold $M$, cap with $[M]$ gives $H^k(M;\mathbb{Z})\cong H_{n-k}(M;\mathbb{Z})$. $\mathbb{Z}/2$-coefficients version: works without orientation. Lefschetz duality for manifolds with boundary. de Rham version: integration pairing on closed oriented smooth manifolds. Consequences: vanishing $\chi$ for odd-dim closed manifolds, signature theorem, Hirzebruch L-genus. Originator: Poincaré Analysis Situs 5th supplement; modern proof Lefschetz 1930 + Eilenberg-Steenrod 1952; de Rham 1931 (smooth version).

  requires: topology.singular-homology, topology.cellular-homology, topology.excision, topology.eilenberg-steenrod

• topology.cap-productopen unit 03.12.17 →

  Cap product

  Front-face/back-face split: $\varphi ⌢\sigma =\varphi (\sigma {|}_{[v_0,\dots ,v_k]})\cdot \sigma {|}_{[v_k,\dots ,v_n]}$. Bilinear $⌢:C^k\otimes C_n\to C_{n-k}$, descends to homology via Leibniz rule $\partial (\varphi ⌢\sigma )=(-1{)}^k(\delta \varphi ⌢\sigma -\varphi ⌢\partial \sigma )$. Makes $H_{\ast }(X)$ a graded $H^{\ast }(X)$-module. Naturality (projection formula): $f_{\ast }(f^{\ast }\varphi ⌢\alpha )=\varphi ⌢f_{\ast }\alpha$. Cap-cup compatibility: $\varphi ⌢(\psi ⌢\alpha )=(\psi ⌣\varphi )⌢\alpha$. Cap with fundamental class $-⌢[M]$ is the Poincaré-duality isomorphism. Originator: Čech 1936 / Whitney 1938 / Lefschetz 1942; axiomatised in Eilenberg-Steenrod 1952.

  requires: topology.singular-homology, topology.cellular-homology

• topology.universal-coefficientopen unit 03.12.18 →

  Universal coefficient theorem (homology and cohomology)

  Homology UCT: split SES $0\to H_n(X)\otimes G\to H_n(X;G)\to \mathrm{Tor}(H_{n-1}(X),G)\to 0$. Cohomology UCT: $0\to \mathrm{Ext}(H_{n-1}(X),G)\to H^n(X;G)\to \mathrm{Hom}(H_n(X),G)\to 0$. Algebraic version for any chain complex of free abelian groups. Tor and Ext as derived functors. Field-coefficient case: Tor and Ext vanish in characteristic 0. Bockstein homomorphism via SES $0\to \mathbb{Z}\to \mathbb{Z}\to \mathbb{Z}/n\to 0$. Originator: Cartan-Eilenberg 1956 (algebraic); Eilenberg-Steenrod 1952 (topological).

  requires: topology.singular-homology, topology.cellular-homology

• symplectic-geometry.ags-convexityopen unit 05.04.03 →

  Atiyah-Guillemin-Sternberg convexity theorem

  For closed connected $(M,\omega )$ with Hamiltonian $T=T^k$ action and moment map $\mu :M\to {\mathfrak{t}}^{\ast }$: (i) $\mu (M)$ is a convex polytope; (ii) $\mu (M)=\mathrm{conv}(\mu (M^T))$; (iii) every fibre ${\mu }^{-1}(c)$ is connected. Atiyah's proof: Morse-Bott analysis of ${\mu }^{\xi }$, even index/coindex via weight-space decomposition, level-set connectedness via even-index attaching, induction on rank. Examples: $S^2$ height function, ${\mathbb{CP}}^n$ standard polytope, coadjoint orbits / Schur-Horn permutohedron. Originators: Atiyah 1982 Convexity and commuting Hamiltonians; Guillemin-Sternberg 1982 Convexity properties of the moment mapping.

  requires: symplectic-geometry.moment-map, symplectic-geometry.symplectic-manifold

• topology.hurewicz-theoremopen unit 03.12.19 →

  Hurewicz theorem

  Hurewicz map $h_n:{\pi }_n(X)\to H_n(X)$, $[\sigma ]\mapsto {\sigma }_{\ast }[S^n]$. Low-dim form (Hatcher 2A.1): $h_1:{\pi }_1\to H_1$ is abelianisation map. Higher-dim form (Hatcher 4.32): if $X$ is $(n-1)$-connected then $H_k(X)=0$ for $1\le k<n$ and $h_n$ is iso. Relative form (Hatcher 4.37). ${\pi }_n(S^n)=\mathbb{Z}$ as a basic computation. Hopf map shows non-injectivity of $h_3:{\pi }_3(S^2)\to H_3(S^2)=0$. Originator: Witold Hurewicz 1935-36 four-paper series.

  requires: topology.homotopy, topology.singular-homology

• topology.whitehead-theoremopen unit 03.12.20 →

  Whitehead's theorem

  A continuous map $f:X\to Y$ between CW complexes inducing iso on all ${\pi }_n$ is a homotopy equivalence (Hatcher 4.5). Proof via cellular approximation + HEP for CW pairs + skeleton-by-skeleton inductive obstruction-theoretic construction of homotopy inverse. Mapping-cylinder reformulation. Hurewicz + Whitehead corollary: simply-connected CW complexes with iso on all $H_n$ are homotopy-equivalent. Warsaw circle is the standard counterexample for the CW hypothesis. Originator: J.H.C. Whitehead 1949.

  requires: topology.cw-complex, topology.homotopy, topology.hurewicz-theorem

• symplectic-geometry.contact-manifoldopen unit 05.10.01 →

  Contact manifold

  Odd-dimensional manifold $M^{2n+1}$ with maximally non-integrable codimension-1 hyperplane field $\xi \subset TM$, locally $\xi =\ker \alpha$ with $\alpha \wedge (d\alpha {)}^n\ne 0$. Co-orientable (global $\alpha$) vs non-co-orientable. Reeb vector field $R$: ${\iota }_R\alpha =1,{\iota }_{R}d\alpha =0$. Standard examples: ${\mathbb{R}}^{2n+1}$ with $dz-\sum y_{i}d{x}_i$; $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ Sasakian; $S(T^{\ast }M)$ unit-cotangent (Reeb = geodesic flow); $J^1(M,\mathbb{R})$. Contact Darboux via Moser. Symplectisation $(M\times \mathbb{R},d(e^t\alpha ))$. Legendrian = $n$-dim submanifold tangent to $\xi$. Gray's theorem (1959): isotopic contact structures are diffeomorphic. Tight/overtwisted dichotomy (Eliashberg 1989).

  requires: symplectic-geometry.symplectic-manifold, diffgeo.exterior-derivative

• symplectic-geometry.generating-functionsopen unit 05.05.03 →

  Generating functions for symplectomorphisms

  Graph of symplectomorphism $\varphi :T^{\ast }M\to T^{\ast }M$ as a Lagrangian in $T^{\ast }M\times T^{\ast }M\cong T^{\ast }(M\times M)$. Type-1 generating function $F:M\times M\to \mathbb{R}$. Four classical types (physics literature). Generating function for exact Lagrangian = scalar potential $S:M\to \mathbb{R}$. Hörmander 1971 GFQI: every closed Lagrangian admits a generating function quadratic at infinity. Sikorav-Viterbo theorem: GFQIs for Hamiltonian-isotopic Lagrangians are equivalent up to stabilisation/fibre-change/constants — gives spectral invariants. Discrete action principle: critical points = fixed points / Lagrangian intersections. Conley-Zehnder used GFQI for the torus Arnold conjecture (1983).

  requires: symplectic-geometry.lagrangian-submanifold, symplectic-geometry.weinstein-neighbourhood, symplectic-geometry.symplectic-manifold

• symplectic-geometry.delzant-theoremopen unit 05.04.04 →

  Delzant theorem (symplectic toric classification)

  Bijection between symplectic toric manifolds (closed, effective Hamiltonian half-dim torus action) and Delzant polytopes (simple + rational + smooth). Five-step construction: facet primitive normals $v_i$; surjection $\beta :{\mathbb{R}}^d\to {\mathbb{R}}^n$; kernel torus $K$; standard $T^d$-action on ${\mathbb{C}}^d$ with diagonal moment map; symplectic reduction $M_{\Delta }={\mu }_K^{-1}(0)/K$. Examples: standard simplex = ${\mathbb{CP}}^n$ (Fubini-Study); standard cube = $({\mathbb{CP}}^1{)}^n$; trapezoid = Hirzebruch ${\mathbb{F}}_n$. Originator: Delzant 1988.

  requires: symplectic-geometry.moment-map, symplectic-geometry.ags-convexity, symplectic-geometry.symplectic-reduction

• symplectic-geometry.duistermaat-heckmanopen unit 05.04.05 →

  Duistermaat-Heckman theorem

  Pushforward of Liouville volume ${\omega }^n/n!$ under moment map $\mu :M\to {\mathfrak{t}}^{\ast }$ has piecewise-polynomial density $\rho (t)$ of degree $\le n-k$ on chambers of $\mu (M)\setminus \mathcal{W}$, walls $\mathcal{W}$ being moment-map images of fixed-point components of proper subtori. Three formulations: (1) pushforward density polynomial; (2) equivariant integration ${\int }_M{e}^{i⟨t,\mu ⟩}{\omega }^n/n!={\sum }_F{\int }_F{e}^{i⟨t,\mu (F)⟩}/e_T({\nu }_F)$ over fixed-point components; (3) reduced symplectic volume $\mathrm{vol}(M_t,{\omega }_t)$ polynomial in $t$. Proof spine: linear variation $[{\omega }_t]=[{\omega }_{t_0}]+⟨t-t_0,c_T⟩$ in cohomology of reduced space; binomial-theorem expansion gives polynomial of degree $\le n-k$. Toric corollary: for symplectic toric $M$, ${\int }_M{\omega }^n/n!=(2\pi {)}^n{\mathrm{vol}}_{\mathrm{Leb}}(\Delta )$. Worked examples: $S^2$ rotation (constant $\rho =2\pi$); ${\mathbb{CP}}^n$ toric (constant $\rho =(2\pi {)}^n$); coadjoint orbit of $U(n)$ (Harish-Chandra-Itzykson-Zuber integral via Weyl-image fixed points). Prototype of equivariant localisation, subsumed by Atiyah-Bott / Berline-Vergne 1982-1984; non-Abelian generalisation Witten 1992. Originators: Duistermaat-Heckman 1982; Atiyah-Bott 1984 (abstract).

  requires: symplectic-geometry.moment-map, symplectic-geometry.symplectic-reduction, symplectic-geometry.ags-convexity

• symplectic-geometry.symplectic-blowupopen unit 05.04.06 →

  Symplectic blow-up and symplectic cut

  Symplectic analogue of the algebraic blow-up: replace a Darboux ball $B(p,\delta )\subset (M,\omega )$ around a point $p$ with a tubular neighbourhood of ${\mathbb{CP}}^{n-1}$ (the exceptional divisor, scaled by $\delta$). Two equivalent constructions: (i) Lerman's symplectic cut at level $\mu =\delta$ for the rotational $S^1$-action with moment map $\mu (z)=\pi \Vert z{\Vert }^2$ on the Darboux ball, giving $M_{\delta }={\mu }^{-1}([0,\delta ])/\sim$ where the $\delta$-level is collapsed by the $S^1$-action; (ii) explicit gluing via the tautological line bundle ${\mathcal{O}}_{{\mathbb{P}}^{n-1}}(-1)$ over ${\mathbb{CP}}^{n-1}$, with the Hopf-style projection making the symplectic form patch smoothly. Properties: $H^{\ast }(\tilde{M})=H^{\ast }(M)\oplus H^{\ast }({\mathbb{CP}}^{n-1}{)}_{\ge 1}$ additively; for surfaces ($n=2$) introduces a $(-1)$-curve, a symplectic 2-sphere of self-intersection $-1$; symplectic volume $\mathrm{vol}(\tilde{M})=\mathrm{vol}(M)-\mathrm{vol}(B(p,\delta ))+\mathrm{vol}({\mathbb{CP}}_{\delta }^{n-1})=\mathrm{vol}(M)-O({\delta }^{2n})$. Examples: blow-up of ${\mathbb{CP}}^2$ at a point = Hirzebruch surface ${\mathbb{F}}_1$; blow-up of ${\mathbb{CP}}^1\times {\mathbb{CP}}^1$ at a point; symplectic cut on toric varieties chops the Delzant polytope along a hyperplane. Inverse: Castelnuovo's contractibility theorem (a $(-1)$-curve in a smooth projective surface contracts to a smooth point) has a symplectic analogue. Applications: birational classification of complex surfaces (Castelnuovo-Beauville-Bombieri); symplectic-embedding flexibility / obstructions (McDuff-Polterovich 1994); compactification of Donaldson instanton moduli spaces (ideal instantons via blow-ups). Originator: algebraic side, Cremona / Italian school 19th c.; symplectic side, Gromov 1985 (sketched) and Lerman 1995 (cut construction made explicit); McDuff-Polterovich 1994 systematic embedding applications.

  requires: symplectic-geometry.symplectic-reduction, symplectic-geometry.delzant-theorem, symplectic-geometry.almost-complex

• symplectic-geometry.symplectisationopen unit 05.10.02 →

  Symplectisation of a contact manifold

  Symplectisation $(M\times \mathbb{R},d(e^t\alpha ))$ of a co-orientable contact manifold. Verification via $(e^t\alpha {)}^{n+1}=e^{(n+1)t}dt\wedge \alpha \wedge (d\alpha {)}^n$. Independence of contact-form choice via $(p,t)\mapsto (p,t-\log f(p))$. Liouville structure with primitive $\lambda =e^t\alpha$ and Liouville vector field ${\partial }_t$. Reeb-flow lift. Floer-theoretic relevance: SFT (Eliashberg-Givental-Hofer 2000) and ECH (Hutchings 2002). Worked example: $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ symplectisation = ${\mathbb{C}}^{n+1}\setminus \{0\}$.

  requires: symplectic-geometry.contact-manifold, symplectic-geometry.symplectic-manifold

• symplectic-geometry.gray-theoremopen unit 05.10.03 →

  Gray's stability theorem

  A smooth path of contact structures on a closed manifold is induced by an isotopy. Proof: Moser-trick adaptation. Choose contact forms ${\alpha }_t$, seek ${\psi }_t^{\ast }{\alpha }_t={\lambda }_t{\alpha }_0$. The contact condition makes the tangent equation ${\iota }_{X_t}d{\alpha }_t{|}_{{\xi }_t}=-{\dot{\alpha }}_t{|}_{{\xi }_t}$ uniquely solvable for $X_t\in {\xi }_t$. Integrate. Consequences: classification up to homotopy = up to diffeomorphism on closed manifolds; Reeb-dynamics stability; foundation for SFT/ECH. 3D extra: tight/overtwisted dichotomy (Eliashberg 1989). Originator: J.W. Gray 1959.

  requires: symplectic-geometry.contact-manifold, symplectic-geometry.moser-trick

• symplectic-geometry.contact-topology-surveyopen unit 05.10.04 →

  Contact topology and Reeb dynamics (survey)

  Survey unit (Cannas P4 optional pointer expanded). Tight vs overtwisted dichotomy (Eliashberg 1989): on closed contact 3-manifolds, overtwisted structures are classified up to isotopy by homotopy classes of plane fields; tight structures are subtle (Bennequin 1983 standard ${\mathbb{R}}^3$ tight, distinct from the overtwisted same-homotopy-class form). Reeb dynamics: closed orbits of $R_{\alpha }$. Weinstein conjecture (1979): every Reeb vector field on a closed contact manifold has at least one closed orbit; proven dim 3 by Taubes 2007 via Seiberg-Witten Floer. Floer-theoretic invariants from pseudoholomorphic curves in symplectisations: Symplectic Field Theory (Eliashberg-Givental-Hofer 2000), Embedded Contact Homology (Hutchings 2002), cylindrical contact homology. Legendrian knots: classical invariants (Thurston-Bennequin, rotation number); refined invariants (Chekanov-Eliashberg DGA 1997, Legendrian contact homology). Convex surface theory (Giroux 1991): characteristic foliation gives discrete neighbourhood data; foundation for cut-and-paste. Open-book decompositions and Giroux correspondence (2002): closed oriented contact 3-manifolds ↔ stable equivalence classes of open books. Higher-dimensional contact topology: Borman-Eliashberg-Murphy 2015 existence h-principle for overtwisted; Cieliebak-Eliashberg 2012 Weinstein-domain technology. Open problems: cardinality of tight structures, ECH spectral invariants, Stein-vs-Weinstein boundary rigidity, contact mapping class groups. Originators: Bennequin 1983; Eliashberg 1989-92; Giroux 2002; Taubes 2007.

  requires: symplectic-geometry.contact-manifold, symplectic-geometry.symplectisation, symplectic-geometry.gray-theorem

• symplectic-geometry.kam-theoremopen unit 05.09.01 →

  Kolmogorov-Arnold-Moser theorem

  APEX UNIT. Persistence of invariant tori under Hamiltonian perturbation. Setup: integrable $H_0(I)$ with action-angle coords; perturbation $H_{ϵ}=H_0+ϵH_1$. Diophantine condition $|⟨k,{\omega }^{\ast }⟩|\ge \gamma /|k{|}^{\tau }$. Kolmogorov non-degeneracy $\det {\partial }^2{H}_0/\partial I^2\ne 0$. Theorem: Diophantine torus survives for small $ϵ$. Newton-iteration / KAM scheme: cohomological equation, Fourier decomposition, super-exponential convergence with controlled analyticity loss. Measure conclusion: $1-C\sqrt{ϵ}$ of phase space remains invariant. Refinements: Moser twist 1962, Pöschel $C^k$ 1982, lower-dim tori, Nekhoroshev exponential stability, Aubry-Mather. Applications: solar-system stability, magnetic confinement, beam dynamics. Originators: Kolmogorov 1954 (4-page Dokl. announcement); Arnold 1963 (full proof); Moser 1962 (smooth twist version).

  requires: symplectic-geometry.integrable-system, symplectic-geometry.action-angle-coordinates, symplectic-geometry.symplectic-manifold, symplectic-geometry.generating-functions

• classical-mechanics.galilean-newtonian-setupopen unit 05.00.06 →

  Galilean group and Newtonian mechanics

  Galilean spacetime as affine 4-space ${\mathbb{A}}^4$ with absolute time $t:{\mathbb{A}}^4\to \mathbb{R}$ and Euclidean structure on each time-fibre. Galilean group $\mathrm{Gal}(4)$ as 10-parameter group of affine transformations preserving time differences and fibre Euclidean structure: spatial translations ${\mathbb{R}}^3$, time translation $\mathbb{R}$, rotations $\mathrm{SO}(3)$, Galilean boosts ${\mathbb{R}}^3$, with semi-direct product structure. Inertial frames = Galilean coordinate systems; Galilean relativity principle. Newton's laws restated geometrically on a configuration manifold $Q$ with Newton's 2nd $m_i{\ddot{q}}_i=F_i(q,\dot{q},t)$ and determinism principle (state $(q,\dot{q},t)$ determines the future). Conservative forces $F=-\nabla V$; Lagrangian $L=T-V$ as the bridge to the Lagrangian formalism. Examples: free particle (Newton's 1st), two-body Kepler, $N$-body system with Galilean invariance giving energy / momentum / angular momentum / centre-of-mass conservation laws. Lie algebra $\mathfrak{gal}(4)$ 10-dim with brackets among translations / boosts / rotations. Inönü-Wigner contraction $c\to \infty$ from Poincaré group. Bargmann central extension by $\mathbb{R}$ governs projective QM representations; mass = central charge (Lévy-Leblond 1963). Originator: Galileo 1632 Dialogue (relativity principle); Newton 1687 Principia; modern group-theoretic framing Bargmann 1954, Souriau 1970.

  requires: manifolds.smooth-manifold

• classical-mechanics.lagrangian-on-tmopen unit 05.00.01 →

  Lagrangian mechanics on the tangent bundle

  Configuration space $Q$, tangent bundle $TQ=\{(q,\dot{q})\}$, Lagrangian $L:TQ\to \mathbb{R}$ (often $T-V$). Action $S[\gamma ]=\int Ldt$. Euler-Lagrange equations $\frac{d}{dt}(\partial L/\partial {\dot{q}}^i)-\partial L/\partial q^i=0$ as critical-point condition. Coordinate-free framing via Poincaré-Cartan one-form ${\theta }_L=(\partial L/\partial {\dot{q}}^i)d{q}^i$. Energy $E_L={\dot{q}}^i(\partial L/\partial {\dot{q}}^i)-L$ as Hamilton-prefiguration. Examples: free particle, particle-in-potential = Newton's 2nd law, geodesics, pendulum. Regularity / hyper-regularity controlling Legendre transform. Originator: Lagrange 1788; modern coordinate-free framing mid-20th-c.

  requires: manifolds.smooth-manifold

• classical-mechanics.hamilton-principleopen unit 05.00.02 →

  Hamilton's principle of least action

  Path is a physical trajectory iff it is a critical point of the action functional $S[\gamma ]=\int L(\gamma ,\dot{\gamma })dt$ among paths with fixed endpoints. First-variation formula: $\delta S=\int (\partial L/\partial q-d/dt(\partial L/\partial \dot{q}))\delta qdt$ + boundary, which vanishes by endpoint condition. Vanishing for arbitrary $\delta q$ ⇔ Euler-Lagrange. Equivalence with Newton's 2nd law for $L=T-V$. Maupertuis reparametrisation pitfall and Jacobi metric. D'Alembert principle for non-conservative forces. Holonomic constraints via Lagrange multipliers. Field-theory generalisation: Klein-Gordon, Maxwell, Yang-Mills, Einstein-Hilbert. Originator: Hamilton 1834; Maupertuis 1744 less-precise predecessor.

  requires: classical-mechanics.lagrangian-on-tm

• classical-mechanics.legendre-transformopen unit 05.00.03 →

  Legendre transform

  Convex transform $f^{\ast }(p)={\sup }_q(⟨p,q⟩-f(q))$. Fenchel-Moreau involution $(f^{\ast }{)}^{\ast }=f$. Differential form $\nabla f^{\ast }(p)=q(p)$ inverse of $\nabla f$. Fibre Legendre transform $\mathbb{F}L:TQ\to T^{\ast }Q$, $\mathbb{F}L(q,\dot{q})=(q,\partial L/\partial \dot{q})$. Regularity = local diffeomorphism (Hessian non-singular); hyper-regularity = global diffeomorphism. Hamiltonian $H(q,p)=⟨p,\dot{q}⟩-L$. Equivalence of EL on $TQ$ with Hamilton's equations on $T^{\ast }Q$. Cotangent bundle as natural symplectic phase space. Singular Lagrangians and Dirac-Bergmann constraint analysis (gauge theories, GR). Originator: Legendre 1787; mechanics application Hamilton 1834; modern framing Abraham-Marsden 1978.

  requires: classical-mechanics.lagrangian-on-tm

• classical-mechanics.noether-theoremopen unit 05.00.04 →

  Noether's theorem

  Every smooth one-parameter family of symmetries of the action $S$ gives a conserved quantity along EL flow. Setup: vector field $X$ on $Q$ with prolongation $X^{(1)}$ on $TQ$; invariance ${\mathcal{L}}_{X^{(1)}}L=0$. Noether charge $J_X=⟨\partial L/\partial \dot{q},X⟩$. Examples: time-translation → energy; space-translation → momentum; rotation → angular momentum. Hamiltonian-side: Poisson-commute condition $\{f,H\}=0$. Lifts to moment-map theory: $G$-action on $(M,\omega )$ with $\mu :M\to {\mathfrak{g}}^{\ast }$. Field-theory generalisation: Noether currents ${\partial }_{\mu }{J}_X^{\mu }=0$. Inverse Noether (Cartan-Lie). Originator: Emmy Noether 1918.

  requires: classical-mechanics.lagrangian-on-tm, classical-mechanics.hamilton-principle

• symplectic-geometry.geodesic-flow-hamiltonianopen unit 05.02.06 →

  Geodesic flow as a Hamiltonian flow

  Kinetic-energy Hamiltonian $H(q,p)=\frac{1}{2}{g}^{ij}{p}_i{p}_j$ on $T^{\ast }M$. Hamilton's equations recover the geodesic equation ${\ddot{q}}^i+{\Gamma }_{jk}^i{\dot{q}}^j{\dot{q}}^k=0$. Unit cotangent bundle $S(T^{\ast }M)$ is contact; Reeb = geodesic spray. Killing vector fields → Noether-conserved quantities. Maupertuis-Jacobi reformulation: mechanics with potential $V$ on energy level $H=E$ ↔ pure geodesic flow of Jacobi metric $\tilde{g}=(E-V)g$. Examples: flat ${\mathbb{R}}^n$ (straight lines), round $S^n$ (great circles, integrable), hyperbolic plane (Anosov), Jacobi-integrable ellipsoid. Originator: Jacobi 1837.

  requires: symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.cotangent-bundle, manifolds.smooth-manifold

• symplectic-geometry.euler-arnold-equationsopen unit 05.09.05 →

  Euler-Arnold equations

  Body-frame projection of geodesic flow on a Lie group $G$ with left-invariant Riemannian metric to the dual Lie algebra ${\mathfrak{g}}^{\ast }$. Setup: inertia operator $I:\mathfrak{g}\to {\mathfrak{g}}^{\ast }$, kinetic-energy Hamiltonian $H({\xi }^{\ast })=\frac{1}{2}⟨{\xi }^{\ast },I^{-1}{\xi }^{\ast }⟩$. Equation: ${\dot{\xi }}^{\ast }=-{\mathrm{ad}}_{I^{-1}{\xi }^{\ast }}^{\ast }{\xi }^{\ast }$. Hamiltonian flow of $H$ for the Lie-Poisson bracket on ${\mathfrak{g}}^{\ast }$; coadjoint orbits are the symplectic leaves with KKS form. Conservation: energy + Casimirs (= ${\mathrm{Ad}}_G^{\ast }$-invariant functions). Examples: $\mathrm{SO}(3)$ rigid-body Euler equations (Euler 1758, Liouville-integrable, Poinsot construction); $\mathrm{SO}(n)$ rigid body (Manakov 1976, Lax pair with spectral parameter); $\mathrm{SDiff}(M)$ ideal-fluid Euler equations (Arnold 1966, $L^2$ inertia); $\mathrm{Diff}(S^1)$ Bott-Virasoro group with $L^2$ metric → KdV, with $H^1$ metric → Camassa-Holm (Misiołek 1998). Tennis-racket theorem: stability of long/short-axis rotation, instability of medium-axis. Geodesic completeness: full for compact finite-dim $G$, blow-up possible for infinite-dim (Beale-Kato-Majda 1984 vorticity criterion for 3D Euler). Mishchenko-Fomenko argument-shift method (1978) gives Liouville-integrability on semisimple $\mathfrak{g}$. Heavy top via semidirect product $\mathrm{SO}(3)⋉{\mathbb{R}}^3$. Originators: Euler 1758 (rigid body), Arnold 1966 (general Lie group + ideal fluid).

  requires: symplectic-geometry.coadjoint-orbit, symplectic-geometry.geodesic-flow-hamiltonian, lie-theory.lie-group

• topology.blakers-masseyopen unit 03.12.21 →

  Blakers-Massey theorem

  Homotopy excision for CW triads $X=A\cup B$ with $C=A\cap B$ path-connected and connectivity hypotheses on the inclusions. ${\pi }_k(A,C)\to {\pi }_k(X,B)$ is iso for $k<m+n$, surjective at $m+n$. Homology-excision-up-to-stable-range. Freudenthal suspension theorem as corollary: ${\pi }_k(X)\to {\pi }_{k+1}(\Sigma X)$ iso for $k<2n-1$ when $X$ is $(n-1)$-connected. Foundation of stable homotopy theory and Adams spectral sequence. ∞-categorical generalisation in modal homotopy type theory.

  requires: topology.homotopy, topology.cw-complex, topology.hurewicz-theorem

• topology.euler-characteristicopen unit 03.12.23 →

  Euler characteristic

  Cellular form $\chi (X)=\sum (-1{)}^n{c}_n$. Homological form $\chi (X)=\sum (-1{)}^n\mathrm{rk}{H}_n$. Cellular = homological proof. Multiplicativity via Künneth. Euler-Poincaré for fibre bundles $\chi (E)=\chi (F)\chi (B)$. Vanishes on closed odd-dim orientable manifolds (Poincaré duality + alternating sum). Gauss-Bonnet $\int KdA=2\pi \chi$ on closed surfaces; Chern-Gauss-Bonnet via Pfaffian in even dim. Poincaré-Hopf $\sum {\mathrm{ind}}_{p}V=\chi (M)$. Lefschetz fixed-point. Examples: $S^n,{\Sigma }_g,{\mathbb{RP}}^n,{\mathbb{CP}}^n,T^n$. Originator: Euler 1758 Elementa doctrinae solidorum.

  requires: topology.cw-complex, topology.cellular-homology, topology.poincare-duality

• classical-mechanics.hamilton-jacobiopen unit 05.05.04 →

  Hamilton-Jacobi equation

  $\partial S/\partial t+H(q,\partial S/\partial q,t)=0$ for action $S(q,t)$; time-indep $H(q,\partial W/\partial q)=E$. Generating-function-type-2 making $H\mapsto 0$. Method of characteristics: characteristics = Hamilton flow. Complete integrals giving full integration. Separation of variables (central potentials, Stäckel framework). Action-angle coordinates from HJ. Geometric: solutions = Lagrangian submanifolds of $T^{\ast }M$. WKB / eikonal limit of Schrödinger. Viscosity-solution theory (Crandall-Lions 1983) for caustics + optimal control. Originator: Hamilton 1834; Jacobi 1866 computational.

  requires: symplectic-geometry.generating-functions, symplectic-geometry.cotangent-bundle, classical-mechanics.legendre-transform

• classical-mechanics.liouville-volumeopen unit 05.02.07 →

  Liouville's volume theorem

  Hamiltonian flows preserve symplectic volume ${\omega }^n/n!$. Proof via Cartan: ${\mathcal{L}}_{X_H}\omega =0\Rightarrow {\mathcal{L}}_{X_H}{\omega }^n=0$. Darboux-coordinate divergence-free form. Liouville equation ${\partial }_t\rho +\{\rho ,H\}=0$ for phase-space density. Equilibrium measures as $f(H)$. Volume-rigid + length-flexible characterising symplectic geometry. Counterexample: gradient flows are NOT volume-preserving. Foundation for Poincaré recurrence and statistical mechanics. Originator: Liouville 1838.

  requires: symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.symplectic-manifold

• classical-mechanics.poincare-recurrenceopen unit 05.02.08 →

  Poincaré recurrence theorem

  On finite measure space with measure-preserving $T:X\to X$, every measurable $A$ with $\mu (A)>0$ has a.e. point returning infinitely often. Pigeonhole proof on $T^{-n}(A)$ disjoint iterates. Mean recurrence time = $\mu (X)/\mu (A)$ (Kac's lemma). Hamiltonian application via Liouville volume. Boltzmann/Zermelo timescale resolution of H-theorem tension. Quantum recurrence in finite-dim Hilbert space; failure in infinite-dim. Foundation of ergodic theory (ergodicity, mixing, K-systems, Bernoulli). Originator: Poincaré 1890; Carathéodory 1919 abstract.

  requires: classical-mechanics.liouville-volume, symplectic-geometry.hamiltonian-vector-field

• symplectic-geometry.poincare-cartan-invariantsopen unit 05.02.09 →

  Poincaré-Cartan integral invariants

  Extended phase space $\tilde{M}=T^{\ast }Q\times \mathbb{R}$. Poincaré-Cartan one-form $\theta =pdq-Hdt$. Differential $\omega =d\theta =dp\wedge dq-dH\wedge dt$ has rank $2n$ and one-dimensional kernel; suspended Hamiltonian vector field ${\tilde{X}}_H=X_H+{\partial }_t$ is the kernel direction. First integral invariant ${\oint }_{\gamma }\theta$ flow-invariant on closed one-cycles (Poincaré 1890). Higher invariants ${\omega }^k$ for $k=1,\dots ,n$ — Cartan 1922 graded family with Liouville volume ${\omega }^n/n!$ as top degree. Cartan-formula proof: ${\iota }_{{\tilde{X}}_H}\omega =0$ plus closedness of $\omega$ gives ${\mathcal{L}}_{{\tilde{X}}_H}{\omega }^k=0$. Tube-of-trajectories reformulation: integrals on two rims of any tube agree by Stokes. Action variables $I_k=\frac{1}{2\pi }{\oint }_{{\gamma }_k}pdq$ on a Liouville torus are first invariants on basis loops; angles canonical-conjugate (Liouville-Arnold). Maupertuis principle on energy level $H=E$ as reduction of Hamilton's principle to spatial $\delta \int pdq=0$. Cartan's relative-versus-absolute distinction. Originator: Poincaré 1890/1899; Cartan 1922 systematic theory.

  requires: symplectic-geometry.symplectic-manifold, symplectic-geometry.hamiltonian-vector-field, classical-mechanics.liouville-volume

• symplectic-geometry.adiabatic-invariantsopen unit 05.09.02 →

  Adiabatic invariants

  Slowly-varying Hamiltonian $H(q,p;\lambda (ϵt))$. Adiabatic theorem (Burgers/Ehrenfest 1916, 1D): action $I=(1/2\pi )\oint pdq$ conserved up to $O(ϵ)$ over time $O({ϵ}^{-1})$. Geometric proof: average over fast angle. Higher-dim issues with resonant tori. Magnetic mirror $\mu =m{v}_{\perp }^2/(2B)$ adiabatic invariant for charged-particle motion (foundational for tokamaks). Quantum adiabatic theorem (Born-Fock 1928). Berry phase as adiabatic-correction holonomy. Nekhoroshev exponential-stability extension. Connection to KAM. Originator: Ehrenfest 1913-16.

  requires: symplectic-geometry.action-angle-coordinates, symplectic-geometry.integrable-system, symplectic-geometry.symplectic-manifold

• symplectic-geometry.birkhoff-normal-formopen unit 05.09.03 →

  Birkhoff normal form

  Hamiltonian system near an elliptic equilibrium $z_0$ with $H_2=\frac{1}{2}\sum {\omega }_j(q_j^2+p_j^2)$. If $\omega =({\omega }_1,\dots ,{\omega }_n)$ is non-resonant up to order $N$ — no integer relation $\sum k_j{\omega }_j=0$ with $|k|\le N$ — then a sequence of canonical (Lie-series) transformations puts $H=H_2+Z_4+Z_6+\cdots +Z_N+R_{N+1}$ where each $Z_{2k}$ depends only on the actions $I_j=(q_j^2+p_j^2)/2$ and $R_{N+1}=O(\Vert z{\Vert }^{N+1})$. Construction: at each order $k$, generating function $F_k$ with $\{H_2,F_k\}$ killing the non-resonant part of $H_k$; resonant terms become $Z_k$. Small-divisor / Poincaré non-integrability: full series generally diverges; Diophantine condition gives Gevrey-class smoothness. Examples: 1D (action-angle), 2D non-resonant (full Birkhoff), 2D 1:1 resonance (focus/saddle/centre classification), Lyapunov families near triangular Lagrange points in restricted three-body problem. Refinements: Birkhoff-Gustavson algorithm; KAM as the convergent island in the Birkhoff scheme; Nekhoroshev stability bounds derived from finite-order Birkhoff truncation. Originator: Birkhoff 1927; Siegel 1942 small divisors; Moser 1956 analytic refinement; Pöschel 1989 Gevrey.

  requires: symplectic-geometry.hamiltonian-vector-field, symplectic-geometry.action-angle-coordinates, symplectic-geometry.symplectic-manifold

• symplectic-geometry.nekhoroshevopen unit 05.09.06 →

  Nekhoroshev estimates

  Near-integrable Hamiltonian $H=H_0(I)+ϵH_1(I,\theta )$ with $H_0$ analytic and steep (a generic non-degeneracy condition stronger than non-resonance, weaker than convexity). Theorem (Nekhoroshev 1977): for all initial conditions and all $|t|\le T(ϵ)\asymp \exp (c{ϵ}^{-a})$, $|I(t)-I(0)|\le C{ϵ}^b$ for explicit Nekhoroshev exponents $a,b$ depending on $H_0$ and dimension. Comparison with KAM. KAM gives $|I(t)-I(0)|=0$ on a positive-measure Cantor set of Diophantine initial conditions for all time; Nekhoroshev gives $|I(t)-I(0)|=O({ϵ}^b)$ for every initial condition but only over an exponentially long interval. Steepness condition. $H_0$ is steep at $I^{\ast }$ if for every linear subspace $L\subseteq {\mathbb{R}}^n$ through $I^{\ast }$, the restriction $H_0{|}_L$ has only a degenerate isolated extremum at $I^{\ast }$ along directions in some lower-dim sublattice. Generic in $C^{\infty }$ topology, weaker than convexity, but excludes degenerate $H_0$. Proof outline (Lochak 1992 / Pöschel 1993). Block Birkhoff normal form on resonance regions: each resonance class ${\mathcal{M}}_K=\{⟨k,\omega ⟩=0:k\in K\}$ gets its own normal form, the non-resonant part is killed up to order $\sim 1/ϵ$, and steepness controls geometric diffusion across resonance regions. Iteration over resonance lattices gives the exponential time bound. Examples and exponents. Convex $H_0$ (Lochak): $a=1/(2n)$, $b=1/(2n)$. Quasi-convex (Pöschel 1993): same exponents — convex on energy levels suffices. Generic steep: smaller exponents, same shape. Solar system (Niederman 2007): explicit Nekhoroshev-type stability bounds; Arnold conjectured Lyapunov stability of the planetary motion at the perihelion-precession level over Hubble timescales. Arnold diffusion. For non-convex $H_0$ in dimensions $n\ge 3$, Nekhoroshev is essentially optimal: there exist initial conditions with $|I(t)-I(0)|\to \infty$. Arnold 1964 Instability of dynamical systems with several degrees of freedom (Doklady Akad. Nauk SSSR 156) gave the first explicit example via heteroclinic chains of whiskered tori. Mather, Berti-Bolle, Cheng-Yan made the diffusion mechanism rigorous in various settings. Combined picture (Bourgain-Kuksin and beyond). KAM gives positive-measure permanent stability; Nekhoroshev gives global polynomial stability over exponentially long times; Arnold diffusion gives the exceptional unstable orbits in the resonance gaps. Originator: Nekhoroshev 1977; Lochak 1992 simplification via simultaneous Diophantine approximation; Pöschel 1993 explicit constants.

  requires: symplectic-geometry.kam-theorem, symplectic-geometry.action-angle-coordinates, symplectic-geometry.birkhoff-normal-form

• symplectic-geometry.williamson-normal-formopen unit 05.09.04 →

  Williamson normal form for quadratic Hamiltonians

  Symplectic congruence classification of quadratic Hamiltonians $H(z)=\frac{1}{2}{z}^{T}Az$ on $({\mathbb{R}}^{2n},{\omega }_0)$ with $A$ real symmetric. Positive-definite case (the headline statement): there exists $S\in \mathrm{Sp}(2n,\mathbb{R})$ with $S^{T}AS=\mathrm{diag}({\omega }_1,\dots ,{\omega }_n,{\omega }_1,\dots ,{\omega }_n)$, where the positive numbers $\{{\omega }_1,\dots ,{\omega }_n\}$ — the symplectic eigenvalues or Williamson invariants of $A$ — are read off as the absolute values of the eigenvalues of $JA$ (which are $\pm i{\omega }_k$, all purely imaginary). The unordered multiset $\{{\omega }_k\}$ is a complete symplectic-conjugacy invariant. Proof: $JA$ is skew-self-adjoint for the inner product $⟨u,v{⟩}_A=u^{T}Av$ (using $A^T=A$, $J^T=-J$), so its spectrum is purely imaginary; pair complex eigenvectors with their conjugates to build a real symplectic basis that simultaneously diagonalises $A$. Indefinite case (full Williamson 1936 classification): $JA$-spectrum decomposes under joint $\pm$ and $\bar{\cdot }$ symmetries into orbits of four types — purely imaginary pair (elliptic, $H=\omega (q^2+p^2)/2$ with a $\pm$ Krein sign), real pair (saddle, $H=\omega qp$), complex quadruple (loxodromic), and zero (parabolic / nilpotent). Long 1971 refinement at degenerate strata (Williamson-Long form). Connections: quadratic linearisation underlying the Birkhoff normal form near an elliptic equilibrium; symplectic eigenvalues are the frequency vector entering the KAM Diophantine condition; symplectic capacity of an ellipsoid $\{z^{T}Az\le 1\}$ is $\pi /{\omega }_{\max }$; metaplectic quantisation has spectrum $\sum (n_k+1/2){\omega }_k$; Robertson-Schrödinger uncertainty principle bounds each symplectic eigenvalue of a Gaussian covariance matrix below by $\hslash /2$; Krein theory of strong stability identifies elliptic-block sign collisions as the linear birth of parametric resonance instabilities (e.g. triangular Lagrange points at the critical mass ratio of the restricted three-body problem). Originator: Williamson 1936 On the algebraic problem concerning the normal forms of linear dynamical systems (Amer. J. Math. 58); Long 1971 refinement; Arnold App. 6 mechanics-flavoured exposition.

  requires: symplectic-geometry.symplectic-vector-space, symplectic-geometry.symplectic-manifold, symplectic-geometry.symplectic-group

• topology.cofibrationopen unit 02.01.08 →

  Cofibration and homotopy extension property

  Map $i:A\to X$ with the homotopy extension property: any homotopy $H:A\times [0,1]\to Z$ compatible with $f:X\to Z$ extends to $\tilde{H}:X\times [0,1]\to Z$. Equivalent retract characterisation: $X\times \{0\}{\cup }_{A\times \{0\}}A\times [0,1]$ is a retract of $X\times [0,1]$. Mapping-cylinder factorisation $X\hookrightarrow M_f\stackrel{\simeq }{\to }Y$. CW pair inclusions are cofibrations. Eckmann-Hilton dual to fibration; cofibre sequence $A\to X\to X/A\to \Sigma A\to \cdots$. Originator: Borsuk 1931 (ANR theory); modern HEP framing Strom 1968 / Steenrod 1967.

  requires: topology.topological-space, topology.continuous-map, topology.quotient-topology, topology.homotopy

• topology.compact-open-topologyopen unit 02.01.09 →

  Compact-open topology and function spaces

  Compact-open topology on $C(X,Y)$: subbasis $W(K,U)=\{f:f(K)\subseteq U\}$ for $K$ compact, $U$ open. Evaluation $C(X,Y)\times X\to Y$ continuous when $X$ is locally compact Hausdorff. Exponential law $C(X\times Y,Z)\cong C(X,C(Y,Z))$ as a homeomorphism. Compactly-generated weak Hausdorff (CGWH) = Steenrod's convenient category. Loop space $\Omega X$, suspension-loop adjunction $[\Sigma X,Y]\cong [X,\Omega Y]$. Originator: Fox 1945; Steenrod 1967 A convenient category of topological spaces.

  requires: topology.topological-space, topology.continuous-map

• topology.cw-complexopen unit 03.12.10 →

  CW complex

  Inductive skeleton construction $X^{(n)}=X^{(n-1)}{\cup }_{\Phi }{⨆}_{\alpha }{D}_{\alpha }^n$ with attaching map $\Phi :{⨆}_{\alpha }{S}_{\alpha }^{n-1}\to X^{(n-1)}$. Weak (colimit) topology on $X=\mathrm{colim}{X}^{(n)}$. Cellular pushout square. Standard examples: $S^n$, ${\mathbb{RP}}^n$, ${\mathbb{CP}}^n$, classifying spaces, Lie groups. CW pair inclusions are cofibrations. Cellular approximation theorem; Whitehead's theorem (homotopy-equivalence detection by ${\pi }_n$). Originator: J.H.C. Whitehead 1949 Combinatorial homotopy I.

  requires: topology.topological-space, topology.quotient-topology, topology.homotopy

• topology.fibrationopen unit 02.01.07 →

  Fibration (Hurewicz and Serre)

  Hurewicz fibration: HLP for all spaces. Serre fibration: HLP for CW pairs / discs. Standard examples: covering maps, fibre bundles over paracompact bases, path-space fibration $PX\to X$ with fibre $\Omega X$. Long exact sequence of homotopy groups ${\pi }_n(F)\to {\pi }_n(E)\to {\pi }_n(B)\to {\pi }_{n-1}(F)$. Fibration replacement: any map factors through a Hurewicz fibration. Hopf fibration $S^1\to S^3\to S^2$ as the originator example. Loop-space adjunction: ${\pi }_n(\Omega X)\cong {\pi }_{n+1}(X)$. Connection to Leray-Serre spectral sequence.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.quotient-topologyopen unit 02.01.06 →

  Quotient and identification topology

  Identification topology on $Y=X/\sim$: $U$ open iff $q^{-1}(U)$ open in $X$. Universal property: continuous maps $Y\to Z$ are continuous maps $X\to Z$ constant on equivalence classes. Standard quotients: cone $CX$, suspension $SX$ ($\Sigma S^n=S^{n+1}$), mapping cylinder $M_f$, mapping cone $C_f$, adjunction space $Y{\cup }_{f}X$, wedge $X\vee Y$, smash $X\wedge Y$. Cellular pushout for CW complex skeleton attachments. Quotient by group action gives covering spaces when properly discontinuous.

  requires: topology.topological-space, topology.continuous-map

• topology.seifert-van-kampenopen unit 03.12.09 →

  Seifert-van Kampen theorem

  Classical group form: ${\pi }_1(U\cup V,x_0)={\pi }_1(U,x_0){\ast }_{{\pi }_1(U\cap V,x_0)}{\pi }_1(V,x_0)$ when $U$, $V$, $U\cap V$ path-connected. Brown's groupoid form: pushout in $\mathbf{Groupoid}$ for ${\pi }_1(X,A)$ with $A$ meeting every path-component of $U$, $V$, $U\cap V$ — no connectedness hypothesis on $U\cap V$. Lebesgue-number subdivision argument. Key computations: figure eight ($F_2$), sphere ($0$), genus-$g$ surface ($⟨a_i,b_i\mid \prod [a_i,b_i]⟩$).

  requires: topology.homotopy, topology.fundamental-groupoid, topology.topological-space

• rep-theory.cartan-weyl-classificationopen unit 07.04.01 →

  Cartan-Weyl classification

  Bijection {irreducible Dynkin diagrams} ↔ {simple complex Lie algebras}. Four infinite families $A_n,B_n,C_n,D_n$ and five exceptionals $G_2,F_4,E_6,E_7,E_8$. Root systems with crystallographic condition. Cartan matrix and Serre relations. Compact real forms. Langlands duality on root systems. Connections to Kac-Moody algebras, buildings (Tits), McKay correspondence, $E_8$ in lattice theory and string theory.

  requires: rep-theory.group-representation, rep-theory.highest-weight-representation, lie.lie-algebra

• rep-theory.highest-weight-representationopen unit 07.03.01 →

  Highest weight representation

  Cartan + root decomposition $\mathfrak{g}=\mathfrak{h}\oplus {\mathfrak{n}}^{+}\oplus {\mathfrak{n}}^{-}$. Weight space decomposition. Highest weight vector annihilated by ${\mathfrak{n}}^{+}$. Verma modules $M(\lambda )$, irreducible quotient $L(\lambda )$. Bijection between dominant integral weights and finite-dim irreducibles. Weyl character formula and Weyl dimension formula. Borel-Weil(-Bott) realisation. Category $\mathcal{O}$, Kazhdan-Lusztig polynomials, crystals, quantum groups.

  requires: rep-theory.group-representation, rep-theory.schur-lemma, lie.lie-algebra

• rep-theory.schur-lemmaopen unit 07.01.02 →

  Schur's lemma

  (1) An equivariant linear map between irreducibles is zero or an isomorphism. (2) Over an algebraically closed field, ${\mathrm{End}}_G(V)=k\cdot {\mathrm{id}}_V$ for irreducible $V$. Powers character orthogonality, dimension formula $\sum (\dim V_i{)}^2=|G|$. Generalises to simple modules over algebras and to abelian categories. von Neumann's commutant theorem extends to unitary representations. Schur-Weyl duality and categorical Schur.

  requires: rep-theory.group-representation, linalg.vector-space

• rep-theory.group-representationopen unit 07.01.01 →

  Group representation

  Homomorphism $\rho :G\to \mathrm{GL}(V)$ equivalently a $kG$-module structure on $V$. Subrepresentations, irreducibility, semisimplicity, intertwiners. Direct sum, tensor product, dual, Hom. Maschke's theorem (complete reducibility in char 0 or coprime to $|G|$). Characters, orthogonality relations. Regular representation. Frobenius reciprocity. Connections to modular and categorical representation theory.

  requires: groups.group, linalg.vector-space

• rep-theory.symmetric-group-representationopen unit 07.05.01 →

  Symmetric group representation

  Bijection {partitions of $n$} ↔ {irreducible $S_n$-reps over $\mathbb{C}$}. Frobenius character formula via Vandermonde × power-sum determinant. Young symmetriser $c_{\lambda }\in \mathbb{C}[S_n]$ realising $V^{\lambda }=\mathbb{C}[S_n]\cdot c_{\lambda }$ as a left ideal. Hook length formula $\dim V^{\lambda }=n!/\prod h(i,j)$. Murnaghan-Nakayama border-strip rule. Frobenius characteristic map to symmetric functions. Schur-Weyl duality with ${\mathrm{GL}}_d$. RSK correspondence. Connections to Hecke algebras, Brauer algebras, crystal bases, Schubert calculus.

  requires: rep-theory.group-representation, rep-theory.schur-lemma

• rep-theory.young-diagramopen unit 07.05.02 →

  Young diagram and tableau

  Young diagram $[\lambda ]$ as left-justified array of cells encoding partition $\lambda ⊢n$. Standard Young tableau (SYT): bijective filling strictly increasing along rows and columns; $f^{\lambda }=|\mathrm{SYT}(\lambda )|$. Semistandard Young tableau (SSYT): weak rows, strict columns, with content. Hook length formula $f^{\lambda }=n!/\prod h(i,j)$ (Frame-Robinson-Thrall 1954); Greene-Nijenhuis-Wilf hook-walk proof; Novelli-Pak-Stoyanovskii bijective proof. Schur polynomials $s_{\lambda }$ as generating functions of SSYTs; basis of $\Lambda$. RSK correspondence $S_n\leftrightarrow {⨆}_{\lambda }\mathrm{SYT}(\lambda {)}^2$. Littlewood-Richardson rule. Plancherel measure asymptotics (Vershik-Kerov-Logan-Shepp; Baik-Deift-Johansson Tracy-Widom). Crystal bases (Kashiwara). Schubert calculus on Grassmannians.

  requires: rep-theory.symmetric-group-representation

• rep-theory.specht-moduleopen unit 07.05.03 →

  Specht module

  Permutation module $M^{\lambda }=k\{\text{tabloids}\}={\mathrm{Ind}}_{S_{\lambda }}^{S_n}(\mathrm{triv})$. Polytabloid $e_T={\sum }_{\sigma \in C(T)}\mathrm{sgn}(\sigma )\{\sigma T\}$ (column antisymmetrisation). Specht module $S^{\lambda }=k⟨e_T⟩\subseteq M^{\lambda }$. Standard polytabloids $\{e_T:T\in \mathrm{SYT}(\lambda )\}$ form a $\mathbb{Z}$-basis. Theorem: in char 0, $\{S^{\lambda }{\}}_{\lambda ⊢n}$ are exactly the irreducible $k{S}_n$-modules; in char $p$, $D^{\lambda }=S^{\lambda }/\mathrm{rad}(S^{\lambda })$ for $p$-regular $\lambda$ are the irreducible modular representations (James 1976). James submodule theorem: any submodule $U\subseteq M^{\lambda }$ contains $S^{\lambda }$ or sits in $(S^{\lambda }{)}^{\perp }$. Garnir relations. Branching rule via removable corners. Mullineux involution $D^{\lambda }\otimes \mathrm{sgn}\cong D^{m_p(\lambda )}$ (Ford-Kleshchev 1997). Connections to Hecke algebras, $q$-Schur algebras, LLT algorithm, KLR categorification, Cellular algebras (Graham-Lehrer).

  requires: rep-theory.symmetric-group-representation, rep-theory.young-diagram

• complex-analysis.riemann-roch-compact-rsopen unit 06.04.01 →

  Riemann-Roch theorem for compact Riemann surfaces

  $\ell (D)-i(D)=\mathrm{deg}(D)-g+1$ on compact Riemann surface of genus $g$. Analytic version of algebraic Riemann-Roch [04.04.01]; equivalent by Serre's GAGA. Index of speciality $i(D)=\dim \{\omega \in {\Omega }_{\mathrm{hol}}^1:(\omega )\ge D\}$. Serre duality identifies $i(D)=\ell (K-D)$. Hodge decomposition powers the analytic proof. Riemann-Hurwitz, Brill-Noether, Clifford generalisations. Hirzebruch-Riemann-Roch and Grothendieck-Riemann-Roch generalisations.

  requires: complex-analysis.holomorphic-function, complex-analysis.riemann-surface, alg-geom.riemann-roch-curves

• complex-analysis.riemann-surfaceopen unit 06.03.01 →

  Riemann surface

  1-dimensional complex manifold (real dimension 2). Charts to $\mathbb{C}$ with holomorphic transitions. Examples: $\mathbb{C}$, Riemann sphere $\hat{\mathbb{C}}$, tori $\mathbb{C}/\Lambda$, smooth complex projective curves. Genus and Euler characteristic. Uniformization: every simply connected Riemann surface is $\hat{\mathbb{C}}$, $\mathbb{C}$, or $\mathbb{H}$. GAGA: compact Riemann surfaces $=$ smooth complex projective curves. Sheaves ${\mathcal{O}}_X$, ${\omega }_X$, $\mathcal{O}(D)$. Hodge decomposition, Abel-Jacobi, Teichmüller and moduli spaces.

  requires: complex-analysis.holomorphic-function, manifolds.smooth-manifold, topology.topological-space

• complex-analysis.holomorphic-functionopen unit 06.01.01 →

  Holomorphic function

  Complex differentiability $f^{\prime }(z_0)={\lim }_{h\to 0}(f(z_0+h)-f(z_0))/h$ direction-independently. Cauchy-Riemann equations $u_x=v_y$, $u_y=-v_x$ equivalent to $\bar{\partial }f=0$. Analyticity (local power series). Cauchy's theorem, Cauchy integral formula, residue theorem, Liouville, maximum modulus, identity theorem. Conformal maps, open mapping theorem, Schwarz lemma, Riemann mapping theorem, Picard, Mittag-Leffler.

  requires: topology.topological-space, topology.continuous-map

• alg-geom.cartier-divisoropen unit 04.05.04 →

  Cartier divisor

  Section of ${\mathcal{K}}^{\times }/{\mathcal{O}}^{\times }$ — system of local equations $\{(U_i,f_i)\}$ with $f_i/f_j\in {\mathcal{O}}^{\times }$. $\mathrm{CaCl}(X)=\mathrm{Pic}(X)$ exactly. On smooth/locally-factorial schemes, $\mathrm{CaDiv}=\mathrm{Div}$. Effective Cartier divisor = locally principal codimension-1 closed subscheme. Failure of Weil = Cartier on cone over conic. Base-change-friendly, the natural relative-setting divisor. Cartier dual of group schemes. Arakelov arithmetic.

  requires: alg-geom.weil-divisor, alg-geom.line-bundle-scheme, alg-geom.coherent-sheaf

• alg-geom.line-bundle-schemeopen unit 04.05.03 →

  Line bundle on a scheme

  Locally free coherent sheaf of rank 1. Transition functions in ${\mathcal{O}}^{\times }$. $\mathcal{L}\otimes {\mathcal{L}}^{-1}={\mathcal{O}}_X$. Picard group $\mathrm{Pic}(X)=H^1(X;{\mathcal{O}}^{\times })$ via Čech cohomology. Equivalence with Cartier divisors. On locally factorial schemes, $\mathrm{Pic}=\mathrm{Cl}$. Twisting sheaves $\mathcal{O}(d)$ on ${\mathbb{P}}^n$. Exponential sequence and first Chern class. Picard scheme. Ample/nef/big positivity classes.

  requires: alg-geom.coherent-sheaf, alg-geom.quasi-coherent-sheaf, alg-geom.weil-divisor

• alg-geom.weil-divisoropen unit 04.05.01 →

  Weil divisor

  Formal $\mathbb{Z}$-linear combinations of codimension-1 prime divisors. $\mathrm{div}(f)=\sum v_Y(f)[Y]$ for rational $f$. Principal divisors and divisor class group $\mathrm{Cl}(X)$. Linear equivalence. $\mathrm{Cl}({\mathbb{P}}^n)=\mathbb{Z}$ via degree. ${\mathrm{Cl}}^0$ for curves equals Jacobian. Connection to ideal class group of number fields. Equality $\mathrm{Cl}=\mathrm{Pic}$ for locally factorial schemes. Excision sequence for class groups. Foundation for Chow rings.

  requires: alg-geom.scheme, alg-geom.affine-scheme, alg-geom.coherent-sheaf

• alg-geom.coherent-sheafopen unit 04.06.02 →

  Coherent sheaf

  Quasi-coherent + locally finitely generated. On Noetherian $\mathrm{Spec}(R)$: ${\mathbf{Mod}}^{\mathrm{fg}}(R)\cong \mathbf{Coh}(\mathrm{Spec}(R))$. Closed under kernel/cokernel/tensor/Hom/pullback. Pushforward coherent under proper morphisms (Grothendieck EGA III). Finiteness theorem: $H^i$ on projective scheme is finitely generated. Hilbert polynomial, Castelnuovo-Mumford regularity. Resolution by locally free sheaves on regular schemes.

  requires: alg-geom.quasi-coherent-sheaf, alg-geom.scheme

• alg-geom.quasi-coherent-sheafopen unit 04.06.01 →

  Quasi-coherent sheaf

  On $\mathrm{Spec}(R)$, quasi-coherent sheaves $\tilde{M}$ correspond to $R$-modules $M$ via $\tilde{M}(D(f))=M[f^{-1}]$. Equivalence $\mathbf{Mod}(R)\cong \mathbf{QCoh}(\mathrm{Spec}(R))$. Closed under kernels, cokernels, tensor product, pullback, pushforward (q-c-q-s). Serre's vanishing on affines: $H^i(\mathrm{Spec}(R);\tilde{M})=0$ for $i\ge 1$. On Proj: $\mathbf{grMod}(S)/\mathbf{Tor}\cong \mathbf{QCoh}(\mathrm{Proj}(S))$.

  requires: alg-geom.sheaf, alg-geom.scheme, alg-geom.affine-scheme

• alg-geom.affine-schemeopen unit 04.02.02 →

  Affine scheme

  $\mathrm{Spec}(R)=\{\text{prime ideals of }R\}$ with Zariski topology and structure sheaf $\mathcal{O}$. Stalks are localisations $R_{\mathfrak{p}}$. Locally ringed space. Spec/Γ adjunction: ${\mathbf{CRing}}^{\mathrm{op}}\cong \mathbf{AffSch}$. Distinguished opens $D(f)$, sections $\mathcal{O}(D(f))=R_f$. Hilbert Nullstellensatz reformulation. Generic vs closed points. Non-reduced schemes via nilpotents. $\mathrm{Spec}(\mathbb{Z})$ as arithmetic line.

  requires: alg-geom.sheaf, alg-geom.scheme, algebra.associative-algebra, algebra.ideal

• alg-geom.projective-schemeopen unit 04.02.03 →

  Projective scheme

  $\mathrm{Proj}(S)$ for graded commutative ring $S$; homogeneous primes excluding the irrelevant ideal $S_{+}$. Affine cover by distinguished opens $D(f)\cong \mathrm{Spec}((S[f^{-1}]{)}_0)$. Twisting sheaves $\mathcal{O}(d)$. Every projective $k$-scheme embeds in ${\mathbb{P}}_k^N$. Veronese and Segre embeddings. Closed subschemes correspond to homogeneous ideals. Coherent sheaves correspond to graded modules modulo torsion. Hilbert polynomial / regularity / Hilbert scheme.

  requires: alg-geom.sheaf, alg-geom.scheme, alg-geom.affine-scheme, alg-geom.projective-space

• alg-geom.projective-spaceopen unit 04.07.01 →

  Projective space

  ${\mathbb{P}}_k^n=(k^{n+1}\setminus 0)/k^{\times }=\mathrm{Proj}k[x_0,\dots ,x_n]$. Open affine cover by $U_i=\{x_i\ne 0\}$. Twisting sheaves $\mathcal{O}(d)$, foundational cohomology computation. $\mathrm{Pic}({\mathbb{P}}^n)=\mathbb{Z}$. Canonical $\omega =\mathcal{O}(-n-1)$. Functor of points: surjections $R^{n+1}\to L$. Toric perspective. Plücker embedding of Grassmannians. Bézout's theorem.

  requires: alg-geom.scheme, linalg.vector-space, linalg.field

• alg-geom.riemann-roch-curvesopen unit 04.04.01 →

  Riemann-Roch theorem for curves

  $\ell (D)-\ell (K-D)=\mathrm{deg}(D)-g+1$ on smooth projective curve of genus $g$. Equivalently $\chi (\mathcal{L})=\mathrm{deg}(\mathcal{L})+1-g$. Serre duality $H^1(C;\mathcal{L})\cong H^0(C;{\omega }_C\otimes {\mathcal{L}}^{-1}{)}^{\vee }$. Canonical divisor of degree $2g-2$. Inductive proof via skyscraper short exact sequences. Hirzebruch and Grothendieck generalisations. Brill-Noether theory and Clifford's theorem for special divisors.

  requires: alg-geom.sheaf, alg-geom.scheme, alg-geom.sheaf-cohomology

• alg-geom.hurwitz-formulaopen unit 04.04.02 →

  Hurwitz formula

  $2g_Y-2=n(2g_X-2)+\mathrm{deg}R$ for a finite separable morphism $\pi :Y\to X$ of smooth projective curves of degree $n$, with ramification divisor $R={\sum }_y(e_y-1)y$ in the tame case ($\mathrm{char}k\nmid e_y$). Proof via ${\pi }^{\ast }{K}_X=K_Y-R$ and degree comparison. Worked examples: $t\mapsto t^n$ on ${\mathbb{P}}^1$, hyperelliptic double cover ramified at $2g+2$ points, elliptic double cover with four branch points. Wild ramification in positive characteristic replaces $\sum (e_y-1)$ with the different ${\mathfrak{d}}_{Y/X}$. Castelnuovo-Severi inequality bounds $g_Y$ in terms of $g_X$, $n$, and ramification.

  requires: alg-geom.riemann-roch-curves

• alg-geom.elliptic-curvesopen unit 04.04.03 →

  Elliptic curves

  A smooth projective curve of genus 1 over $k$ equipped with a $k$-rational point $O$. Equivalently a 1-dimensional abelian variety over $k$. Weierstrass form $y^2=x^3+ax+b$ with $\Delta =-16(4a^3+27b^2)\ne 0$ (in $\mathrm{char}k\ne 2,3$); $j$-invariant $j(E)=1728\cdot 4a^3/(4a^3+27b^2)$ classifies $E$ over $\bar{k}$. Group law via chord-and-tangent construction or via $E(k)\cong {\mathrm{Pic}}^0(E)$ from Riemann-Roch on a genus-1 curve. Mordell-Weil theorem: $E(K)$ is finitely generated for every number field $K$ (Mordell 1922 for $\mathbb{Q}$, Weil 1929 for general number fields). Mazur's torsion theorem 1977: $E(\mathbb{Q}{)}_{\mathrm{tors}}$ is one of 15 explicit groups. Modularity theorem (Wiles-Taylor-Breuil-Conrad-Diamond 1995-2001): every $E/\mathbb{Q}$ is modular. BSD conjecture: ${\mathrm{ord}}_{s=1}L(E,s)=\mathrm{rank}(E(\mathbb{Q}))$. Hasse bound over ${\mathbb{F}}_q$: $|\#E({\mathbb{F}}_q)-q-1|\le 2\sqrt{q}$. Tate's algorithm for reduction types. CM theory and class field theory of imaginary quadratic fields. Heegner points and Gross-Zagier-Kolyvagin for rank-$\le 1$ BSD. Modular curves $X_0(N),X_1(N)$ as moduli of elliptic curves with level structure.

  requires: alg-geom.riemann-roch-curves, alg-geom.hurwitz-formula, alg-geom.picard-group

• alg-geom.sheaf-cohomologyopen unit 04.03.01 →

  Sheaf cohomology

  $H^i(X;\mathcal{F})=R^i\Gamma (X,-)$ as right-derived functors of global sections. Long exact sequence in cohomology from short exact sequence of sheaves. Čech cohomology and comparison theorem. Acyclic resolutions: flabby, soft, fine. Serre's vanishing theorem on affine schemes. Grothendieck vanishing in degrees above dimension. Leray spectral sequence. Hodge decomposition for Kähler manifolds. Coherent cohomology of ${\mathbb{P}}^n$ via $\mathcal{O}(d)$.

  requires: alg-geom.sheaf, alg-geom.scheme, topology.de-rham-cohomology

• alg-geom.schemeopen unit 04.02.01 →

  Scheme

  Locally ringed space locally isomorphic to $\mathrm{Spec}(R)$. Anti-equivalence ${\mathbf{CRing}}^{\mathrm{op}}\cong \mathbf{AffSch}$. Zariski topology, structure sheaf, projective and affine schemes. Functor of points. Reduced vs non-reduced (nilpotents).

  requires: alg-geom.sheaf, algebra.associative-algebra, algebra.ideal, topology.topological-space

• alg-geom.sheafopen unit 04.01.01 →

  Sheaf

  Presheaf as contravariant functor from open sets; sheaf axioms (locality + gluing); sheafification; stalks; morphisms; pushforward and pullback adjunction; étale space construction; sheaves form a topos.

  requires: topology.topological-space, topology.continuous-map

• alg-geom.picard-groupopen unit 04.05.02 →

  Picard group

  $\mathrm{Pic}(X)=\{\text{line bundles on }X\}/\cong$ under tensor product. Three descriptions: $H^1(X;{\mathcal{O}}_X^{\times })$ via Čech cocycles; Cartier divisor classes $\mathrm{CaCl}(X)$; Weil divisor classes $\mathrm{Cl}(X)$ on locally factorial schemes. Functorial pullback. Picard scheme ${\mathrm{Pic}}_{X/S}$ representing the relative Picard functor (Grothendieck FGA 1962). ${\mathrm{Pic}}^0(X)$ as connected component, abelian variety for smooth projective $X$. Néron-Severi $\mathrm{NS}(X)=\mathrm{Pic}(X)/{\mathrm{Pic}}^0(X)$, finitely generated. Jacobian of a curve as ${\mathrm{Pic}}^0(C)$. Examples: $\mathrm{Pic}({\mathbb{P}}^n)=\mathbb{Z}$, $\mathrm{Pic}(\mathrm{Spec}{\mathcal{O}}_K)=\mathrm{Cl}({\mathcal{O}}_K)$. Picard scheme of an abelian variety as dual abelian variety with Poincaré bundle (Mumford).

  requires: alg-geom.weil-divisor, alg-geom.line-bundle-scheme, alg-geom.cartier-divisor

• alg-geom.ample-line-bundleopen unit 04.05.05 →

  Ample and very ample line bundle

  $\mathcal{L}$ very ample if global sections give a closed immersion $X\hookrightarrow {\mathbb{P}}^N$; ample if some power ${\mathcal{L}}^{\otimes m}$ is very ample. Cartan-Serre-Grothendieck: $\mathcal{L}$ ample iff for every coherent $\mathcal{F}$, $\mathcal{F}\otimes {\mathcal{L}}^{\otimes m}$ is globally generated for $m\gg 0$ iff $H^i(X;\mathcal{F}\otimes {\mathcal{L}}^{\otimes m})=0$ for $i\ge 1$, $m\gg 0$. Numerical (Nakai-Moishezon): on a complete scheme, $\mathcal{L}$ ample iff ${\mathcal{L}}^{\dim Y}\cdot Y>0$ for every irreducible closed $Y$. Kleiman criterion for nef. Ample cone in $\mathrm{NS}(X)\otimes \mathbb{R}$, dual to Mori cone of curves. Kodaira embedding theorem (analytic ample = positive line bundle). MMP and the cone theorem.

  requires: alg-geom.line-bundle-scheme, alg-geom.coherent-sheaf, alg-geom.projective-space

• alg-geom.stalk-of-sheafopen unit 04.01.02 →

  Stalk of a sheaf

  ${\mathcal{F}}_x={\underrightarrow{\lim }}_{U\ni x}\mathcal{F}(U)$, the colimit over open neighbourhoods of $x$. Equivalently equivalence classes of pairs $(U,s)$ with $s\in \mathcal{F}(U)$, modulo agreement on a smaller neighbourhood. Stalks of ${\mathcal{O}}_X$ are local rings. Sheafification produces a sheaf with the same stalks as the input presheaf. Morphisms of sheaves are isomorphisms iff each stalk map is an isomorphism. Skyscraper sheaves. Étale space $\acute{\mathrm{E}}\mathrm{t}(\mathcal{F})={⨆}_x{\mathcal{F}}_x$ with local-section topology. Leray's 1946 introduction in Oflag XVII-A.

  requires: alg-geom.sheaf

• alg-geom.sheafificationopen unit 04.01.03 →

  Sheafification

  Left adjoint $(-{)}^{+}:\mathbf{PSh}(X)\to \mathbf{Sh}(X)$ to the inclusion. Construction via étale-space: ${\mathcal{F}}^{+}(U)=\{$continuous sections $s:U\to \acute{\mathrm{E}}\mathrm{t}(\mathcal{F})\}$. Universal property: for any sheaf $\mathcal{G}$ and presheaf morphism $\mathcal{F}\to \mathcal{G}$, factors uniquely through ${\mathcal{F}}^{+}$. Stalks unchanged: ${\mathcal{F}}_x^{+}={\mathcal{F}}_x$ for all $x$. Sheafification is exact. For sheaf categories: kernels are presheaf kernels, cokernels and images require sheafification. Foundational tool throughout sheaf theory. Cartan-Serre exposés 1948–55; systematised in Godement 1958.

  requires: alg-geom.sheaf, alg-geom.stalk-of-sheaf

• alg-geom.direct-inverse-imageopen unit 04.01.04 →

  Direct and inverse image of sheaves

  For $f:X\to Y$ continuous, pushforward $f_{\ast }\mathcal{F}(V)=\mathcal{F}(f^{-1}(V))$. Inverse image $f^{-1}\mathcal{G}$ is the sheafification of $V\mapsto {\underrightarrow{\lim }}_{U\supseteq f(V)}\mathcal{G}(U)$. Adjunction $f^{-1}⊣f_{\ast }$ on sheaves of sets. For ringed spaces: pullback $f^{\ast }\mathcal{G}=f^{-1}\mathcal{G}{\otimes }_{f^{-1}{\mathcal{O}}_Y}{\mathcal{O}}_X$, adjoint to $f_{\ast }$ on $\mathcal{O}$-modules. $f^{-1}$ exact; $f_{\ast }$ left exact only. Right derived $R^i{f}_{\ast }$ measure failure. Six functor formalism precursor: Grothendieck Tôhoku 1957. Proper base change for $f$ proper, projection formula. Modern: Lurie's $\infty$-categorical six functors; Grothendieck duality $f^{!}$.

  requires: alg-geom.sheaf

• alg-geom.morphism-of-schemesopen unit 04.02.04 →

  Morphism of schemes

  Morphism of locally ringed spaces $(f,f^{♯}):(X,{\mathcal{O}}_X)\to (Y,{\mathcal{O}}_Y)$ with $f$ continuous and $f^{♯}:{\mathcal{O}}_Y\to f_{\ast }{\mathcal{O}}_X$ a sheaf-of-rings map inducing local-ring maps on stalks. For affine schemes $\mathrm{Hom}(\mathrm{Spec}A,\mathrm{Spec}B)=\mathrm{Hom}(B,A)$ (anti-equivalence). Properties (EGA IV): finite type, finite presentation, separated, proper, flat, smooth, étale, finite, affine, projective, quasi-finite, quasi-compact. Closed/open immersions. Base change $X{\times }_{Y}Z$. Diagonal $\Delta :X\to X{\times }_{Y}X$. Valuative criteria for separatedness and properness. Functor of points: $h_X(T)=\mathrm{Hom}(T,X)$. Galois descent and faithfully flat descent for morphisms.

  requires: alg-geom.scheme, alg-geom.affine-scheme

• alg-geom.smooth-etale-unramifiedopen unit 04.02.05 →

  Smooth, étale, and unramified morphisms

  Three local properties of $f:X\to Y$ at $x\in X$, $y=f(x)$. Unramified: $f$ locally of finite type with ${\Omega }_{X/Y,x}^1=0$, equivalently ${\mathfrak{m}}_y{\mathcal{O}}_{X,x}={\mathfrak{m}}_x$ and $\kappa (x)/\kappa (y)$ finite separable. Smooth (relative dim $n$): $f$ locally of finite presentation, flat at $x$, with regular geometric fibre $X_{\bar{y}}$ at $\bar{x}$; equivalently ${\Omega }_{X/Y}^1$ locally free of rank $n$ near $x$ (EGA IV.17.5.1). Étale: smooth of relative dimension $0$ = flat + locally finitely presented + unramified. Jacobian criterion: for $X=\mathrm{Spec}(A[t_1,\dots ,t_n]/(g_1,\dots ,g_c))$ over $\mathrm{Spec}(A)$, smooth at $x$ of relative dim $n-c$ iff $\mathrm{rank}(\partial g_i/\partial t_j)(x)=c$. Stable under composition and base change. Examples: $\mathrm{Spec}(k[x,y]/(y^2-x))\to \mathrm{Spec}(k[x])$ étale on $x\ne 0$ (char $\ne 2$); Frobenius $F:X\to X$ in char $p$ never étale (purely inseparable); $z\mapsto z^n$ on ${\mathbb{G}}_m$ étale iff $n$ invertible in base. Local structure (EGA IV.18.4.6): every unramified morphism factors étale-locally as a closed immersion into an étale cover. Formal characterisations: smooth = formally smooth + locally finitely presented (lifting along square-zero ideals surjective); étale = formally étale (bijective lifting); unramified = formally unramified (injective lifting). Étale fundamental group ${\pi }_1^{\text{eˊt}}(X,\bar{x})$ classifies finite étale covers (SGA 1 Exposé V); profinite completion of ${\pi }_1^{\text{top}}(X^{\text{an}})$ for smooth complex $X$ (SGA 1 Exposé XII, Riemann existence). Henselisation / strict henselisation as étale-local rings; cotangent complex $L_{X/Y}$ as derived refinement (Illusie 1971–72); Kunz 1969: regular ⟺ Frobenius flat in char $p$. Originators: Auslander-Buchsbaum-Serre 1956–58 (regularity criterion); Grothendieck-Dieudonné EGA IV (1965–67); Grothendieck SGA 1 (1960–61, étale fundamental group).

  requires: alg-geom.morphism-of-schemes, alg-geom.sheaf-of-differentials

• alg-geom.nullstellensatz-dimensionopen unit 04.02.07 →

  Nullstellensatz and dimension theory

  Over algebraically closed $k$ and $R=k[x_1,\dots ,x_n]$: Weak Nullstellensatz — every proper ideal $\mathfrak{a}⊊R$ has $V(\mathfrak{a})\ne \varnothing$. Strong Nullstellensatz — $I(V(\mathfrak{a}))=\sqrt{\mathfrak{a}}$, equivalently $f\in I(V(\mathfrak{a}))\iff f^N\in \mathfrak{a}$. Maximal-ideal form — every maximal ideal $\mathfrak{m}\subset R$ has the form ${\mathfrak{m}}_a=(x_1-a_1,\dots ,x_n-a_n)$ for a unique $a\in k^n$. Proof: weak via Zariski lemma (a field finitely generated as a $k$-algebra equals $k$ when $k$ is algebraically closed) and Noether normalisation; weak ⇒ strong via the Rabinowitsch trick (introduce $y$ and use $(\mathfrak{a},1-yf)$). Krull dimension $\dim R=\sup$ length of prime chains ${\mathfrak{p}}_0⊊\cdots ⊊{\mathfrak{p}}_d$. For irreducible affine variety $X$ over $k$: $\dim X={\dim }_{\mathrm{Krull}}k[X]={tr.deg}_{k}k(X)$. For noetherian local $(R,\mathfrak{m})$: $\dim R=\mathrm{deg}{P}_{\mathfrak{q}}(n)$ (Hilbert-Samuel polynomial of $\mathrm{length}(R/{\mathfrak{q}}^n)$ for $\mathfrak{q}$ $\mathfrak{m}$-primary) = min generators of an $\mathfrak{m}$-primary ideal (system of parameters). Krull's Hauptidealsatz / height theorem (1928): in noetherian $R$, every minimal prime over $(f)$ has height $\le 1$ (= 1 if $f$ is a non-zero-divisor non-unit); minimal primes over $(f_1,\dots ,f_r)$ have height $\le r$. Geometrically: hypersurface in ${\mathbb{A}}^n$ has dimension $n-1$. Cohen-Seidenberg going-up / going-down for integral extensions; catenary rings and the dimension formula $\mathrm{ht}(\mathfrak{p})+\dim (R/\mathfrak{p})=\dim R$; finitely generated $k$-algebras universally catenary; Nagata 1956 non-catenary domain. Cohen-Macaulay rings (depth = dimension); Krull's intersection theorem; effective Nullstellensatz (Brownawell 1987 with $N\le n{d}^n$, Kollár 1988 sharpening to $d^n$); real Nullstellensatz / Positivstellensatz (Stengle 1974, Krivine 1964); arithmetic Nullstellensatz over $\mathbb{Z}$; model-theoretic content as quantifier elimination for ${\mathrm{ACF}}_p$; Stillman's conjecture (Ananyan-Hochster 2020). Originators: Hilbert 1893 (Math. Ann. 42, appendix); Lasker 1905 (primary decomposition); Noether 1921 (noetherian foundations); Krull 1928 (Hauptidealsatz); Rainich/Rabinowitsch 1929 (auxiliary-variable trick); Zariski 1947 (modern proof).

  requires: alg-geom.affine-scheme, alg-geom.morphism-of-schemes

• alg-geom.blowupopen unit 04.07.02 →

  Blowup

  ${\mathrm{Bl}}_{Z}X={\mathrm{Proj}}_X{⨁}_{n\ge 0}{\mathcal{I}}^n$, the relative Proj of the Rees algebra. Universal property: minimal surgery making the ideal sheaf invertible. Exceptional divisor as effective Cartier divisor; projective bundle $\mathbb{P}({\mathcal{N}}_{Z/X}^{\vee })$ for smooth centres. Hironaka 1964 resolution theorem in characteristic 0; weak factorisation; Castelnuovo contraction for surfaces. Open in characteristic $p$ for dimension $\ge 4$.

  requires: alg-geom.scheme, alg-geom.projective-space, alg-geom.cartier-divisor

• alg-geom.sheaf-of-differentialsopen unit 04.08.01 →

  Sheaf of differentials

  ${\Omega }_{X/Y}^1$ defined by the universal ${\mathcal{O}}_Y$-linear derivation $d:{\mathcal{O}}_X\to {\Omega }_{X/Y}^1$. Corresponds to Kähler differentials ${\Omega }_{R/S}^1$ on affines. Conormal/normal sheaf exact sequences for closed immersions and smooth morphisms. First fundamental exact sequence $f^{\ast }{\Omega }_{Y/Z}^1\to {\Omega }_{X/Z}^1\to {\Omega }_{X/Y}^1\to 0$. Locally free of rank equal to relative dimension on smooth morphisms. Foundation for canonical sheaf and Serre duality.

  requires: alg-geom.scheme, alg-geom.coherent-sheaf, alg-geom.morphism-of-schemes

• alg-geom.canonical-sheafopen unit 04.08.02 →

  Canonical sheaf

  ${\omega }_X=\det {\Omega }_X^1={\bigwedge }^n{\Omega }_X^1$ for smooth variety of dimension $n$. Canonical divisor class $K_X$. $\mathrm{deg}(K_C)=2g-2$ on a smooth projective curve. Adjunction formula ${\omega }_Y=({\omega }_X\otimes \mathcal{O}(D)){|}_Y$ for smooth divisor $Y\subset X$. Dualising sheaf and Serre duality. Riemann's implicit canonical divisor (1857) via everywhere-holomorphic 1-forms. Kodaira dimension classifies birational geometry by canonical-sheaf positivity.

  requires: alg-geom.sheaf-of-differentials, alg-geom.line-bundle-scheme

• alg-geom.serre-dualityopen unit 04.08.03 →

  Serre duality

  For smooth projective $X$ of dimension $n$ over a field, $H^i(X;\mathcal{F})\cong H^{n-i}(X;{\mathcal{F}}^{\vee }\otimes {\omega }_X{)}^{\vee }$ for locally free $\mathcal{F}$. Trace map $H^n(X;{\omega }_X)\to k$. Serre 1955 Un théorème de dualité. Grothendieck duality (1966) generalises to proper morphisms via dualising complex ${\omega }_X^{\bullet }$. On curves: $H^1(\mathcal{L})\cong H^0(\omega \otimes {\mathcal{L}}^{-1}{)}^{\vee }$, powering Riemann-Roch.

  requires: alg-geom.sheaf-cohomology, alg-geom.canonical-sheaf, alg-geom.coherent-sheaf

• alg-geom.serre-duality-curvesopen unit 06.04.04 →

  Serre duality on a curve

  For a line bundle $L$ on a smooth projective curve $X$ of genus $g$ over an algebraically closed field $k$, the residue trace $\mathrm{tr}:H^1(X;K_X)\to k$ pairs $H^1(X;L)\times H^0(X;K_X\otimes L^{-1})\to k$ non-degenerately, giving $H^1(X;L)\cong H^0(X;K_X\otimes L^{-1}{)}^{\ast }$. Combined with Riemann-Roch ($\dim H^0(L)-\dim H^1(L)=\mathrm{deg}L+1-g$) yields $\dim H^0(L)-\dim H^0(K\otimes L^{-1})=\mathrm{deg}L+1-g$. Vanishing $H^1(L)=0$ for $\mathrm{deg}L>2g-2$. Specialisation of Grothendieck duality to dimension 1; the dualising sheaf is $K_X={\omega }_X$. Geometric content: $H^0(K\otimes L^{-1})$ measures specialty (Brill-Noether). Originator: Serre 1955 (Comm. Math. Helv. 29).

  requires: complex-analysis.holomorphic-line-bundle, complex-analysis.riemann-roch-compact-rs, alg-geom.canonical-sheaf

• alg-geom.hodge-decompositionopen unit 04.09.01 →

  Hodge decomposition

  For compact Kähler $X$, $H^n(X;\mathbb{C})={⨁}_{p+q=n}{H}^{p,q}(X)$ with $H^{p,q}=H^q(X;{\Omega }^p)$ and $\overline{H^{p,q}}=H^{q,p}$. Hodge 1941 Theory and Applications of Harmonic Integrals — harmonic representatives. Hodge-to-de-Rham degeneration. Algebraic proof: Deligne-Illusie 1987 via reduction mod $p$. Polarised Hodge structures and period domains. Bridges algebra and topology.

  requires: alg-geom.sheaf-cohomology, topology.de-rham-cohomology

• alg-geom.hodge-decomposition-curvesopen unit 06.04.03 →

  Hodge decomposition on a compact Riemann surface

  For a compact Riemann surface $X$ (= smooth projective complex curve) of genus $g$, complex conjugation on $C^{\infty }$ forms intertwines the Dolbeault bidegree decomposition with the Kähler-harmonic theorem to give $H^1(X;\mathbb{C})=H^{1,0}(X)\oplus H^{0,1}(X)$, with $H^{1,0}(X)=H^0(X;{\Omega }_X^1)$ (holomorphic 1-forms), $H^{0,1}(X)=H^1(X;{\mathcal{O}}_X)$, and $\overline{H^{1,0}}=H^{0,1}$. Dimensions $h^{1,0}=h^{0,1}=g$, total $2g$ matching topology. Period matrix $\Pi =(A\mid B)$ over a symplectic homology basis satisfies Riemann's bilinear relations $A{B}^T=B{A}^T$, $\mathrm{Im}(B{A}^{-1})>0$, exhibiting the Jacobian $\mathrm{Jac}(X)=H^0(X;{\Omega }_X^1{)}^{\ast }/H_1(X;\mathbb{Z})$ as a principally polarised abelian variety. Curve case of the general Kähler Hodge $(p,q)$-decomposition (Hodge 1941); originator content traces to Riemann's 1857 bilinear relations. Direct prerequisite for Serre duality on curves and for the bilinear-relations and Schottky machinery on Jacobians.

  requires: complex-analysis.holomorphic-line-bundle, complex-analysis.riemann-roch-compact-rs, alg-geom.canonical-sheaf, topology.de-rham-cohomology

• alg-geom.jacobi-inversionopen unit 06.06.06 →

  Jacobi inversion theorem

  For a smooth projective compact Riemann surface $X$ of genus $g\ge 1$, fix a basis ${\omega }_1,\dots ,{\omega }_g$ of $H^0(X;{\Omega }_X^1)$, a symplectic basis of $H_1(X;\mathbb{Z})$, and a reference point $p_0\in X$. The Abel-Jacobi map $\alpha :{\mathrm{Sym}}^g(X)\to \mathrm{Jac}(X)$, $\alpha (p_1+\cdots +p_g)=({\sum }_i{\int }_{p_0}^{p_i}{\omega }_j{)}_j\mathrm{mod}\Lambda$, is surjective and birational (Jacobi 1834 — Considerationes generales de transcendentibus Abelianis, Crelle 9). Riemann 1857 refines this: the exceptional locus, after translation by the Riemann constant $\kappa$, coincides with the theta divisor $\Theta =\alpha (W_{g-1})\subset \mathrm{Jac}(X)$, equivalently with the zero locus of the Riemann theta function $\theta (z;\tau )={\sum }_{n\in {\mathbb{Z}}^g}\exp (\pi i{n}^T\tau n+2\pi i{n}^{T}z)$ (Riemann's vanishing theorem). Proof via four steps: image is closed (compactness of ${\mathrm{Sym}}^g(X)$), image has dimension $g$ (differential = Brill-Noether matrix, full-rank by Riemann-Roch + Serre duality on non-special divisors), image equals the full Jacobian (closed + full-dim in connected target), generic fibre is a point (Riemann-Roch on a generic line bundle of degree $g$ gives $h^0=1$). Combined with Abel's theorem yields the structural identification ${\mathrm{Pic}}^0(X)\cong \mathrm{Jac}(X)$ as complex Lie groups. Brill-Noether stratification $W_d^r\subset {\mathrm{Pic}}^d(X)$ extends the inversion to all degrees: $\dim W_d^r\ge g-(r+1)(g-d+r)$ when non-negative (Kempf 1971, Kleiman-Laksov 1972), with equality on a general curve (Griffiths-Harris 1980, Gieseker 1982). Torelli's theorem (Torelli 1913) recovers $X$ from the principally polarised Jacobian $(\mathrm{Jac}(X),\Theta )$. Schottky problem characterises Jacobi loci inside ${\mathcal{A}}_g$; Schottky 1888 ($g=4$), Welters trisecant (Krichever 2006), Novikov-Shiota KP characterisation (Shiota 1986).

  requires: complex-analysis.jacobian-variety, alg-geom.serre-duality-curves, alg-geom.hodge-decomposition-curves

• alg-geom.riemann-bilinearopen unit 06.06.07 →

  Riemann's bilinear relations

  For a compact Riemann surface $X$ of genus $g$, fix a symplectic basis $a_1,\dots ,a_g,b_1,\dots ,b_g$ of $H_1(X;\mathbb{Z})$ ($a_i\cdot b_j={\delta }_{ij}$, $a_i\cdot a_j=b_i\cdot b_j=0$) and a basis ${\omega }_1,\dots ,{\omega }_g$ of $H^0(X;{\Omega }_X^1)$ normalised so that ${\int }_{a_j}{\omega }_i={\delta }_{ij}$. The period matrix ${\tau }_{ij}={\int }_{b_j}{\omega }_i$ satisfies (RB1) ${\tau }_{ij}={\tau }_{ji}$ (symmetry) and (RB2) $\mathrm{Im}\tau$ positive definite. Equivalently, in non-normalised form $(A\mid B)$, $A{B}^T=B{A}^T$ and $\mathrm{Im}(B{A}^{-1})>0$. Proof: Riemann's bilinear identity ${\int }_X\omega \wedge \eta ={\sum }_k[{\int }_{a_k}\omega \cdot {\int }_{b_k}\eta -{\int }_{b_k}\omega \cdot {\int }_{a_k}\eta ]$ (cut $X$ along the symplectic basis to a $4g$-gon, apply Stokes); for $\omega ,\eta$ holomorphic, $\omega \wedge \eta \in {\Omega }^{2,0}(X)=0$ on a curve, giving (RB1). For (RB2), positivity of $\frac{i}{2}{\int }_X\omega \wedge \bar{\omega }$ (the Kähler form $\frac{i}{2}fdz\wedge d\bar{z}=|f{|}^{2}dV$) plus the bilinear identity gives $\mathrm{Im}\tau$ as the Gram matrix of a positive-definite Hermitian form on $H^0(X;{\Omega }_X^1)$. Geometric content: the period matrix lies in the Siegel upper half space ${\mathfrak{H}}_g=\{$symmetric $g\times g$ complex matrices with positive-definite imaginary part$\}$; conversely every $\tau \in {\mathfrak{H}}_g$ defines a principally polarised abelian variety (PPAV). The Schottky problem (Schottky 1888 for $g=4$ via a quartic modular relation; Welters-Shiota for the KP / Novikov characterisation; Krichever 2006 for the trisecant conjecture) asks which points of ${\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ come from curves: the Jacobi locus has dimension $3g-3$ vs. $g(g+1)/2$ for ${\mathfrak{H}}_g$, so the inclusion is far from surjective for $g\ge 4$. Riemann theta function $\theta (z;\tau )$ on ${\mathbb{C}}^g\times {\mathfrak{H}}_g$ uses the bilinear relations directly in its quasi-periodicity. Foundation of Riemann's 1857 theta-function theory and of the modern theory of Abelian integrals, complex multiplication, Heegner points, modular curves.

  requires: complex-analysis.jacobi-inversion, alg-geom.hodge-decomposition-curves, complex-analysis.period-matrix

• alg-geom.schottky-problemopen unit 06.06.08 →

  Schottky problem

  Identify the period matrices of compact Riemann surfaces among all symmetric positive-imaginary $g\times g$ complex matrices, modulo $\mathrm{Sp}(2g,\mathbb{Z})$. Setup: Siegel upper half-space ${\mathfrak{H}}_g=\{\tau \in M_g(\mathbb{C}):\tau ={\tau }^T,\mathrm{Im}\tau >0\}$ of dimension $g(g+1)/2$; quotient ${\mathcal{A}}_g={\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ = moduli space of principally polarised abelian varieties; period mapping $\mathrm{Per}:{\mathcal{M}}_g\to {\mathcal{A}}_g$, $X\mapsto \mathrm{Jac}(X)$; Jacobi locus ${\mathcal{J}}_g=\mathrm{Per}({\mathcal{M}}_g)\subset {\mathcal{A}}_g$ of dimension $3g-3$ for $g\ge 2$; codimension $(g-2)(g-3)/2$, so ${\mathcal{J}}_g={\mathcal{A}}_g$ iff $g\le 3$ and strict subvariety for $g\ge 4$. Five characterisations of ${\mathcal{J}}_g$: (1) Schottky 1888 — single explicit polynomial $S(\theta )=0$ in even theta-constants of degree $4$ on $12$ specific characteristics, irreducible hypersurface (Igusa 1980); Schottky-Jung 1909 extends via Pryms; (2) Andreotti-Mayer 1967 — ${\mathcal{J}}_g\subseteq {\mathcal{N}}_{g-4}$ where ${\mathcal{N}}_k=\{\dim \mathrm{Sing}(\Theta )\ge k\}$, with the reverse containment conjectural and false in general (Beauville-Debarre 1986, Debarre 1992 identify additional components); (3) Novikov-Shiota 1979/1986 — $(A,\Theta )$ is a Jacobian iff $u(x,y,t)=2{\partial }_x^2\log \theta (xV+yU+tW+z_0;\tau )$ solves the KP equation $\frac{3}{4}{u}_{yy}=(u_t-\frac{3}{2}u{u}_x-\frac{1}{4}{u}_{xxx}{)}_x$, descending from Krichever's 1977 construction of KP solutions from algebraic curves via the Baker-Akhiezer function; alternative algebro-geometric proof in Arbarello-De Concini 1987; (4) Welters 1984 / Krichever 2010 — $(A,\Theta )$ is a Jacobian iff its Kummer variety $K(\tau )\subset {\mathbb{P}}^{2^g-1}$ admits a trisecant line, descending from the Fay 1973 trisecant identity for Jacobians; (5) modular forms / Siegel-Igusa-Tsuyumine ring of theta-constants $\theta [m,m^{\prime }]$ of weight $1/2$ on $\Gamma (2)$. Generalisations: Prym Schottky problem (Beauville 1977, Donagi 1981); hyperelliptic Schottky (Mumford Tata II §IIIb); $k$-gonal strata; $p$-adic Schottky uniformisation (Mumford 1972, Mumford curves). Modern surveys: Donagi 1988, Grushevsky 2009. Foundation for the integration of curve geometry, theta-function theory, Sato-Grassmannian / Hirota tau-function integrable systems, projective geometry of Kummer varieties, and Siegel modular forms.

  requires: alg-geom.riemann-bilinear, alg-geom.vhs-jacobian, alg-geom.jacobi-inversion

• alg-geom.gauss-maninopen unit 06.08.01 →

  Gauss-Manin connection

  For a smooth proper morphism $\pi :\mathcal{X}\to S$ with compact Kähler fibres $X_s$, the cohomology bundle ${\mathcal{H}}^k=R^k{\pi }_{\ast }{\mathbb{C}}_{\mathcal{X}}$ is a local system on $S$ with fibre $H^k(X_s;\mathbb{C})$, and the Gauss-Manin connection $\nabla :{\mathcal{H}}^k\to {\mathcal{H}}^k\otimes {\Omega }_S^1$ is the canonical flat connection whose horizontal sections are locally-constant cycles. Three equivalent formulations: (i) trivial connection on the local system encoding monodromy; (ii) Katz-Oda algebraic construction on the de Rham cohomology bundle of a smooth proper morphism; (iii) Čech-cocycle level differentiation of locally lifted cocycles. Periods $\Pi (s)={\int }_{{\gamma }_s}\omega$ are multivalued holomorphic functions on $S$, solutions of an algebraic Picard-Fuchs ODE with regular singularities at boundary divisors. Worked examples: family of elliptic curves with Picard-Fuchs = Gauss hypergeometric / Legendre equation (Euler-Gauss); quintic threefold $\sum z_i^5=5\psi z_1\cdots z_5$ whose Picard-Fuchs is the mirror-symmetry quintic equation (Candelas-de la Ossa-Green-Parkes 1991, encoding genus-zero Gromov-Witten invariants of the mirror via Givental-Kontsevich); family of Riemann surfaces over ${\mathcal{M}}_g$ producing the variation of Hodge structure (VHS) on $H^1$. Hodge filtration $F^p{\mathcal{H}}^k$ varies holomorphically but is not preserved by $\nabla$; instead Griffiths transversality $\nabla F^p\subseteq F^{p-1}\otimes {\Omega }_S^1$ holds. Period mapping $\mathcal{P}:S\to \mathcal{D}/\Gamma$ to the period domain modulo monodromy is horizontal in the transversality distribution. Modern context: mirror-symmetry identification with quantum cohomology connection; Shimura-variety automorphic theory; $p$-adic Hodge comparison theorems (Berthelot-Ogus-Faltings); Dubrovin's Frobenius-manifold framework. Originator: Manin 1958 (Izv. Akad. Nauk SSSR 22), with Picard-Fuchs antecedents in Gauss's hypergeometric work and Picard-Fuchs 1891-1928; modern algebraic framework Katz-Oda 1968 (J. Math. Kyoto Univ. 8); Hodge-theoretic interpretation Griffiths 1968.

  requires: alg-geom.hodge-decomposition-curves, alg-geom.serre-duality-curves, alg-geom.sheaf-cohomology, topology.de-rham-cohomology

• alg-geom.vhs-jacobianopen unit 06.08.02 →

  Variation of Hodge structure on the Jacobian

  For a smooth proper family $\pi :\mathcal{X}\to S$ of compact Riemann surfaces of genus $g$, each fibre cohomology $H^1(X_s;\mathbb{C})$ carries a polarised pure Hodge structure of weight $1$ with Hodge filtration $F^1{H}^1=H^{1,0}(X_s)\subset H^1(X_s;\mathbb{C})$ of rank $g$ inside the rank-$2g$ total space; the Hodge subbundle $F^1{\mathcal{H}}^1={\pi }_{\ast }{\Omega }_{\mathcal{X}/S}^1$ varies holomorphically over $S$. The relative Jacobian $\mathrm{Jac}(\mathcal{X}/S)=(F^1{\mathcal{H}}^1{)}^{\ast }/R^1{\pi }_{\ast }\mathbb{Z}$ is a smooth proper family of principally polarised abelian varieties (PPAV) over $S$, and the variation of Hodge structure is the data $(R^1{\pi }_{\ast }\mathbb{Z},F^1{\mathcal{H}}^1,Q)$ with the symplectic intersection form $Q$ providing the polarisation. The period mapping $\mathcal{P}:S\to {\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ to the Siegel upper half-space modulo the symplectic monodromy is holomorphic and horizontal: Griffiths transversality $\nabla F^1\subseteq F^0\otimes {\Omega }_S^1={\mathcal{H}}^1\otimes {\Omega }_S^1$ is the differential constraint that holds automatically in weight $1$, while the substantive content is the Cauchy-Riemann compatibility of the period matrix $\Pi (s)$ via the Gauss-Manin connection (Griffiths 1968 Amer. J. Math. 90). The period domain $\mathcal{D}$ for weight-1 PHS is the symmetric domain $\mathrm{Sp}(2g,\mathbb{R})/U(g)$ realised as ${\mathfrak{H}}_g$, the Siegel upper half-space; the quotient ${\mathfrak{H}}_g/\mathrm{Sp}(2g,\mathbb{Z})$ is the moduli space ${\mathcal{A}}_g$ of PPAV. Torelli's theorem (Andreotti 1958) asserts the period mapping ${\mathcal{M}}_g\to {\mathcal{A}}_g$ is injective: a curve is determined by its principally polarised Jacobian. The Schottky problem (Schottky 1888 for $g=4$ via a single explicit modular relation; Shiota 1986 / Novikov conjecture: KP-equation characterisation) asks for the image — the Schottky locus ${\mathcal{J}}_g$ of dimension $3g-3$ inside ${\mathcal{A}}_g$ of dimension $g(g+1)/2$, of codimension $g(g-1)/2-(3g-3)$ for $g\ge 4$. The Riemann theta function $\theta (z;\tau )$ provides quasi-periodic sections of the principal polarisation line bundle on $\mathrm{Jac}(X_s)$; Jacobian theta functions satisfy the KP hierarchy (Shiota's theorem). Modular interpretation: $g=1$ recovers the upper half-plane ${\mathfrak{H}}_1$ and modular forms for $\mathrm{SL}(2,\mathbb{Z})$, the bridge to number theory. Generalisations: VHS of higher weight (multi-step Hodge filtration with substantive Griffiths transversality), mixed Hodge structures (Deligne 1971-74) for non-compact / singular fibres, $\mathbb{R}$-VHS on homogeneous period domains $G(\mathbb{R})/V$, and $p$-adic VHS (Faltings, Berthelot rigid cohomology). Originator: Griffiths 1968-70 four-paper series Periods of integrals on algebraic manifolds (Inventiones, Amer. J. Math. 90, Publ. Math. IHÉS 38); Andreotti 1958 (Torelli for curves); Schottky 1888 (genus-4 Schottky); Shiota 1986 (general Schottky via KP, Invent. Math. 83).

  requires: alg-geom.gauss-manin, alg-geom.jacobi-inversion, alg-geom.hodge-decomposition-curves

• alg-geom.moduli-of-riemann-surfacesopen unit 06.08.03 →

  Moduli of Riemann surfaces

  The moduli space ${\mathcal{M}}_g$ of compact Riemann surfaces of genus $g$ is the parameter space of all isomorphism classes of smooth projective complex curves of fixed genus. It is naturally a smooth Deligne-Mumford stack of complex dimension $3g-3$ for $g\ge 2$ (Riemann's count, rigorously realised in Mumford 1965 GIT), $\dim {\mathcal{M}}_1=1$ (the $j$-line), $\dim {\mathcal{M}}_0=0$ (a single point with stack structure $B\mathrm{PGL}(2)$). The coarse moduli space $M_g$ is a quasi-projective complex variety; the stack structure records non-trivial automorphisms of curves (hyperelliptic involutions, Klein quartic $\mathrm{PSL}(2,{\mathbb{F}}_7)$ etc.) that prevent ${\mathcal{M}}_g$ from being a scheme. Three non-tautological constructions: Teichmüller-theoretic (${\mathcal{T}}_g$ contractible, $\cong {\mathbb{R}}^{6g-6}$, ${\mathcal{M}}_g={\mathcal{T}}_g/\mathrm{MCG}(g)$ with $\mathrm{MCG}(g)={\pi }_0({\mathrm{Diff}}^{+}({\Sigma }_g))$); algebraic-geometric (Mumford GIT on the Hilbert scheme of pluri-canonically embedded curves); period-mapping ($\mathcal{P}:{\mathcal{M}}_g\to {\mathcal{A}}_g$ via the Jacobian, image the Schottky locus ${\mathcal{J}}_g$, Torelli injectivity by Andreotti 1958). For $g\ge 2$ a fourth description by complete hyperbolic metrics (Fenchel-Nielsen length-twist coordinates) realises ${\mathcal{T}}_g\cong {\mathbb{R}}^{6g-6}$. Compactification: ${\overline{\mathcal{M}}}_g$ adds stable nodal curves (Deligne-Mumford 1969 Publ. IHÉS 36); a smooth proper Deligne-Mumford stack of dimension $3g-3$ with normal-crossings boundary $\partial {\overline{\mathcal{M}}}_g={\Delta }_0\cup {\Delta }_1\cup \cdots \cup {\Delta }_{\lfloor g/2\rfloor }$. Marked-point version ${\mathcal{M}}_{g,n}$ has dimension $3g-3+n$ (when $2g-2+n>0$). Tautological ring: classes ${\kappa }_i,{\psi }_i,{\lambda }_i\in H^{\ast }({\overline{\mathcal{M}}}_{g,n};\mathbb{Q})$, generated by Mumford-Morita-Miller / Hodge / cotangent-line classes; Mumford's relation ${\lambda }_1={\kappa }_1/12$ via Grothendieck-Riemann-Roch on the universal curve; Faber's conjecture (1999) on the Gorenstein structure of $R^{\ast }({\mathcal{M}}_g)$ (open in general). Witten conjecture (1990) / Kontsevich theorem (1992 Comm. Math. Phys. 147): the generating function of $\psi$-class intersection numbers on ${\overline{\mathcal{M}}}_{g,n}$ is a tau-function of the KdV hierarchy, proved via the matrix Airy integral and Strebel ribbon-graph combinatorics. Madsen-Weiss 2007 Ann. of Math. 165: ${\mathrm{colim}}_g{H}^{\ast }({\mathcal{M}}_g;\mathbb{Q})\cong \mathbb{Q}[{\kappa }_1,{\kappa }_2,\dots ]$ (Mumford's conjecture), via cobordism-category methods (Galatius-Madsen-Tillmann-Weiss 2009 Acta Math. 202). Applications: Gromov-Witten invariants integrate over ${\overline{\mathcal{M}}}_{g,n}(X,\beta )$ pushing to $\psi$-classes on ${\overline{\mathcal{M}}}_{g,n}$ (Kontsevich-Manin 1994); closed-bosonic-string scattering amplitudes are integrals of vertex operators against the Weil-Petersson measure on ${\overline{\mathcal{M}}}_{g,n}$ (Polyakov 1981); Hurwitz numbers via the ELSV formula (Ekedahl-Lando-Shapiro-Vainshtein 2001) translate combinatorial enumeration of branched covers to $\lambda$- and $\psi$-class integrals. Arithmetic moduli: ${\mathcal{M}}_g$ has a model over $\mathrm{Spec}\mathbb{Z}$, with Galois action of $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ on $\widehat{{\pi }_1({\mathcal{M}}_g\times \overline{\mathbb{Q}})}$ and the Grothendieck-Teichmüller programme (Grothendieck Esquisse 1984; Drinfeld 1990). Open problems: Faber's conjecture, Schottky problem in higher genus, explicit description of $H^{\ast }({\overline{\mathcal{M}}}_g)$ for large $g$, gauge-theoretic / quantum-field-theoretic interpretations. Originator: Riemann 1857 (parameter count); Mumford 1965 (algebraic-geometric construction); Deligne-Mumford 1969 (stack and stable-curves compactification); Teichmüller 1939-44 (foundations of ${\mathcal{T}}_g$).

  requires: alg-geom.vhs-jacobian, alg-geom.jacobi-inversion, alg-geom.moduli-of-curves

• alg-geom.kodaira-vanishingopen unit 04.09.02 →

  Kodaira vanishing theorem

  $H^i(X;{\omega }_X\otimes L)=0$ for $i>0$, $L$ ample on smooth projective complex $X$. Kodaira 1953 (transcendental). Akizuki-Nakano generalisation to ${\Omega }^p$. Algebraic proof via Deligne-Illusie reduction mod $p$. Kawamata-Viehweg generalisation to nef + big. Kollár's injectivity. Foundational for the minimal model program.

  requires: alg-geom.sheaf-cohomology, alg-geom.ample-line-bundle, alg-geom.hodge-decomposition

• alg-geom.intersection-pairing-surfacesopen unit 04.05.06 →

  Intersection pairing on a surface

  Symmetric bilinear pairing $\mathrm{Pic}(X)\times \mathrm{Pic}(X)\to \mathbb{Z}$, $(D,D^{\prime })\mapsto D\cdot D^{\prime }$, on a smooth projective surface $X$ over algebraically closed $k$. Three equivalent definitions: geometric (transverse intersection count after moving lemma), cohomological ($D\cdot D^{\prime }=\chi ({\mathcal{O}}_X)-\chi (\mathcal{O}(-D))-\chi (\mathcal{O}(-D^{\prime }))+\chi (\mathcal{O}(-D-D^{\prime }))$, Hartshorne V.1.4), cup-product (cycle-class map $\mathrm{Pic}(X)\to H^2(X,\mathbb{Z})$ intertwines intersection with $⌣$). Self-intersection $D^2$. Examples: line $L^2=1$ on ${\mathbb{P}}^2$, exceptional divisor $E^2=-1$, ruling $f^2=0$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$. Italian school (Castelnuovo-Enriques 1880–1910); Hartshorne V scheme-theoretic framing; Fulton's $A^{\ast }(X)$-deepening. Load-bearing for adjunction, Riemann-Roch on surfaces, Hodge index, and the Castelnuovo-Beauville-Bombieri-Kodaira classification of surfaces.

  requires: alg-geom.scheme, alg-geom.weil-divisor, alg-geom.cartier-divisor, alg-geom.line-bundle-scheme, alg-geom.picard-group

• alg-geom.adjunction-formulaopen unit 04.05.07 →

  Adjunction formula on a surface

  For a smooth projective curve $C$ on a smooth projective surface $X$ over algebraically closed $k$, the adjunction formula in canonical-restriction form: ${\omega }_C\cong ({\omega }_X\otimes {\mathcal{O}}_X(C)){|}_C$. Genus form: $2g_C-2=K_X\cdot C+C^2$, equivalently $g_C=1+\frac{1}{2}(K_X\cdot C+C^2)$. Proof via the conormal exact sequence $0\to {\mathcal{N}}_{C/X}^{\vee }\to {\Omega }_X^1{|}_C\to {\Omega }_C^1\to 0$, taking determinants, and using ${\mathcal{N}}_{C/X}^{\vee }\cong {\mathcal{O}}_X(-C){|}_C$. Codim-$1$ case of the general adjunction ${\omega }_Y\cong ({\omega }_X\otimes \det {\mathcal{N}}_{Y/X}){|}_Y$. Recovers Plücker's plane-curve genus formula $g=(d-1)(d-2)/2$, the bidegree-$(a,b)$ genus $(a-1)(b-1)$ on ${\mathbb{P}}^1\times {\mathbb{P}}^1$, the K3 inequality $C^2\ge -2$, the Castelnuovo $(-1)$-curve diagnostic, and Noether's formula $K_X^2+e(X)=12\chi ({\mathcal{O}}_X)$. Picard 1897 originator-text; Castelnuovo 1892 and Severi 1921 Italian-school synthesis; Hartshorne V.1.5 modern scheme-theoretic framing.

  requires: alg-geom.intersection-pairing-surfaces, alg-geom.canonical-sheaf, alg-geom.sheaf-of-differentials, alg-geom.riemann-roch-curves

• alg-geom.riemann-roch-surfacesopen unit 04.05.08 →

  Riemann-Roch theorem for surfaces

  Theorem (Hartshorne V.1.6). For a smooth projective surface $S$ over algebraically closed $k$ and a divisor $D$ on $S$, $\chi ({\mathcal{O}}_S(D))=\chi ({\mathcal{O}}_S)+\frac{1}{2}D\cdot (D-K_S)$, where $K_S$ is the canonical divisor and $\chi ({\mathcal{O}}_S)=1-q+p_g$ with $q=h^1({\mathcal{O}}_S)$ the irregularity and $p_g=h^0({\omega }_S)=h^2({\mathcal{O}}_S)$ the geometric genus. Noether's formula (Max Noether 1883): $12\chi ({\mathcal{O}}_S)=K_S^2+c_2(S)$ couples the holomorphic and topological Euler characteristics on $S$. Hirzebruch-Riemann-Roch derivation: surface Riemann-Roch is the degree-two specialisation of $\chi (X,\mathcal{F})={\int }_X\mathrm{ch}(\mathcal{F})\cdot \mathrm{td}(T_X)$ at $\mathcal{F}={\mathcal{O}}_S(D)$, with $\mathrm{ch}({\mathcal{O}}_S(D))=1+D+D^2/2$ and $\mathrm{td}(T_S)=1-K_S/2+(K_S^2+c_2)/12$ truncated on a surface. Direct (Italian-school) proof: short exact sequence $0\to {\mathcal{O}}_S(D-C)\to {\mathcal{O}}_S(D)\to {\mathcal{O}}_C(D{|}_C)\to 0$ for a smooth curve $C\subset S$, take Euler characteristics, apply curve Riemann-Roch on $C$, and use adjunction to identify the genus. Worked examples. (1) $S={\mathbb{P}}^2$ with $K_S=-3H$, $H^2=1$: $\chi (\mathcal{O}(d))=d(d+3)/2+1=\left(\genfrac{}{}{0ex}{}{d+2}{2}\right)$, recovering the dimension count for plane curves of degree $d$. (2) $S={\mathbb{P}}^1\times {\mathbb{P}}^1$ with $K_S=-2H_1-2H_2$ and bidegree $(a,b)$: $\chi =(a+1)(b+1)$. Hodge index theorem (Hartshorne V.1.9): the intersection form on $\mathrm{NS}(S)\otimes \mathbb{R}$ has signature $(1,\rho -1)$, equivalently the intersection form on $H^{1,1}(S,\mathbb{R})$ has signature $(1,h^{1,1}-1)$. Castelnuovo's contractibility criterion (Hartshorne V.5.7): a smooth rational curve $E$ with $E^2=-1$ on a smooth projective surface is the exceptional divisor of a blow-up at a smooth point, with adjunction giving $E\cdot K_S=-1$ as the dual diagnostic. Geography of surfaces: Noether inequality $K_S^2\ge 2p_g-4$ (Noether 1875); Bogomolov-Miyaoka-Yau inequality $K_S^2\le 9\chi ({\mathcal{O}}_S)$ (Yau 1977, Miyaoka 1977) for surfaces of general type; the strip $2\chi -6\le K_S^2\le 9\chi$ is the realisation region. Enriques-Kodaira classification: every smooth projective surface is birational to a unique minimal model with no $(-1)$-curves, and minimal models fall into four Kodaira-dimension classes: $\kappa =-\infty$ (rational/ruled), $\kappa =0$ (K3, Enriques, abelian, bielliptic), $\kappa =1$ (properly elliptic), $\kappa =2$ (general type). Castelnuovo-Enriques-Severi (1900s-1920s) for char zero, Bombieri-Mumford (1976-77) for char $p$. Originators: Max Noether 1883 (the topological correction term); Castelnuovo, Enriques, Severi 1890s-1920s (Italian-school synthesis); Severi 1926 Trattato (refined version); Enriques 1949 Le superficie algebriche (synthesis); Hirzebruch 1956 (modern HRR proof); Hartshorne 1977 §V.1 (canonical scheme-theoretic framing).

  requires: alg-geom.intersection-pairing-surfaces, alg-geom.adjunction-formula, alg-geom.riemann-roch-curves, alg-geom.serre-vanishing-finiteness

• alg-geom.moduli-of-curvesopen unit 04.10.01 →

  Moduli of curves

  ${\mathcal{M}}_g$ moduli space of smooth genus-$g$ curves; quasi-projective scheme of dimension $3g-3$ for $g\ge 2$ (Riemann's count, Mumford's GIT construction). Deligne-Mumford compactification $\overline{{\mathcal{M}}_g}$ via stable curves (1969). Tautological classes $\kappa ,\psi ,\lambda$. Witten's conjecture (Kontsevich theorem). ${\mathcal{M}}_{g,n}$ with marked points. Connection to teichmüller theory and string topology.

  requires: alg-geom.riemann-roch-curves, alg-geom.coherent-sheaf

• alg-geom.gitopen unit 04.10.02 →

  Geometric invariant theory

  Mumford 1965 Geometric Invariant Theory. GIT quotient $X//G=\mathrm{Proj}{⨁}_n{H}^0(X;L^n{)}^G$ for linearised ample $L$. Stable, semistable, unstable points; Hilbert-Mumford numerical criterion. Symplectic reduction correspondence (Kempf-Ness). Foundational for moduli of vector bundles, varieties, sheaves. Kirwan's stratification of unstable locus. Variation of GIT.

  requires: alg-geom.scheme, lie-groups.lie-group, algebra.group-action

• topology.spectrumopen unit 03.12.04 →

  Spectrum

  Sequential spectra with structure maps $\Sigma E_n\to E_{n+1}$. $\Omega$-spectra. Stable homotopy groups indexed over $\mathbb{Z}$. Brown representability. Stable homotopy category as triangulated, symmetric monoidal. Examples: sphere, Eilenberg-MacLane, K-theory, bordism. Connection to generalised cohomology theories.

  requires: topology.topological-space, topology.homotopy, topology.suspension

• topology.suspensionopen unit 03.12.03 →

  Suspension

  Unreduced and reduced suspension. $\Sigma S^n\cong S^{n+1}$. Suspension-loop adjunction. Freudenthal suspension theorem. Stable homotopy as the limit of iterated suspension. Smash product: $\Sigma X=S^1\wedge X$.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.covering-spaceopen unit 03.12.02 →

  Covering space

  Local-product covering maps with discrete fibres. Path-lifting, homotopy-lifting, deck transformation group. Universal cover. Galois correspondence between subgroups of ${\pi }_1$ and connected covers. Spin double cover $\mathrm{Spin}(n)\to \mathrm{SO}(n)$ as a fundamental example.

  requires: topology.topological-space, topology.continuous-map, topology.homotopy

• topology.metric-spaceopen unit 02.01.05 →

  Metric space

  Sets equipped with a distance function satisfying positive-definiteness, symmetry, and triangle inequality. Metric topology, Cauchy sequences and completeness, equivalence of metrics generating the same topology, completion. Banach fixed-point theorem, Heine-Borel, Arzelà-Ascoli, Stone-Weierstrass, Baire category. Bridge from topology to functional analysis.

  requires: topology.topological-space

• functional-analysis.banach-spaceopen unit 02.11.04 →

  Banach space

  Complete normed vector spaces; foundation for bounded linear operators, compact operators, Fredholm theory, and PDE estimates.

  requires: linalg.vector-space

• functional-analysis.normed-vector-spaceopen unit 02.11.06 →

  Normed vector space

  Norm axioms, induced metric, norm topology, continuous linear maps, equivalent norms in finite dimension, and foundation for Banach spaces.

  requires: linalg.vector-space, topology.metric-space

• functional-analysis.inner-product-spaceopen unit 02.11.07 →

  Inner product space

  Inner products, induced norm, Cauchy-Schwarz inequality, orthogonality, projection geometry, parallelogram identity, and foundation for Hilbert spaces.

  requires: functional-analysis.normed-vector-space, linalg.bilinear-form

• functional-analysis.hilbert-spaceopen unit 02.11.08 →

  Hilbert space

  Complete inner-product spaces; supplies orthogonality, projection, adjoints, and state-space language for quantum theory and CFT.

  requires: linalg.vector-space, linalg.bilinear-form, functional-analysis.banach-space

• functional-analysis.unbounded-self-adjointopen unit 02.11.03 →

  Unbounded self-adjoint operators

  Densely defined unbounded operators on Hilbert spaces, adjoints, symmetric versus self-adjoint operators, closed graph, deficiency spaces, and spectral calculus.

  requires: functional-analysis.hilbert-space

• functional-analysis.bounded-operatorsopen unit 02.11.01 →

  Bounded linear operators

  Linear operators between normed spaces, operator norm, equivalence of boundedness and continuity, Banach-space structure of $\mathcal{B}(X,Y)$, submultiplicativity, Banach algebra structure. Banach-Steinhaus, open mapping, closed graph. Adjoints in Hilbert space and the C*-identity. Spectrum and spectral radius.

  requires: functional-analysis.banach-space, linalg.vector-space

• functional-analysis.compact-operatorsopen unit 02.11.05 →

  Compact operators

  Bounded linear operators sending bounded sets to relatively compact sets. Closed two-sided ideal in $\mathcal{B}(X)$. Density of finite-rank operators in $\mathcal{K}(\mathcal{H})$ (Hilbert spaces). Schauder's theorem (compactness of adjoint). Riesz-Schauder spectral theorem. Hilbert-Schmidt and trace-class refinements (Schatten ideals). Calkin algebra $\mathcal{B}/\mathcal{K}$ as the home of Fredholm theory.

  requires: functional-analysis.bounded-operators, functional-analysis.banach-space, linalg.vector-space

• functional-analysis.fredholm.operatorsopen unit 03.09.06 →

  Fredholm operators

  Bounded operators with finite-dimensional kernel, closed range, finite-dimensional cokernel. Index = $\dim \ker -\dim \mathrm{coker}$ is the central invariant. Atkinson: Fredholm iff invertible modulo compacts. Stable under compact and small-norm perturbations. Foundation for the Atiyah-Singer index theorem.

  requires: functional-analysis.bounded-linear-operators, functional-analysis.banach-space, functional-analysis.compact-operators

• char-classes.chern-weil.homomorphismopen unit 03.06.06 →

  Chern-Weil homomorphism

  Invariant polynomials on a Lie algebra evaluated on curvature forms. Produces closed de Rham classes independent of connection; natural under pullback. Gateway from connections and gauge curvature to characteristic classes.

  requires: bundle.principal-bundle, diffgeo.connection.connection, diffgeo.connection.curvature, lie-algebra.lie-algebra, topology.de-rham-cohomology

• bundle.principal-bundleopen unit 03.05.01 →

  Principal bundle

  Smooth principal right $G$-bundles, free transitive fiber action, local product charts, transition functions and cocycles, associated bundles, pullbacks, reductions of structure group, and gauge transformations. Connections and curvature are separate units.

  requires: topology.topological-space, diffgeo.smooth-manifold, lie-groups.lie-group

• bundle.vector-bundleopen unit 03.05.02 →

  Vector bundle

  Finite-rank real and complex vector bundles, local linear product charts, sections, transition functions into ${\mathrm{GL}}_n$, frame bundles, associated-bundle equivalence, pullback, direct sum, tensor product, duals, and relation to K-theory and characteristic classes.

  requires: linalg.vector-space, topology.topological-space, diffgeo.smooth-manifold

• bundle.sphere-bundle-hopf-indexno unit yet

  Sphere bundle, the global angular form, and the Hopf index theorem

  Oriented sphere bundle as the unit-sphere bundle of an oriented rank-$r$ real vector bundle with $SO(r)$ structure group. Global angular form $\psi \in {\Omega }^{r-1}(S(E))$ characterised by fibre-integral $1$ and $d\psi =-{\pi }^{\ast }e(E)$. Euler class of an oriented sphere bundle equals the Euler class of the associated vector bundle. Hopf index theorem ${\sum }_p{\mathrm{ind}}_p(V)=\chi (M)$ for a vector field with isolated zeros. Poincaré-Hopf as the Morse-function specialisation. Worked examples: $e(T{S}^{2n})$ as twice the generator, Hopf bundle Euler class $-h$ on $\mathbb{C}{P}^1$. Bott-Tu §11 originator-text for the global-angular-form derivation; Hopf 1926 Vektorfelder in $n$-dimensionalen Mannigfaltigkeiten (Math. Ann. 96) for the original index theorem.

  requires: bundle.vector-bundle, bundle.frame-bundle.orthonormal, char-classes.pontryagin-chern.definitions, diffgeo.de-rham.thom-cv-cohomology

• diffgeo.connection.vector-bundle-connectionopen unit 03.05.04 →

  Connection on a vector bundle

  Covariant derivatives on vector bundles, Leibniz rule, local connection forms, affine space modeled on End(E)-valued one-forms, induced connections, and curvature.

  requires: bundle.vector-bundle, manifold.smooth, diffgeo.differential-forms

• bundle.complex-vector-bundleopen unit 03.05.08 →

  Complex vector bundle

  Smooth complex vector bundles via $J^2=-\mathrm{id}$ structure or via ${\mathrm{GL}}_n(\mathbb{C})$ cocycles. Hermitian metrics and reduction to $U(n)$. Direct sum, tensor, dual, conjugate, complexification, realification. Self-conjugacy of $V_{\mathbb{C}}$ and 2-torsion of odd Chern classes (motivating Pontryagin classes). Holomorphic bundles, Chern connections, K-theory.

  requires: bundle.vector-bundle, linalg.vector-space

• bundle.connection.curvatureopen unit 03.05.09 →

  Curvature of a connection

  Curvature 2-form on a principal bundle, Cartan structure equation, Bianchi identity. Vector-bundle curvature as $\mathrm{End}(E)$-valued 2-form. Gauge covariance of curvature. Frobenius integrability of horizontal distribution. Ambrose-Singer holonomy theorem. Riemann tensor / Ricci / scalar curvature as the metric-affine special case.

  requires: bundle.principal-bundle, diffgeo.connection.vector-bundle-connection, diffgeo.connection.connection, lie-algebra.lie-algebra, topology.de-rham-cohomology

• diffgeo.connection.connectionopen unit 03.05.07 →

  Principal bundle with connection

  Principal connections as Lie-algebra-valued one-forms or equivariant horizontal distributions; local gauge potentials, gauge transformations, associated vector-bundle connections, and curvature.

  requires: bundle.principal-bundle, lie-groups.lie-group, lie-algebra.lie-algebra, diffgeo.differential-forms

• topology.de-rham-cohomologyopen unit 03.04.06 →

  De Rham cohomology

  Defines $H_{\mathrm{dR}}^k(M)=\ker d/\mathrm{im}d$, exact forms, closed forms, functorial pullback, wedge product, Poincare lemma, de Rham theorem, and integration pairing with cycles. Provides the cohomology target for Chern-Weil representatives.

  requires: diffgeo.smooth-manifold, diffgeo.differential-forms, diffgeo.exterior-derivative, diffgeo.stokes-theorem

• diffgeo.integration-on-manifoldsopen unit 03.04.03 →

  Integration on manifolds

  Integration of compactly supported top-degree forms on oriented manifolds, partitions of unity, change of variables, orientation reversal, boundary orientation, and Stokes-compatible formalism.

  requires: diffgeo.smooth-manifold

• diffgeo.variational-calculusopen unit 03.04.08 →

  Variational calculus on manifolds

  Action functionals, variations, first variation, Euler-Lagrange equations, integration by parts, boundary terms, and gauge-theoretic variational formulas.

  requires: diffgeo.integration-on-manifolds, diffgeo.smooth-manifold

• char-classes.pontryagin-chern.definitionsopen unit 03.06.04 →

  Pontryagin and Chern classes

  Chern classes for complex vector bundles, Pontryagin classes for real bundles via complexification, splitting principle, Whitney product formula, Chern-Weil representatives, and standard examples such as $T{\mathbb{CP}}^n$.

  requires: char-classes.chern-weil.homomorphism, bundle.vector-bundle, topology.de-rham-cohomology, linalg.vector-space

• k-theory.classifying-spacesopen unit 03.08.04 →

  Classifying space

  Universal principal bundles, pullback classification, homotopy invariance, numerable bundles, and the role of $BU(n)$ and $BO(n)$ in characteristic classes and K-theory.

  requires: bundle.principal-bundle, topology.homotopy, topology.topological-space

• k-theory.vector-bundlesopen unit 03.08.01 →

  Topological K-theory

  Grothendieck group of vector bundles, direct sum, virtual bundles, reduced K-theory, pullback functoriality, and preparation for Bott periodicity.

  requires: bundle.vector-bundle, k-theory.classifying-spaces, topology.topological-space

• topology.eilenberg-maclaneopen unit 03.12.05 →

  Eilenberg-MacLane space

  Spaces $K(A,n)$ with ${\pi }_n=A$ and other ${\pi }_k=0$. Uniqueness up to weak equivalence. Loop-space characterisation. Representability of ordinary cohomology: $H^n(X;A)\cong [X,K(A,n)]$. Postnikov towers. Cohomology operations and the Steenrod algebra.

  requires: topology.topological-space, topology.homotopy, topology.spectrum

• homotopy.stable-homotopyopen unit 03.08.06 →

  Stable homotopy

  Suspension, stable homotopy groups, spectra-level intuition, Freudenthal stabilization, stable phenomena behind generalized cohomology theories, and the stable classical-group context for Bott periodicity.

  requires: topology.homotopy, topology.suspension, topology.sphere

• k-theory.bott.periodicityopen unit 03.08.07 →

  Bott periodicity

  Complex K-theory has period $2$ and real K-theory has period $8$. The unit treats the classifying-space form, coefficient tables, Bott elements, Clifford-module source of real periodicity, and the role of Bott periodicity in the topological index.

  requires: k-theory.vector-bundles, k-theory.classifying-spaces, spin-geometry.clifford.clifford-algebra, char-classes.pontryagin-chern.definitions

• gauge-theory.yang-mills.actionopen unit 03.07.05 →

  Yang-Mills action

  Defines $\mathrm{YM}(A)=\frac{1}{2}\Vert F_A{\Vert }_{L^2}^2$, proves gauge invariance, derives $d_A^{\ast }{F}_A=0$, relates four-dimensional self-duality to Yang-Mills, and separates metric-dependent action from Chern-Weil topological classes.

  requires: bundle.principal-bundle, diffgeo.connection.connection, diffgeo.connection.curvature, lie-algebra.lie-algebra, topology.de-rham-cohomology, char-classes.chern-weil.homomorphism

• physics.cft.basicsopen unit 03.10.02 →

  CFT basics

  Introduces two-dimensional conformal symmetry, primary fields, stress tensor, Witt and Virasoro algebras, central charge, OPEs, radial quantization, and the state-operator correspondence.

  requires: functional-analysis.hilbert-space, lie-algebra.central-extension, lie-algebra.infinite-dimensional-representations, lie-algebra.virasoro

• index-theory.atiyah-singer.index-theoremopen unit 03.09.10 →

  Atiyah-Singer index theorem

  States ${\mathrm{ind}}_a(D)={\mathrm{ind}}_t([\sigma (D)])$, explains symbol K-theory and Bott-periodic topological index, specializes to the spin Dirac formula $⟨\hat{A}(TM)\mathrm{ch}(E),[M]⟩$, and connects heat-kernel local index theory to Chern-Weil forms.

  requires: functional-analysis.fredholm.operators, spin-geometry.dirac.dirac-operator, k-theory.bott.periodicity, char-classes.pontryagin-chern.definitions, char-classes.chern-weil.homomorphism, diffgeo.elliptic-operators

• spin-geometry.spin-groupopen unit 03.09.03 →

  Spin group $\mathrm{Spin}(n)$

  Connected double cover of $\mathrm{SO}(n)$ for $n\ge 3$. Even part of the Pin group. Realizes universal cover for $n=3$. Spin(4) is SU(2) × SU(2). Spin(6) is SU(4).

  requires: linalg.bilinear-form, spin-geometry.clifford.clifford-algebra, lie-groups.connected-double-cover

• diffgeo.operator.symbolopen unit 03.09.07 →

  Symbol of a differential operator

  Principal symbols of linear differential operators, order filtration, lower-order quotient, composition rule, cotangent-bundle interpretation, and Dirac symbol.

  requires: linalg.vector-space, diffgeo.smooth-manifold, bundle.vector-bundle

• diffgeo.elliptic-operatorsopen unit 03.09.09 →

  Elliptic operators on a manifold

  Elliptic differential operators, invertible principal symbols off the zero section, Laplacian and Dirac examples, parametrices, elliptic regularity, Fredholmness, and symbol K-theory class.

  requires: diffgeo.operator.symbol, functional-analysis.unbounded-self-adjoint, bundle.vector-bundle, diffgeo.smooth-manifold

• char-classes.invariant-polynomial.adjoint-invariantopen unit 03.06.05 →

  Invariant polynomial on a Lie algebra

  Symmetric multilinear $G$-invariant functions on a Lie algebra; the algebra $S^{\bullet }({\mathfrak{g}}^{\ast }{)}^G$. Group/Lie-algebra invariance equivalence for connected $G$. Generators for matrix Lie algebras: $\mathrm{tr}(A^k)$, elementary symmetric polynomials of eigenvalues, Pfaffian for $\mathfrak{so}$. Chevalley's restriction theorem. The input to Chern-Weil theory.

  requires: lie-algebra.lie-algebra, linalg.vector-space, linalg.bilinear-form

• lie-algebra.lie-algebraopen unit 03.04.01 →

  Lie algebra

  Lie algebras as vector spaces with a bilinear, antisymmetric, Jacobi-respecting bracket; classical examples ${\mathfrak{gl}}_n$, ${\mathfrak{sl}}_n$, ${\mathfrak{so}}_n$, ${\mathfrak{u}}_n$; the adjoint representation; ideals and homomorphisms; the commutator-on-an-associative-algebra construction. The Killing form, simple/semisimple/solvable structure, Cartan classification, and the exponential map / BCH formula are at Master tier.

  requires: linalg.vector-space

• lie-algebra.central-extensionopen unit 03.11.01 →

  Central extension of a Lie algebra

  Central extensions of Lie algebras, one-dimensional extensions by 2-cocycles, coboundary equivalence, relation to projective representations, and Virasoro central charge.

  requires: lie-algebra.lie-algebra

• lie-algebra.infinite-dimensional-representationsopen unit 03.11.02 →

  Infinite-dimensional Lie algebra representations

  Lie algebra representations on infinite-dimensional vector spaces, modules, highest-weight representations, central elements acting by scalars under irreducibility hypotheses, and the representation-theoretic bridge from central extensions to CFT.

  requires: lie-algebra.lie-algebra, lie-algebra.central-extension, functional-analysis.hilbert-space

• lie-algebra.virasoroopen unit 03.11.03 →

  Virasoro algebra

  Witt algebra of Laurent vector fields, Virasoro central extension, Gelfand-Fuchs cocycle, central charge, highest-weight modules, and the CFT stress-tensor mode algebra.

  requires: lie-algebra.central-extension, lie-algebra.infinite-dimensional-representations

• symplectic-geometry.symplectic-vector-spaceopen unit 05.01.01 →

  Symplectic vector space

  A vector space with a nondegenerate skew form. Central theorem: standard symplectic basis theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 01.01.03, 01.01.15

• symplectic-geometry.symplectic-manifoldopen unit 05.01.02 →

  Symplectic manifold

  A smooth manifold with a closed nondegenerate 2-form. Central theorem: symplectic volume form theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.02.01, 03.04.02, 03.04.04, 05.01.01

• symplectic-geometry.symplectic-groupopen unit 05.01.03 →

  Symplectic group

  The linear transformations preserving a symplectic form. Central theorem: matrix criterion for the symplectic group. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.03.01, 03.03.03, 05.01.01

• symplectic-geometry.darboux-theoremopen unit 05.01.04 →

  Darboux's theorem

  The local normal form for every symplectic form. Central theorem: Darboux local coordinate theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.04.04

• symplectic-geometry.moser-trickopen unit 05.01.05 →

  Moser's trick

  Path-method proof technique. Given ${\omega }_t$ a path of cohomologous symplectic forms with ${\dot{\omega }}_t=d{\alpha }_t$, define $X_t$ by ${\iota }_{X_t}{\omega }_t=-{\alpha }_t$; the time-1 flow ${\psi }_1$ satisfies ${\psi }_1^{\ast }{\omega }_1={\omega }_0$. Originator-anchor for Darboux's theorem (modern proof), Weinstein neighbourhood theorem, equivariant Darboux, and smooth structure of regular symplectic reduction.

  requires: symplectic-geometry.symplectic-manifold, symplectic-geometry.darboux-theorem

• symplectic-geometry.weinstein-neighbourhoodopen unit 05.05.02 →

  Weinstein Lagrangian neighbourhood theorem

  Closed Lagrangian $L\subset (M,\omega )$ has a tubular neighbourhood symplectomorphic to a neighbourhood of the zero section in $T^{\ast }L$ with the canonical form $-d\lambda$. Three-step proof: Lagrangian splitting + normal bundle = $T^{\ast }L$; tubular diffeomorphism via exponential map; Moser's trick on path of forms vanishing on $L$. Equivariant version. Generating-function bridge to symplectomorphisms. Foundational for Floer-theoretic comparison and Arnold-Givental conjectures. Originator: Weinstein 1971 Symplectic manifolds and their Lagrangian submanifolds (Adv. Math. 6).

  requires: symplectic-geometry.lagrangian-submanifold, symplectic-geometry.symplectic-manifold, symplectic-geometry.moser-trick

• symplectic-geometry.hamiltonian-vector-fieldopen unit 05.02.01 →

  Hamiltonian vector field

  The vector field determined by a function and the symplectic form. Central theorem: Hamiltonian flow preserves the symplectic form. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.04.04

• symplectic-geometry.poisson-bracketopen unit 05.02.02 →

  Poisson bracket and Poisson manifold

  A lie bracket on functions encoding hamiltonian dynamics. Central theorem: Poisson bracket satisfies Jacobi identity. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.01, 03.04.01

• symplectic-geometry.integrable-systemopen unit 05.02.03 →

  Integrable system

  Many commuting conserved quantities on a symplectic manifold. Central theorem: commuting Hamiltonians preserve common level sets. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.01, 05.02.02

• symplectic-geometry.action-angle-coordinatesopen unit 05.02.04 →

  Action-angle coordinates

  Canonical coordinates near compact invariant tori. Central theorem: Liouville-Arnold local normal form. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.03, 05.01.04

• symplectic-geometry.cotangent-bundleopen unit 05.02.05 →

  Cotangent bundle as canonical symplectic manifold

  The natural symplectic form on a cotangent bundle. Central theorem: canonical one-form gives the cotangent symplectic form. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.02.01, 03.04.02, 03.04.04, 05.01.02

• symplectic-geometry.coadjoint-orbitopen unit 05.03.01 →

  Coadjoint orbit

  An orbit in the dual of a lie algebra with a natural symplectic form. Central theorem: Kirillov-Kostant-Souriau form is symplectic. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.03.01, 03.04.01, 05.02.02

• symplectic-geometry.moment-mapopen unit 05.04.01 →

  Moment map

  A map whose components generate an infinitesimal group action. Central theorem: moment map components are Hamiltonians for fundamental fields. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 03.03.02, 05.02.01, 05.02.02

• symplectic-geometry.symplectic-reductionopen unit 05.04.02 →

  Marsden-Weinstein symplectic reduction

  A quotient construction producing a smaller symplectic manifold. Central theorem: regular reduction theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.04.01, 03.03.02, 05.01.02

• symplectic-geometry.lagrangian-submanifoldopen unit 05.05.01 →

  Lagrangian submanifold

  A half-dimensional submanifold on which the symplectic form vanishes. Central theorem: graph of a closed one-form is Lagrangian. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.02.01

• symplectic-geometry.almost-complex-structureopen unit 05.06.01 →

  Almost-complex structure on a symplectic manifold

  An endomorphism squaring to minus identity compatible with the symplectic form. Central theorem: compatible almost-complex structures exist. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 03.05.02

• symplectic-geometry.pseudoholomorphic-curveopen unit 05.06.02 →

  Pseudoholomorphic curve

  A surface whose tangent map intertwines complex structures. Central theorem: energy-area identity for pseudoholomorphic curves. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.06.01, 03.02.01

• complex-geometry.newlander-nirenbergopen unit 05.06.03 →

  Newlander-Nirenberg integrability theorem

  An almost-complex structure $J$ on $M$ comes from a complex-manifold structure if and only if its Nijenhuis tensor $N_J(X,Y)=[JX,JY]-J[JX,Y]-J[X,JY]-[X,Y]$ vanishes. Equivalent formulations: involutivity of $T^{1,0}M$ under the Lie bracket; ${\bar{\partial }}^2=0$ on ${\Lambda }^{\ast }$; local existence of $n$ smooth $\mathbb{C}$-valued functions $z_1,\dots ,z_n$ with $\bar{\partial }{z}_i=0$ and $d{z}_1\wedge \cdots \wedge d{z}_n\ne 0$. Originator: Newlander-Nirenberg 1957 (smooth case via Hörmander-style $L^2$-estimates for $\bar{\partial }$); predecessors Korn 1914 / Lichtenstein 1916 (real-dimension 2), Cartan-Kähler 1934 (real-analytic case via Frobenius), Eckmann-Frölicher 1951 (formal version); modern simplifications Malgrange 1969, Webster 1989. Real-dimension cases: dim 2 automatic; dim 4 generically obstructed; $S^4$ admits no almost-complex structure (Wu / Massey); $S^6$ integrability is the famous open problem on the $G_2$-invariant structure. Cross-strand bridge between symplectic almost-complex geometry and complex geometry; prerequisite for Kodaira-Spencer deformation theory of complex structures.

  requires: symplectic-geometry.almost-complex-structure, symplectic-geometry.pseudoholomorphic-curve

• symplectic-geometry.non-squeezingopen unit 05.07.01 →

  Gromov non-squeezing theorem

  The rigidity theorem forbidding symplectic squeezing of a ball into a thin cylinder. Central theorem: Gromov non-squeezing theorem. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.02, 05.06.02

• symplectic-geometry.symplectic-capacityopen unit 05.07.02 →

  Symplectic capacity

  A numerical invariant measuring symplectic size. Central theorem: capacity monotonicity under symplectic embeddings. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.07.01, 05.01.02

• symplectic-geometry.arnold-conjectureopen unit 05.08.01 →

  Arnold conjecture and Floer homology setup

  Fixed points of hamiltonian diffeomorphisms counted through floer theory. Central theorem: Arnold fixed-point lower bound statement. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.02.01, 05.05.01, 05.06.02, 05.07.02

• symplectic-geometry.floer-homologyopen unit 05.08.02 →

  Floer homology

  A homology theory generated by hamiltonian orbits or lagrangian intersections. Central theorem: boundary squares to zero under compactness and gluing. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.08.01, 05.06.02

• symplectic-geometry.maslov-indexopen unit 05.08.03 →

  Maslov index

  An integer measuring winding of lagrangian subspaces. Central theorem: Maslov index is homotopy invariant with fixed endpoints. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.05.01, 05.01.01

• symplectic-geometry.conley-zehnder-indexopen unit 05.08.04 →

  Conley-Zehnder index

  An index grading nondegenerate hamiltonian periodic orbits. Central theorem: Conley-Zehnder index changes by Maslov index under loop composition. Used in the v0.5 symplectic geometry strand for Hamiltonian dynamics, reduction, rigidity, and Floer-theoretic constructions.

  requires: 05.01.03, 05.08.03, 05.02.01

• stat-mech.partition-functionopen unit 08.01.01 →

  Partition function (statistical mechanics)

  The weighted sum of all allowed states in a statistical system. Central theorem: thermodynamic derivatives of log partition function. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 00.02.05

• stat-mech.ising-modelopen unit 08.01.02 →

  Ising model

  A lattice model whose spins take two values and interact with neighbors. Central theorem: one-dimensional Ising transfer-matrix solution. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01

• stat-mech.boltzmann-distributionopen unit 08.01.03 →

  Boltzmann distribution and canonical ensemble

  The probability rule assigning lower weight to higher energy at fixed temperature. Central theorem: canonical distribution maximizes entropy with fixed mean energy. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01

• stat-mech.free-energyopen unit 08.01.04 →

  Free energy

  The thermodynamic potential obtained from the logarithm of the partition function. Central theorem: free energy generates canonical thermodynamics. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01, 08.01.03

• stat-mech.mean-fieldopen unit 08.02.01 →

  Mean-field theory and Curie-Weiss model

  An approximation replacing many neighbors by an average field. Central theorem: Curie-Weiss self-consistency equation. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.02, 08.01.04

• stat-mech.spontaneous-symmetry-breakingopen unit 08.02.02 →

  Spontaneous symmetry breaking

  The selection of asymmetric equilibrium states from symmetric equations. Central theorem: mean-field double-well criterion for broken symmetry. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.02.01, 01.02.01

• stat-mech.mermin-wagneropen unit 08.02.03 →

  Mermin-Wagner theorem

  A low-dimensional obstruction to breaking continuous symmetries at positive temperature. Central theorem: absence of continuous-symmetry long-range order in two dimensions. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.02.02, 02.01.05

• stat-mech.onsager-solutionopen unit 08.03.01 →

  Onsager solution of the 2D Ising model (transfer matrix)

  The exact solution locating the two-dimensional ising critical point. Central theorem: Onsager critical temperature formula. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.02, 08.03.02

• stat-mech.transfer-matrixopen unit 08.03.02 →

  Transfer matrix

  A linear operator that advances a lattice model one slice at a time. Central theorem: largest eigenvalue controls thermodynamic free energy. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 01.01.03, 02.11.01

• stat-mech.real-space-rgopen unit 08.04.01 →

  Renormalisation group (real-space block decimation)

  A scale-changing transformation on statistical systems. Central theorem: fixed points organize long-distance behavior. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.02, 08.01.04

• stat-mech.wilson-fisheropen unit 08.04.02 →

  Wilson-Fisher fixed point and universality

  The non-gaussian fixed point governing many critical phenomena below four dimensions. Central theorem: epsilon-expansion fixed point to first order. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01, 08.06.01, 08.04.03

• stat-mech.beta-functionopen unit 08.04.03 →

  Beta function (renormalisation group)

  The vector field describing how couplings change with scale. Central theorem: fixed points are zeros of the beta function. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01

• stat-mech.block-spin-decimationopen unit 08.04.04 →

  Block-spin decimation

  A concrete coarse-graining map replacing blocks of spins by effective spins. Central theorem: decimation induces effective couplings. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01

• stat-mech.critical-exponentsopen unit 08.05.01 →

  Critical exponents and scaling laws

  Numbers measuring power-law behavior near a phase transition. Central theorem: Rushbrooke scaling relation. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.02.01, 08.04.02, 08.01.04

• stat-mech.correlation-functionsopen unit 08.05.02 →

  Correlation functions (statistical mechanics)

  Expectations of products of observables at separated points. Central theorem: connected correlations detect fluctuations. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01, 08.05.01

• stat-mech.gaussian-fieldopen unit 08.06.01 →

  Gaussian field theory and free boson

  A field theory whose action is quadratic and whose correlations are determined by a green kernel. Central theorem: Gaussian Wick factorization. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.05.02, 02.11.08

• stat-mech.conformal-criticalityopen unit 08.06.02 →

  Conformal symmetry at criticality

  Scale symmetry enhanced by angle-preserving transformations at critical points. Central theorem: two-dimensional primary-field two-point form. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.05.01, 08.06.01, 03.10.02

• stat-mech.path-integralopen unit 08.07.01 →

  Path integral formulation of statistical mechanics

  A continuum weighted sum over field configurations. Central theorem: saddle-point expansion around a stationary configuration. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.01.01, 08.06.01, 03.04.08

• stat-mech.wilson-lattice-gaugeopen unit 08.08.01 →

  Wilson's lattice gauge theory

  A lattice regularization of gauge fields using group elements on links. Central theorem: Wilson plaquette action approximates Yang-Mills action. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 03.03.01, 03.05.07, 08.08.02

• stat-mech.wilson-actionopen unit 08.08.02 →

  Wilson action

  The plaquette action measuring lattice curvature in gauge theory. Central theorem: small-plaquette expansion recovers curvature squared. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 03.07.05, 03.03.01

• stat-mech.effective-field-theoryopen unit 08.08.03 →

  Effective field theory

  A scale-dependent description retaining operators relevant at the chosen resolution. Central theorem: irrelevant operators are suppressed at long distance. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.04.01, 08.07.01

• stat-mech.wick-rotationopen unit 08.09.01 →

  Quantum-classical correspondence (Wick rotation)

  The relation between quantum time evolution and statistical weights after imaginary-time continuation. Central theorem: thermal trace as imaginary-time path integral. Used in the v0.5 statistical field theory strand for lattice models, criticality, renormalisation, and Euclidean field theory.

  requires: 08.07.01, 02.11.08

• complex-analysis.meromorphic-functionopen unit 06.01.05 →

  Meromorphic function

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.cauchy-integral-formulaopen unit 06.01.02 →

  Cauchy integral formula

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.residue-theoremopen unit 06.01.03 →

  Residue theorem

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.02, 06.01.05

• complex-analysis.analytic-continuationopen unit 06.01.04 →

  Analytic continuation

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.riemann-mapping-theoremopen unit 06.01.06 →

  Riemann mapping theorem

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01, 06.01.02

• complex-analysis.riemann-sphereopen unit 06.01.07 →

  Riemann sphere

  Distinctive Ahlfors item from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering one-point compactification, the two-chart holomorphic atlas, biholomorphism with ${\mathbb{P}}^1(\mathbb{C})$, Möbius automorphism group ${\mathrm{PSL}}_2(\mathbb{C})$, function field $\mathbb{C}(z)$, Riemann-Roch in genus 0, and Grothendieck's splitting of vector bundles on ${\mathbb{P}}^1$.

  requires: complex-analysis.holomorphic-function

• complex-analysis.mobius-transformationsopen unit 06.01.08 →

  Möbius (linear-fractional) transformations

  Distinctive Ahlfors item 7 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the group ${\mathrm{PSL}}_2(\mathbb{C})=\mathrm{Aut}({\mathbb{C}}_{\infty })$, three-point uniqueness via cross-ratio, classification by trace squared (parabolic / elliptic / hyperbolic / loxodromic), circle-and-line preservation, and the bridge to hyperbolic geometry on $\mathbb{H}$ via ${\mathrm{PSL}}_2(\mathbb{R})$ and the modular group ${\mathrm{PSL}}_2(\mathbb{Z})$. Foundational for Schwarz-Pick, Schwarz-Christoffel, and the modular function $\lambda /j$.

  requires: complex-analysis.riemann-sphere

• complex-analysis.cauchy-riemannopen unit 06.01.10 →

  Cauchy-Riemann equations and harmonic conjugate

  Distinctive Ahlfors item 9 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the differential characterisation of holomorphicity: real Jacobian as complex multiplication, equivalence of complex differentiability with the CR system $u_x=v_y$, $u_y=-v_x$, harmonic conjugate existence on simply-connected domains, Wirtinger $\partial /\bar{\partial }$ formalism, Hartogs's separately-holomorphic theorem, and the connection to elliptic-regularity for general elliptic PDE.

  requires: complex-analysis.riemann-sphere

• complex-analysis.harmonic-functionsopen unit 06.01.11 →

  Harmonic functions on the plane

  Distinctive Ahlfors item 10 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the planar Laplace equation $u_{xx}+u_{yy}=0$, the connection $f=u+iv$ holomorphic $\Rightarrow u,v$ harmonic and the converse on simply-connected domains via harmonic conjugate, the mean-value property, the maximum principle, the Poisson integral as the disc Dirichlet solution, the Liouville theorem for bounded plane harmonic functions, and the elliptic-regularity perspective that frames harmonic-function theory as the prototypical scalar elliptic PDE on ${\mathbb{R}}^2$. Foundational for Schwarz reflection, Schwarz lemma, Perron's method on general domains, Hardy-space theory, and the higher-dimensional generalisation $\Delta u=\sum u_{x_i{x}_i}=0$ on ${\mathbb{R}}^n$.

  requires: complex-analysis.cauchy-riemann

• complex-analysis.max-modulus-schwarzopen unit 06.01.12 →

  Maximum modulus + Schwarz lemma

  Distinctive Ahlfors item 11 from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit. Three-tier unit covering the maximum modulus principle on connected open sets (non-constant holomorphic $f$ cannot attain an interior modulus maximum), the Schwarz lemma on the unit disc ($f:\mathbb{D}\to \mathbb{D}$ with $f(0)=0$ satisfies $|f(z)|\le |z|$ and $|f^{\prime }(0)|\le 1$, with rotation as the only equality case), the disc automorphism group $\mathrm{Aut}(\mathbb{D})\cong \mathrm{PSU}(1,1)$, the Schwarz-Pick hyperbolic-metric contraction, Schwarz-Ahlfors-Pick on negatively-curved metrics, Phragmén-Lindelöf on unbounded sectors, Hadamard three-circles convexity, and Cartan's lemma in several complex variables. Foundational for the Riemann mapping theorem's uniqueness statement, Koenigs linearisation in complex dynamics, the modular-function bridge to Picard's theorems, and the Kobayashi-hyperbolic framework on complex manifolds.

  requires: complex-analysis.cauchy-riemann, complex-analysis.harmonic-functions

• complex-analysis.argument-principleopen unit 06.01.13 →

  Argument principle and Rouché's theorem

  Distinctive Ahlfors item from plans/fasttrack/lars-ahlfors-complex-analysis.md §3 audit (P1, Ch. 4 §5). Three-tier unit covering the argument principle ($(2\pi i{)}^{-1}{\oint }_{\gamma }{f}^{\prime }/fdz=N-P$ for meromorphic $f$ with no zeros or poles on $\gamma$, equivalently the winding number of $f\circ \gamma$ around the origin), the generalised argument principle with a holomorphic weight $g$, Rouché's theorem in both the asymmetric form $|g|<|f|$ on $\partial D$ and the symmetric form $|f+g|<|f|+|g|$ on $\partial D$, the open mapping theorem (non-constant holomorphic functions are open maps, with the local $m$-to-$1$ refinement), Hurwitz's theorem on uniform limits of non-vanishing holomorphic functions, and the fundamental theorem of algebra proved cleanly via Rouché on a large circle. Foundational for the local-degree theory of branched coverings ($\mathrm{06.02.02}$) and the Riemann-Hurwitz formula, the existence half of the Riemann mapping theorem ($\mathrm{06.01.06}$, via Hurwitz applied to Montel-extremal sequences), Jensen's formula and Nevanlinna value-distribution theory, and the argument-principle-as-index-theorem perspective that bridges to the Atiyah-Singer index theorem on higher-dimensional elliptic operators.

  requires: complex-analysis.cauchy-riemann

• complex-analysis.branch-point-ramificationopen unit 06.02.01 →

  Branch point and ramification

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.04, 06.03.01

• complex-analysis.branched-coveringsopen unit 06.02.02 →

  Branched coverings of Riemann surfaces

  Forster-distinctive item from plans/fasttrack/forster-riemann-surfaces.md priority 2 (item 9). Three-tier unit organising branched covers as a category, with Riemann's existence theorem as the master equivalence.

  requires: 04.04.02, 06.05.02

• alg-geom.riemann-existence-theoremopen unit 06.02.03 →

  Riemann's existence theorem for algebraic curves

  The converse direction to complex-analysis.branched-coverings (06.02.02): every compact Riemann surface is biholomorphic to the analytification of a smooth projective algebraic curve. Three proof routes (Riemann-Roch + very-ample embedding; function-field generation with $P(f,g)=0$; branched-cover monodromy + GAGA). Master tier covers GAGA, the function-field-of-curves equivalence of categories, Belyi's theorem, and Chow's theorem as the higher-dimensional analogue.

  requires: 06.02.02, 06.04.04

• complex-analysis.genus-riemann-surfaceopen unit 06.03.02 →

  Genus of a Riemann surface

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.03.01

• complex-analysis.uniformization-theoremopen unit 06.03.03 →

  Uniformization theorem

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.03.01, 06.01.06

• complex-analysis.divisor-riemann-surfaceopen unit 06.05.01 →

  Divisor on a Riemann surface

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.05, 06.03.01

• complex-analysis.holomorphic-line-bundleopen unit 06.05.02 →

  Holomorphic line bundle on a Riemann surface

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.05.01, 06.03.01, 03.05.02

• complex-analysis.riemann-hurwitz-formulaopen unit 06.05.03 →

  Riemann-Hurwitz formula

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.02.01, 06.03.02

• complex-analysis.holomorphic-one-formopen unit 06.06.01 →

  Holomorphic 1-form / abelian differential

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.03.01, 03.04.02

• complex-analysis.period-matrixopen unit 06.06.02 →

  Period matrix

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.06.01, 06.03.02

• complex-analysis.jacobian-varietyopen unit 06.06.03 →

  Jacobian variety

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.06.02, 06.06.04

• complex-analysis.abel-jacobi-mapopen unit 06.06.04 →

  Abel-Jacobi map

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.05.01, 06.06.01

• complex-analysis.theta-functionopen unit 06.06.05 →

  Theta function

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.06.02, 06.06.03

• complex-analysis.holomorphic-several-variablesopen unit 06.07.01 →

  Holomorphic functions of several variables

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.01.01

• complex-analysis.hartogs-phenomenonopen unit 06.07.02 →

  Hartogs phenomenon

  Supporting v0.5 Riemann-surface and complex-analysis unit.

  requires: 06.07.01

• rep-theory.lie-algebra-representationopen unit 07.06.01 →

  Lie algebra representation

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.04.01, 01.01.03, 07.01.01

• rep-theory.universal-enveloping-algebraopen unit 07.06.02 →

  Universal enveloping algebra

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.04.01, 03.01.02, 07.06.01

• rep-theory.root-systemopen unit 07.06.03 →

  Root system

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.04.01, 01.01.15, 07.04.01

• rep-theory.weyl-groupopen unit 07.06.04 →

  Weyl group

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.03

• rep-theory.dynkin-diagramopen unit 07.06.05 →

  Dynkin diagram

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.03, 07.06.04

• rep-theory.verma-moduleopen unit 07.06.06 →

  Verma module

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.01, 07.06.02, 07.03.01

• rep-theory.weyl-character-formulaopen unit 07.06.07 →

  Weyl character formula

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.03.01, 07.06.03, 07.06.04

• rep-theory.weyl-dimension-formulaopen unit 07.06.08 →

  Weyl dimension formula

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.07

• rep-theory.borel-weil-theoremopen unit 07.06.09 →

  Borel-Weil theorem

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.06.07, 03.05.02

• rep-theory.compact-lie-group-representationopen unit 07.07.01 →

  Compact Lie group representation

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.03.01, 07.01.01

• rep-theory.peter-weyl-theoremopen unit 07.07.02 →

  Peter-Weyl theorem

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 07.07.01, 02.11.08, 07.01.01

• rep-theory.haar-measureopen unit 07.07.03 →

  Haar measure

  Supporting v0.5 Lie-algebraic and compact-Lie representation unit.

  requires: 03.03.01, 03.04.03

• rep-theory.characteropen unit 07.01.03 →

  Character of a representation

  Class function ${\chi }_{\rho }(g)=\mathrm{tr}(\rho (g))$. Originated by Frobenius via Dedekind's group-determinant problem. Powers orthogonality, dimension formula, decomposition algorithms. Generalises to compact Lie groups via integration against Haar measure.

  requires: rep-theory.group-representation

• rep-theory.character-orthogonalityopen unit 07.01.04 →

  Character orthogonality

  Row and column orthogonality. Gives multiplicity formula $n_V=⟨\chi ,{\chi }_V⟩$ and isomorphism criterion (representations isomorphic iff characters agree). Frobenius's first orthogonality relation; Schur's 1905 derivation via Schur's lemma is the modern textbook approach.

  requires: rep-theory.character, rep-theory.schur-lemma

• rep-theory.regular-representationopen unit 07.01.05 →

  Regular representation

  $\mathbb{C}G$ as a $G$-representation. Contains every irreducible $V$ with multiplicity $\dim V$, giving $|G|={\sum }_V(\dim V{)}^2$. Character ${\chi }_{\mathrm{reg}}(e)=|G|$, ${\chi }_{\mathrm{reg}}(g)=0$ for $g\ne e$. Foundational computational object.

  requires: rep-theory.group-representation

• rep-theory.tensor-product-of-representationsopen unit 07.01.06 →

  Tensor product of representations

  $({\rho }_1\otimes {\rho }_2)(g)={\rho }_1(g)\otimes {\rho }_2(g)$ on $V_1\otimes V_2$. Character ${\chi }_{V_1\otimes V_2}={\chi }_{V_1}\cdot {\chi }_{V_2}$. Decomposition into irreducibles (Clebsch-Gordan). Foundation of Schur-Weyl duality.

  requires: rep-theory.group-representation, linalg.tensor-product

• rep-theory.induced-representationopen unit 07.01.07 →

  Induced representation

  ${\mathrm{Ind}}_H^{G}W=\mathbb{C}G{\otimes }_{\mathbb{C}H}W$. Character formula ${\chi }_{\mathrm{Ind}W}(g)=\frac{1}{|H|}{\sum }_{x\in G:x^{-1}gx\in H}{\chi }_W(x^{-1}gx)$. Foundation of Mackey theory and ${\mathrm{GL}}_n({\mathbb{Q}}_p)$ representations.

  requires: rep-theory.group-representation, groups.group

• rep-theory.frobenius-reciprocityopen unit 07.01.08 →

  Frobenius reciprocity

  Adjunction ${\mathrm{Hom}}_G({\mathrm{Ind}}_H^{G}W,V)\cong {\mathrm{Hom}}_H(W,{\mathrm{Res}}_H^{G}V)$. Prototype of all categorical adjunctions, predating category theory by 50+ years. Powers Mackey theory, Brauer's theorem on induced characters, Langlands correspondences.

  requires: rep-theory.induced-representation, rep-theory.character

• rep-theory.maschke-theoremopen unit 07.02.01 →

  Maschke's theorem

  Complete reducibility of finite-group representations over fields where $|G|$ is invertible. Averaging-projection proof. Failure in characteristic dividing $|G|$ launches modular representation theory (Brauer 1935).

  requires: rep-theory.group-representation

• spin-geometry.heat-kernel-indexopen unit 03.09.20 →

  Heat-kernel proof of the Atiyah-Singer index theorem

  The McKean-Singer formula $\mathrm{ind}(D)=\mathrm{Str}(e^{-t{D}^2})$ holds for every $t>0$. Large-$t$ limit returns the analytic index from the harmonic projection; small-$t$ asymptotic expansion of the heat kernel produces the local index density $\hat{A}({\nabla }^{TM})\mathrm{ch}({\nabla }^E)$ (spin Dirac case). Getzler 1986 rescaling reduces the small-$t$ computation to the harmonic-oscillator heat kernel of Mehler. Alvarez-Gaumé 1983 reproduces the result by supersymmetric path-integral evaluation. The proof package supplies the local form, not just the index integer.

  requires: index-theory.atiyah-singer.index-theorem, diffgeo.elliptic-operators, spin-geometry.dirac.dirac-operator, char-classes.pontryagin-chern.definitions

• spin-geometry.family-equivariant-indexopen unit 03.09.21 →

  Family, equivariant, and Lefschetz fixed-point index theorems

  Family index of an elliptic family $D=\{D_b{\}}_{b\in B}$ is a class $\mathrm{ind}(D)\in K^0(B)$ whose Chern character equals the integral of $\hat{A}\cdot \mathrm{ch}(\sigma )$ over the fibre. Equivariant index of a $G$-equivariant elliptic operator lives in the representation ring $R(G)$. Lefschetz fixed-point formula expresses the equivariant index via local contributions at the fixed-point set $M^g$. The three refinements share a heat-kernel proof: insert the family / group action into the supertrace, invoke localisation.

  requires: index-theory.atiyah-singer.index-theorem, spin-geometry.heat-kernel-index, k-theory.bott.periodicity, bundle.principal-bundle.connection

• spin-geometry.pseudodifferentialopen unit 03.09.22 →

  Sobolev spaces, pseudodifferential operators, and elliptic parametrices

  Sobolev spaces $H^s$ measure regularity by integrability of derivatives. Embedding theorem $H^s\hookrightarrow C^k$ for $s>k+n/2$. Rellich-Kondrachov compactness $H^s\hookrightarrow H^{s^{\prime }}$ for $s>s^{\prime }$ on bounded domains. Pseudodifferential operators of order $m$ are quantisations of symbols $a(x,\xi )\in S^m$. Elliptic symbols admit a parametrix $Q\in {\Psi }^{-m}$ with $QP-I,PQ-I\in {\Psi }^{-\infty }$ smoothing. On a closed manifold smoothing operators are compact between Sobolev spaces; Atkinson then yields Fredholmness of every elliptic operator. Notation: Thom class $\tau \in H^{\ast }$ (cohomology), $\Lambda \in K^{\ast }$ (K-theory) per LM.

  requires: functional-analysis.compact-operators, diffgeo.operator.symbol, diffgeo.elliptic-operators

• spin-geometry.dirac-bundleopen unit 03.09.14 →

  Generalised Dirac bundles and the Bochner-Weitzenböck identity

  A Dirac bundle is a Hermitian vector bundle equipped with a fibrewise Clifford action of $T^{\ast }M$ and a metric-compatible connection whose Levi-Civita-Leibniz rule holds. The Dirac operator is $D=c\circ {\nabla }^{\mathfrak{E}}$. The universal Bochner-Weitzenböck identity reads $D^2=({\nabla }^{\mathfrak{E}}{)}^{\ast }{\nabla }^{\mathfrak{E}}+{\mathcal{R}}^{\mathfrak{E}}$ with ${\mathcal{R}}^{\mathfrak{E}}=\frac{1}{2}{\sum }_{j,k}c(e^j)c(e^k)R^{\mathfrak{E}}(e_j,e_k)$. Specialisations: spinor bundle ($\mathcal{R}=\frac{1}{4}\mathrm{Scal}$, Lichnerowicz 1963), de Rham bundle ($\mathcal{R}$ = Ricci on 1-forms, Bochner 1946; Weitzenböck curvature operator on $k$-forms), twisted spinor bundle ($\mathcal{R}=\frac{1}{4}\mathrm{Scal}+F^E$, Atiyah-Singer twist).

  requires: spin-geometry.spinor-bundle, spin-geometry.dirac.dirac-operator, bundle.principal-bundle.connection, diffgeo.de-rham

• spin-geometry.clk-diracopen unit 03.09.15 →

  Cl_k-linear Dirac operators and the KO-valued index

  A ${\mathrm{Cl}}_k$-linear Dirac bundle carries a graded right ${\mathrm{Cl}}_k$-action commuting with the Clifford action of $T^{\ast }M$. Its Clifford-index $\hat{\alpha }(D)$ lives in ${\hat{\mathfrak{M}}}_k=M_k/i^{\ast }{M}_{k+1}$, the ABS module quotient, which equals $K{O}_k$ via the Atiyah-Bott-Shapiro isomorphism. The Cl_k-linear AS theorem (Atiyah-Singer Index IV) computes the topological side as the KO-pushforward of the symbol class. The $\alpha$-invariant of Hitchin is the special case $\hat{\alpha }(D_{\mathfrak{S}})$ on a spin manifold, the foundational psc obstruction in dimensions where the integer Dirac index vanishes. Notation disambiguation: ${\mathrm{Cl}}_k$ here is the standard real Clifford algebra ${\mathrm{Cl}}_{0,k}$ (positive-definite in the sign convention $v^2=-|v{|}^2$), distinct from the ${\mathrm{Cl}}_{r,s}$ chessboard family of 03.09.11.

  requires: spin-geometry.dirac-bundle, spin-geometry.clifford.clifford-algebra, index-theory.atiyah-singer.index-theorem, k-theory.bott.periodicity

• spin-geometry.exercise-pack-ch1open unit 03.09.E1 →

  Clifford and spin algebra exercise pack (Lawson-Michelsohn Ch. I supplement)

  Twelve exercises covering Clifford chessboard low-rank computations, Pin/Spin extensions, automorphism inner-products, Atiyah-Bott-Shapiro module classifications, K-theory orientations, and exceptional Lie group descriptions. Cross-cuts the existing units 03.09.02, 03.09.03, 03.09.11, and (in low-dim and chessboard exercises) 03.09.13. Difficulty distribution: 4 easy / 5 medium / 3 hard. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: spin-geometry.clifford.clifford-algebra, spin-geometry.spin-group, spin-geometry.clifford-chessboard

• spin-geometry.calibrated-geometriesopen unit 03.09.19 →

  Calibrated geometries — Special Lagrangian, associative, coassociative, Cayley

  A calibration of degree $p$ on a Riemannian manifold is a closed $p$-form of comass at most 1 globally. Submanifolds calibrated by $\phi$ — those whose oriented unit tangent $p$-vector lies in the contact set $\{\xi :\phi (\xi )=1\}$ — are volume-minimisers in their homology class by the Harvey-Lawson fundamental theorem (one chain of inequalities: Stokes plus comass bound). The four named calibrated geometries are: Special Lagrangian on a Calabi-Yau $n$-fold ($\mathrm{Re}\Omega$, holonomy $\mathrm{SU}(n)$, contact set $\mathrm{SU}(n)/\mathrm{SO}(n)$); associative on a $G_2$ 7-fold ($\varphi$, contact set $G_2/\mathrm{SO}(4)$); coassociative on a $G_2$ 7-fold ($\ast \varphi$); Cayley on a Spin(7) 8-fold ($\Phi$). Each calibrating form is the spinor square of a parallel spinor preserved by the special holonomy. McLean 1998 computed the moduli-space dimensions: $b_1$ for SL, $b_{+}^2$ for coassociative, normal-Dirac kernel for associative and Cayley.

  requires: spin-geometry.triality, spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure

• spin-geometry.psc-obstructionopen unit 03.09.16 →

  Positive scalar curvature obstruction theory

  The classical psc obstruction chain spans 1963–1992 in four links. Lichnerowicz 1963: $D^2={\nabla }^{\ast }\nabla +\frac{1}{4}\mathrm{Scal}$ on the spinor bundle, with positive scalar curvature forcing $\ker D=0$ and hence $\hat{A}(M)[M]=0$. Hitchin 1974: the Cl_n-linear refinement, with the α-invariant $\alpha (M)\in K{O}_n$ vanishing under psc; the new content is the $\mathbb{Z}/2$ obstructions in dimensions $n\equiv 1,2\mathrm{mod}8$, detecting exotic spheres without psc. Gromov-Lawson 1980/83: the surgery theorem (psc preserved under codim-$\ge 3$ surgery) and the enlargeable manifold theorem (manifolds admitting arbitrarily small Lipschitz maps to $S^n$ from finite covers admit no psc, ruling out $T^n$). Stolz 1992: in the simply-connected case, $\alpha (M)=0$ is equivalent to psc-existence in dimension $\ge 5$. Lawson is co-originator of Gromov-Lawson. Notation: the α-invariant symbol is $\alpha (M)$ (notation decision #24, pinned in this unit).

  requires: spin-geometry.dirac-bundle, spin-geometry.clk-dirac, spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure

• spin-geometry.witten-positive-massopen unit 03.09.17 →

  Witten positive-mass theorem via spinors

  Witten 1981 reproved the Schoen-Yau positive-mass theorem in three pages by introducing a harmonic spinor with prescribed asymptotic value ${\psi }_{\infty }$, applying the Lichnerowicz formula on the asymptotically flat spin 3-manifold, and identifying the integration-by-parts boundary term at infinity with $-4\pi m_{\mathrm{ADM}}|{\psi }_{\infty }{|}^2$. Result: $4\pi m_{\mathrm{ADM}}={\int }_M|\nabla \psi {|}^2+\frac{1}{4}{\int }_M\mathrm{Scal}|\psi {|}^2\ge 0$ under non-negative scalar curvature; equality forces a parallel spinor and hence flat ${\mathbb{R}}^3$. The Parker-Taubes 1982 reformulation made the existence of the Witten spinor rigorous via weighted Sobolev spaces. The Dirac-Witten operator $\hat{D}=D-\frac{1}{2}{K}^{ij}c(e_i){\nabla }_{e_j}$ extends the argument to the spacetime case (positive-energy theorem with dominant energy condition).

  requires: spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure, spin-geometry.dirac-bundle

• spin-geometry.berger-holonomyopen unit 03.09.18 →

  Berger holonomy classification and parallel spinors

  Berger 1955 classified the restricted holonomy of simply-connected, irreducible, non-symmetric Riemannian manifolds: $\mathrm{SO}(n)$ (generic), $\mathrm{U}(n)$ (Kähler), $\mathrm{SU}(n)$ (Calabi-Yau, Ricci-flat), $\mathrm{Sp}(n)$ (hyperkähler), $\mathrm{Sp}(n)\mathrm{Sp}(1)$ (quaternionic Kähler, Einstein), $G_2$ (dim 7, Ricci-flat), $\mathrm{Spin}(7)$ (dim 8, Ricci-flat). The proof is a representation-theoretic case analysis: which irreducible subalgebras of $\mathfrak{so}(n)$ admit non-zero curvature tensors satisfying the first Bianchi identity? Wang 1989 established the bijection between Berger's special holonomies $\{\mathrm{SU}(n),\mathrm{Sp}(n),G_2,\mathrm{Spin}(7)\}$ and the existence of a non-zero parallel spinor, with parallel-spinor counts 2, $n+1$, 1, 1 respectively. Bryant 1987 constructed local exceptional-holonomy metrics; Joyce 1996/2000 constructed compact $G_2$ and $\mathrm{Spin}(7)$ manifolds. The Wang bijection is the structural foundation of the Harvey-Lawson calibrated-geometry framework: every parallel spinor produces a calibration via spinor squaring.

  requires: spin-geometry.spinor-bundle, spin-geometry.structure.spin-structure, spin-geometry.spin-group

• spin-geometry.exercise-pack-ch4open unit 03.09.E2 →

  Chapter IV applications exercise pack (Lawson-Michelsohn Ch. IV supplement)

  Fourteen exercises covering Chapter IV applications of spin geometry. Distribution: 4 easy / 7 medium / 3 hard. Group I (4 exercises): psc obstruction — Lichnerowicz on flat torus, Â-genus of K3, α-invariant on $\mathbb{H}{P}^2$, enlargeable propagation under product. Group II (3 exercises): Witten positive-mass theorem — ADM mass of Schwarzschild, Witten boundary identity, equality case via parallel-spinor rigidity. Group III (3 exercises): Berger holonomy — list size, parallel-spinor count on K3, Calabi-Yau Ricci-flatness. Group IV (4 exercises): calibrated geometries — comass of volume form, Wirtinger inequality, Special Lagrangian lines, $G_2$ associative 3-planes via the cross product. Cross-cuts the four Batch-2 units 03.09.16–03.09.19. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: spin-geometry.psc-obstruction, spin-geometry.witten-positive-mass, spin-geometry.berger-holonomy, spin-geometry.calibrated-geometries

• diffgeo.de-rham.kunnethopen unit 03.04.12 →

  Künneth formula for de Rham cohomology — two proofs

  $H_{\mathrm{dR}}^{\ast }(M\times F)\cong H_{\mathrm{dR}}^{\ast }(M)\otimes H_{\mathrm{dR}}^{\ast }(F)$ for finite-type $M$, proved twice. First proof: Mayer-Vietoris induction over a good cover, with the cross-product map ${\pi }_M^{\ast }\alpha \wedge {\pi }_F^{\ast }\beta$ as the natural transformation, base case the Poincaré lemma on ${\mathbb{R}}^n\times F$, inductive step via the five lemma on the MV ladder. Second proof: tic-tac-toe ascent on the Čech-de Rham double complex of the product cover $\mathcal{U}\times \mathcal{V}$, factoring as Čech-of-constant-coefficients tensored. The dual-proof presentation is Bott-Tu's pedagogical signature — one theorem, told twice; the second telling is shorter because the first installed the machinery. Failure mode: infinite-type manifolds; cross-product is injective but the tensor-product splitting fails. Consequences: $H^{\ast }(T^n)={\Lambda }^{\ast }[{\theta }_1,\dots ,{\theta }_n]$ exterior algebra of dimension $2^n$; multiplicativity of Euler characteristic $\chi (M\times F)=\chi (M)\chi (F)$; Poincaré-polynomial multiplication $P_{M\times F}(t)=P_M(t)P_F(t)$.

  requires: diffgeo.de-rham.mayer-vietoris, diffgeo.de-rham.good-cover-induction, diffgeo.de-rham.cech-de-rham-double-complex, topology.de-rham-cohomology

• alg-top.singular-cohomologyopen unit 03.04.13 →

  Singular cohomology and the de Rham theorem (with $\mathbb{Z}$ coefficients)

  Singular complex $C_{\ast }(X)={⨁}_n\mathbb{Z}⟨\sigma :{\Delta }^n\to X⟩$ with face boundary $\partial$; singular cohomology $H_{\mathrm{sing}}^{\ast }(X;A)=H^{\ast }(\mathrm{Hom}(C_{\ast },A))$. Key constructions: cone construction — every chain on a contractible space is a boundary, hence acyclic; small chains — for an open cover, every chain is chain-equivalent to a sum of chains supported in single opens, giving a singular MV long exact sequence. De Rham theorem $H_{\mathrm{dR}}^k(M)\cong H_{\mathrm{sing}}^k(M;\mathbb{R})$ via three routes: (i) MV induction over a good cover comparing the de Rham and singular MV functors via the integration pairing; (ii) Čech-de Rham double complex collapses to both Čech and de Rham, identifying both with ${\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})$; (iii) sheaf-cohomology via Leray on the constant sheaf $\underline{\mathbb{R}}$ with the Poincaré-lemma fine resolution. The integer-coefficient version requires the singular complex (de Rham only sees $\mathbb{R}$ coefficients). Eilenberg-Steenrod axioms 1952 axiomatised the construction; de Rham 1931 proved the analytic equivalence directly.

  requires: diffgeo.de-rham.mayer-vietoris, topology.de-rham-cohomology, diffgeo.de-rham.cech-de-rham-double-complex, alg-geom.sheaf-cohomology

• alg-geom.cohomology.local-system-monodromyopen unit 04.03.02 →

  Local systems, monodromy, and twisted cohomology

  A local system on $X$ is a locally constant sheaf $\mathcal{L}$ of $A$-modules — concretely, a sheaf such that every point has a neighborhood where $\mathcal{L}$ is the constant sheaf with stalk a fixed $A$-module $V$. On a connected, locally simply-connected $X$, the category of local systems is equivalent to the category of ${\pi }_1(X,x_0)$-representations: parallel transport along loops gives the monodromy representation $\rho :{\pi }_1(X)\to \mathrm{Aut}(V)$. Examples: constant sheaf (trivial monodromy); orientation local system ${\mathrm{or}}_M$ on a non-orientable manifold (monodromy $\mathbb{Z}/2$ via the orientation double cover); Möbius local system on $S^1$. Twisted cohomology $H^{\ast }(X;\mathcal{L})$ is the cohomology with local-system coefficients; for orientation-twisted coefficients on a non-orientable manifold, Poincaré duality recovers via $H^k(M;\underline{\mathbb{R}})\cong H_c^{n-k}(M;{\mathrm{or}}_M)$ (the twisted Poincaré duality of Bott-Tu §7). Cohomology $H^{\ast }(X;\mathcal{L})$ on $K(\pi ,1)$-spaces equals group cohomology $H^{\ast }(\pi ;V)$ with $V$ a $\pi$-module via $\rho$.

  requires: alg-geom.sheaf-cohomology, topology.cover.double-cover, topology.covering-space, topology.homotopy

• diffgeo.de-rham.exercise-packs.chapter-iopen unit 03.04.E1 →

  Mayer-Vietoris and degree-theory exercise pack (Bott-Tu Ch. I supplement)

  Fourteen exercises covering Chapter I of Bott-Tu — Mayer-Vietoris computation, degree theory, Hopf invariant, sphere cohomology by induction, ${\mathbb{R}}^n$-minus-$k$-points, the punctured torus, Stokes-on-manifolds-with-boundary applications. Distribution: 4 easy / 6 medium / 4 hard. Each exercise carries a hint and full solution in <details> blocks. Cross-cuts N1 (MV), N2 (good cover), N4 (Thom and degree), and the deepening D1 of 03.04.06. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: diffgeo.de-rham.mayer-vietoris, diffgeo.de-rham.good-cover-induction, diffgeo.de-rham.thom-cv-cohomology, topology.de-rham-cohomology

• diffgeo.de-rham.mayer-vietorisopen unit 03.04.07 →

  Mayer-Vietoris sequence for de Rham cohomology

  For a smooth manifold $M=U\cup V$, the short exact sequence $0\to {\Omega }^{\ast }(M)\to {\Omega }^{\ast }(U)\oplus {\Omega }^{\ast }(V)\to {\Omega }^{\ast }(U\cap V)\to 0$ produces the long exact Mayer-Vietoris sequence in de Rham cohomology. Surjectivity is the partition-of-unity argument; the connecting homomorphism $\delta :H^q(U\cap V)\to H^{q+1}(M)$ is the canonical zig-zag of homological algebra. Compactly-supported variant has reversed maps; both extend to arbitrary good covers via the generalised Mayer-Vietoris sequence (the row exactness of the Čech-de Rham double complex). The canonical inductive computation of $H^{\ast }(S^n)$ runs through this machinery — Bott-Tu's pedagogical heart. Closes Bott-Tu §1 → §2 within-chapter sequencing.

  requires: topology.de-rham-cohomology, diffgeo.exterior-derivative, diffgeo.differential-forms, diffgeo.smooth-manifold

• diffgeo.de-rham.good-cover-inductionopen unit 03.04.10 →

  Good covers, finite-dimensionality of de Rham cohomology, and the Mayer-Vietoris induction

  A good cover of a smooth manifold is an open cover $\mathcal{U}$ such that every finite intersection $U_{{\alpha }_0\cdots {\alpha }_p}$ is diffeomorphic to a Euclidean ball (in particular contractible). Existence: choose a Riemannian metric and use the fact that geodesically convex normal-coordinate neighborhoods exist around every point and have geodesically convex finite intersections. Finite good cover exists on every compact manifold; countable good cover on every paracompact manifold. The Mayer-Vietoris induction over a finite good cover proves: $H_{\mathrm{dR}}^{\ast }(M)$ is finite-dimensional in each degree on a compact manifold, vanishes above $\dim M$, and admits explicit inductive computation. Foundation for both the Čech-de Rham double-complex theorem and the Künneth / Poincaré-duality dual proofs.

  requires: diffgeo.de-rham.mayer-vietoris, topology.de-rham-cohomology, diffgeo.smooth-manifold

• diffgeo.de-rham.cech-de-rham-double-complexopen unit 03.04.11 →

  Čech-de Rham double complex and the tic-tac-toe principle

  Bigraded complex $K^{p,q}(\mathcal{U})={\prod }_{{\alpha }_0<\cdots <{\alpha }_p}{\Omega }^q(U_{{\alpha }_0\cdots {\alpha }_p})$ on a cover $\mathcal{U}$, with Čech differential $\delta$ horizontal (alternating-sum restriction), de Rham differential $d$ vertical (componentwise exterior derivative), and total differential $D=d+(-1{)}^p\delta$ on bidegree $(p,q)$. Total complex ${\mathrm{Tot}}^n(K)={⨁}_{p+q=n}{K}^{p,q}$. On a good cover, both row-collapse (yielding $H_{\mathrm{dR}}^{\ast }(M)$) and column-collapse (yielding ${\check{H}}^{\ast }(\mathcal{U};\underline{\mathbb{R}})$) compute the same total cohomology — Weil's 1952 proof of the de Rham theorem. Tic-tac-toe ascent is the diagonal-staircase algorithm that makes the equivalence explicit. Generalised Mayer-Vietoris is the row exactness; tic-tac-toe Künneth and Poincaré duality follow from the bigraded structure. Prototype of the spectral sequence of a filtered complex (Bott-Tu §14). Notation $K^{p,q}$, ${\mathrm{Tot}}^n$, $\delta$ vs $d$, $D=d+(-1{)}^p\delta$ — Pass 4 §3.4 decisions #12, #13, #27, #28.

  requires: diffgeo.de-rham.good-cover-induction, topology.de-rham-cohomology, diffgeo.de-rham.mayer-vietoris, alg-geom.sheaf-cohomology

• diffgeo.de-rham.thom-cv-cohomologyopen unit 03.04.09 →

  Compactly-supported cohomology, integration along the fiber, and the de Rham Thom isomorphism

  ${\Omega }_c^{\ast }(M)$ compactly supported forms; $H_c^{\ast }(M)$ compactly supported cohomology. For a vector bundle $\pi :E\to M$, ${\Omega }_{cv}^{\ast }(E)$ is the compactly-vertical complex (forms whose support meets every fiber in a compact set; Bott-Tu coinage), and $H_{cv}^{\ast }(E)$ its cohomology. Integration along the fiber ${\pi }_{\ast }:{\Omega }_{cv}^{p+r}(E)\to {\Omega }^p(M)$ for a rank-$r$ oriented bundle commutes with $d$ and induces the Thom isomorphism $H^p(M)\cong H_{cv}^{p+r}(E)$, with inverse $\omega \mapsto {\pi }^{\ast }\omega \wedge \Phi$ for the Thom class $\Phi \in H_{cv}^r(E)$ characterised by ${\int }_{\mathrm{fiber}}\Phi =1$. Global angular form $\psi$ on the unit-sphere bundle of an oriented rank-$r$ bundle, with Bott-Tu sign convention $d\psi =-{\pi }^{\ast }e(E)$. Provides the de Rham model of the Euler class as the obstruction to a global section, and dual proof (via Čech-de Rham of [03.04.11]) of the Thom isomorphism. Notation ${\Omega }_{cv}^{\ast }$, $H_{cv}^{\ast }$, $\Phi$, $\psi$ — Pass 4 §3.4 decisions #4, #8, #9, #21, #22.

  requires: diffgeo.de-rham.mayer-vietoris, topology.de-rham-cohomology, bundle.vector-bundle, diffgeo.stokes-theorem

• homotopy.spectral-sequence.filtered-complexopen unit 03.13.01 →

  Spectral sequences — exact couples, filtered complexes, double complexes

  A spectral sequence is a tower of bigraded pages $\{E_r^{p,q},d_r{\}}_{r\ge 1}$ with $d_r$ of bidegree $(r,1-r)$ (notation decision #15), $H(E_r,d_r)=E_{r+1}$, and limiting page $E_{\infty }$ recovering an associated graded of a filtration on a target $H^{\ast }$. The convergence symbol $\Rightarrow$ (decision #30) denotes this abutment. Three equivalent presentations: Massey 1952 exact couples (the algebraic structure that drives the page-advance), filtered cochain complexes (the geometric source — bounded filtrations give the cleanest convergence), and double complexes with two anticommuting differentials $d^{\prime },d^{\prime \prime }$ producing two spectral sequences ${}_{I}E$ and ${}_{II}E$ (decision #29) both abutting to $H^{\ast }(\mathrm{Tot})$. The Čech-de Rham double complex of [03.04.11] is the concrete prototype: its two collapsing spectral sequences prove the de Rham theorem on a good cover. Multiplicative structure: a filtered DGA gives each page the structure of a bigraded ring, $d_r$ a derivation, with $E_{\infty }$ the associated graded of the cup product. Edge homomorphisms and transgression supply the partial maps used in low-degree calculations. Master section channels Leray 1946 directly: invented at Oflag XVII-A, the prisoner-of-war camp where Leray hid his fluid-mechanics expertise behind the topological work that became sheaf theory and spectral sequences.

  requires: topology.de-rham-cohomology, diffgeo.de-rham.cech-de-rham-double-complex

• homotopy.spectral-sequence.serreopen unit 03.13.02 →

  Leray-Serre spectral sequence and the Gysin sequence

  For a fibration $F\to E\stackrel{\pi }{\to }B$ with $B$ simply connected (or a local system describing monodromy on ${\pi }_1(B)$), the Leray-Serre spectral sequence has $E_2^{p,q}=H^p(B;H^q(F))$ and converges multiplicatively to $H^{p+q}(E)$. Three canonical computations following Bott-Tu: (i) the Hopf fibration $S^1\to S^3\to S^2$ — non-trivial $d_2$ kills classes in degrees 1 and 2, giving $H^{\ast }(S^3)$ with classes only in 0 and 3; (ii) the trivial $S^1\to T^2\to S^1$ where the spectral sequence collapses at $E_2$ and recovers the Künneth product; (iii) the unit-circle bundle of $\mathcal{O}(1)\to \mathbb{C}{P}^k$ giving the cohomology of $S^{2k+1}$ via Gysin. The Gysin sequence is the long exact sequence

  requires: homotopy.spectral-sequence.filtered-complex, diffgeo.de-rham.mayer-vietoris, bundle.vector-bundle

• homotopy.spectral-sequence.leray-hirschopen unit 03.13.03 →

  Leray-Hirsch theorem and the splitting principle for vector bundles

  Leray-Hirsch theorem. For a fibre bundle $F\to E\stackrel{\pi }{\to }B$ with $H^{\ast }(F)$ free over the coefficient ring and finitely generated, if there exist global classes $e_1,\dots ,e_k\in H^{\ast }(E)$ whose restrictions to each fibre form a basis of $H^{\ast }(F)$, then $H^{\ast }(E)$ is a free $H^{\ast }(B)$-module on $\{e_i\}$:

  requires: homotopy.spectral-sequence.serre, char-classes.pontryagin-chern.definitions, bundle.vector-bundle

• homotopy.spectral-sequence.exercise-packopen unit 03.13.E1 →

  Spectral-sequence computation exercise pack (Bott-Tu Ch. III supplement)

  Eighteen exercises covering general spectral-sequence machinery, Leray-Serre and Gysin computations, Leray-Hirsch and the splitting principle, and Eilenberg-Moore on the path-loop fibration. Distribution: 5 easy / 9 medium / 4 hard. Group I (5 easy): bidegree of $d_r$, convergence symbol semantics, exact-couple from a SES, two-filtration $E_2$ identification, collapse-at-$E_2$ from a row/column hypothesis. Group II (4 medium): Hopf fibration $S^3\to S^2$ Serre SS, $S^1\to S^3\to S^2$ Gysin, $H^{\ast }(\mathbb{C}{P}^n)$ via Borel construction, Borel computation of $H^{\ast }(BU(n))$. Group III (5 medium): Leray-Hirsch on the projectivization $\mathbb{P}(E)\to M$, splitting principle for Pontryagin classes, $H^{\ast }(\mathrm{Gr}(2,4))$, Whitney via splitting, $H^{\ast }(\Omega S^n)$ via Serre. Group IV (4 hard): ${\pi }_4(S^3)=\mathbb{Z}/2$ via Postnikov truncation of $K(\mathbb{Z},3)$, Eilenberg-Moore on the path-loop fibration, transgression in a specific bundle, multiplicative structure. Cross-cuts the three Batch units 03.13.01–03. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: homotopy.spectral-sequence.filtered-complex, homotopy.spectral-sequence.serre, homotopy.spectral-sequence.leray-hirsch

• homotopy.rational.sullivan-minimal-modelsopen unit 03.12.06 →

  Sullivan minimal models and rational homotopy theory

  For a simply-connected space $X$ of finite rational type, the Sullivan minimal model is a quasi-isomorphism $\phi :(\Lambda V,d)\stackrel{\sim }{\to }{A}_{PL}(X)$ where $\Lambda V$ is a free graded-commutative algebra on a graded $\mathbb{Q}$-vector space $V$, the differential $d$ satisfies $dV\subseteq {\Lambda }^{\ge 2}V$ (minimality), and $A_{PL}$ is the piecewise-polynomial de Rham functor sending $X$ to compatible polynomial forms over each simplex. Sullivan's main theorem (1977): $V^n\cong {\mathrm{Hom}}_{\mathbb{Z}}({\pi }_n(X),\mathbb{Q})$, and the differential encodes Whitehead products and higher Massey products. Existence and uniqueness via the lifting lemma for minimal Sullivan algebras. Worked examples: $S^{2k}$ → $\Lambda (x,y)$ with $\mathrm{deg}x=2k$, $\mathrm{deg}y=4k-1$, $dy=x^2$; $S^{2k+1}$ → $\Lambda (x)$ with $\mathrm{deg}x=2k+1$, $dx=0$; $\mathbb{C}{P}^n$ → $\Lambda (x,y)$ with $\mathrm{deg}x=2$, $\mathrm{deg}y=2n+1$, $dy=x^{n+1}$. Halperin's algorithm computes the minimal model of a fibration $F\to E\to B$ as a perturbed tensor product $M_B\otimes \Lambda V_F$ with transgression encoded by the perturbation $\tau$. Formality theorem (Deligne-Griffiths-Morgan-Sullivan 1975): simply-connected compact Kähler manifolds have minimal models determined by their cohomology rings via the $d{d}^c$-lemma. Master section channels Sullivan 1977 Publ. IHÉS 47 directly: rational homotopy theory as the differential-form calculus on rational invariants, parallel to Quillen's DG-Lie-algebraic side, with Bott-Tu §19 the canonical pedagogical exposition.

  requires: topology.eilenberg-maclane, topology.de-rham-cohomology, homotopy.spectral-sequence.serre

• homotopy.universal-bundle-borelopen unit 03.08.05 →

  Universal bundle, $H^{\ast }(BU(k))$, and the Borel presentation of flag-manifold cohomology

  The universal complex rank-$k$ bundle ${\gamma }_k\to BU(k)$ is realised in the Grassmannian model $BU(k)=G_k({\mathbb{C}}^{\infty })=\underrightarrow{\lim }{G}_k({\mathbb{C}}^n)$ as the colimit ${\gamma }_k=\underrightarrow{\lim }{\gamma }_k^n$ of the tautological bundles ${\gamma }_k^n\to G_k({\mathbb{C}}^n)$ (notation decision #19). Steenrod's classification theorem identifies $[X,BU(k)]$ with rank-$k$ complex vector bundles on $X$ up to isomorphism, the bijection sending $[f]$ to $f^{\ast }{\gamma }_k$. The cohomology ring $H^{\ast }(BU(k);\mathbb{Z})=\mathbb{Z}[c_1,\dots ,c_k]$ is the polynomial algebra on the universal Chern classes, $\mathrm{deg}{c}_i=2i$; computed via the Leray-Serre spectral sequence of the fibration $\mathbb{C}{P}^{\infty }\to BU(k-1)\to BU(k)$ (or directly from the Schubert cell decomposition of the Grassmannian). Borel presentation (1953): for a compact Lie group $G$ with maximal torus $T$ of rank $r$ and Weyl group $W=N_G(T)/T$,

  requires: k-theory.classifying-spaces, homotopy.spectral-sequence.serre, char-classes.pontryagin-chern.definitions, homotopy.spectral-sequence.leray-hirsch

• homotopy.whitehead-tower-rational-hurewiczopen unit 03.12.07 →

  Whitehead tower, rational Hurewicz theorem, and Serre's finiteness

  The Whitehead tower of a connected space $X$ is a sequence $\cdots \to X⟨3⟩\to X⟨2⟩\to X⟨1⟩\to X$ of fibrations such that $X⟨n⟩$ is $n$-connected and the fibre $X⟨n⟩\to X⟨n-1⟩$ is the Eilenberg-MacLane space $K({\pi }_n(X),n-1)$. Dual to the Postnikov tower (which truncates from above), the Whitehead tower truncates from below: at each stage one kills the lowest non-vanishing homotopy group. The basepoint-loop space $\Omega X$ is the fibre of $PX\to X$ where $PX$ is the based path space (notation decision #32). Hurewicz theorem (1935-36): for a path-connected space $X$ with ${\pi }_i(X)=0$ for $i<n$, the Hurewicz map ${\pi }_n(X)\to H_n(X;\mathbb{Z})$ is an isomorphism (and surjective with abelianisation kernel for $n=1$). Rational Hurewicz: for simply-connected $X$, if ${\pi }_i(X)\otimes \mathbb{Q}=0$ for $1\le i\le n-1$, then ${\pi }_n(X)\otimes \mathbb{Q}\stackrel{\sim }{\to }{H}_n(X;\mathbb{Q})$ is an isomorphism, and ${\pi }_i(X)\otimes \mathbb{Q}\stackrel{\sim }{\to }{H}_i(X;\mathbb{Q})$ is an isomorphism for $1\le i\le 2n-2$. Hopf invariant $H(f)\in \mathbb{Z}$ (notation decision #34) of a map $f:S^{2n-1}\to S^n$: the integer counted by the linking number of two preimage spheres of generic regular values. Adams 1960 proved $H(f)=\pm 1$ only when $n\in \{1,2,4,8\}$. Serre's finiteness theorem (1953): for $k\ne n$ and except ${\pi }_{4k-1}(S^{2k})$ which has a $\mathbb{Z}$-summand from the Hopf invariant, the homotopy groups ${\pi }_k(S^n)$ are finite. Computations via Whitehead tower: ${\pi }_4(S^3)=\mathbb{Z}/2$ via $S^3\to K(\mathbb{Z},3)$ killing ${\pi }_3$; ${\pi }_5(S^3)=\mathbb{Z}/2$ via further truncation. Master section channels J.H.C. Whitehead 1953 + Hurewicz 1935-36 + Serre 1953 directly.

  requires: topology.eilenberg-maclane, homotopy.rational.sullivan-minimal-models, homotopy.spectral-sequence.serre

• homotopy.rational.exercise-packopen unit 03.12.E1 →

  Rational homotopy and Sullivan minimal-model exercise pack (Bott-Tu Ch. III §19 supplement)

  Eight exercises covering the Sullivan minimal-model machinery and rational homotopy. Distribution: 2 easy / 4 medium / 2 hard. Easy: minimal model of $S^{2k}$ (closed case), minimal model of $\mathbb{C}{P}^n$. Medium: minimal model of $\Omega S^n$ via Bott-Samelson; rational Hurewicz for a simply-connected space; minimal model of a fibration via Halperin's algorithm; minimal model of a Lie group as exterior algebra on primitive generators. Hard: Sullivan's solution to Serre's question ${\pi }_{\ast }(S^n)\otimes \mathbb{Q}$; formality of compact Kähler manifolds via the $d{d}^c$-lemma. Cross-cuts N12 and N14. Each exercise carries a hint and full answer in <details> blocks. Exercise-pack-only unit type — slimmed frontmatter with tiers_present: [intermediate], no Lean infrastructure.

  requires: homotopy.rational.sullivan-minimal-models, homotopy.whitehead-tower-rational-hurewicz, topology.de-rham-cohomology

• alg-geom.cech-schemesopen unit 04.03.03 →

  Čech cohomology of sheaves on schemes

  Čech cochain complex $C^p(\mathcal{U},\mathcal{F})={\prod }_{i_0<\cdots <i_p}\mathcal{F}(U_{i_0\cdots i_p})$ with alternating-sum-of-restriction differential $\delta :C^p\to C^{p+1}$; Čech cohomology ${\check{H}}^p(\mathcal{U},\mathcal{F})=\ker {\delta }^p/\mathrm{im}{\delta }^{p-1}$; refinement colimit ${\check{H}}^p(X,\mathcal{F})={\mathrm{colim}}_{\mathcal{U}}{\check{H}}^p(\mathcal{U},\mathcal{F})$. Cartan's comparison theorem: for separated scheme $X$, affine open cover $\mathcal{U}$, and quasi-coherent $\mathcal{F}$, the canonical map ${\check{H}}^{\ast }(\mathcal{U},\mathcal{F})\to H^{\ast }(X,\mathcal{F})$ is an isomorphism in all degrees. Foundation for the cohomology computation on ${\mathbb{P}}_k^n$ via the standard cover $\{D_{+}(x_i)\}$ — Hartshorne III.5. Affine vanishing (Serre): $H^p(\mathrm{Spec}A,\tilde{M})=0$ for $p>0$. Two-set covers reproduce Mayer-Vietoris. Čech-derived spectral sequence $E_2^{p,q}={\check{H}}^p(\mathcal{U},{\mathcal{H}}^q(\mathcal{F}))\Rightarrow H^{p+q}(X,\mathcal{F})$ degenerates when higher derived presheaves vanish on intersections. Originator: Čech 1932 (combinatorial topology); modern scheme version Serre 1955 FAC.

  requires: alg-geom.scheme, alg-geom.affine-scheme, alg-geom.sheaf-cohomology

• alg-geom.cohomology-projectiveopen unit 04.03.04 →

  Cohomology of line bundles on projective space

  Theorem (Hartshorne III.5.1, Serre 1955): for $X={\mathbb{P}}_k^n$ and $\mathcal{O}(d)$ the standard twisting sheaf, $H^0({\mathbb{P}}^n,\mathcal{O}(d))=S_d$ the degree-$d$ piece of $S=k[x_0,\dots ,x_n]$, of dimension $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ for $d\ge 0$ and zero otherwise; $H^n({\mathbb{P}}^n,\mathcal{O}(d))$ has basis Laurent monomials $x_0^{a_0}\cdots x_n^{a_n}$ with all $a_i<0$ and $\sum a_i=d$, of dimension $\left(\genfrac{}{}{0ex}{}{-d-1}{n}\right)$ for $d\le -n-1$ and zero otherwise; $H^p({\mathbb{P}}^n,\mathcal{O}(d))=0$ for $0<p<n$. Proof via Čech on the standard cover $\{D_{+}(x_i){\}}_{i=0}^n$: localised polynomial rings $S_{x_{i_0}\cdots x_{i_p}}$, alternating-sum differential, monomial bookkeeping with negative-support decomposition. Serre duality on ${\mathbb{P}}^n$: $H^n({\mathbb{P}}^n,\mathcal{O}(d))\cong H^0({\mathbb{P}}^n,\mathcal{O}(-d-n-1){)}^{\ast }$ via cup-product pairing, with dual basis matching $a_i+b_i=-1$; canonical sheaf ${\omega }_{{\mathbb{P}}^n}=\mathcal{O}(-n-1)$ read off from this. Euler-Poincaré characteristic: $\chi ({\mathbb{P}}^n,\mathcal{O}(d))=\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ as a polynomial identity in $d$. Hilbert polynomial of ${\mathcal{O}}_{{\mathbb{P}}^n}$ of degree $n$, leading coefficient $1/n!$, constant term $1$. Bott vanishing: $H^p({\mathbb{P}}^n,{\Omega }^q(d))=0$ for $0<p<n$, $d>0$ via Euler exact sequence and the cohomology table. Riemann-Roch on ${\mathbb{P}}^n$ via Chern character and Todd class recovers $\left(\genfrac{}{}{0ex}{}{n+d}{n}\right)$ from $\int e^{dH}\mathrm{td}({\mathbb{P}}^n)$. Originator: Serre 1955 FAC; standard pedagogical reference Hartshorne III.5.

  requires: alg-geom.cech-schemes, alg-geom.projective-scheme

• alg-geom.serre-vanishing-finitenessopen unit 04.03.05 →

  Serre's vanishing and finiteness theorems

  Theorem (Serre 1955; Hartshorne III.5.2). Setup: $A$ a noetherian ring, $X$ a noetherian projective scheme over $A$ with very ample ${\mathcal{O}}_X(1)$, $\mathcal{F}$ a coherent ${\mathcal{O}}_X$-module. (1) Finiteness: $H^i(X,\mathcal{F})$ is a finitely generated $A$-module for every $i\ge 0$, and $H^i(X,\mathcal{F})=0$ for $i>\dim X$ (Grothendieck vanishing). (2) Vanishing: there exists $n_0=n_0(\mathcal{F})$ such that $H^i(X,\mathcal{F}(n))=0$ for every $n\ge n_0$ and every $i>0$. Proof outline (3 ingredients). (i) Reduce to $X={\mathbb{P}}_A^N$ via the closed immersion $i:X\hookrightarrow {\mathbb{P}}_A^N$: $i_{\ast }$ is exact with $R^p{i}_{\ast }=0$ for $p\ge 1$ (closed immersion is affine), so the Leray spectral sequence collapses to $H^p(X,\mathcal{F})\cong H^p({\mathbb{P}}_A^N,i_{\ast }\mathcal{F})$. (ii) Compute $H^p({\mathbb{P}}_A^N,\mathcal{O}(d))$ explicitly via Čech on the standard cover [04.03.04]: $H^0=A[x_0,\dots ,x_N{]}_d$ free of rank $\left(\genfrac{}{}{0ex}{}{N+d}{N}\right)$ for $d\ge 0$, $H^N=$ free $A$-module of rank $\left(\genfrac{}{}{0ex}{}{-d-1}{N}\right)$ for $d\le -N-1$, zero in middle and high degrees. (iii) Reduce general coherent $\mathcal{F}$ to line bundles via Serre's theorem A: every coherent $\mathcal{F}$ on ${\mathbb{P}}_A^N$ admits a surjection $\mathcal{O}(-d{)}^{\oplus k}\twoheadrightarrow \mathcal{F}$, kernel $\mathcal{K}$ coherent, long exact sequence in cohomology, descending induction on $i$ with base case $i>N$ (Grothendieck vanishing). Cohomological characterisation of ampleness (Hartshorne III.5.3): $\mathcal{L}$ ample iff for every coherent $\mathcal{F}$ there is $n_0(\mathcal{F})$ with $H^i(X,\mathcal{F}\otimes {\mathcal{L}}^n)=0$ for $i>0$ and $n\ge n_0$. Castelnuovo-Mumford regularity: $\mathcal{F}$ is $m$-regular iff $H^i({\mathbb{P}}_A^N,\mathcal{F}(m-i))=0$ for $i\ge 1$; the regularity index $\mathrm{reg}(\mathcal{F})$ is the smallest such $m$, finite, monotone ($m$-regular implies $m^{\prime }$-regular for $m^{\prime }\ge m$), and provides the explicit threshold $n_0=\mathrm{reg}(\mathcal{F})$ for Serre vanishing; $\mathcal{F}(\mathrm{reg}(\mathcal{F}))$ is globally generated and the multiplication map $H^0(\mathcal{F}(m))\otimes H^0(\mathcal{O}(d))\to H^0(\mathcal{F}(m+d))$ is surjective (Mumford 1966). Generalisations: Grothendieck's relative finiteness/vanishing for proper morphisms (EGA III §2); Serre duality as the dual statement; Kodaira vanishing $H^i(X,{\omega }_X\otimes \mathcal{L})=0$ for ample $\mathcal{L}$ on smooth projective $X/\mathbb{C}$ (one-twist refinement, characteristic zero); Kawamata-Viehweg vanishing for nef-and-big $\mathbb{Q}$-divisors; Serre's affineness criterion ($X$ affine iff $H^i(X,\mathcal{F})=0$ for every coherent $\mathcal{F}$ and $i\ge 1$). Worked example on ${\mathbb{P}}^2$ with ideal sheaf ${\mathcal{I}}_C$ of a smooth plane curve $C$ of degree $e$ uses the SES $0\to \mathcal{O}(d-e)\to \mathcal{O}(d)\to {\mathcal{O}}_C(d)\to 0$ to compute the explicit threshold. Originator: Serre 1955 FAC; modernised in Hartshorne III.5.2-3 (1977) and EGA III (1961-63).

  requires: alg-geom.cech-schemes, alg-geom.cohomology-projective

• alg-geom.cech-cohomology-line-bundlesopen unit 06.04.02 →

  Čech cohomology of holomorphic line bundles

  For a compact Riemann surface $X$, a holomorphic line bundle $L\to X$ with transition functions $g_{ij}$ on a cover $\mathcal{U}=\{U_i\}$, and the sheaf $\mathcal{O}(L)$ of holomorphic sections, the Čech cochain complex $C^p(\mathcal{U},\mathcal{O}(L))={\prod }_{i_0<\cdots <i_p}\Gamma (U_{i_0\cdots i_p},L)$ with alternating-sum-of-restriction differential gives the Čech cohomology ${\check{H}}^p(X,L)={\mathrm{colim}}_{\mathcal{U}}{\check{H}}^p(\mathcal{U},L)$. Real dimension two forces $H^p(X,L)=0$ for $p\ge 2$, leaving only $H^0$ (global sections) and $H^1$. Dolbeault comparison: the fine resolution $0\to \mathcal{O}(L)\to C^{\infty }(L)\stackrel{\bar{\partial }}{\to }{C}^{\infty }(L\otimes \overline{T^{\ast }X})\to 0$ identifies ${\check{H}}^p(X,L)\cong H_{\bar{\partial }}^{0,p}(X,L)$ canonically. Worked computation on ${\mathbb{P}}^1$ with $L=\mathcal{O}(d)$ on the standard two-set cover $\{U_0,U_1\}$ with transition $z\mapsto z^d$ on ${\mathbb{C}}^{\ast }$: $\dim H^0(\mathcal{O}(d))=d+1$ for $d\ge 0$ and zero for $d\le -1$; by Serre duality $\dim H^1(\mathcal{O}(d))=-d-1$ for $d\le -2$ and zero otherwise. Mittag-Leffler problem (Forster §26): the obstruction to globalising prescribed principal parts on $X$ lives in $H^1(X,L)$, recovering Cousin I (additive principal parts) and Cousin II (multiplicative divisors) as cohomological-obstruction problems. Computational tool: short exact sequences $0\to \mathcal{O}(-D)\to \mathcal{O}\to {\mathcal{O}}_D\to 0$ for a divisor $D$ produce long exact sequences relating $H^{\ast }(X,\mathcal{O}(-D))$ to $H^{\ast }(X,\mathcal{O})$ and the local data of $D$, the inductive backbone of every line-bundle cohomology computation on a curve. Originator: Čech 1932 (combinatorial topology) + Behnke-Stein 1949 (Math. Ann. 120) for holomorphic-bundle data on Riemann surfaces; Serre 1955 FAC for the algebraic-side counterpart on schemes.

  requires: complex-analysis.holomorphic-line-bundle, complex-analysis.riemann-roch-compact-rs

• complex-analysis.stein-riemann-surfacesopen unit 06.09.01 →

  Stein Riemann surfaces

  A non-compact Riemann surface $X$ is Stein when it is holomorphically convex (for every compact $K\subset X$ the holomorphic hull $\hat{K}=\{z:|f(z)|\le {\sup }_K|f|,\forall f\in \mathcal{O}(X)\}$ is compact) and holomorphically separable (for every $p\ne q$ in $X$ there is $f\in \mathcal{O}(X)$ with $f(p)\ne f(q)$). Equivalent characterisations on a non-compact Riemann surface: existence of a strictly subharmonic exhaustion $\varphi :X\to \mathbb{R}$; vanishing $H^q(X,\mathcal{F})=0$ for $q\ge 1$ and every coherent analytic sheaf $\mathcal{F}$ (Theorem B); solvability of the $\bar{\partial }$-equation $\bar{\partial }f=g$ with no compatibility condition; solvability of Cousin I and II problems. Behnke-Stein theorem (1949): every non-compact Riemann surface is Stein — the trivial classification on a curve, replaced by an inequivalent rigidity condition only in higher complex dimension. Cohomological consequences: $H^1(X,\mathcal{O})=0$ (Mittag-Leffler always solvable), $H^1(X,{\mathcal{O}}^{\ast })=0$ for the simply-connected part (every line bundle on a non-compact RS is holomorphically trivial), classical Runge approximation extends to all of $X$ when $X\setminus K$ has no relatively compact components. Examples = the entire class of non-compact Riemann surfaces: $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, $\mathbb{C}\setminus \mathbb{N}$, the upper half-plane $\mathbb{H}$, the open unit disc $\mathbb{D}$, every annulus, the universal cover ${\mathbb{H}}^2$ of a genus-$g\ge 2$ surface, every compact Riemann surface minus a finite point set. Higher-dimensional Stein theory: same axioms produce the Cartan-Serre theorems A and B in arbitrary complex dimension, with Hörmander's $L^2$-method and Grauert's Oka principle as analytic engines. Bridge to symplectic topology: every Stein manifold carries a canonical Weinstein structure compatible with the cotangent symplectic form (Cieliebak-Eliashberg 2012). Bridge to algebraic geometry via Serre's GAGA on the projective side and the affine-versus-Stein dictionary on the open side. Originators: Behnke-Stein 1949 (RS case), Stein 1951 (general Stein-manifold definition), Cartan 1951–53 (Theorems A and B).

  requires: complex-analysis.holomorphic-line-bundle, alg-geom.cech-cohomology-line-bundles

• complex-analysis.theorems-a-and-b-stein-rsopen unit 06.09.02 →

  Theorems A and B for Stein Riemann surfaces

  Theorem A: on a Stein Riemann surface, every coherent analytic sheaf is generated by its global sections. Theorem B: on a Stein Riemann surface, $H^q(X,\mathcal{F})=0$ for every coherent analytic $\mathcal{F}$ and every $q\ge 1$. Forster's 1-d proof: exhaustion by relatively compact Runge opens combined with Schwartz finiteness and a limit-passage argument. Forces Cousin I, Cousin II, and Mittag-Leffler on $X$ from a single vanishing identity. Originators: Cartan 1951–53; Cartan-Serre 1953.

  requires: complex-analysis.stein-riemann-surfaces, alg-geom.cech-cohomology-line-bundles

• complex-analysis.behnke-steinopen unit 06.09.03 →

  Behnke-Stein theorem

  Behnke-Stein theorem (1949). Every connected non-compact Riemann surface is Stein. Cornerstone classification of non-compact Riemann surfaces; combined with the uniformisation theorem on the compact side gives a complete picture of complex one-folds. Equivalent reformulations: (1) every non-compact RS admits a strictly subharmonic exhaustion $\varphi :X\to \mathbb{R}$; (2) for every coherent analytic sheaf $\mathcal{F}$ on a non-compact $X$, $H^q(X,\mathcal{F})=0$ for $q\ge 1$ (Theorem B); (3) $H^1(X,\mathcal{O})=0$, i.e. Mittag-Leffler holds globally; (4) $\mathrm{Pic}(X)=0$, every line bundle holomorphically trivial. Proof outline: construct a Runge exhaustion $K_1⋐K_2⋐\cdots$ with $\bigcup K_n=X$; build a strictly subharmonic exhaustion via a limit-passage Runge approximation; solve $\bar{\partial }f=g$ globally via Hörmander $L^2$ on each pseudoconvex piece + Mittag-Leffler / Fréchet limit passage; deduce cohomological vanishing and holomorphic separability + holomorphic convexity from the exhaustion. Why dim 1 is special. Every domain in $\mathbb{C}$ is automatically holomorphically convex (Runge's theorem 1885 generalises). In higher dim, this fails — ${\mathbb{C}}^2\setminus \{0\}$ is not Stein. The notion of "non-compact + open" implies Stein in dim 1; in higher dim, pseudoconvexity is required (Levi's theorem). The dim-1 proof is therefore much simpler than the general Stein-manifold theory. Higher-dim picture. Stein $\iff$ holomorphically convex $\iff$ pseudoconvex (Bergman 1934 / Oka 1942 / Bremermann 1954 / Norguet 1954) — the Levi problem. In dim $n\ge 2$, not every non-compact connected complex $n$-fold is Stein; Behnke-Stein is the dim-1 "easy" case. Examples (all Stein on the dim-1 side): $\mathbb{C}$, ${\mathbb{C}}^{\ast }$, $\mathbb{C}\setminus K$ for $K$ finite, open Riemann surfaces of any genus, disc, half-plane, annulus, compact RS minus a finite point set, universal cover $\widetilde{{\Sigma }_g}$ of higher-genus ${\Sigma }_g$ (the disc). Connection to Riemann's mapping theorem: every simply connected proper open $\Omega \subset \mathbb{C}$ biholomorphic to the unit disc — special case of Behnke-Stein for simply connected non-compact RS, biholomorphic to $\mathbb{C}$ or the disc by uniformisation. Connection to uniformisation: compact RS (3 types — ${\mathbb{P}}^1$, elliptic, hyperbolic); non-compact (universal cover $\mathbb{C}$ or disc); Behnke-Stein gives the structure of the quotient by deck transformations. Originators. Heinrich Behnke + Karl Stein 1949 Entwicklung analytischer Funktionen auf Riemannschen Flächen (Math. Ann. 120). Karl Stein 1951 abstracted to Stein manifolds in higher dim. Levi problem in higher dim solved by Bremermann + Norguet + Oka in 1954.

  requires: complex-analysis.stein-riemann-surfaces, complex-analysis.theorems-a-and-b-stein-rs

• complex-analysis.cousin-i-additiveopen unit 06.09.04 →

  Cousin I (additive)

  Cousin I problem (additive). Given an open cover $\{U_i\}$ of a non-compact Riemann surface $X$ and meromorphic functions $f_i\in \mathcal{M}(U_i)$ with $f_i-f_j\in \mathcal{O}(U_i\cap U_j)$ on every overlap, find a global meromorphic $f\in \mathcal{M}(X)$ with $f-f_i\in \mathcal{O}(U_i)$ for every $i$. The differences $g_{ij}=f_i-f_j$ form a Čech 1-cocycle in the structure sheaf ${\mathcal{O}}_X$, and solvability is exactly the cohomology class $[\{g_{ij}\}]\in H^1(X,{\mathcal{O}}_X)$ vanishing. By Behnke-Stein 1949 + Cartan-Serre Theorem B, $H^1(X,{\mathcal{O}}_X)=0$ on every non-compact Riemann surface, so Cousin I is unconditionally solvable. Sheaf-theoretic formulation. Use the short exact sequence $0\to {\mathcal{O}}_X\to {\mathcal{M}}_X\to {\mathcal{P}}_X\to 0$ where ${\mathcal{M}}_X$ is the meromorphic-function sheaf and ${\mathcal{P}}_X={\mathcal{M}}_X/{\mathcal{O}}_X$ is the principal-parts sheaf with stalks the Laurent-tail rings. The connecting homomorphism $\delta :\Gamma (X,{\mathcal{P}}_X)\to H^1(X,{\mathcal{O}}_X)$ is the Cousin I obstruction map; the datum is solvable when $\delta (\sigma )=0$. Classical Mittag-Leffler. The 1884 planar case on $\mathbb{C}$: for $a_n\to \infty$ and prescribed Laurent tails $P_n(1/(z-a_n))$, the convergence-factor series ${\sum }_n(P_n(1/(z-a_n))-q_n(z))$ with $q_n$ a partial Taylor expansion is the explicit Čech coboundary trivialisation. Closed form for poles at integers with residue 1: $\pi \cot (\pi z)$. Compact case. On a compact RS of genus $g$, $H^1(X,{\mathcal{O}}_X)\cong {\mathbb{C}}^g$ (Hodge / Serre duality on a curve), so a generic Cousin I datum is not solvable; the obstruction dimension is the genus. Higher-dimensional picture. Cousin posed the problem in 1895 on bidiscs in ${\mathbb{C}}^n$ and solved the polydisc case by iterated Cauchy integrals. Oka 1937 (J. Sci. Hiroshima Univ.) settled Cousin I on every domain of holomorphy in ${\mathbb{C}}^n$. Cartan-Serre 1953 (Cartan séminaire, CRAS 237) extended to abstract Stein manifolds via Theorem B. Hörmander 1965 supplied the modern $L^2$-PDE engine. Failure on non-Stein manifolds. ${\mathbb{C}}^2\setminus \{0\}$: Hartogs extension forces every holomorphic function to extend to all of ${\mathbb{C}}^2$, breaking the Stein hypothesis and producing a non-zero $H^1(X,{\mathcal{O}}_X)$. The dichotomy Cousin-I-solvable-vs-not coincides with Stein-vs-not in arbitrary complex dimension. Cousin II (multiplicative). Replace differences with ratios $f_i/f_j\in {\mathcal{O}}^{\ast }(U_i\cap U_j)$; obstruction in $H^1(X,{\mathcal{O}}_X^{\ast })=\mathrm{Pic}(X)$. On a non-compact RS, both $H^1(X,{\mathcal{O}}_X)$ and $H^2(X,\mathbb{Z})$ vanish, so Cousin II is also unconditionally solvable; on Stein manifolds in higher dimension, Cousin II can fail when $H^2(X,\mathbb{Z})\ne 0$. Originators. Pierre Cousin 1895 (Acta Math. 19) — thesis problem in ${\mathbb{C}}^n$; Mittag-Leffler 1884 (Acta Math. 4) — planar special case; Oka 1937 — domains of holomorphy in ${\mathbb{C}}^n$; Cartan-Serre 1953 — Stein manifolds in arbitrary dimension; Behnke-Stein 1949 — RS case via the dimension-one Stein theorem.

  requires: complex-analysis.theorems-a-and-b-stein-rs, complex-analysis.behnke-stein

• complex-analysis.cousin-ii-multiplicativeopen unit 06.09.05 →

  Cousin II (multiplicative)

  Cousin II problem (multiplicative). Given an open cover $\{U_i\}$ of a Riemann surface $X$ and non-zero meromorphic $f_i\in {\mathcal{M}}^{\ast }(U_i)$ with $f_i/f_j\in {\mathcal{O}}^{\ast }(U_i\cap U_j)$ (holomorphic and non-vanishing) on every overlap, find a global $f\in {\mathcal{M}}^{\ast }(X)$ with $f/f_i\in {\mathcal{O}}^{\ast }(U_i)$ for every $i$. The ratios $g_{ij}=f_i/f_j$ form a multiplicative Čech 1-cocycle in ${\mathcal{O}}_X^{\ast }$, and solvability is exactly the cohomology class $[\{g_{ij}\}]\in H^1(X,{\mathcal{O}}_X^{\ast })=\mathrm{Pic}(X)$ vanishing — equivalently, the holomorphic line bundle determined by the cocycle is holomorphically trivial. Exponential sheaf sequence. $0\to 2\pi i\mathbb{Z}\to {\mathcal{O}}_X\stackrel{\exp }{\to }{\mathcal{O}}_X^{\ast }\to 0$ produces the long exact sequence segment $H^1(X,{\mathcal{O}}_X)\to \mathrm{Pic}(X)\stackrel{c_1}{\to }{H}^2(X,\mathbb{Z})\to H^2(X,{\mathcal{O}}_X)$ where $c_1$ is the first Chern class. Non-compact RS case. On a connected non-compact Riemann surface, $H^1(X,{\mathcal{O}}_X)=0$ (Behnke-Stein + Theorem B) and $H^2(X,\mathbb{Z})=0$ (top-degree integer cohomology of a connected non-compact 2-manifold), so $\mathrm{Pic}(X)=0$ and Cousin II is unconditionally solvable. The two-fold vanishing chain — analytic + topological — distinguishes Cousin II from Cousin I, which only needs the analytic part. Sheaf-theoretic formulation. Short exact sequence $0\to {\mathcal{O}}_X^{\ast }\to {\mathcal{M}}_X^{\ast }\to {\mathcal{D}}_X\to 0$ where ${\mathcal{D}}_X={\mathcal{M}}_X^{\ast }/{\mathcal{O}}_X^{\ast }$ is the divisor sheaf; connecting map $\delta :\Gamma (X,{\mathcal{D}}_X)\to \mathrm{Pic}(X)$ is the Cousin II obstruction. Classical Weierstrass product. The 1876 planar case on $\mathbb{C}$: for $a_n\to \infty$ and multiplicities $m_n$, the product ${\prod }_n{E}_{p_n}(z/a_n{)}^{m_n}$ with elementary factors $E_p(z)=(1-z)\exp (z+z^2/2+\cdots +z^p/p)$ is the explicit multiplicative Čech coboundary trivialisation. Closed form for simple zeros at every integer: $\sin (\pi z)$. Compact case. On a compact RS of genus $g$, $\mathrm{Pic}(X)$ sits in $0\to {\mathrm{Pic}}^0(X)\to \mathrm{Pic}(X)\stackrel{\mathrm{deg}}{\to }\mathbb{Z}\to 0$ with ${\mathrm{Pic}}^0(X)$ the Jacobian, a $g$-dim complex torus. Generic Cousin II data are not solvable; obstruction is the line bundle's degree (topological) plus its class in ${\mathrm{Pic}}^0$ (analytic). Higher-dimensional picture. Cousin posed Cousin II in 1895 on polydiscs in ${\mathbb{C}}^n$ and solved them directly (contractible). Oka 1939 (J. Sci. Hiroshima Univ. Ser. A 9) gave the first counterexample on a domain in ${\mathbb{C}}^2$ with non-trivial $H^2(\mathbb{Z})$ and formulated the Oka principle: on a Stein manifold, every continuous Cousin II datum has a holomorphic solution iff the topological obstruction vanishes. Grauert 1958 (Math. Ann. 135) extended to vector bundles of arbitrary rank — the Oka-Grauert principle: holomorphic and topological classifications of complex vector bundles on a Stein manifold agree as bijections. Cousin I vs Cousin II. Cousin I obstruction in $H^1(\mathcal{O})$, purely analytic; Cousin II obstruction in $\mathrm{Pic}$, sandwiched between analytic and topological via the exponential sequence. In dim 1 non-compact, both vanish; in dim ≥ 2 Stein, Cousin I always solvable but Cousin II conditional on topology. Originators. Pierre Cousin 1895 (Acta Math. 19) — thesis problem in ${\mathbb{C}}^n$; Weierstrass 1876 — planar special case (entire functions with prescribed zeros); Oka 1939 — counterexample + Oka principle; Grauert 1958 — Oka-Grauert principle for vector bundles; Cartan-Serre 1953 — cohomological reformulation via the exponential sequence on Stein manifolds; Behnke-Stein 1949 — RS case via dimension-one Stein theorem.

  requires: complex-analysis.cousin-i-additive, complex-analysis.theorems-a-and-b-stein-rs

• alg-geom.sheaf-cohomology-surveyopen unit 06.04.07 →

  Survey of sheaf cohomology on Riemann surfaces

  Synoptic survey gathering the four pictures of sheaf cohomology on a Riemann surface and the comparison theorems that make them equivalent. (1) Čech: ${\check{H}}^{\ast }(X,\mathcal{F})$ via the alternating-sum-of-restriction complex on open covers and the colimit over refinements (computational; dovetails with the line-bundle case in 06.04.02). (2) Dolbeault / harmonic: $H^q(X,\mathcal{F})\cong {\mathcal{H}}^q(X,\mathcal{F})$ via the harmonic kernel of the Hodge-Laplace, with the Bochner-Kodaira-Nakano refinement (analytic; the framework of 06.04.05). (3) Derived-functor: $H^{\ast }(X,\mathcal{F})=R^{\ast }\Gamma (X,-)$ in the abelian category of sheaves of ${\mathcal{O}}_X$-modules (Grothendieck 1957); most flexible, least computational. (4) Singular / topological for constant coefficient sheaves: $H^{\ast }(X,\mathbb{Z})$ via singular cochains on the underlying topological surface, matched with $H_{dR}^{\ast }(X,\mathbb{C})$ via de Rham. Comparison theorems. Cartan-Leray: Čech equals derived-functor on a paracompact Hausdorff space. Dolbeault: fine resolution by smooth $(0,q)$-forms identifies derived-functor cohomology with $\bar{\partial }$-cohomology, with harmonic representatives via the Hodge theorem on compact Kähler. Hodge: $H^k(X,\mathbb{C})={⨁}_{p+q=k}{H}^{p,q}(X)$ with $H^{p,q}$ the harmonic $(p,q)$-forms; on a Riemann surface this is the genus identification $H^1(X,\mathbb{C})=H^{1,0}(X)\oplus H^{0,1}(X)={\mathbb{C}}^g\oplus \overline{{\mathbb{C}}^g}$. Serre duality (Serre 1955 Un théorème de dualité): $H^q(X,\mathcal{F})\cong H^{n-q}(X,{\mathcal{F}}^{\vee }\otimes {\omega }_X{)}^{\ast }$ for coherent $\mathcal{F}$ on a smooth proper $n$-dimensional scheme; the curve case ($n=1$) is 06.04.04. GAGA (Serre 1956 Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier 6): on a smooth projective variety over $\mathbb{C}$, algebraic and analytic coherent cohomology agree, identifying the analytic Čech / Dolbeault picture with the algebraic Hartshorne III picture. Computational toolkit. Long exact sequence in cohomology from a short exact sequence of sheaves; Mayer-Vietoris from a two-set cover; twisting / shift via $0\to \mathcal{O}(L-p)\to \mathcal{O}(L)\to {\mathbb{C}}_p\to 0$; Schwartz finiteness $\dim H^q(X,\mathcal{F})<\infty$ for coherent $\mathcal{F}$ on compact $X$; Riemann-Roch $\chi (L)=\mathrm{deg}L+1-g$ for line bundles on a curve; Serre vanishing $H^1(X,L)=0$ for $\mathrm{deg}L\ge 2g-1$. Standard cohomology table (compact RS of genus $g$): $h^0({\mathcal{O}}_X)=1$, $h^1({\mathcal{O}}_X)=g$; $h^0({\omega }_X)=g$, $h^1({\omega }_X)=1$; for $\mathcal{O}(D)$ with $\mathrm{deg}D>2g-2$, $h^0=\mathrm{deg}D-g+1$ and $h^1=0$ (non-special range); for $\mathrm{deg}D<0$, $h^0=0$ and $h^1=h^0({\omega }_X\otimes \mathcal{O}(-D))$; for $0\le \mathrm{deg}D\le 2g-2$, both sides require Riemann-Roch + Serre + speciality bookkeeping. Higher-dimensional generalisation. Cartan-Serre Theorems A and B on Stein manifolds; Kodaira vanishing on compact Kähler with positive line bundles; Grothendieck cohomology on noetherian schemes via injective resolutions; the four-pictures-agree mantra extends with Hodge replaced by mixed Hodge (Deligne) on singular varieties and de Rham replaced by crystalline / étale on positive-characteristic schemes. Failure modes. Non-Kähler compact complex manifolds (Hopf surface): Hodge symmetry $h^{p,q}=h^{q,p}$ fails, so harmonic-vs-Dolbeault identification is incomplete. Non-paracompact spaces: Čech-vs-derived-functor can differ (counterexamples in Godement). Non-coherent sheaves on a Riemann surface: $H^{\ast }$ may be infinite-dimensional, and harmonic theory does not directly apply. Originators: Čech 1932 (combinatorial cohomology); Cartan 1953-55 (sheaf-theoretic refinement, séminaire); Serre 1955 FAC (algebraic-side coherent cohomology); Hodge 1941 (harmonic representatives); Dolbeault 1953 (Dolbeault cohomology); Hörmander 1965 ($L^2$ existence); Grothendieck 1957 Tôhoku (derived-functor cohomology in an abelian category).

  requires: alg-geom.cech-cohomology-line-bundles, complex-analysis.dbar-hilbert-pde, alg-geom.serre-duality-curves

• complex-analysis.dbar-hilbert-pdeopen unit 06.04.05 →

  Hilbert-space PDE for $\bar{\partial }$

  Setup: compact Riemann surface $X$ with Hermitian metric $\omega$ and Hermitian holomorphic line bundle $(L,h)$. The $L^2$ space of $L$-valued $(0,q)$-forms is the completion of ${\Omega }^{0,q}(X,L)$ under $⟨s,t⟩={\int }_X⟨s,t{⟩}_{h}d{V}_{\omega }$. The Cauchy-Riemann operator $\bar{\partial }:L_{(0,q)}^2(X,L)\to L_{(0,q+1)}^2(X,L)$ is a closed densely-defined unbounded operator with Hilbert adjoint ${\bar{\partial }}^{\ast }$. The Hodge-Laplace ${\Delta }_{\bar{\partial }}=\bar{\partial }{\bar{\partial }}^{\ast }+{\bar{\partial }}^{\ast }\bar{\partial }$ is self-adjoint, elliptic of order two, non-negative. Hodge-Dolbeault theorem on a compact Riemann surface: $L_{(0,q)}^2(X,L)={\mathcal{H}}^q(X,L)\oplus \overline{\mathrm{im}\bar{\partial }}\oplus \overline{\mathrm{im}{\bar{\partial }}^{\ast }}$ orthogonally; ${\mathcal{H}}^q(X,L)=\ker {\Delta }_{\bar{\partial }}$ is finite-dimensional and canonically isomorphic to $H^q(X,L)$ via the Dolbeault fine resolution. Solvability: $\bar{\partial }f=g$ for $g\in L_{(0,1)}^2(X,L)$ closed has a solution iff $g\perp {\mathcal{H}}^1(X,L)$; the canonical solution is $f={\bar{\partial }}^{\ast }Gg$, the Bergman / Green-operator output. Bochner-Kodaira-Nakano formula ${\Delta }_{\bar{\partial }}={\Delta }_{\partial }+[i\Theta (L,h),\Lambda ]$ on $L$-valued forms gives the curvature lower bound $⟨{\Delta }_{\bar{\partial }}\alpha ,\alpha ⟩\ge {\int }_X(i\Theta /\omega )|\alpha {|}^{2}dV$ on $(0,1)$-forms; positive curvature forces ${\mathcal{H}}^1=0$, recovering Kodaira vanishing $H^1(X,L)=0$ for $\mathrm{deg}L>2g-2$. Hörmander's $L^2$ existence theorem: on a complete Kähler manifold with curvature $i\Theta (L,h)\ge c\omega$, every $\bar{\partial }$-closed $g\in L_{(0,q)}^2$ admits a solution with $\Vert f{\Vert }^2\le c^{-1}\Vert g{\Vert }^2$ — foundation of complex analysis on Stein manifolds. Schwartz finiteness theorem $\dim H^q(X,\mathcal{F})<\infty$ for coherent $\mathcal{F}$ on compact $X$ via $\bar{\partial }$-Hodge theory + ellipticity of ${\Delta }_{\bar{\partial }}$. Spectral discreteness on compact $X$: eigenvalues $0={\lambda }_0<{\lambda }_1\le {\lambda }_2\le \cdots \to \infty$ with finite multiplicities; the Bergman kernel of $\ker \bar{\partial }$ encodes geometric data (Tian 1990, Zelditch 1998). Worked examples: ${\mathbb{P}}^1$ with $\mathcal{O}(d)$ (Kodaira positive for $d\ge 1$ gives $H^1=0$); elliptic curve $\mathbb{C}/\Lambda$ trivial bundle (harmonic $(0,1)$-form $d\bar{z}$ generates $H^1(X,\mathcal{O})\cong \mathbb{C}$); higher genus with $\mathrm{deg}L\ge 2g-1$ gives $H^1=0$. Originators: Hörmander 1965 (Acta Math. 113); Andreotti-Vesentini 1965 (compact Kähler $L^2$); Hodge 1941 (harmonic representatives); Kodaira 1953 (vanishing).

  requires: alg-geom.cech-cohomology-line-bundles, functional-analysis.bounded-operators

• analysis.real-number-axiomsopen unit 02.02.01 →

  Real-number axioms (ordered field)

  The standard axiomatic characterisation of $\mathbb{R}$ as a complete ordered field. Thirteen axioms in three families: nine field axioms (closure, associativity, commutativity of $+,\cdot$; identity elements $0\ne 1$; additive inverse; multiplicative inverse for nonzero; distributivity), three order axioms (trichotomy, transitivity, compatibility of $<$ with $+$ and $\cdot$), and one completeness axiom (every non-empty bounded-above subset has a least upper bound). Apostol's Axiom 11 is the standard textbook framing. *Categoricity:** any two complete ordered fields are uniquely order-isomorphic, so $\mathbb{R}$ is uniquely characterised up to isomorphism. Archimedean property as theorem from completeness: for every $x\in \mathbb{R}$ some $n\in \mathbb{N}$ has $n>x$; proof by contradiction via $\sup \mathbb{N}$. Equivalent reformulations of (C): Cauchy completeness plus Archimedean, nested-interval property, monotone convergence theorem, Bolzano-Weierstrass. Independence of axioms demonstrated by witnessing structures: $\mathbb{Z}$ violates (F6), $\mathbb{C}$ admits no compatible order, $\mathbb{Q}$ violates (C). Constructive vs classical completeness (Bishop-Bridges 1967) — Cauchy completeness is intuitionistically valid; LUB completeness needs LEM. Non-Archimedean extensions (Robinson 1966) — hyperreals $^\mathbb{R}$relaxArchimedean,restoringinfinitesimals.\ast \ast Tarski1948\ast \ast —thefirst-ordertheoryofrealclosedfieldsisdecidablebyquantifierelimination,contrastingwiththeundecidabilityof$\mathbb{N}$. Foundational for [02.02.02] sup/inf, [02.03.01] sequence convergence, [02.04.04] FTC, and the entire single-variable analysis strand. Originators: Hilbert 1899 (Grundlagen der Geometrie) for the axiomatisation; Apostol 1967 for the canonical pedagogical presentation; Dedekind 1872 / Cantor 1872 for the construction-from-rationals alternative; Bourbaki 1940- for the modern formalisation; Tarski 1948 for the elementary theory's decision procedure.

  requires: set-theory.function

• analysis.cauchy-bolzano-weierstrassopen unit 02.03.02 →

  Cauchy sequences and Bolzano-Weierstrass

  Two convergence criteria for sequences in $\mathbb{R}$ that together encode the metric form of completeness. Cauchy sequence: $(a_n)$ is Cauchy iff for every $\epsilon >0$ some $N$ has $|a_m-a_n|<\epsilon$ whenever $m,n\ge N$ — terms eventually cluster tightly against one another rather than against a named limit. Bolzano-Weierstrass theorem: every bounded sequence in $\mathbb{R}$ has a convergent subsequence, proved by repeated bisection of a bounding interval and the nested-interval consequence of completeness. Cauchy criterion theorem: a real sequence converges iff it is Cauchy, proved by combining the triangle inequality (convergent implies Cauchy) with boundedness plus Bolzano-Weierstrass plus the Cauchy condition (Cauchy implies convergent). The two theorems package the metric form of the LUB axiom: completeness in the sense of "every Cauchy sequence converges" is equivalent over the Archimedean ordered field axioms to the least-upper-bound completeness of [02.02.01]. Worked example: $a_n=(-1{)}^n/n$ is bounded between $-1$ and $1$; the even-index subsequence converges to $0$; the full sequence converges to $0$; Bolzano-Weierstrass guarantees the subsequence's existence from boundedness alone. Master scope: metric-space completeness (every Cauchy sequence converges); the Cauchy completion functor (every metric space embeds densely into a complete one, generalising the construction of $\mathbb{R}$ from $\mathbb{Q}$); sequential compactness (every sequence has a convergent subsequence, equivalent to compactness on metric spaces but not in general); Heine-Borel in ${\mathbb{R}}^n$ (closed and bounded equals compact equals sequentially compact); failures in infinite dimensions — the closed unit ball of ${\ell }^2$ is bounded but the standard-basis sequence $(e_n)$ has no convergent subsequence, so Bolzano-Weierstrass fails; Banach's contraction principle uses Cauchy completeness as input. Originators: Bolzano 1817 Rein analytischer Beweis for the original intermediate-value-style clustering argument; Weierstrass 1860s Berlin lectures (published in his Werke 1894-1895) for the bisection proof now standard; Cauchy 1821 Cours d'analyse for the Cauchy criterion as a working definition (without a proof of completeness); modern unified treatment in Apostol Calculus Vol. 1 Ch. 10 and Rudin Principles of Mathematical Analysis Ch. 3. Foundational for [02.01.05] metric-space completeness, [02.03.03] series-convergence tests via the Cauchy criterion for partial sums, [02.04.02] the extreme-value theorem and the IVT via compactness, [02.05.01] multi-variable limits using Cauchy and sequential compactness, and [02.11.04] Banach-space completeness.

  requires: analysis.real-number-axioms

• analysis.multi-variable-limit-continuityopen unit 02.05.01 →

  Multi-variable limit and continuity

  Limit and continuity for $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$. Open ball $B_r(a)=\{x:\Vert x-a\Vert <r\}$ from the Euclidean norm. Definition: ${\lim }_{x\to a}f(x)=L$ iff for every $\epsilon >0$ there exists $\delta >0$ with $0<\Vert x-a\Vert <\delta \Rightarrow \Vert f(x)-L\Vert <\epsilon$; $f$ is continuous at $a$ iff ${\lim }_{x\to a}f(x)=f(a)$; continuous iff continuous at every point of its domain. Worked example contrast: $f(x,y)=x^2+y^2$ is continuous at the origin, but $g(x,y)=xy/(x^2+y^2)$ has no limit at the origin — the approach $y=x$ gives $1/2$ while $y=0$ gives $0$, so directional limits disagree. Sequential characterisation theorem: $f$ is continuous at $a$ iff for every sequence $x_k\to a$, the image sequence $f(x_k)\to f(a)$. The forward direction picks $\delta$ from $\epsilon$ and $N$ from $\delta$; the contrapositive of the converse builds a witness sequence with $\Vert x_k-a\Vert <1/k$ but $\Vert f(x_k)-f(a)\Vert \ge {\epsilon }_0$. Bridge to the general-topology characterisation (preimage of open is open, [02.01.02]); to uniform continuity and the multivariable Heine-Cantor theorem on compact domains; to the differential structure built on top of continuity in [02.05.02]; to first-countability — sequential continuity equals topological continuity exactly when the domain has countable neighbourhood bases. The path-independence requirement is the central insight: in ${\mathbb{R}}^n$ for $n\ge 2$, "$x$ approaches $a$" allows infinitely many directions, so a candidate limit must agree along all of them. Master scope: multivariable Heine-Cantor (continuous on compact equals uniformly continuous); composition continuity via preimage axioms; Tietze extension on normal spaces; Banach fixed-point theorem on complete metric spaces; first-countability and the gap to general topological continuity. Originators: Cauchy 1821 (Cours d'analyse) for single-variable $\epsilon$-$\delta$; Riemann's lectures of the 1850s for the multivariable case in pedagogical practice; Heine 1872 and Cantor 1872 (independently) for uniform continuity on compact sets; Fréchet's 1906 thesis introduced the metric / abstract-space framework that subsumes both single- and multi-variable limits. Foundational for [02.05.02] partial derivative and the differential, [02.05.03] chain rule, [02.05.04] inverse and implicit function theorems, and [02.05.05] Taylor and extrema.

  requires: analysis.real-number-axioms

• analysis.multivariable-chain-ruleopen unit 02.05.03 →

  Chain rule for multi-variable functions

  The composition rule for differentiation between Euclidean spaces. Setup: $g:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^m$ differentiable at $a$ with derivative $Dg(a):{\mathbb{R}}^n\to {\mathbb{R}}^m$ (the Jacobian) and $f:V\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^p$ differentiable at $b=g(a)$ with derivative $Df(b)$. Statement: $f\circ g$ is differentiable at $a$ and $D(f\circ g)(a)=Df(g(a))\circ Dg(a)$; in Jacobian-matrix form, $J_{f\circ g}(a)=J_f(g(a))\cdot J_g(a)$, an ordinary $p\times m$ by $m\times n$ matrix product yielding a $p\times n$ matrix. Worked example: $g(t)=(\cos t,\sin t)$ and $f(x,y)=x^2+y^2$ give $(f\circ g)(t)=1$; direct derivative is $0$; chain-rule gives $(2\cos t,2\sin t)\cdot (-\sin t,\cos t{)}^T=0$. Proof structure (linear-approximation form): define remainders ${\rho }_g(h)=g(a+h)-g(a)-Dg(a)h$ and ${\rho }_f(k)=f(b+k)-f(b)-Df(b)k$, both $o(\Vert h\Vert )$ and $o(\Vert k\Vert )$ respectively; set $k(h)=Dg(a)h+{\rho }_g(h)$; then $(f\circ g)(a+h)=f(g(a))+Df(b)\cdot Dg(a)h+Df(b){\rho }_g(h)+{\rho }_f(k(h))$; bound $\Vert Df(b){\rho }_g(h)\Vert \le \Vert Df(b){\Vert }_{\mathrm{op}}\Vert {\rho }_g(h)\Vert$ and $\Vert k(h)\Vert \le (\Vert Dg(a){\Vert }_{\mathrm{op}}+1)\Vert h\Vert$ on a small ball; conclude the two remainder terms are each $o(\Vert h\Vert )$. Bridge to the implicit and inverse function theorems via $D{g}^{-1}(b)=Dg(a{)}^{-1}$ (chain rule applied to $g^{-1}\circ g=\mathrm{id}$); to the differential structure of smooth manifolds where the chain rule is functoriality of the tangent functor; to the de Rham complex where pullback of forms satisfies $(\psi \circ \varphi {)}^{\ast }={\varphi }^{\ast }\circ {\psi }^{\ast }$; to the change-of-variables formula in multi-variable integration with $|\det Dg|$ correction factor. Master scope: Banach-space chain rule (same proof, Fréchet derivative in normed spaces); pushforward on tangent vectors with $(\psi \circ \varphi {)}_{\ast }={\psi }_{\ast }\circ {\varphi }_{\ast }$ (functoriality); pullback on differential forms with $(\psi \circ \varphi {)}^{\ast }={\varphi }^{\ast }\circ {\psi }^{\ast }$ (contravariance); Faà di Bruno formula for the $n$-th derivative of a composition with sum over set partitions; Itô formula in stochastic calculus with the half-Hessian quadratic-variation correction term; categorical view of differentiation as a functor from smooth manifolds to vector bundles. Originators: Leibniz 1684 (Nova methodus pro maximis et minimis, Acta Eruditorum) — originator of the single-variable rule $dy/dx\cdot dx/dt$; Cauchy and Lagrange (early nineteenth century) for rigorous proofs; Cartan c. 1900 for the intrinsic differential / coordinate-free form; Apostol Vol. 2 1969 for the canonical pedagogical multi-variable presentation; Faà di Bruno 1855 for the combinatorial higher-order formula; Itô 1944 for the stochastic chain rule. Foundational for [02.05.04] implicit and inverse function theorems, [02.05.05] Taylor and extrema (multi-variable Taylor uses the chain rule iteratively), [02.06.*] ODE systems (variational equations via the chain rule), [03.02.01] smooth manifolds (chain rule is the structural reason coordinate changes patch together), [03.04.04] exterior derivative (pullback functoriality is the chain rule), and the change-of-variables formula in multi-variable integration.

  requires: analysis.multi-variable-limit-continuity

• analysis.implicit-inverse-function-theoremsopen unit 02.05.04 →

  Implicit and inverse function theorems

  The two foundational local-invertibility theorems for multi-variable maps. Inverse function theorem: if $f:U\subseteq {\mathbb{R}}^n\to {\mathbb{R}}^n$ is $C^k$ on open $U$ and $Df(a)$ is invertible at $a\in U$, then there are open neighbourhoods $V\ni a$ and $W\ni f(a)$ with $f{|}_V:V\to W$ a $C^k$ bijection with $C^k$ inverse, and $D(f^{-1})(f(a))=(Df(a){)}^{-1}$. Implicit function theorem: if $g:U\subseteq {\mathbb{R}}^{n+m}\to {\mathbb{R}}^m$ is $C^k$ with $g(a,b)=0$ and $D_{2}g(a,b)$ (the Jacobian in the last $m$ variables) is invertible, then there is a neighbourhood $V\ni a$ and a $C^k$ map $h:V\to {\mathbb{R}}^m$ with $h(a)=b$ and $g(x,h(x))=0$ for $x\in V$, with derivative $Dh(a)=-D_{2}g(a,b{)}^{-1}{D}_{1}g(a,b)$. Worked example (inverse): $f(x,y)=(x^2-y^2,2xy)$ at $(1,0)$: $Df=\mathrm{diag}(2,2)$, determinant $4$, invertible, so local inverse exists (globally $f$ is the complex squaring map, 2-to-1). Worked example (implicit): $g(x,y)=x^2+y^2-1$ at $(1/\sqrt{2},1/\sqrt{2})$: $g_y=2y=\sqrt{2}\ne 0$, so the unit circle is locally a graph $y=\sqrt{1-x^2}$. Proof structure (inverse implies implicit): define $F(x,y)=(x,g(x,y))$; then $DF(a,b)$ is block lower-triangular with diagonal blocks $I_n$ and $D_{2}g(a,b)$, hence invertible iff $D_{2}g(a,b)$ is invertible; apply inverse function theorem to $F$ and read off $h$ from the second component of the inverse. Proof structure (inverse function theorem): define $T_y(x)=x-Df(a{)}^{-1}(f(x)-y)$; this is a contraction on a small closed ball for $y$ near $f(a)$, with contraction constant $1/2$ from continuity of $Df$; Banach contraction principle gives a unique fixed point — the inverse value $f^{-1}(y)$. Bridge to the manifold structure on level sets (a regular level set of a $C^k$ submersion is a $C^k$ submanifold — the regular-value theorem of [03.02.01]); to the constant-rank theorem (a $C^k$ map of constant rank is locally equivalent to a linear projection $(x_1,\dots ,x_n)\mapsto (x_1,\dots ,x_r,0,\dots ,0)$ in suitable coordinates); to the Banach-space inverse function theorem with the same contraction-mapping proof under a topological-isomorphism hypothesis; to the holomorphic inverse function theorem powering the local theory of Riemann surfaces and the definition of étale morphisms in algebraic geometry; to the real-analytic inverse function theorem via Cauchy-Kovalevskaya majorant series; to the Nash-Moser hard implicit function theorem on tame Fréchet spaces with loss-of-derivatives compensation, used in KAM theory and the Riemannian embedding problem. Master scope: Banach-space form with topological-isomorphism hypothesis; failure of inverse on Banach spaces when $Df(a)$ is algebraically bijective but lacks continuous inverse (witness $T(x_n)=(x_n/n)$ on $c_0$); holomorphic form and étale morphisms; real-analytic form; constant rank theorem subsuming both inverse and implicit theorems; Nash-Moser smoothed Newton iteration on tame Fréchet spaces; applications to KAM, isometric Riemannian embedding, and nonlinear PDE. Originators: Newton's Method of Fluxions (1671, published 1736) for single-variable series-form forerunner; Lagrange Théorie des fonctions analytiques 1797 for multi-variable series form; Cauchy 1831 Turin lectures for the analytic majorant-series form; Dini 1877 Lezioni di analisi infinitesimale for the modern ${\mathbb{R}}^n$ form under continuous-partials hypothesis; Goursat 1903 Bulletin de la Société Mathématique de France 31 for the contraction-mapping proof under $C^1$ hypothesis; Banach 1922 Fundamenta Mathematicae 3 for the contraction-mapping principle as an abstract tool; Dieudonné 1960 Foundations of Modern Analysis for the Banach-space textbook presentation; Apostol 1969 Vol. 2 Ch. 13 for the canonical pedagogical presentation; Nash 1956 Annals of Mathematics 63 for the smoothed Newton iteration in the Riemannian embedding problem; Moser 1966 Annali Scuola Norm. Sup. Pisa 20 for the Nash-Moser refinement and KAM applications; Hamilton 1982 Bulletin of the AMS 7 for the tame-Fréchet-space framework. Foundational for [02.05.05] Taylor and extrema (Lagrange multiplier rule is a direct corollary), [03.02.01] smooth manifolds (regular-value theorem makes level sets into submanifolds), [03.02.02] the constant-rank theorem and submanifold structure, [04.] étale morphisms in algebraic geometry, [05.09.] KAM theory via the Nash-Moser hard IFT, [06.*] Riemann surfaces via the holomorphic IFT, and the Picard-Lindelöf existence theorem for ODEs which shares the Banach contraction engine.

  requires: analysis.multivariable-chain-rule

• analysis.multivariable-taylor-extremaopen unit 02.05.05 →

  Taylor's theorem and extrema in several variables

  Multi-variable Taylor expansion plus the second-derivative test for extrema. Taylor's theorem (multi-variable, Lagrange form): for $f:U\to \mathbb{R}$ of class $C^{k+1}$ on a convex open $U\subseteq {\mathbb{R}}^n$, $a\in U$, $a+h\in U$, there exists $c$ on the segment from $a$ to $a+h$ with $f(a+h)={\sum }_{|\alpha |\le k}{D}^{\alpha }f(a)h^{\alpha }/\alpha !+{\sum }_{|\alpha |=k+1}{D}^{\alpha }f(c)h^{\alpha }/\alpha !$, using multi-index notation $\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{N}}^n$, $|\alpha |=\sum {\alpha }_i$, $\alpha !={\alpha }_1!\cdots {\alpha }_n!$, $h^{\alpha }=h_1^{{\alpha }_1}\cdots h_n^{{\alpha }_n}$, $D^{\alpha }={\partial }^{|\alpha |}/(\partial x_1^{{\alpha }_1}\cdots \partial x_n^{{\alpha }_n})$. The second-order specialisation reads $f(a+h)=f(a)+\nabla f(a)\cdot h+\frac{1}{2}{h}^T{H}_f(a)h+o(\Vert h{\Vert }^2)$ where $H_f(a)=({\partial }^{2}f/\partial x_i\partial x_j(a))$ is the Hessian matrix, symmetric by Clairaut-Schwarz for $C^2$ functions. Second-derivative test theorem: for a $C^2$ function $f$ with $\nabla f(a)=0$ and $H=H_f(a)$ the Hessian: (1) $H$ positive definite ⇒ strict local min; (2) $H$ negative definite ⇒ strict local max; (3) $H$ indefinite (both positive and negative eigenvalues) ⇒ saddle; (4) $H$ semidefinite with a zero eigenvalue ⇒ inconclusive. Worked examples: $f(x,y)=x^2+y^2$, critical point $(0,0)$, Hessian $\mathrm{diag}(2,2)$, both eigenvalues positive, local minimum; $g(x,y)=x^2-y^2$, critical point $(0,0)$, Hessian $\mathrm{diag}(2,-2)$, mixed signs, saddle. Proof structure (case (1)): Taylor expansion at $a$ with $\nabla f(a)=0$ gives $f(a+h)-f(a)=\frac{1}{2}{h}^T{H}_f(c(h))h$ for $c(h)$ on the segment; by continuity of $H_f$, $h^T{H}_f(c(h))h\ge \frac{1}{2}{\lambda }_{\min }(H)\Vert h{\Vert }^2$ for small $\Vert h\Vert$, with ${\lambda }_{\min }>0$ the smallest eigenvalue of $H$; the Rayleigh-quotient bound dominates the remainder, giving strict positivity. Counterexamples for case (4): $f(x,y)=x^4+y^4$ has zero Hessian at origin and is a strict local min; $g(x,y)=x^4-y^4$ has zero Hessian at origin and is a saddle; $h(x,y)=x^2-y^3$ has positive semidefinite Hessian (one zero eigenvalue) but the origin is not a local min. Bridge to Morse theory — a Morse function has only non-degenerate critical points, each labelled by an index (number of negative Hessian eigenvalues), and the topology of sublevel sets changes by attaching a cell of dimension equal to the index as $c$ passes a critical value; foundation for the h-cobordism theorem and Floer homology. To Lagrange multipliers via the implicit function theorem [02.05.04]: constrained extrema are critical points of the Lagrangian $L(x,\lambda )=f(x)-\lambda g(x)$, and the bordered-Hessian classifies them. To catastrophe theory (Thom 1972): generic degenerate critical points in $r$-parameter families with $r\le 4$ have one of seven normal forms (fold, cusp, swallowtail, butterfly, three umbilics). To Laplace's method: $\int e^{-Nf(x)}dx\sim e^{-Nf(a^{\ast })}(2\pi /N{)}^{n/2}/\sqrt{\det H_f(a^{\ast })}$ as $N\to \infty$ for a non-degenerate minimum $a^{\ast }$. Master scope: integral form of the Taylor remainder; Morse lemma (local quadratic normal form at non-degenerate critical points via parametric inverse function theorem); Lagrange multiplier rule with bordered-Hessian classification; Thom's elementary catastrophes; Laplace's method and the saddle-point method in complex analysis; multi-jet classification of degenerate critical points. Originators: Taylor 1715 for the single-variable formula; Lagrange 1797 for the remainder form; Cauchy 1821 Cours d'analyse for the first rigorous $\epsilon$-$\delta$ proof; Hesse 1857 for the Hessian determinant in algebraic geometry; Morse 1925 for Morse theory; Milnor 1963 Morse Theory for the modern textbook presentation; Smale 1961 for the h-cobordism theorem application; Thom 1972 for catastrophe theory; Mather 1968–1971 for the rigorous proofs; Apostol 1969 Vol. 2 Ch. 9 for the canonical undergraduate pedagogical presentation. Foundational for Morse theory and the topology of manifolds (pending unit in 03.12.), Lagrange multipliers in constrained optimisation, the saddle-point method in complex analysis (pending unit in 06.), the stationary-phase asymptotic in oscillatory-integral theory, and partition-function expansions in statistical mechanics.

  requires: analysis.implicit-inverse-function-theorems