Sources
The Fast Track booklist. 92 primary texts anchor the curriculum. Each shipped Master-tier unit cites at least one of these books, with page references in the bibliography.
12
Free / open
1
Public domain
76
Acquire
13
Audits done
Audits available
Audits derive a punch-list of curriculum gap-units from the book's table of contents. Each new audit expands the queue. 13 of 92 books audited so far.
- Arnold — *Mathematical Methods of Classical Mechanics* (Fast Track 1.11) — Audit + Gap Plan · FT 1.11
- Bott-Tu — *Differential Forms in Algebraic Topology* — Fast Track Equivalence Plan · FT 3.27
- Brown — *Topology and Groupoids* (Fast Track 1.05) — Audit + Gap Plan · FT 1.05
- Cannas da Silva — *Lectures on Symplectic Geometry* (Fast Track 1.11) — Audit + Gap Plan · FT 1.11
- Donaldson — *Riemann Surfaces* (Fast Track 1.07) — Audit + Gap Plan · FT 1.07
- Forster — *Lectures on Riemann Surfaces* (Fast Track 1.07-paired) — Audit + Gap Plan · FT 1.07
- Hartshorne — *Algebraic Geometry* (Fast Track reference anchor) — Audit + Gap Plan
- Hatcher — *Algebraic Topology* (Fast Track reference anchor) — Audit + Gap Plan
- Ahlfors — *Complex Analysis* (Fast Track 1.04) — Audit + Gap Plan · FT 1.04
- Lawson-Michelsohn — *Spin Geometry* — Fast Track Equivalence Plan
- Lang — *Basic Mathematics* (Fast Track 0.1) — Audit + Gap Plan · FT 0.1
- Apostol — *Calculus, Vol. 2: Multi-Variable Calculus and Linear Algebra, with Applications to Differential Equations and Probability* (Fast Track 0.3) — Audit + Gap Plan · FT 0.3
- Apostol — *Calculus, Vol. 1: One-Variable Calculus, with an Introduction to Linear Algebra* (Fast Track 0.2) — Audit + Gap Plan · FT 0.2
0. Prerequisites (assumed before Section 1)
| FT | Title | Author | Topic | Status | Audit |
|---|---|---|---|---|---|
0.1 | Basic Mathematics | Serge Lang | Pre-calc foundations | BUY | open |
0.2 | Calculus, Vol 1 (Single Variable) | Tom M. Apostol | Single-variable calculus, proof-based | BUY | open |
0.3 | Calculus, Vol 2 (Multi-Variable) | Tom M. Apostol | Multivariable calculus, linear algebra intro | BUY | open |
0.4 | Ordinary Differential Equations | Vladimir Arnold | ODEs via geometric/qualitative perspective | BUY | open |
1. Fundamental Ideas in Mechanics, Fields, and Geometry
| FT | Title | Author | Topic | Status | Audit |
|---|---|---|---|---|---|
1.01 | Linear Algebra | Georgiy Shilov | Linear algebra fundamentals, proof-based | PD | — |
1.02 | Mechanics | Landau, Lifshitz | Classical mechanics, Lagrangian/Hamiltonian | BUY | — |
1.03 | The Classical Theory of Fields | Landau, Lifshitz | SR, EM, GR | BUY | — |
1.04 | Complex Analysis | Lars Ahlfors | Contour integration, residues | BUY | open |
1.05 | Topology and Groupoids | Ronald Brown | Algebraic topology, groupoids | FREE | open |
1.06 | Riemann Surfaces | Ahlfors, Sario | Homology, Riemann surface theory | BUY | — |
1.07 | Riemann Surfaces | Simon Donaldson | Modern: theta functions, bundles, Gauss-Manin | BUY | open |
1.08 | Lie Groupoids and Lie Algebroids in Diff Geometry | Kirill Mackenzie | Lie groupoids, geometric structures | BUY | — |
1.09 | General Theory of Lie Groupoids and Lie Algebroids | Kirill Mackenzie | Advanced Lie groupoid theory | BUY | — |
1.10 | Lectures on Differential Geometry | Shlomo Sternberg | Diff geom, multilinear algebra, Lie groups | FREE | — |
1.11 | Dynamical Systems IV (Symplectic Geometry) | Arnol'd, Dubrovin | Symplectic geometry | BUY | open |
1.12 | Topological Methods in Hydrodynamics | Vladimir Arnol'd | Diff geom in fluid mechanics | BUY | open |
1.13 | Geometric Integration Theory | Hassler Whitney | Analytical geometric integration | BUY | — |
1.14 | Curvature in Mathematics and Physics | Shlomo Sternberg | GR, gauge theory applications | BUY | — |
1.15 | Group Theory and Physics | Shlomo Sternberg | Rep theory, quantum mechanics | BUY | — |
1.16 | Applications of Lie Groups to Differential Equations | Peter Olver | Lie symmetries of PDEs | BUY | — |
1.17 | Differential Forms in Algebraic Topology | Bott, Tu | Cohomology, differential forms, De Rham | BUY | open |
1.18 | Multivariable Mathematics (Ch. 8) | Theodore Shifrin | Differential forms, Stokes' theorem | BUY | — |
1.19 | Rational Homotopy Theory and Differential Forms | Griffiths, Morgan | Differential forms in homotopy theory | BUY | — |
1.20 | Finite Element Exterior Calculus | Douglas N. Arnold | Numerical PDEs via FE | FREE | open |
2. Quantum Theory and Statistical Physics
| FT | Title | Author | Topic | Status | Audit |
|---|---|---|---|---|---|
2.01 | Quantum Mechanics: Non-Relativistic Theory | Landau, Lifshitz | QM fundamentals | BUY | — |
2.02 | Quantum Theory, Groups and Representations | Peter Woit | QM via rep theory, geometric quantization | FREE | — |
2.03 | QFT Lecture Notes | Sourav Chatterjee | Second quantization, basic QFT | FREE | — |
2.04 | Generalized Functions Vol. 1 | Gel'fand, Vilenkin | Delta functions, Fourier | BUY | — |
2.05 | Generalized Functions Vol. 2 | Gel'fand, Vilenkin | Schwartz spaces, TVS | BUY | — |
2.06 | Generalized Functions Vol. 3 | Gel'fand, Vilenkin | Distributions for ODE/PDE | BUY | — |
2.07 | Generalized Functions Vol. 4 | Gel'fand, Vilenkin | Self-adjoint operators, unitary reps | BUY | — |
2.08 | Generalized Functions Vol. 5 | Gel'fand, Vilenkin | Integral geometry, rep theory | BUY | — |
2.09 | Generalized Functions Vol. 6 | Gel'fand, Vilenkin | Adeles, automorphic forms, zetas | BUY | — |
2.10 | Statistical Physics 1 | Landau, Lifshitz | Stat mech, phase transitions | BUY | — |
2.11 | Structure of Dynamical Systems | Jean-Marie Souriau | Symplectic formulation of stat mech | BUY | — |
2.12 | Exactly Solved Models in Statistical Mechanics | R.J. Baxter | Lattice statistical models | BUY | — |
2.13 | Statistical Field Theory Vol. 1 | Itzykson, Drouffe | Stat field theory foundations | BUY | — |
2.14 | Statistical Field Theory Vol. 2 | Itzykson, Drouffe | Advanced stat field theory | BUY | — |
2.15 | MIT OCW: Statistical Mechanics II | MIT (Kardar et al) | Stat field theory course | FREE | — |
2.16 | Stochastic Quantization | Parisi et al | Stochastic quantization | OTHER | — |
2.17 | The Quantum Theory of Fields Vol. 1 | Steven Weinberg | QFT foundations | BUY | — |
2.18 | The Quantum Theory of Fields Vol. 2 | Steven Weinberg | Gauge theory, electroweak, instantons | BUY | — |
2.19 | The Quantum Theory of Fields Vol. 3 | Steven Weinberg | Supersymmetry | BUY | — |
2.20 | Quantum Electrodynamics | Landau, Lifshitz | QED | BUY | — |
3. Modern Geometry, Algebraic Topology, Mathematical Foundations
| FT | Title | Author | Topic | Status | Audit |
|---|---|---|---|---|---|
3.01 | Algebra | Serge Lang | Algebraic structures, rep theory, Galois | BUY | — |
3.02 | Homological Algebra (Algebra V) | Gel'fand, Manin | Derived categories, D-modules, Riemann-Hilbert | BUY | — |
3.03 | Morse Theory | John Milnor | Critical points, Bott periodicity | FREE | — |
3.04 | Morse Homology | Matthias Schwarz | Homology via Morse functions | BUY | — |
3.05 | Lectures on Field Theory and Topology | Daniel Freed | TQFT, cobordism | OTHER | — |
3.06 | Floer Homology Groups in Yang-Mills Theory | S.K. Donaldson | Infinite-dim Morse, gauge theory | BUY | open |
3.07 | Lectures on the H-Cobordism Theorem | John Milnor | h-cobordism theory | BUY | — |
3.08 | Characteristic Classes | Milnor, Stasheff | Topological invariants, vector bundles | BUY | — |
3.09 | Geometric Quantization | N.M.J. Woodhouse | Symplectic, Hamilton-Jacobi, quantization | BUY | — |
3.10 | K Theory | Michael Atiyah | Vector bundles, elliptic operators | BUY | — |
3.11 | Representation Theory: A First Course | Fulton, Harris | Finite groups, Lie groups, Dynkin | BUY | — |
3.12 | Complex Semisimple Lie Algebras | Jean-Pierre Serre | Semisimple classification | BUY | — |
3.13 | Lie Algebras and Lie Groups | Jean-Pierre Serre | p-adic, homological | BUY | — |
3.14 | A Course in Arithmetic | Jean-Pierre Serre | Number theory, Ostrowski | BUY | — |
3.15 | Linear Representations of Finite Groups | Jean-Pierre Serre | Finite group rep theory | BUY | — |
3.16 | Probability and Representation Theory | Persi Diaconis | Probability via rep theory | OTHER | — |
3.17 | Differential Geometry, Lie Groups, and Symmetric Spaces | Sigurdur Helgason | Real forms, symmetric spaces | BUY | — |
3.18 | Foundations of Differential Geometry Vol. 1 | Kobayashi, Nomizu | Connections, curvature | BUY | — |
3.19 | Foundations of Differential Geometry Vol. 2 | Kobayashi, Nomizu | Complex manifolds, Chern-Weil | BUY | — |
3.20 | Geometry of Yang-Mills Fields | Michael Atiyah | Yang-Mills, instantons | BUY | — |
3.21 | Algebraic Geometry | Robin Hartshorne | Varieties, schemes, sheaves | BUY | open |
3.22 | Geometry of Algebraic Curves | Griffiths, Harris | Algebraic curve theory | BUY | — |
3.23 | Function Theory of Several Complex Variables | Steven G. Krantz | Multivariable complex analysis | BUY | — |
3.24 | Introduction to Holomorphic Functions of Several Variables | R.C. Gunning | Several complex variable theory | BUY | — |
3.25 | Topological Methods in Algebraic Geometry | Friedrich Hirzebruch | Diff geom in alg geometry | BUY | — |
3.26 | Manifolds and Modular Forms | Friedrich Hirzebruch | Modular forms, topology | BUY | — |
3.27 | Hodge Theory and Complex Algebraic Geometry | Claire Voisin | Hodge theory via complex geometry | BUY | open |
3.28 | Riemannian Geometry and Geometric Analysis | Jürgen Jost | Diff geom approach to Hodge | BUY | — |
3.29 | Heat Kernels and Dirac Operators | Bismut, Ghys, Quillen | Index theorems via heat kernels | BUY | — |
3.30 | Moduli of Curves | Joe Harris | Deformation/moduli theory | BUY | — |
3.31 | Geometric Invariant Theory | David Mumford | Invariant theory, moduli | BUY | — |
3.32 | Introduction to Toric Varieties | William Fulton | Toric geometry, momentum maps | BUY | — |
3.33 | Tropical Geometry and Mirror Symmetry | Mark Gross | Tropical methods, string theory | BUY | — |
3.34 | Introduction to Modern Number Theory | Yuri I. Manin | Number theory, algebraic geometry | BUY | — |
3.35 | Spin Geometry | Lawson, Michelsohn | Clifford algebras, spinors, index theorem | BUY | open |
3.36 | Spinors and Space-Time Vol. 1 | Penrose, Rindler | Spinors in Lorentzian geometry | BUY | — |
3.37 | Spinors and Space-Time Vol. 2 | Penrose, Rindler | Twistor space | BUY | — |
3.38 | A Concise Course in Algebraic Topology | J. Peter May | Algebraic topology survey | FREE | — |
3.39 | More Concise Algebraic Topology | May, Ponto | Localization, completion, model categories | FREE | — |
3.40 | Simplicial Objects in Algebraic Topology | J. Peter May | Simplicial methods | FREE | — |
3.41 | Simplicial Homotopy Theory | Goerss, Jardine | Homotopy via simplicial objects | BUY | — |
3.42 | Complex Cobordism and Stable Homotopy Groups of Spheres | Doug Ravenel | Cobordism, stable homotopy | FREE | — |
3.43 | Quantum Fields and Strings Vols. 1-2 | Deligne, Kazhdan, Etingof, Morgan, Freed, Morrison, Jeffrey, Witten | Mathematical QFT, gauge theory, strings | FREE | — |
3.44 | The Global Approach to Quantum Field Theory Vols. 1-2 | Bryce S. DeWitt | Global QFT, curved spacetime | BUY | — |
3.45 | Microlocal Analysis of QF on Curved Spacetimes | Christian Gérard | Distribution theory for QFT | BUY | — |
3.46 | Differential Forms and Cohomology | Isu Vaisman | Sheaves, Čech cohomology | BUY | — |
3.47 | Loop Spaces, Characteristic Classes, Geometric Quantization | Jean-Luc Brylinski | Higher cohomology, stacks | BUY | — |
3.48 | Einstein Manifolds | Arthur L. Besse | Curvature, GR, Einstein metrics | BUY | — |