Learning Paths (Postdoc / Quals Programs)

Compact ordered reading programs through existing (or soon-to-exist) pages. Each bullet is a step: open the page, extract the named theorems, then proceed. Not a course syllabus — a dependency-respecting map for prelims and research onboarding.

See also How to use · Coverage · Home.

Algebra quals path

Standard algebra prelim: groups through Galois, then commutative and homological toolkit.

  1. Group theory — actions, Sylow, f.g. abelian, Jordan–Hölder.
  2. Rings and fields — ideals, Noetherian, localization, \(\operatorname{Spec}\) intuition.
  3. Modules — free/projective/injective; structure over PID; Hom/\(\otimes\) exactness.
  4. Galois theory — FTGT, cyclotomic, solvability by radicals.
  5. Commutative algebra foundations — dimension, depth, primary decomposition, Serre criteria.
  6. Homological commutative algebra — pd, Koszul, Auslander–Buchsbaum.
  7. Homological algebra foundations — Ext/Tor, resolutions, abelian categories.
  8. Derived categories — \(D(\mathcal{A})\), triangles (quals-plus).
  9. Representation theory foundations — characters; Maschke; Schur.
  10. Category theory foundations — adjunctions/Yoneda as organizing language.
  11. Algebraic geometry foundations — varieties vs schemes entrance.
  12. Cheatsheet: abstract algebra / Ext–Tor — formula drill before the exam.

Analysis quals path

Real, complex, measure, FA, and Fourier/PDE language as one program.

  1. Metric spaces — completeness, compactness, Baire.
  2. Real analysis — uniform convergence, interchange, multivariable interface.
  3. Measure theory foundations — measures, integral, \(L^p\), Fubini, RN.
  4. Functional analysis foundations — Banach/Hilbert, HB, open mapping, spectra.
  5. Complex analysis foundations — holomorphy, Cauchy theory setup.
  6. Analytic functions — power series, identities, singularities sketch.
  7. Complex integration · Residue theory — contour package.
  8. Fourier analysis foundations — Plancherel, inversion, Poisson.
  9. PDE foundations — type classification, weak solutions, energy methods.
  10. DE foundations — Picard, linear systems, Sturm–Liouville.
  11. Microlocal analysis foundations — wavefront sets, \(\Psi\)DOs (quals-plus).
  12. Calculus multivariable — Stokes package recall if rusty.
  13. Cheatsheet: FA / complex / DE — spectral radius, residues, characteristics.

Geometry / topology path

From point-set and classical geometry to algebraic topology, DG, and geometric analysis.

  1. Topology foundations — open sets, separation, compactness.
  2. Metric spaces — completeness criteria used everywhere downstream.
  3. Algebraic topology — \(\pi_1\), coverings, \(H_*\), SVK, MV.
  4. Homological algebra foundations — UCT/Künneth engine.
  5. Differential geometry — manifolds, connections, curvature, geodesics.
  6. Non-Euclidean geometry — model spaces; curvature sign.
  7. Complex geometry foundations — complex manifolds, Hodge entrance.
  8. Kähler geometry — Kähler identities; Calabi–Yau pointer.
  9. Geometric measure theory foundations — rectifiability, currents.
  10. Minimal surfaces — GMT applications.
  11. Spectral sequences foundations — Serre/Leray computational tool.
  12. Category theory foundations — functors of homotopy/homology.
  13. Geometry index — Euclidean recall map if needed.

Arithmetic geometry / Langlands path

From elementary arithmetic through ANT/ANT analytic to schemes, Galois reps, and Langlands.

  1. Number theory foundations — primes, arithmetic functions, Dirichlet.
  2. Modular arithmetic — CRT, units, Euler–Fermat.
  3. Galois theory — finite Galois; cyclotomic template.
  4. Algebraic number theory foundations — \(\mathcal{O}_K\), Dedekind, splitting.
  5. Class field theory — global/local CFT; Artin map.
  6. Analytic number theory foundations — \(\zeta\), PNT package.
  7. L-functions — Euler products, functional equations.
  8. Commutative algebra foundations — Spec, dimension for schemes.
  9. Algebraic geometry foundations · Schemes — arithmetic base schemes.
  10. Cohomology of schemes — coherent; étale pointer.
  11. Arithmetic geometry foundations — points, heights, Faltings sketch.
  12. Langlands foundations — global reciprocity philosophy; modularity.
  13. Local Langlands — Weil–Deligne \(\leftrightarrow\) smooth reps of \(\mathrm{GL}_n\).
  14. Representation theory foundations — automorphic side prerequisites.
  15. K-theory algebraic — \(K\)-groups in arithmetic (optional branch).

Probability / stochastic path

Measure-theoretic probability through martingales, SDE, and infinite-dimensional noise.

  1. Measure theory foundations — \(L^p\), product measures, RN.
  2. Probability foundations — Kolmogorov, conditional expectation, convergence modes.
  3. Distributions — laws, characteristic functions, CLT/LLN packaging.
  4. Stochastic processes foundations — filtrations, BM, Markov, Itô setup.
  5. Martingales — OST, convergence, maximal inequalities.
  6. SPDE foundations — mild solutions; noise regularity.
  7. Rough paths foundations — \(p\)-variation, signature, Lyons lift.
  8. Ergodic theory foundations — Birkhoff, mixing, entropy.
  9. PDE foundations — Fokker–Planck / generators interface.
  10. Functional analysis foundations — semigroups; Hilbert methods for SPDE.
  11. Information theory foundations — entropy / LD rate functions (optional).
  12. Probability · Inferential — applied interface.
  13. Cheatsheet: stochastic / SDE — Itô, Girsanov, generators.

Mathematical physics path

Geometry + analysis prerequisites, then classical/quantum fields and gauge theory.

  1. Differential geometry — metrics, curvature, geodesics, connections.
  2. Functional analysis foundations — unbounded operators, spectral theorem.
  3. PDE foundations — hyperbolic/elliptic field equations.
  4. Representation theory · Lie groups · Lie algebras — symmetry.
  5. Mathematical physics foundations — Lagrangian/Hamiltonian; Noether; quantization paths; QFT axioms pointer.
  6. Gauge theory — principal bundles, YM, instantons, Chern–Weil.
  7. Quantum — Hilbert QM, \(C^*\)/KMS, scattering sketch.
  8. Complex geometry foundations — Kähler/Calabi–Yau contact with SUSY/strings.
  9. Fourier / microlocal · Microlocal foundations — propagators, wavefront sets.
  10. Stochastic processes foundations · SPDE — Euclidean / stochastic quantization.
  11. Topology algebraic — characteristic classes, index topological side.
  12. Cheatsheet: symplectic (new block) — Hamiltonian identities at a glance.

How to run a path

Step Action
1 Skim Job / Core tables only on each page
2 Attempt Check questions before reading answers
3 Follow Next only when a dependency blocks you
4 Use glossary / cheatsheet for named formula recall
5 Jump to monographs in Refs when a theorem is needed in full

Paths are modular: the algebra and analysis quals paths feed every research path above.