All About Mathematics¶

A doctoral / postdoctoral reference: compact, precise notes across pure and applied mathematics. Undergraduate-titled pages are recall only.

Audience¶

PhD students and postdocs — prelims, seminars, paper preliminaries. Not a first undergraduate course. See writing style and learning paths.

Domain	Emphasis
Abstract algebra	Groups, rings, modules, Galois
Commutative algebra	Spec, dimension, local cohomology
Representation theory	Finite groups, Lie algebras/groups
Homological algebra	Ext/Tor, derived categories
Spectral sequences	Filtrations, Serre/Leray/Grothendieck
Algebraic geometry	Schemes, coherent cohomology
Derived AG	Derived rings, cotangent complex
Spectral AG	\(E_\infty\)-rings, spectral schemes
Motivic homotopy	\(\mathbb{A}^1\)-homotopy, \(\mathrm{SH}\)
Arithmetic geometry	Points, heights, étale
Langlands · Geometric Langlands	Automorphic \(\leftrightarrow\) Galois / Bun\(_G\)
Analytic / Algebraic NT	\(\zeta\), \(L\), class field
\(K\)-theory	Algebraic / topological \(K\)
Tropical · Mirror	Combinatorial / HMS–SYZ
Symplectic · Contact	Hamiltonian, Floer, Reeb
Noncommutative geometry	Spectral triples, cyclic coh.
Cryptography	Hardness, PQ, ZK/MPC, protocols

Domain	Emphasis
Prismatic cohomology	Prisms, \(A_{\inf}\), \(p\)-adic Hodge
Condensed mathematics	Condensed sets, solid/liquid, analytic rings
Higher Teichmüller	Anosov, Hitchin, maximal components

Domain	Emphasis
Topology · Alg. topology	Spaces, \(\pi_1\), \(H_*\)
Measure · FA	Integration, operators
Microlocal · GMT	WF sets, rectifiability
Complex · SCV · Complex geom.	Holomorphy → Hodge/Kähler
PDE · Dynamics · Ergodic	Weak solutions, chaos, Birkhoff
Fourier	Plancherel, harmonics

Domain	Emphasis
Probability	Measure-theoretic
Stochastic · SPDE · Rough paths	Martingales → SPDE → Lyons
Random matrices · Free probability	Wigner, freeness
Statistics · Optimization · Numerics	Inference, convex, algorithms
ML mathematics	Approximation, NTK, generalization
Information theory · Game theory	Shannon, equilibria

Domain	Emphasis
Logic · Category	FOL, ZFC, Yoneda
Model categories · Higher cats · ∞-topoi	Homotopy, \((\infty,1)\)
Topos · HoTT	Sheaves, univalence
Mathematical physics · Axiomatic QFT	Gauge, QM, Wightman/OS
Discrete math	Combinatorics, graphs

Domain	Role
Linear algebra	Spectral / structure toolkit
Calculus	Recall → multilinear use
Geometry (classical)	Euclidean recall + non-Euclidean
Arithmetic / Algebra (elem.) / Trig	Recall cards only