Mathematics Lecture Notes
โWhat we know is not much. What we do not know is immense.โ
38 subjects ยท 325 chapters ยท 2059 pages
Foundations
Undergraduate Core
Calculus
Single and multivariable calculus, sequences, series, and vector calculus
Linear Algebra
Vector spaces, linear transformations, eigenvalues, inner product spaces, and canonical forms
Abstract Algebra
Groups, rings, fields, Galois theory, and module theory
Real Analysis
Rigorous treatment of limits, continuity, differentiation, integration, and metric spaces
Complex Analysis
Analytic functions, contour integration, residues, and conformal mappings
Ordinary Differential Equations
First and higher order ODEs, systems, stability, and qualitative theory
Probability & Statistics
Probability theory, random variables, distributions, estimation, and hypothesis testing
Elementary Number Theory
Divisibility, primes, congruences, quadratic reciprocity, and arithmetic functions
Point-Set Topology
Topological spaces, continuity, compactness, connectedness, and metrization
Discrete Mathematics
Logic, counting, graph theory, recurrences, and Boolean algebra
Undergraduate Elective
Combinatorics
Enumerative combinatorics, generating functions, Ramsey theory, and extremal combinatorics
Graph Theory
Graphs, trees, matchings, colorings, planarity, and network flows
Numerical Analysis
Interpolation, numerical integration, root finding, and numerical linear algebra
Geometry
Euclidean, affine, projective, and hyperbolic geometry
Dynamical Systems
Discrete and continuous dynamics, chaos, bifurcations, and ergodic theory
Knot Theory
Knot invariants, links, braids, and applications to 3-manifolds
Mathematical Physics
Classical mechanics, quantum mechanics, and mathematical foundations of physics
Graduate Core
Measure Theory
Lebesgue measure, abstract measures, integration, and Lp spaces
Functional Analysis
Banach spaces, Hilbert spaces, operators, spectral theory, and distributions
Algebraic Topology
Fundamental group, covering spaces, homology, cohomology, and homotopy theory
Differential Geometry
Curves, surfaces, manifolds, Riemannian geometry, and connections
Commutative Algebra
Rings, modules, localization, primary decomposition, and completions
Homological Algebra
Chain complexes, derived functors, Ext, Tor, and spectral sequences
Representation Theory
Group representations, characters, modules over group algebras, and Lie algebra representations
Graduate Advanced
Algebraic Number Theory
Number fields, rings of integers, ideal theory, class groups, and p-adic numbers
Analytic Number Theory
Zeta functions, L-functions, prime distribution, and sieve methods
Partial Differential Equations
Classification, wave, heat, and Laplace equations, Sobolev spaces, and weak solutions
Lie Groups & Lie Algebras
Matrix Lie groups, Lie algebras, root systems, and representation theory
Stochastic Processes
Markov chains, martingales, Brownian motion, and stochastic calculus
Symplectic Geometry
Symplectic manifolds, Hamiltonian mechanics, and moment maps
Mirror Symmetry
Calabi-Yau manifolds, homological mirror symmetry, and SYZ conjecture
Research Level
Algebraic Geometry
Varieties, schemes, sheaves, cohomology, and moduli spaces
Category Theory
Categories, functors, natural transformations, adjunctions, and limits
Higher Category Theory
Infinity-categories, simplicial sets, model categories, and higher topos theory
Algebraic Stacks
Fibered categories, descent, algebraic stacks, and moduli problems
Algebraic K-Theory
K-groups, Quillen construction, Waldhausen categories, and motivic cohomology