Combinatorics & Graph Theory
Enumeration and generating functions, posets and lattices, symmetric functions and RSK, graph theory (connectivity, matchings, colouring), extremal and Ramsey theory, designs and codes, the probabilistic method, and analytic combinatorics. Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 59 units
Dynamical Systems & Ergodic Theory
Topological and symbolic dynamics, hyperbolicity and structural stability, the ergodic theorems, mixing and spectral theory, entropy, and smooth ergodic theory. Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 76 units
Information & Coding Theory
Shannon entropy, mutual information, and Kullback-Leibler divergence; source coding (Kraft inequality, Huffman, arithmetic, Lempel-Ziv, rate-distortion); channel capacity and Shannon's noisy-channel coding theorem (discrete memoryless, Gaussian, feedback); information-theoretic statistics (Stein's lemma, Chernoff information, method of types, I-projection and Blahut-Arimoto); side-information problems (Slepian-Wolf, Wyner-Ziv, Gelfand-Pinsker, Costa dirty-paper); network information theory (multiple-access, broadcast, relay, interference channels); algebraic coding (weight enumerators, MacWilliams identity, algebraic-geometry codes, list decoding, expander codes); and modern capacity-achieving codes (LDPC with belief propagation, turbo codes, polar codes). Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 118 units
Foundations, Logic & Category Theory
Mathematical logic and the foundations of mathematics: first-order logic and completeness, model theory, set theory and forcing, computability and the degrees, and category theory (limits, adjunctions, Yoneda, monads, Kan extensions). Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 64 units
Numerical Analysis & Scientific Computing
Floating-point arithmetic and conditioning, root-finding for nonlinear equations, direct and iterative solvers for linear systems, least squares and QR, the SVD and low-rank approximation, eigenvalue algorithms, Krylov subspace methods, interpolation and approximation, numerical quadrature, time-stepping for ODEs, and finite-difference schemes for PDEs, alongside finite-element exterior calculus. Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 168 units
Operator Algebras & NCG
C*-algebras and von Neumann algebras, K-theory and AF algebras, Tomita-Takesaki modular theory, nuclearity and exactness, and Connes' noncommutative geometry (spectral triples, cyclic cohomology). Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 79 units
Optimization & Control
Convex sets and functions, convex duality and the KKT conditions, unconstrained optimization (gradient, Newton, and quasi-Newton methods with line search and trust regions), constrained nonlinear programming (sequential quadratic programming, interior-point, augmented Lagrangian), conic and semidefinite programming, first-order and large-scale methods (subgradient, proximal, accelerated, ADMM, stochastic gradient), optimal control via the calculus of variations and Pontryagin's maximum principle, and dynamic programming with the Bellman equation and Hamilton-Jacobi-Bellman theory. Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 155 units
Probability & Stochastics
Measure-theoretic probability, limit theorems, martingales, Markov chains, stochastic calculus, large deviations, and random matrices. Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 151 units
Statistics & Learning Theory
Statistical decision theory and point estimation (sufficiency, exponential families, maximum likelihood, the Cramer-Rao bound, UMVU estimators), hypothesis testing and confidence sets (Neyman-Pearson, UMP and likelihood-ratio tests), Bayesian inference, asymptotic statistics (consistency, asymptotic normality, the delta method, M- and Z-estimators, local asymptotic normality), empirical processes and nonparametrics (the bootstrap, kernel density estimation, U-statistics), high-dimensional and regularized regression (ridge, the LASSO, oracle inequalities, model selection), and statistical learning theory (empirical risk minimization, VC dimension, Rademacher complexity, generalization bounds, kernels and support vector machines, boosting, and the EM algorithm). Prerequisites are pulled in automatically so the result is always a complete, learnable path. · 190 units
Theoretical Computer Science
Formal languages and automata theory (regular, context-free, and Turing-recognizable languages; DFA/NFA equivalence; pumping lemmas; Chomsky hierarchy), computational complexity (P, NP, co-NP, PSPACE, EXPTIME, the polynomial hierarchy, circuit complexity, and probabilistic classes BPP and RP), advanced complexity (oracle machines, relativization, the PCP theorem and hardness of approximation, interactive proofs and IP=PSPACE, counting classes and Toda's theorem), algorithm design and analysis (divide-and-conquer, dynamic programming, greedy algorithms, amortized analysis, graph algorithms, NP-hardness reductions), randomized algorithms (Las Vegas, Monte Carlo, probabilistic method, hashing, Karger's min-cut), and cryptographic foundations (one-way functions, pseudorandom generators, zero-knowledge proofs, semantic security, RSA, Diffie-Hellman). Cross-refs computability from section 42 and graph theory from section 40. · 66 units