Track · Mathematics
The other kind of rigor.
Twenty-two subjects in the order they were taught at Moscow's Mekh-Mat and VMK: analysis, linear algebra, analytic geometry, discrete math, differential equations, complex analysis, differential geometry, functional analysis, probability and statistics, math physics, math logic, neural networks, big data and ML, game theory, math economics, risk theory, discrete optimization, calculus of variations, optimal control, convex analysis, differential games, complex systems.
What you will study
I · Analysis
Mathematical Analysis
Mathematical analysis: limits, derivatives, integrals, series, multivariable functions, and applications in economics and physics
II · Algebra
Linear Algebra
Linear algebra: vector spaces, matrices, determinants, eigenvalues, linear transformations, and ML applications
III · Analytic Geom.
Analytic Geometry
Analytic geometry: coordinate systems, lines and planes, second-order curves, vector algebra
IV · Discrete Math
Discrete Mathematics
Discrete mathematics: logic, set theory, combinatorics, graph theory, algorithms, and number theory
V · Diff. Equations
Differential Equations
Differential equations: first- and second-order ODEs, systems, PDEs, and applications in physics and engineering
VI · Complex An.
Complex Analysis
Complex analysis: complex numbers, analytic functions, Cauchy integrals, Laurent series, and residues
VII · Diff. Geometry
Differential Geometry
Differential geometry: curves and surfaces, curvature, geodesics, tensor calculus, and manifolds
VIII · Func. Analysis
Functional Analysis
Functional analysis: normed spaces, Banach and Hilbert spaces, operators, and spectral theory
IX · Probability
Probability & Statistics
Probability theory and mathematical statistics: probability spaces, random variables, distributions, CLT, and statistical tests
X · Math. Physics
Mathematical Physics Equations
Equations of mathematical physics: heat, wave, Laplace equations, Fourier method, and Green's functions
XI · Math Logic
Math Logic & Algorithms
Mathematical logic: predicate calculus, Gödel's theorems, computability theory, Turing machines, and complexity
XII · Neural Nets
Neural Networks & Deep Learning
Neural networks: perceptron, backpropagation, CNNs and RNNs, transformers, and modern architectures
XIII · Big Data & ML
Big Data & Machine Learning
Big data: Hadoop, Spark, streaming, feature engineering, GNN, MLOps, and responsible AI
XIV · Game Theory
Game Theory
Game theory: Nash equilibrium, cooperative games, incomplete information games, and applications in economics and politics
XV · Math. Economics
Mathematical Methods in Economics
Mathematical economics: consumer and producer theory, Walrasian equilibrium, optimization, growth, and dynamic models
XVI · Risk Theory
Risk Theory & Actuarial Math
Actuarial mathematics: risk theory, life and property insurance, survival models, and risk management
XVII · Disc. Optimization
Discrete Optimization
Discrete optimization: integer programming, branch and bound, network flows, metaheuristics, and approximation algorithms
XVIII · Calc. Variations
Calculus of Variations
Calculus of variations: functionals, Euler-Lagrange equation, second-order conditions, and links to mechanics and optimal transport
XIX · Optim. Control
Optimal Control
Optimal control: Pontryagin maximum principle, LQR, MPC, stochastic control, and reinforcement learning
XX · Convex Analysis
Convex Analysis & Optimization
Convex analysis: convex sets and functions, duality, KKT conditions, SDP, first-order algorithms, and ML applications
XXI · Diff. Games
Differential Games
Differential games: Isaacs equation, pursuit problems, N-player games, cooperative games, MFG, and applications in economics and robotics
XXII · Complex Systems
Complex Systems Theory
Complex systems theory: nonlinear dynamics, chaos, network theory, agent-based modeling, critical phenomena, and economic applications
"There is no royal road to geometry."