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§ Track III · Mathematics
Twenty-two subjects in the order they were taught at Mekh-Mat — from linear algebra through differential equations to optimal control. Articles group into clusters; each closes with a quiz, dialogue or flashcards.
22 courses
§ Year I — Classical foundations
Analytic Geometry
Vectors and Operations on Them
Analytic Geometry
Line and Plane in Space
Analytic Geometry
Curves and Surfaces of the Second Order
Analytic Geometry
Complex Numbers: Algebra and Geometry
Algebra
Matrices: Operations, Types, Rank
Algebra
Analytic Geometry
Уравнение плоскости: нормальный вектор и расстояния
Analytic Geometry
Прямая в пространстве: параметрическое и каноническое уравнения
Analytic Geometry
Взаимное расположение прямых и плоскостей
Analytic Geometry
Determinants: Properties and Calculation
Algebra
Algebra
Матрицы: операции, типы и свойства
Algebra
Определитель: вычисление и геометрический смысл
Algebra
Обратная матрица: существование и нахождение
Algebra
Coordinate Systems and Transformations
Analytic Geometry
Algebra
Groups: Definition, Examples, Theorems
Algebra
Rings and Fields
Algebra
Systems of Linear Equations: The Kronecker–Capelli Theorem
Algebra
Polar and Spherical Coordinates
Analytic Geometry
Algebra
Собственные числа и собственные векторы матриц
Algebra
Диагонализация матриц: условия и алгоритм
Algebra
Жорданова нормальная форма
Algebra
Quadratic Forms and Classification of Surfaces
Analytic Geometry
Analytic Geometry
Поверхности второго порядка: эллипсоид, гиперболоид, параболоид
Analytic Geometry
Евклидово пространство: скалярное произведение и ортогональность
Analytic Geometry
Ортогональные и ортонормированные базисы: процесс Грама–Шмидта
Analytic Geometry
Vector Spaces: Basic Concepts
Algebra
Calculus
Axioms of Real Numbers and Set Theory
Mathematical Analysis
Limit of a Numerical Sequence: Definition and Properties
Mathematical Analysis
Principle of Completeness and the Bolzano–Weierstrass Theorem
Mathematical Analysis
Linear Transformations and Matrices
Algebra
Calculus
Непрерывность функции: определение и свойства
Mathematical Analysis
Производная: определение, геометрический смысл, правила
Mathematical Analysis
Дифференциал и линейное приближение
Mathematical Analysis
Limit of a Function: Cauchy's Definition and Its Corollaries
Mathematical Analysis
Calculus
Continuity of Functions and Theorems About Their Properties
Mathematical Analysis
Theorems on the Mean and Local Properties of Functions
Mathematical Analysis
Формула Тейлора: разложение и оценка остатка
Mathematical Analysis
Eigenvalues and Eigenvectors
Algebra
Calculus
Монотонность, экстремумы и точки перегиба
Mathematical Analysis
Выпуклость функции и неравенство Йенсена
Mathematical Analysis
Derivative: Definition and Geometric Meaning
Mathematical Analysis
Focal Properties of Conic Sections
Analytic Geometry
Calculus
Extrema and the Study of Functions
Mathematical Analysis
Taylor Formula and Applications
Mathematical Analysis
Интегрирование рациональных дробей
Mathematical Analysis
Поля и расширения: алгебраические числа
Algebra
Calculus
Определённый интеграл: суммы Римана и интегрируемость
Mathematical Analysis
Формула Ньютона–Лейбница и вычисление интегралов
Mathematical Analysis
The Riemann Integral: Definition and Existence
Mathematical Analysis
Reduction of Curves and Surfaces to Canonical Form
Analytic Geometry
Discrete Mathematics
Set Theory and Operations
Discrete Mathematics
Binary Relations and Their Properties
Discrete Mathematics
Lattices and Boolean Algebras
Discrete Mathematics
Invariant Subspaces and the Jordan Form
Algebra
Discrete Mathematics
Размещения, перестановки, сочетания: основные формулы
Discrete Mathematics
Принцип включений-исключений
Discrete Mathematics
Normal Forms and Post's Theorem
Discrete Mathematics
Canonical Forms: Rational and Real
Algebra
Discrete Mathematics
Combinatorics: The Principle of Inclusion-Exclusion
Discrete Mathematics
Recurrence Relations
Discrete Mathematics
Планарные графы и теорема Эйлера для многогранников
Discrete Mathematics
Bilinear and Quadratic Forms
Algebra
Discrete Mathematics
Рекуррентные соотношения: решение и характеристический корень
Discrete Mathematics
Числа Фибоначчи, Каталана и Бернулли
Discrete Mathematics
Graphs: Basic Concepts and Theorems
Discrete Mathematics
Newton–Leibniz Theorem and Techniques of Integration
Mathematical Analysis
Discrete Mathematics
Planarity and Graph Coloring
Discrete Mathematics
Eulerian and Hamiltonian Graphs. Flows in Networks
Discrete Mathematics
Китайская теорема об остатках
Discrete Mathematics
Конечные поля GF(p^n) и их приложения
Algebra
Discrete Mathematics
Булева алгебра: решётки, дистрибутивность, нормальные формы
Discrete Mathematics
Конечные автоматы и регулярные языки
Discrete Mathematics
Enumerative Combinatorics
Discrete Mathematics
Коды с исправлением ошибок: линейные коды и Хэмминг
Algebra
§ Year II — Analysis and differential equations
Calculus
Improper Integrals and Their Convergence
Mathematical Analysis
Разложения элементарных функций в ряды Тейлора
Mathematical Analysis
Ряды Фурье: тригонометрическое разложение и сходимость
Mathematical Analysis
Introduction to Differential Equations
Differential Equations
Calculus
Numerical Series and Convergence Tests
Mathematical Analysis
Functional Series and Uniform Convergence
Mathematical Analysis
Fourier Series
Mathematical Analysis
Криволинейный интеграл 2-го рода: работа поля
Mathematical Analysis
Calculus
Поверхностный интеграл 1-го рода: площадь и масса
Mathematical Analysis
Поверхностный интеграл 2-го рода: поток поля
Mathematical Analysis
Partial Derivatives and the Total Differential
Mathematical Analysis
Exact Equations and Integrating Factor
Differential Equations
Calculus
Multiple Integrals and Their Calculation
Mathematical Analysis
Theorems on Implicit Functions and Inverse Mappings
Mathematical Analysis
Экстремумы функций двух переменных и метод Лагранжа
Mathematical Analysis
Singular Solutions and Envelopes
Differential Equations
Differential Equations
ОДУ с разделяющимися переменными и однородные уравнения
Differential Equations
Линейные ОДУ первого порядка: формула Бернулли
Differential Equations
Уравнения в полных дифференциалах и интегрирующий множитель
Differential Equations
Теорема Пикара и теорема существования и единственности
Mathematical Analysis
Differential Equations
Picard's Theorem: Existence and Uniqueness
Differential Equations
Dependence on Initial Conditions and Parameters
Differential Equations
Peano's Theorem and the Non-Lipschitz Case
Differential Equations
Пространства Банаха и принцип сжимающих отображений
Mathematical Analysis
Differential Equations
Системы ОДУ: матричная форма и фундаментальная матрица
Differential Equations
Автономные системы и фазовый портрет
Differential Equations
Устойчивость по Ляпунову: функции Ляпунова
Differential Equations
Euclidean Space and Orthogonalization
Algebra
Differential Equations
Linear ODEs: Structure of the General Solution
Differential Equations
Method of Variation of Constants
Differential Equations
Euler Equations and Order Reduction
Differential Equations
Curvilinear Integrals
Mathematical Analysis
Differential Equations
Преобразование Лапласа: таблица и применение к ОДУ
Differential Equations
Преобразование Фурье и спектральный анализ
Differential Equations
Функции Грина и задачи с граничными условиями
Differential Equations
Surface Integrals and Stokes' Theorem
Mathematical Analysis
Differential Equations
Systems of Linear ODEs
Differential Equations
Phase Portraits of Two-Dimensional Systems
Differential Equations
Nonlinear Systems and Autonomous Equations
Differential Equations
Differential Forms and the General Stokes' Theorem
Mathematical Analysis
Differential Equations
Метод Эйлера и методы Рунге–Кутта
Differential Equations
Жёсткие системы ОДУ и неявные методы
Differential Equations
Конечно-разностные схемы для УЧП
Differential Equations
Погрешность численного интегрирования
Mathematical Analysis
Differential Equations
Lyapunov Stability
Differential Equations
Lyapunov Functions and Stability Criteria
Differential Equations
Bifurcations in Dynamical Systems
Differential Equations
Гамма-функция, бета-функция и их свойства
Mathematical Analysis
Probability Theory
Probability Space and Kolmogorov's Axioms
Probability Theory
Conditional Probability and Independence
Probability Theory
Geometric Probabilities and Continuous Spaces
Probability Theory
Generating Functions and Their Applications
Discrete Mathematics
Probability Theory
Дискретные распределения: Бернулли, Пуассон, биномиальное
Probability Theory
Непрерывные распределения: равномерное, нормальное, экспоненциальное
Probability Theory
Discrete Distributions
Probability Theory
Extremal Combinatorics
Discrete Mathematics
Probability Theory
Continuous Distributions
Probability Theory
Multivariate Distributions and Copulas
Probability Theory
Центральная предельная теорема и нормальное приближение
Probability Theory
Теория меры Лебега и интеграл Лебега
Mathematical Analysis
Probability Theory
Mathematical Expectation and Moments
Probability Theory
Generating Functions and Characteristic Functions
Probability Theory
Probability Theory Inequalities
Probability Theory
Энтропия и теория информации Шеннона
Discrete Mathematics
Probability Theory
Случайные процессы: стационарность и эргодичность
Probability Theory
Пуассоновский процесс и процессы восстановления
Probability Theory
Law of Large Numbers
Probability Theory
Теория массового обслуживания: очереди M/M/1
Discrete Mathematics
Probability Theory
Central Limit Theorem
Probability Theory
Convergences of Random Variables and Large Deviations
Probability Theory
Проверка статистических гипотез: p-value и мощность критерия
Probability Theory
Случайные графы: модель Эрдёша–Реньи и пороговые явления
Discrete Mathematics
§ Year III — Advanced analysis
Complex Analysis
Functions of a Complex Variable: Basic Concepts
Complex Analysis
Conformal Mappings
Complex Analysis
The Fundamental Theorem of Algebra via Complex Function Theory
Complex Analysis
Lebesgue Measure and the Lebesgue Integral
Mathematical Analysis
Complex Analysis
Интегральная теорема Коши и формула Коши
Complex Analysis
Cauchy Integral and Its Consequences
Complex Analysis
Taylor Series and Analyticity of Holomorphic Functions
Complex Analysis
Analytic Continuation
Complex Analysis
Complex Analysis
Мероморфные функции и теорема Вейерштрасса
Complex Analysis
Laurent Series and Isolated Singular Points
Complex Analysis
Residue Theory
Complex Analysis
Lp Spaces and Fubini’s Theorem
Mathematical Analysis
Calculus
Fourier Transform and Plancherel's Theorem
Mathematical Analysis
Касательное пространство и дифференциальные формы
Mathematical Analysis
Интегрирование форм и теорема Стокса в общем виде
Mathematical Analysis
Principle of the Argument and Its Applications
Complex Analysis
Complex Analysis
Дзета-функция Римана: аналитическое продолжение и функциональное уравнение
Complex Analysis
Additional Methods for Calculating Integrals
Complex Analysis
Special Functions via Complex Analysis
Complex Analysis
Finite Automata and Regular Languages
Discrete Mathematics
Discrete Mathematics
Context-Free Languages and Grammars
Discrete Mathematics
Turing Machines and Computational Complexity
Discrete Mathematics
Univalent Functions and Theorems on Mappings
Complex Analysis
Седловая точка и метод Лапласа
Mathematical Analysis
Differential Geometry
Кривые в ℝ³: кривизна, кручение и трёхгранник Френе
Differential Geometry
Поверхности: первая и вторая квадратичные формы
Differential Geometry
Гауссова и средняя кривизна поверхности
Differential Geometry
Связь конформных отображений и метрики
Complex Analysis
Differential Geometry
Тензоры: определение, операции, свёртка
Differential Geometry
Риманова метрика и связность Леви-Чивиты
Differential Geometry
Геодезические: вариационный и уравнения Кристоффеля
Differential Geometry
Тензор Римана и тензор кривизны
Differential Geometry
Differential Geometry
Группы Ли: определение и базовые примеры (SO, SU, GL)
Differential Geometry
Алгебра Ли и экспоненциальное отображение
Differential Geometry
Главные расслоения и связности
Differential Geometry
Self-Adjoint and Unitary Operators
Algebra
Differential Geometry
Топология: открытые множества и непрерывность
Differential Geometry
Компактность, связность и сепарабельность
Differential Geometry
Фундаментальная группа и накрытия
Differential Geometry
Weierstrass Factorization Theorem
Complex Analysis
Differential Geometry
Симплициальная гомология и числа Бетти
Differential Geometry
Когомология де Рама и теорема де Рама
Differential Geometry
Характеристические классы и формула Гаусса–Бонне
Differential Geometry
Mittag-Leffler's Theorem and Meromorphic Functions
Complex Analysis
Differential Geometry
Уравнения Эйнштейна и геометрическая картина гравитации
Differential Geometry
Чёрные дыры: геодезические в метрике Шварцшильда
Differential Geometry
Космологические модели и уравнения Фридмана
Differential Geometry
Laplace Transform and Complex Function Theory
Complex Analysis
Functional Analysis
Metric Spaces: Basic Concepts and Examples
Functional Analysis
Normed Spaces and Banach Spaces
Functional Analysis
Compactness and Bounded Linear Operators
Functional Analysis
Parametric Estimation
Probability Theory
Functional Analysis
Гильбертовы пространства: скалярное произведение и ортогональные разложения
Functional Analysis
Hilbert Spaces and Orthogonality
Functional Analysis
Orthonormal Bases and Fourier Series
Functional Analysis
Nonparametric Methods and Bootstrap
Probability Theory
Functional Analysis
Linear Functionals and Dual Spaces
Functional Analysis
Ограниченные линейные операторы и их нормы
Functional Analysis
Spectral Theory of Linear Operators
Functional Analysis
Sufficient Statistics and the Rao-Blackwell Theorem
Probability Theory
Functional Analysis
Compact Operators and Fredholm Equations
Functional Analysis
Operator Semigroups and Evolution Equations
Functional Analysis
Спектральная теорема для самосопряжённых операторов
Functional Analysis
Операторы ковариации и KL-разложение
Probability Theory
Functional Analysis
Sobolev Spaces and Weak Derivatives
Functional Analysis
The Finite Element Method as an Implementation of the Variational Approach
Functional Analysis
Optimization in Functional Spaces
Functional Analysis
УЧП в пространствах распределений
Differential Equations
Functional Analysis
Пространства Соболева W^{k,p}: вложения и следы
Functional Analysis
Слабые производные и слабые решения граничных задач
Functional Analysis
Fourier Transform in $L^1$ and $L^2$
Functional Analysis
Регулярность слабых решений эллиптических задач
Differential Equations
§ Year IV — Applied mathematics
Mathematical Physics
Lagrangian Mechanics and Equations of Motion
Mathematical Physics
Hamiltonian Mechanics and Poisson Brackets
Mathematical Physics
Canonical Transformations and Action-Angle Variables
Mathematical Physics
Schwartz Distribution Theory
Functional Analysis
Mathematical Physics
Уравнение волн: принцип суперпозиции и дисперсия
Mathematical Physics
Maxwell's Equations in Differential Form
Mathematical Physics
Special Theory of Relativity and 4-Tensors
Mathematical Physics
Wavelets and Applications of Functional Analysis
Functional Analysis
Mathematical Physics
Electromagnetic Waves and Radiation
Mathematical Physics
Гамильтонов формализм и канонические уравнения
Mathematical Physics
Postulates of Quantum Mechanics
Mathematical Physics
Интегральные уравнения Вольтерра и Фредгольма
Differential Equations
Probability Theory
Hypothesis Testing
Probability Theory
Linear Regression and Analysis of Variance
Probability Theory
Goodness-of-Fit Tests
Probability Theory
Exact Solutions: Oscillator and Hydrogen Atom
Mathematical Physics
Probability Theory
Анализ главных компонент (PCA) и SVD-разложение
Probability Theory
Расстояние Куллбека–Лейблера и взаимная информация
Probability Theory
Discrete-Time Markov Chains
Probability Theory
Perturbation Theory and Transitions
Mathematical Physics
Probability Theory
Poisson Process and Continuous-Time Markov Chains
Probability Theory
Martingales and the Theory of Optimal Stopping
Probability Theory
СДУ: существование решений и связь с УЧП
Probability Theory
Уравнение Фоккера–Планка и диффузионные процессы
Mathematical Physics
Mathematical Logic
Syntax and Semantics of Propositional Logic
Mathematical Logic
Normal Forms and the DPLL Algorithm
Mathematical Logic
Axiom Systems and Theorems of Completeness
Mathematical Logic
Машины Тьюринга и проблема остановки
Discrete Mathematics
Mathematical Logic
Syntax and Semantics of First-Order Logic
Mathematical Logic
The Resolution Method and Unification
Mathematical Logic
Gödel's Theorems and First-Order Boundaries
Mathematical Logic
Аппроксимационные алгоритмы и нижние оценки
Discrete Mathematics
Neural Systems
Perceptron and Multilayer Neural Networks
Neural Systems
Convolutional Neural Networks (CNN)
Neural Systems
Regularization and Prevention of Overfitting
Neural Systems
VC-размерность и обучаемость по Вапнику
Mathematical Logic
Big Data and Deep Learning
Gradient Boosting and Ensemble Methods
Big Data & Deep Learning
Reinforcement Learning
Big Data & Deep Learning
Automatic Architecture Search and AutoML
Big Data & Deep Learning
Training Deep Networks: Problems and Solutions
Neural Systems
Big Data and Deep Learning
Approximation Theory and Deep Networks
Big Data & Deep Learning
Optimization in Deep Learning
Big Data & Deep Learning
Transformers and Attention Architectures
Big Data & Deep Learning
Explainable AI and Representation Analysis
Neural Systems
Big Data and Deep Learning
High-Dimensional Geometry and Concentration
Big Data & Deep Learning
Sparsity and Compressed Sensing Theory
Big Data & Deep Learning
Generative Models: VAE and Diffusion Models
Big Data & Deep Learning
Graph Neural Networks (GNN)
Neural Systems
Game Theory
Fundamentals of Game Theory: Players, Strategies, Payoffs
Game Theory
Nash Equilibrium: Existence and Computation
Game Theory
Dominant Strategies and Iterative Elimination
Game Theory
Consumer Theory: Utility Functions and Demand
Mathematical Economics
Game Theory
Dynamic Games and Subgame Perfect Equilibrium
Game Theory
Incomplete Information and Bayesian Equilibrium
Game Theory
Repeated Games and the Folk Theorem
Game Theory
General Equilibrium and the Welfare Theorems
Mathematical Economics
Game Theory
Эволюционная теория игр: эволюционно стабильные стратегии
Game Theory
Core, Shapley Value, and Cooperative Concepts
Game Theory
Nucleolus, Stable Coalitions, and Matching Theory
Game Theory
Game Theory: Nash Equilibrium, Strategic Interactions, and Mechanism Design
Mathematical Economics
Mathematical Economics
Monopoly, Oligopoly, and Strategic Pricing
Mathematical Economics
Production Theory and Firm Efficiency
Mathematical Economics
Integrability of Demand and Consumer Surplus
Mathematical Economics
Two-Sided Markets and Matching Theory
Game Theory
Mathematical Economics
Ожидаемая полезность: аксиомы фон Неймана–Моргенштерна
Mathematical Economics
Dynamic Programming in Economics
Mathematical Economics
Overlapping Generations and Pension Systems
Mathematical Economics
Стохастическое доминирование и сравнение рисков
Probability Theory
Mathematical Economics
Asset Pricing and No-Arbitrage
Mathematical Economics
Мартингалы и теорема об отсутствии арбитража
Mathematical Economics
Asymmetric Information: Moral Hazard and Adverse Selection
Mathematical Economics
Меры риска: VaR, CVaR и когерентность
Probability Theory
§ Year V — Optimization and control
Calculus of Variations
Functionals and the Euler-Lagrange Equation
Calculus of Variations
Second-Order Conditions and the Jacobi Condition
Calculus of Variations
Isoperimetric Problems and Lagrange Multipliers
Calculus of Variations
Functionals and the Euler–Lagrange Equation
Optimal Control
Optimal Control
Second Order Conditions and Sufficient Conditions for Extremum
Optimal Control
Variational Principles in Mechanics and Physics
Optimal Control
Formulation of the Optimal Control Problem
Optimal Control
Bolza Problem and Transversality Condition
Calculus of Variations
Optimal Control
Maximum Principle: Proof and Applications
Optimal Control
Problems with Finite and Infinite Horizon
Optimal Control
LQR-задача: явное решение и матричное уравнение Риккати
Optimal Control
Variational Problems with Constraints and Connections
Calculus of Variations
Optimal Control
Выпуклые множества: полуплоскости, конусы и теорема разделения
Convex Analysis
Выпуклые функции: определение, сопряжённые и субградиенты
Convex Analysis
Опорная функция выпуклого множества
Convex Analysis
Principle of Optimality and the Bellman Equation
Optimal Control
Optimal Control
Градиентный спуск: сходимость для L-гладких функций
Convex Analysis
Проксимальный алгоритм и метод ADMM
Convex Analysis
Ускоренный градиентный метод Нестерова
Convex Analysis
Discrete Dynamic Programming and Numerical Methods
Optimal Control
Optimal Control
DP in Economics: Accumulation, Resources, Growth
Optimal Control
Фильтр Калмана: оценка состояния в линейных системах
Optimal Control
Linear Systems: Controllability and Observability
Optimal Control
Hamilton's Principle and Analytical Mechanics
Calculus of Variations
Risk Theory
The Nature of Risk and Classification
Risk & Actuarial Theory
Risk Measures: VaR, CVaR, and Coherent Measures
Risk & Actuarial Theory
Reinsurance and Risk Portfolio Management
Risk & Actuarial Theory
Brownian Motion: Construction and Properties
Probability Theory
Risk Theory
Theory of Survival and Demographic Tables
Risk & Actuarial Theory
Lifetime Annuities and Life Insurance
Risk & Actuarial Theory
Pension Systems and Long-Term Liabilities
Risk & Actuarial Theory
Ito's Lemma and Stochastic Differential Equations
Probability Theory
Discrete Optimization
Introduction to Discrete Optimization: Problems, Models, Complexity
Discrete Optimization
Graph Theory in Optimization: Flows and Matchings
Discrete Optimization
Linear Programming as the Foundation of ILP
Discrete Optimization
LP-релаксация и двойственная задача
Convex Analysis
Discrete Optimization
Branch and Bound Method
Discrete Optimization
Cutting Planes and Branch-and-Cut
Discrete Optimization
Dynamic Programming in Discrete Optimization
Discrete Optimization
Cone programming и задачи второго порядка (SOCP)
Convex Analysis
Differential Games
Игры преследования и уклонения: теорема Айзекса
Differential Games
Уравнение Айзекса–Гамильтона–Якоби в теории дифференциальных игр
Differential Games
Кооперативные дифференциальные игры и распределение выигрыша
Differential Games
PID Control and Regulator Synthesis
Optimal Control
Complex Systems
What Is a Complex System: Definition, Properties, Examples
Complex Systems
Nonlinear Dynamics: Attractors, Bifurcations, and Chaos
Complex Systems
Network Theory: Small World, Scale-Free Networks, Clustering
Complex Systems
Игры среднего поля: предел бесконечного числа агентов
Differential Games