Wikipedia
- Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

Discrete differential geometry

Wikipedia
- Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics, geometry processing and topological combinatorics.
YouTube / Lecture videos for an intro course on Discrete Differential Geometry at Carnegie Mellon University (15-458)
Course notes on Discrete Differential Geometry, covering Laplace operators, parametric surfaces, discrete mappings, conformal geometry, and optimal transport

Trigonometry

The Origins of Trigonometric Functions (sine, cosine, tangent, cotangent, secant, and cosecant) by Sunny Labh
YouTube
- Уроки Шаталова / Тригонометрия

Topology

iXBT / Математика и лабиринты: как найти выход из любой головоломки с помощью топологии
YouTube / Physics for the Birds / The Topological Problem with Voting
Cantor’s Paradise / The Topology of Matrix Multiplication Is Beautiful by Kasper Müller

Mathematical analysis

Wikipedia
- Major topics in mathematical analysis
- Principles of Mathematical Analysis colloquially known as "PMA" or "Baby Rudin," is an undergraduate real analysis textbook written by Walter Rudin.
Calculus Made Easy, by Silvanus P. Thompson, 1910
Calculus. A Guide to Developing Lasting Intuition by Kalid Azad
Зорич «Математический анализ»
Courses
- Открытое образование
  - Введение в математический анализ для поступающих в магистратуру
  - Высшая математика. Математический анализ
YouTube / The Math Sorcerer / The Easiest Calculus Book In The World

Integration

Courses
- Открытое образование
  - Многомерный анализ, интегралы и ряды для поступающих в магистратуру
  - Кратные интегралы и гармонический анализ для поступающих в магистратуру

Differential equations

Courses
- Открытое образование
  - Дифференциальные уравнения: простейшие типы, линейные уравнения и системы, задачи Коши
  - Дифференциальные уравнения: продолжимость решений, линейные уравнения II-го порядка с переменным коэффициентами, элементы вариационного исчисления

Теория вероятностей и математическая статистика

Ю.В. Прохоров, Л.С. Пономаренко, Лекции по теории вероятностей и математической статистике
Гмурман В.Е. Теория вероятностей и математическая статистика
Introduction to Probability, Statistics, and Random Processes by Hossein Pishro-Nik

Probability theory

Wikipedia
- Rule of succession used, particularly to estimate underlying probabilities when there are few observations or events that have not been observed to occur at all in (finite) sample data.
Боровков А.А. Теория вероятностей
Courses
- Открытое образование
  - Введение в теорию вероятностей
  - Случайные процессы
SecurityLab.ru / Простая формула, меняющая мир: как теорема Байеса помогает нам заглянуть в будущее

Statistics

Wikipedia
- Template:Statistics
- Шаблон:Среднее
- Категория:Средние величины
- Template:Quantitative forecasting methods
- Moving average is commonly used with time series data to smooth out short-term fluctuations and highlight longer-term trends or cycles.
  - Скользящая средняя обычно используются с данными временных рядов для сглаживания краткосрочных колебаний и выделения основных тенденций или циклов
- Time series
  - Временной ряд
- Категория:Анализ временных рядов
- Template:Statistical software
- OpenBUGS is a software application for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods.
- Power law is a functional relationship between two quantities, where a relative change in one quantity results in a relative change in the other quantity proportional to a power of the change, independent of the initial size of those quantities: one quantity varies as a power of another.
YouTube
- Статистика Для Начинающих
Журнал «Код» / Что такое нормальное распределение и как оно предсказывает будущее
Skillbox Media / Нормальное распределение (или распределение Гаусса): что это, от чего зависит и по какому закону работает

Queueing theory

Wikipedia
- Queueing theory is the mathematical study of waiting lines, or queues.
  - Теория массового обслуживания (теория очередей) раздел теории вероятностей, целью исследований которого является рациональный выбор структуры системы обслуживания и процесса обслуживания на основе изучения потоков требований на обслуживание, поступающих в систему и выходящих из неё, длительности ожидания и длины очередей. В теории массового обслуживания используются методы теории вероятностей и математической статистики.
- Категория:Теория массового обслуживания
twirpx.com / Академическая и специальная литература / Математика / Теория вероятностей и математическая статистика / Теория вероятностей / Теория массового обслуживания (теория очередей)

O.R. (operations research, operational research)

Wikipedia
- Категория:Исследование операций

Algebra

freeCodeCamp.org / Free College Algebra Course (with Python Code!)
- YouTube
YouTube / @TheMathSorcerer / Super Hardcore Algebra
The Grammar of Algebra by E. Justin Hills
Wikipedia
- Algebraic structure
nLab
- algebra

Linear algebra

Wikipedia
- Template:Linear algebra
- Householder transformation is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.
Devpractice Team. Линейная алгебра на Python - 114 страниц
Immersive Math This is a nice interactive free textbook on linear algebra
Courses
- Открытое образование
  - Линейная алгебра
  - Высшая математика. Линейная алгебра и элементы топологии
- freeCodeCamp.org
  - Linear Algebra Crash Course - Mathematics for Machine Learning and Generative AI
YouTube / @TheMathSorcerer / Linear Algebra in 4 Weeks
A “Grammar Lesson” for Linear Algebra by E. L. Lady
TowardsDataScience.com / A bird’s eye view of linear algebra: The measure of a map — determinant by Rohit Pandey

Numerical linear algebra / applied linear algebra

OpenBLAS – это одна из реализаций библиотеки базовых подпрограмм линейной алгебры (BLAS) с открытым исходным кодом. Отличительной особенностью библиотеки, широко используемой в академической среде, является высокий уровень оптимизации кода под различные процессорные архитектуры. Другими словами, она позволяет «выжимать максимум» из каждого поддерживаемого процессора при ее использовании для выполнения операций с векторами и матрицами.
Wikipedia
- Armadillo (C++ library)
- Eigen
Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares guides the reader through practical use-cases

Computer algebra / algebraic computation / symbolic computation

Wikipedia
- Computer algebra
  - Символьные вычисления
- Computer algebra system
  - Система компьютерной алгебры
- List of computer algebra systems
- Maxima (software) is a computer algebra system (CAS)
Courses
- Открытое образование
  - Wolfram: система компьютерной алгебры

Geometric algebra / real Clifford algebra

Wikipedia
- Geometric algebra is an extension of elementary algebra to work with geometrical objects such as vectors.
Comparison of vector algebra and geometric algebra
Rigid Geometric Algebra

Group theory

SecurityLab.ru / Теория групп: как простая математика объясняет устройство вселенной

Mathematical optimization / mathematical programming

Wikipedia
Mathematical Optimization and Economic Analysis by Mikulás Luptácik - 294 pages
Building and Solving Mathematical Programming Models. 50 Practical Examples by José Manuel García Sánchez - 401 pages
- The book presents the construction and resolution of 50 optimization problems, divided into five levels of difficulty, although assigning a level of difficulty is subjective in some cases. The first levels are dedicated toward problems oriented to learning the model construction methodology. The difficulty increases with the levels, up to the complex optimization environments presented in the last level. Despite this, understanding all the problems, including the last levels, is feasible after knowing the modeling methodology and progressing with analysis of the problems from the beginning.
Optimization Methods in Business Analytics by James B. Orlin

Linear programming (LP) / linear optimization

Wikipedia
- Linear programming is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships.
GitHub
- Maxime Labonne
  - Articles
    - Introduction to Constraint Programming in Python
  - mlabonne/linear-programming-course: Hands-on course about linear programming and mathematical optimization.

Discrete optimization

Combinatorial optimization

Wikipedia
- Combinatorial optimization
- Travelling salesman problem NP-hard problem in combinatorial optimization
MIT OpenCourseWare / Mathematics / Combinatorial Optimization
Papers With Code / Combinatorial Optimization
Brilliant Math & Science Wiki / Combinatorial Optimization
YouTube
- Solving Combinatorial Optimization Problems with Constraint Programming and OscaR

Game theory

The Evolution of Trust – an interactive guide to the game theory of why & how we trust each other
Battleship. Optimal Sunk-cost Strategy
- Радио-Т Подкаст
Wikipedia
- Template:Game theory
Courses
- Открытое образование
  - Теория игр
- dev.by / курс по теории игр Стенфордского университета для IT-специалистов

Combinatorics

Wikipedia
- Комбинаторика
- Комбинаторное мышление
Problem-Solving Methods in Combinatorics. An Approach to Olympiad Problems by Pablo Soberón - 174 pages
Courses
- Открытое образование
  - Основы комбинаторики

Mathematical logic

Wikipedia
GitHub
- Z3 Theorem Prover
- vprover/vampire: The Vampire Theorem Prover
Communications of the ACM / April 2014 / Formally Verified Mathematics
- one of the most significant advances in mathematical logic around the turn of the 20th century was the realization that ordinary mathematical arguments can be represented in formal axiomatic systems in such a way their correctness can be verified mechanically, at least in principle
Courses
- Открытое образование
  - Математическая логика

Numerical analysis

Rigorous computation

Wikipedia
- Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification is numerics including mathematically strict error (rounding error, truncation error, discretization error) evaluation, and it is one field of numerical analysis.
- Interval arithmetic
  - Интервальная арифметика
Boost Interval Arithmetic Library Extends the usual arithmetic functions to mathematical intervals.
JuliaIntervals
- Interval arithmetic offers a mathematical framework to perform rigorous computations. This allows you to deal with numerical approximation errors or uncertainty in your model in a rigorous way.

Software

SageMath Open-Source Mathematical Software System
- Documentation
  - Reference
    - Linear Algebra
      - Matrices and Spaces of Matrices
        
        General matrix Constructor and display options
  - Tutorial
    - A Guided Tour
      - Parents, Conversion and Coercion
IBM
- Optimization Software
- Minds of Modern Mathematics
  - IBM Research Blog / Free iPad App from IBM Reinvents Iconic '60s Era Exhibit on History of Mathematics
  - WIRED / New IBM App Presents Nearly 1,000 Years of Math History
    - The app is based on an original, 50-foot-long "Men of Modern Mathematics" installation created in 1964 by Charles and Ray Eames.
    - you can always visit the large-scale timeline on display at the New York Hall of Science in Queens, NY and The Museum of Science in Boston, MA
Wolfram
MathWorks - Makers of MATLAB and Simulink
- Documentation
  - Applications
    - Mathematics and Optimization
      - Optimization Toolbox
        
        Solver-Based Optimization Problem Setup

Apps

Приложения в Google Play
- GeoGebra

Animations

Journal

Convergence MAA's online peer-reviewed journal where history, mathematics, and teaching meet
- Index to Mathematical Treasures
- Mathematical Treasure: IBM’s Men of Modern Mathematics Poster

Exhibition

Mathematica: A World of Numbers... and Beyond
- Wikipedia
- Eames Office (design team)
  - Men of Modern Mathematics The Eames Office produced the 12-foot timeline for IBM, who, for decades, distributed the teaching tool to schools across the nation.
  - Vintage Men of Modern Mathematics A 1960s timeline of the math that shapes our world.
- Charles Eames Graphic Time Line on the History of Mathematicians -- Prepared For IBM (Barry Lawrence Ruderman Antique Maps Inc.)

Blogs

The n-Category Café
- Structure-Like Stuff

Extremums

YouTube / The Math Sorcerer / Hardest Math Course
r/math
- Hardest mathematics book?
  - Algebraic Geometry, Hartshorne
  - Real Analysis by Folland
  - Rudin's book on real and complex analysis
  - PDE by Evans
  - "Set Theory: An Introduction to Independence Proofs" by Kenneth Kunen
  - Thurston's "Three Dimensional Geometry and Topology"
  - Baby Rudin
  - Principles of Program Analysis by Flemming Nielson, Hanne R. Nielson, Chris Hankin
PhysicsForums.com / What is the most difficult text on mathematics?
- Principia Mathematica by Whitehead and Russell
- notebooks of Ramanujan
Quora / What would you say are the hardest graduate textbooks in mathematics?
- UC Berkeley 2018 Fall MATH 225A Metamathematics taught by Thomas Scanlon
  - uses Model Theory : An Introduction (Graduate Texts in Mathematics): 9781441931573 as a starting point.
  - his notes Math 225A – Model Theory – Draft were documented by Martin Speirs.
- The next course is 2019 Spring MATH 225B Metamathematics taught by Antonio Montalban
  - will use the textbook he is writing, see Computable Structure Theory: Within the arithmetic

Career

Skillbox Media / Наиболее востребованными среди российских учителей оказались математики 6 окт 2023
Т—Ж / Как живет студент и репетитор по математике в Москве с доходом 160 000 ₽ 06.05.24

Teaching

Cambridge Day / Teach math like we teach music

Monetization

Хабр / Как мы заработали 250 000 руб, запустив ИИ приложение для обучения математике в США