All topics
Basic Algebra
Elementary algebra from the ground up — variables, equations, lines, polynomials, factoring, fractions, and radicals, building to the Fundamental Theorem of Algebra.
- Variables & Expressions
- Order of Operations
- Signed Numbers
- Types of Numbers
- Fractions, Decimals & Percents
- Linear Equations
- Literal Equations
- Ratios & Proportions
- Functions & Function Notation
- Direct & Inverse Variation
- Percent & Mixture Problems
- The Coordinate Plane
- Slope of a Line
- Graphing Lines
- Systems of Equations
- Word Problems
- Linear Inequalities in One Variable
- Compound Inequalities
- Absolute Value Equations & Inequalities
- Linear Inequalities in Two Variables
- Systems of Linear Inequalities
- Exponent Rules
- Negative & Zero Exponents
- Scientific Notation
- Polynomial Basics
- Polynomial Expansion (FOIL)
- Polynomial Division
- Special Products
- Difference of Squares
- Factoring Polynomials
- Factoring Trinomials
- Rational Expressions
- Radicals & Roots
- Midpoint & Distance
- Rational Exponents
- Radical Equations
- Solving Quadratics by Factoring
- Completing the Square
- Quadratic Equation
- Complex Numbers
- Fundamental Theorem of Algebra
Basic Geometry
Standard school geometry: points and angles, triangles and the Pythagorean theorem, polygons and area, similarity, circles, solids, and transformations.
- Points, Lines & Planes
- Angles & Angle Pairs
- Geometric Proof
- Triangle Basics
- Congruent Triangles
- Isosceles & Equilateral Triangles
- Triangle Inequality
- Pythagorean Theorem
- Special Right Triangles
- Right-Triangle Trigonometry
- Parallel Lines & Transversals
- Angles in Polygons
- Quadrilaterals
- Midsegments & Medians
- Regular Polygons & Apothem
- Area & Perimeter
- Similar Triangles
- Proportional Segments
- Areas of Similar Figures
- Central Angles & Arcs
- Inscribed Angle Theorem
- Tangents, Chords & Secants
- Circumference, Arc Length & Sectors
- Equation of a Circle
- Volume & Surface Area
- Solid Geometry
- Locus
- Transformations
- Constructions
Geometry
Advanced and olympiad geometry: triangle centers and the classical theorems, power of a point, cyclic quadrilaterals, inversion, and non-Euclidean geometry.
- Triangle Centers
- Heron's Formula
- Ceva's Theorem
- Menelaus's Theorem
- Simson Line
- Nine-Point Circle
- Power of a Point
- Cyclic Quadrilaterals
- Ptolemy's Theorem
- Pick's Theorem
- Inversive Geometry (Inversion)
- Platonic Solids
- Non-Euclidean Geometry
Precalculus
Functions, logs, trig, sequences, and limits — the toolkit you need before calculus makes sense.
- Function Basics
- Function Composition & Inverse
- Inverse Functions (2)
- Logarithm Rules
- Log Change of Base
- Radians & the Unit Circle
- Trig Functions
- Trig Values
- Trig Identities
- Law of Cosines
- Law of Sines
- Sequences & Series
- Limits
- Limits at Infinity
- Graph Transformations
- Polynomial Functions & Graphs
- Rational Functions & Asymptotes
- Exponential Functions & Equations
- Conic Sections
- Polar Coordinates
- Vectors (2D)
Discrete Math
The language of proof, counting without listing, number theory classics, and the vocabulary of graphs.
- Set Operations
- Power Sets & Partitions
- Relations
- Equivalence Relations
- Partial Orders
- Logical Connectives
- Quantifiers
- Truth Tables
- Logical Equivalence
- Valid Arguments
- Injective & Surjective Functions
- Cardinality
- Big-O Notation
- Direct Proof
- Proof by Contrapositive
- Proof by Contradiction
- Induction
- Pigeonhole Principle
- Permutations & Combinations
- Inclusion–Exclusion
- Stars and Bars
- Binomial Coefficients
- Constructing Bijections
- Counting Principles
- Factorials
- Recurrence Relations
- Integer Partitions
- Lattices
- Boolean Algebra
- Karnaugh Maps
- Logic Gates & Circuits
- Graph Terminology & Representations
- Trees
- Graph Coloring & Chromatic Number
- Finite-State Machines & Regular Expressions
Graph Theory
Vertices and edges: connectivity, traversals, planarity, directed graphs and shortest paths, and the tree data structures built on them.
- Graph Connectivity
- Bipartite Graphs
- Graph Coloring & Cliques
- Eulerian & Hamiltonian Graphs
- Planar Graphs
- Directed Graphs
- Topological Sort
- Shortest Paths
- Binary Trees
- Binary Search Trees
- Heaps
- Huffman Coding
- Matchings & Vertex Covers
- Hall's Marriage Theorem
- Spanning Tree Count (Cayley & Matrix-Tree)
- Edge Coloring & Vizing's Theorem
- Menger's Theorem
- Graph Isomorphism
- Graphic Sequences (Erdős–Gallai)
- Random Graphs (Erdős–Rényi)
Number Theory
The integers up close: divisibility and primes, gcd via the Euclidean algorithm, modular arithmetic and congruences, and the classical proofs.
- Divisibility & Primes
- Euclidean Algorithm
- Bézout's Identity
- Fundamental Theorem of Arithmetic
- Modular Arithmetic
- Linear Congruences
- Fermat's Little Theorem
- √2 is Irrational
- Infinitely Many Primes
- Euler's Totient Function φ(n)
- Euler's Totient Theorem
- Wilson's Theorem
- Order of an Element & Primitive Roots
- Quadratic Residues & Legendre Symbol
- Quadratic Reciprocity
- Divisor Functions τ(n) and σ(n)
- Möbius Function & Inversion
- Perfect Numbers & Mersenne Primes
Calculus
Single-variable calculus: derivatives, integrals, the classic theorems, and their applications.
- Derivatives Warmup & Power Rule
- Quotient Rule
- Special Derivatives
- Product Rule
- Chain Rule
- Continuity
- Intermediate Value Theorem
- Extreme Value Theorem
- Rolle's Theorem
- Mean Value Theorem
- L'Hôpital's Rule
- Convex Functions
- Integration
- Definite Integral
- Fundamental Theorem of Calculus
- Taylor Series
- Increasing & Decreasing
- Tangents & Normals
- Second Derivative
- Local Maxima & Minima
- Kinematics
- Implicit Differentiation
- Integration Techniques
- Areas & Volumes of Revolution
- Differential Equations
- Related Rates
- Linear Approximation & Differentials
- Newton's Method
- Improper Integrals
- Average Value of a Function
- Surface Area of Revolution
- Parametric Curves & Calculus
- Polar Coordinates & Area
- Series Convergence Tests & Power Series
Real Analysis
Calculus made rigorous: the completeness of ℝ and the supremum, the ε–N limit, sequences and series, compactness, and the Riemann integral.
- Supremum & Completeness
- Nested Interval Property
- Open & Closed Sets
- ε-N Convergence
- Monotone Convergence Theorem
- Cauchy Sequences
- Limit Superior & Inferior
- Series Convergence Tests
- Radius of Convergence
- Bolzano–Weierstrass Theorem
- Compact Sets (Heine–Borel)
- Uniform Continuity
- Uniform Convergence
- Riemann Integrability
Multivariate Real Analysis
Rigorous analysis in ℝ^n: the total derivative as a linear map, the inverse and implicit function theorems, rigorous multivariable integration, and differential forms up to the general Stokes theorem.
- Normed & Metric Spaces
- Total Derivative
- Jacobian Matrix
- Chain Rule as Composition
- Clairaut's Theorem
- Contraction Mapping Theorem
- Inverse Function Theorem
- Implicit Function Theorem
- Multivariable Taylor & Hessian Test
- Measure Zero & Lebesgue Criterion
- Fubini's Theorem
- Partitions of Unity
- Differential Forms
- Stokes' Theorem on Manifolds
- σ-Algebras
- Measures
- Measurable Functions
- The Lebesgue Integral
- Convergence Theorems
- L^p Spaces
- Product Measures
- Radon-Nikodym
Linear Algebra
Vectors, matrices, determinants, eigenvalues, and the structural identities of linear maps.
- Vectors & Dot Products
- Linear Independence
- Span & Basis
- Linear Transformations
- Change of Basis
- Matrix Terminology
- Matrix Multiplication
- Row Reduction
- Matrix Inverse
- Determinant Basics
- Determinant Properties
- Laplace Expansion
- Cramer's Rule
- Eigenvalues Basics
- Characteristic Polynomial
- Eigenvectors & Eigenspaces
- Diagonalization
- Spectral Theorem
- Matrix Rank
- Null Space & Nullity
- Rank-Nullity Theorem
- Operator Norm
- Cauchy-Binet Identity
- Sylvester's Determinant Identity
- Hadamard's Inequality
- Weyl's Inequality
- Sherman-Morrison
- Newton's Identities
- Orthogonality & Orthonormal Sets
- Inner Product Spaces
- Orthogonal Projections
- Gram–Schmidt Process
- Least Squares
- QR Factorization
- Singular Value Decomposition (SVD)
- Positive Definite Matrices
- Orthogonal & Unitary Matrices
- Jordan Normal Form
Probability
From basic event probabilities through expectation and variance to concentration inequalities.
- Probability Basics
- Random Variables
- Conditional Probability
- Markov Chains
- Independence
- Expected Value
- Linearity of Expectation
- Variance
- Law of Total Probability
- Bayes' Theorem
- Law of Large Numbers
- Monty Hall
- Secretary Problem
- Markov's Inequality
- Chebyshev's Inequality (Prob.)
- Chernoff Bound
- Hoeffding's Inequality
- Wald's Identity
- Azuma-Hoeffding
- McDiarmid's Inequality
- Central Limit Theorem
- Binomial Distribution
- Normal Distribution
- Poisson Distribution
- Geometric Distribution
- Exponential Distribution
- Covariance & Correlation
- Joint & Marginal Distributions
- Conditional Expectation
Combinatorics
Counting identities plus the probabilistic and extremal methods built on them.
- Binomial Theorem
- Vandermonde's Identity
- Catalan Numbers
- Generating Functions
- Ramsey's Theorem
- Turán's Theorem
- Expander Mixing Lemma
- Lovász Local Lemma
- Derangements
- Stirling Numbers and the Twelvefold Way
- Mobius Inversion on Posets
- Dilworth's Theorem
- Sperner's Theorem
- Van der Waerden's Theorem
- Sperner's Lemma
Multivariable Calculus
Partial derivatives, gradients, multiple integrals, and the three great theorems of vector calculus.
- Partial Derivatives
- Multivariable Chain Rule
- Gradient
- Lagrange Multipliers
- Double Integrals
- Triple Integrals
- Vector Fields (div, curl)
- Line Integrals
- Surface Integrals
- Green's Theorem
- Stokes' Theorem
- Divergence Theorem
- Tangent Planes & Linear Approximation
- Directional Derivatives
- Change of Variables & the Jacobian
- Parametric Surfaces & Surface Area
- Conservative Fields & Potential Functions
- Limits & Continuity of Multivariable Functions
- 3D Vectors & the Cross Product
- Arc Length & Curvature of Space Curves
Abstract Algebra
Groups, rings, and fields — a Dummit & Foote arc from the group axioms to Galois theory.
- Groups & Group Actions
- Cyclic & Dihedral Groups
- Symmetric Groups
- Subgroups & Lagrange
- Cosets & Normal Subgroups
- Quotient Groups
- Homomorphisms & Iso Theorems
- The Class Equation
- Sylow Theorems
- Finite Abelian Groups
- The Orbit–Stabilizer Theorem
- Burnside's Lemma
- Rings & Ideals
- Integral Domains & Fields
- Polynomial Irreducibility
- Euclidean Domains, PIDs & UFDs
- Chinese Remainder Theorem
- Field Extensions
- Finite Fields
- Splitting Fields
- Cyclotomic Polynomials
- Galois Theory
- Semidirect Products
- Direct Products of Groups
- Solvable Groups
- Nilpotent Groups
- Jordan–Hölder Theorem
- Modules over a Ring
- Structure Theorem for Modules over a PID
- Rational Canonical Form
- Separable & Inseparable Extensions
- Solvability by Radicals
- Free Groups & Presentations
Combinatorial Optimization
Linear programming, duality, and submodular optimization over discrete structures.
- Polyhedra
- Linear Programming
- Simplex Method
- LP Duality
- Complementary Slackness
- Submodularity
- Ford–Fulkerson / Augmenting Paths
- Bipartite Matching & Hall's Theorem
- Assignment Problem / Hungarian Algorithm
- Matroids & the Greedy Algorithm
- Integer Programming & LP Relaxation
- Total Unimodularity
- Branch & Bound
- The Traveling Salesman Problem
- Knapsack & DP / FPTAS
- Approximation Algorithms & Ratio
Inequalities
Classical and analytic inequalities at the heart of competition math and analysis.
- Triangle Inequality (Norms)
- AM-GM
- Cauchy-Schwarz
- Jensen's Inequality
- Power Mean
- Rearrangement
- Schur's Inequality
- Muirhead's Inequality
- Sum of Squares (SOS)
- Hölder's Inequality
- Minkowski's Inequality
- Chebyshev's Sum Inequality
- Karamata's Inequality (Majorization)
- Young's Inequality
- Bernoulli's Inequality
- Tangent Line Trick
- Abel Summation
- Schur-Convexity
- Hilbert's Inequality
- Ravi Substitution
- Smoothing / Mixing Variables
Logic
Self-reference, diagonalization, and the limits of formal systems.
- Self-Reference Paradoxes
- Cantor's Diagonal Argument
- Gödel's Incompleteness
- Functional Completeness of Connectives
- Compactness Theorem (Sentential)
- Quantifiers & First-Order Translation
- Models & Satisfaction
- Natural Deduction
- Soundness & Completeness
- Löwenheim–Skolem Theorem
- Nonstandard Models of Arithmetic
- Decidability & Recursive Sets
AI & ML
Regression, gradient descent, classification, and neural networks — the mathematical core of machine learning.
- Linear Regression
- Gradient Descent
- Logistic Regression
- Cross-Entropy Loss
- Bias–Variance Tradeoff
- Neural Networks & Backprop
- K-Means Clustering
- Principal Component Analysis
- Decision Trees
- Ridge & Lasso Regularization
- Support Vector Machines & Kernels
- Cross-Validation & Model Selection
- Naive Bayes Classifier
- Random Forests & Bagging
- Gradient Boosting
- k-Nearest Neighbors
- ROC Curves & AUC
- EM & Gaussian Mixture Models
- Regularization for Deep Nets (Dropout & Weight Decay)
- Convolutional Neural Networks
- Maximum Likelihood Estimation
Information Theory
Entropy, mutual information, KL divergence, and the limits of communication.
- Shannon Entropy
- Conditional Entropy
- Mutual Information
- Noisy Channels
- MAP vs. ML Decoding
- KL Divergence
- Log-Sum Inequality
- Data Processing Inequality
- Fano's Inequality
- Asymptotic Equipartition Property
- Typical Sets
- Kraft Inequality & Prefix Codes
- Source Coding Theorem
- Channel Capacity Theorem
- Differential Entropy
- Gaussian Channel Capacity
- Maximum Entropy Distributions
- Method of Types & Sanov's Theorem
- Rate-Distortion Function
- Entropy Rate of a Markov Chain
- Lempel-Ziv Universal Coding
Coding Theory
Bits, parity, and error-correcting codes from repetition to Hamming.
- Binary & XOR
- Repetition Code
- Singleton Bound
- Parity Check Matrix
- Hat Puzzle
- Hamming Code
- Hamming Distance & Minimum Distance
- Linear Codes & Generator Matrix
- Syndrome Decoding
- Dual Codes
- Hamming / Sphere-Packing Bound
- Perfect Codes
- Cyclic Codes
- Reed–Solomon Codes
- Gilbert–Varshamov Bound
Game Theory & Social Choice
Strategic play, voting, and the paradoxes of collective choice.
- Minimax Theorem
- Braess' Paradox
- Arrow's Impossibility
- Nash Equilibrium (pure strategies)
- Dominant & Dominated Strategies
- The Prisoner's Dilemma
- Mixed-Strategy Nash Equilibrium
- Pareto Efficiency
- Extensive-Form Games & Backward Induction
- Subgame-Perfect Equilibrium
- Repeated Games & the Folk Theorem
- Nash Bargaining Solution
- The Shapley Value
- The Core
- Stable Matching / Gale–Shapley
- Condorcet Paradox
Competition Math
The contest toolbox: Vieta's formulas, telescoping, invariants, the extremal principle, coloring arguments, and functional equations.
- Vieta's Formulas
- Telescoping Sums
- Invariants & Monovariants
- The Extremal Principle
- Coloring Arguments
- Functional Equations
- Nim and Sprague-Grundy Values
- Infinite Descent
- Double Counting
- Roots of Unity Filter
- Chicken McNugget (Frobenius) Theorem
- Substitution in Inequalities
- Wythoff's Game
Topology
Standard topology topics from the canonical curriculum.
- Topological Spaces & Open Sets
- Basis for a Topology
- Subspace, Product & Quotient Topologies
- Closure, Interior & Limit Points
- Continuity in Topological Spaces
- Homeomorphisms & Topological Equivalence
- Metric Topology
- Connectedness
- Compactness
- Separation Axioms (Hausdorff & beyond)
- Homotopy of Maps & Paths
- The Fundamental Group
- Covering Spaces
- Euler Characteristic & Surface Classification
- Hairy Ball Theorem
Complex Analysis
Standard complex analysis topics from the canonical curriculum.
- Polar & Exponential Form
- De Moivre & Roots of Unity
- Cauchy-Riemann Equations
- Holomorphic & Analytic Functions
- Contour Integration
- Cauchy's Integral Theorem
- Cauchy's Integral Formula
- Laurent Series
- Singularities & Poles
- Residue Theorem
- Residues & Real Integrals
- Argument Principle & Rouché
- Möbius Transformations
- Liouville's Theorem
- Maximum Modulus Principle
- Harmonic Functions & Conjugates
Algorithms
Standard algorithms topics from the canonical curriculum.
- Divide and Conquer
- Dynamic Programming
- Greedy Algorithms
- Binary Search
- Merge Sort and Quicksort
- Master Theorem
- Minimum Spanning Tree
- Hash Tables
- Amortized Analysis
- Union-Find (Disjoint Sets)
- Max-Flow Min-Cut
- NP-Completeness
- Approximation Algorithms
- Graph Traversal (BFS and DFS)
Differential Geometry
Standard differential geometry topics from the canonical curriculum.
- Arc Length & Reparametrization
- Curvature of a Plane Curve
- Torsion & the Frenet-Serret Formulas
- Regular Surfaces & Tangent Planes
- First Fundamental Form
- The Gauss Map & Second Fundamental Form
- Principal Curvatures & Normal Curvature
- Gaussian & Mean Curvature
- Geodesics
- Covariant Derivative & Parallel Transport
- Theorema Egregium
- Gauss-Bonnet Theorem
- Minimal Surfaces
Theory of Computation
Standard theory of computation topics from the canonical curriculum.
- Finite Automata (DFA & NFA)
- Regular Expressions & Languages
- Pumping Lemma for Regular Languages
- Myhill-Nerode Theorem & DFA Minimization
- Context-Free Grammars
- Pushdown Automata
- Pumping Lemma for Context-Free Languages
- Turing Machines
- Church-Turing Thesis
- Decidability & the Halting Problem
- Reducibility & Undecidability
- Rice's Theorem
- The Classes P and NP
- NP-Completeness & Cook-Levin
- Space Complexity & Savitch's Theorem
Computational Complexity
Standard computational complexity topics from the canonical curriculum.
- Time Complexity Classes (P)
- The Class NP & Verifiers
- Polynomial-Time Reductions
- Proving NP-Hardness (Gadget Reductions)
- Space Complexity & PSPACE
- L, NL & Savitch's Theorem
- Time & Space Hierarchy Theorems
- The Polynomial Hierarchy
- Boolean Circuits & P/poly
- Randomized Complexity (BPP, RP)
- Approximation & PCP Theorem
Stochastic Processes
Randomness evolving in time — random walks and Markov-chain behavior through Poisson processes, martingales, Brownian motion, and Itô calculus.
- Random Walks
- Gambler's Ruin
- Poisson Processes
- Stationary Distributions
- Martingales
- Optional Stopping Theorem
- Brownian Motion
- Quadratic Variation
- Itô Integral
- Itô's Lemma
- Stochastic Differential Equations
- Martingale Convergence
- Reflection Principle
- Feynman-Kac Formula
- Girsanov's Theorem
- Martingale Representation
- Compound Poisson Processes
- The Lévy–Khinchin Formula
- Donsker's Invariance Principle
- Skorokhod Embedding
- Bessel Processes
- Fractional Brownian Motion
- The Dirichlet Problem
- Lévy Processes
- Multidimensional Brownian Motion
- The Infinitesimal Generator
- Local Martingales
Mathematics of Finance
The mathematics of money and markets — interest and time value, bond valuation, portfolio risk, CAPM, utility, and growth-optimal betting.
- Interest & Compounding
- Time Value of Money
- Annuities & Loan Amortization
- NPV & IRR
- Bond Pricing & Yield
- Portfolio Return & Risk
- The Efficient Frontier
- CAPM & Beta
- Expected Utility & Risk Aversion
- Kelly Criterion
- Arbitrage & the Law of One Price
- Forwards & Futures
- Options & Put–Call Parity
- Binomial Option Pricing
- Risk-Neutral Pricing
- Black–Scholes
- The Greeks
Convex Optimization
Why convex problems are the ones we can actually solve — recognizing convexity, the standard problem families, Lagrange duality and KKT, and the algorithms that find the optimum.
- Convex Sets, Hulls & Cones
- Operations That Preserve Convexity
- Separating & Supporting Hyperplanes
- Convexity in Several Variables
- The Conjugate Function
- Quasiconvex Functions
- Dual Cones
- Convex Optimization Problems
- Quadratic Programs & QCQP
- Second-Order Cone & Semidefinite Programs
- Geometric Programming
- The Lagrange Dual
- Strong Duality & Slater's Condition
- The KKT Conditions
- Sensitivity & Shadow Prices
- Descent Methods & Line Search
- Newton's Method for Minimization
- The Barrier Method
- Norm Approximation & Regularization
Algebraic Topology
Turning shape into algebra — cell complexes, the fundamental group by pieces, homology and cohomology, and the classical fixed-point theorems they prove.
- Homotopy Equivalence & Deformation Retracts
- CW Complexes
- Van Kampen's Theorem
- Classification of Covering Spaces
- Graphs, Trees & Free Groups
- Δ-Complexes & Simplicial Homology
- Singular Homology
- Exact Sequences & the Long Exact Sequence
- Excision & Mayer–Vietoris
- Cellular Homology
- Degree of a Map
- Brouwer & Borsuk–Ulam
- Cohomology & Universal Coefficients
- Cup Product & the Cohomology Ring
- Poincaré Duality
- Higher Homotopy Groups & Hurewicz
Category Theory
The mathematics of structure itself — objects and arrows, functors and naturality, universal properties and limits, the Yoneda lemma, and adjunctions.
- Categories & Examples
- Isomorphisms, Monos & Epis
- Functors
- Natural Transformations
- Duality & Opposite Categories
- Equivalence of Categories
- Initial & Terminal Objects
- Products & Coproducts
- Equalizers & Coequalizers
- Pullbacks & Pushouts
- Limits & Colimits
- Representable Functors
- The Yoneda Lemma
- Adjoint Functors
- Units, Counits & Triangle Identities
- Adjoints Preserve Limits