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370 concepts

ADMM · The Alternating Direction Method of Multipliers

ADMM (Alternating Direction Method of Multipliers) explained: precise statement, convergence proof via Douglas–Rachford splitting, the scaled-dual for

Optimization

AM-GM Inequality · (Σxᵢ)/n ≥ (Πxᵢ)^(1/n) — arithmetic mean ≥ geometric mean

AM-GM inequality: (x₁ + … + xₙ)/n ≥ (x₁·…·xₙ)^(1/n) for non-negative reals. Equality iff all xᵢ are equal. Foundation of convex optimization, Jensen's

Inequalities

Abel Summation · Turning Sums into Integrals

Abel summation (summation by parts) turns arithmetic sums ∑ aₙf(n) into a boundary term minus ∫ A(t)f′(t)dt — the discrete integration-by-parts at the

Number Theory

Adjoint Functors · The Free-Forgetful Duality

Adjoint functors explained: the precise Hom-set bijection Hom(FX,Y)≅Hom(X,GY), unit/counit, the free-forgetful duality, worked free-group example, and

Category Theory

Analytic Continuation · If two analytic functions agree on a connected open set, they agree wherever both are defined

Analytic continuation is the process of extending the domain of an analytic function f, originally defined on some open set U ⊂ ℂ, to a larger set V ⊃

Complex Analysis

Arnoldi Iteration · Orthonormal basis of the Krylov subspace span{v, Av, A²v, ...} for non-symmetric A

Arnoldi iteration builds an orthonormal basis Q of the Krylov subspace span{v, Av, A²v, ...} via modified Gram–Schmidt, producing an upper-Hessenberg

Iterative Methods

Axiom of Choice · The axiom that decided ZFC

The axiom of choice (AC) states that for every collection of non-empty sets there exists a function picking one element from each set. The statement s

Set Theory

Baire Category Theorem · A complete metric space is not the countable union of nowhere-dense sets — completeness is fat

The Baire category theorem says a complete metric space cannot be written as a countable union of nowhere-dense sets. It powers the open mapping, clos

Topology

Banach Fixed-Point Theorem · Every k-contraction (k < 1) on a complete metric space has exactly one fixed point

The Banach fixed-point theorem: every contraction T: X → X with |T(x) − T(y)| ≤ k|x − y| (k < 1) on a complete metric space has a unique fixed point.

Analysis

Banach Space · A vector space (V, ‖·‖) where every Cauchy sequence converges — generalizes Euclidean ℝⁿ to infinite dimensions

A Banach space is a vector space V (over ℝ or ℂ) with a norm ‖·‖ such that V is complete with respect to the metric d(x, y) = ‖x − y‖ — every Cauchy s

Functional Analysis

Bayes' Theorem · P(A|B) = P(B|A)

Update beliefs when new evidence arrives. A positive medical test on a rare disease gives a surprisingly low probability of actual disease — Bayes cor

Probability

Bessel Functions · The cylindrical sines and cosines — drumhead modes, FM sidebands, diffraction rings

Bessel functions J_n and Y_n solve x²y'' + xy' + (x² − n²)y = 0. They describe vibrating drumhead modes, cylindrical waveguides, and FM radio sideband

Special Functions

Beta Distribution · The conjugate prior for proportions — Bayesian inference's two-parameter workhorse

The Beta distribution Beta(α, β) on [0, 1] is the conjugate prior for Bernoulli/binomial proportions. α and β act as pseudo-counts of prior successes

Probability

Binomial Distribution · n independent yes/no trials with success probability p — the foundation of statistics, from coin flips to clinical trials

The Binomial distribution counts the number of successes in n independent trials, each with the same probability p of success. Coin flips, election po

Probability

Bisection Method · Halve [a,b] where f(a)·f(b)<0 and squeeze a root with guaranteed linear convergence

The bisection method finds a root of f(x) = 0 by repeatedly halving an interval [a,b] where f changes sign. Each iteration shrinks the bracket by exac

Numerical Analysis

Bolzano-Weierstrass Theorem · In ℝⁿ, every bounded sequence has at least one convergent subsequence

The Bolzano-Weierstrass theorem states: every bounded sequence in ℝⁿ has a convergent subsequence. Proven independently by Bernard Bolzano (1817, in h

Real Analysis

Brouwer Fixed-Point Theorem · Any continuous f: D^n → D^n on the closed unit ball has a point with f(x) = x

Brouwer's fixed-point theorem: every continuous function f from the closed n-dimensional ball D^n to itself has at least one fixed point — a point x w

Topology

Brownian Motion · B(t) — continuous-time path with independent Gaussian increments, almost-surely nowhere differentiable

Brownian motion (Wiener process) is a continuous-time stochastic process (B(t))t≥0 satisfying: (1) B(0) = 0, (2) increments B(t) − B(s) ~ N(0, t−s) fo

Probability

Burnside's Lemma · |X/G| = (1/|G|) Σ_{g∈G} |Fix(g)| — average over the group of fixed-point counts

Burnside's lemma (sometimes "Cauchy-Frobenius"; Burnside's 1897 book popularized it but Cauchy proved it earlier and Frobenius generalized): for a fin

Combinatorics

Bézout's Theorem · Counting Intersections of Plane Curves

Bézout's Theorem states two projective plane curves of degree m and n meet in exactly mn points counted with multiplicity over an algebraically closed

Algebraic Geometry

CW Complexes · Building Spaces by Attaching Cells

CW complexes explained: the precise definition, attaching maps, the closure-finite weak topology, cellular homology, Whitehead's theorem, and why ever

Algebraic Topology

Cantor's Diagonalization · The proof technique that revealed uncountable infinity, the halting problem, and Gödel's incompleteness — all from one off-diagonal flip

In 1891 Georg Cantor showed there are more real numbers than natural numbers using a single trick: assume the reals are listable, walk down the diagon

Set Theory

Cardinality of Infinite Sets · Cantor's hierarchy of infinities

Two sets have the same cardinality when there is a bijection between them. Cantor (1874-1891) discovered that this seemingly modest definition produce

Set Theory

Catalan Numbers · C_n = (2n choose n)/(n+1) — the count that appears in 60+ disguises, from balanced parentheses to triangulated polygons

The Catalan numbers 1, 1, 2, 5, 14, 42, 132, 429 form one of the most ubiquitous sequences in combinatorics. The same formula counts balanced parenthe

Combinatorics

Cauchy Integral Formula · f(a) = (1/2πi) ∮ f(z)/(z−a) dz — a holomorphic function is determined by its boundary values

The Cauchy Integral Formula: if f is holomorphic on a simply connected domain D and γ is a positively oriented simple closed contour in D enclosing th

Complex Analysis

Cauchy Sequence · (aₙ) is Cauchy iff for every ε > 0, ∃N: |aₘ − aₙ| < ε for all m, n ≥ N

A Cauchy sequence is one whose terms grow arbitrarily close together: for every ε > 0, there exists N such that |aₘ − aₙ| < ε for all m, n ≥ N.

Real Analysis

Cauchy-Riemann Equations · For f = u + iv, holomorphicity ⇔ ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x

A complex function f(x + iy) = u(x, y) + iv(x, y) is holomorphic at a point iff its real and imaginary parts u, v satisfy the Cauchy-Riemann equations

Complex Analysis

Cauchy-Schwarz Inequality · |⟨u, v⟩| ≤ ‖u‖·‖v‖ — universal in every inner product space, equality iff u and v are collinear

The Cauchy-Schwarz inequality says the absolute inner product of two vectors is at most the product of their norms. It holds in every inner product sp

Inequalities

Cayley-Hamilton Theorem · Every matrix satisfies its characteristic polynomial: p_A(A) = 0

The Cayley-Hamilton theorem states: if A is an n×n matrix over a commutative ring and p_A(λ) = det(λI − A) is its characteristic polynomial, then subs

Linear Algebra

Central Limit Theorem · Sums Become Normal

Averages of any distribution converge to a normal distribution as sample size grows. Why the bell curve appears everywhere — and statistics can work a

Statistics

Chain Rule · (f(g(x)))' = f'(g(x))

When functions are nested, the derivative is the product of outer and inner derivatives. The foundation of backpropagation in neural networks.

Calculus

Chebyshev Polynomials · Equioscillation, minimax approximation, and the death of the Runge phenomenon

Chebyshev polynomials T_n satisfy T_n(cos θ) = cos(nθ). They equioscillate between ±1 and give the minimax-optimal polynomial approximation on [−1, 1]

Special Functions

Chebyshev's Inequality · At most 1/k² of any distribution sits more than k standard deviations from the mean

Chebyshev's inequality: P(|X − μ| ≥ kσ) ≤ 1/k². The tightest distribution-free bound on tail probability — at most 25% of any distribution's mass sits

Probability

Cheeger's Inequality · The Spectral Gap Bounds Graph Expansion

Cheeger's inequality for graphs, stated precisely: λ₂/2 ≤ h(G) ≤ √(2λ₂). The spectral gap of the normalized Laplacian bounds graph conductance. Proof

Graph Theory

Chi-Squared Distribution · Sum of k squared standard normals — mean k, variance 2k

The chi-squared distribution χ²(k): the sum of k independent squared standard-normal random variables. Mean k, variance 2k. Backbone of goodness-of-fi

Statistics

Chinese Remainder Theorem · Solve a system of modular congruences with pairwise-coprime moduli — and recover a unique answer mod the product

When you know a number's remainder under several pairwise-coprime moduli, the Chinese Remainder Theorem reconstructs the number uniquely modulo the pr

Number Theory

Cholesky Decomposition · The square root of a positive-definite matrix — twice as fast as LU

A symmetric positive-definite matrix A factors uniquely as A = LL^T where L is lower triangular. Twice as fast as LU, half the memory, no pivoting nee

Linear Algebra

Cohomology · The dual of homology — equipped with a cup product that makes H(X) a graded ring*

Cohomology Hⁿ(X) is the dual of homology, defined via cochains and coboundaries δⁿ. It measures n-dimensional holes but with an extra cup-product ⌣ ma

Algebraic Topology

Compact Set · The topology that makes maxima exist

A subset K of a topological space is compact if every collection of open sets that covers K has a finite subcollection that still covers K. In metric

Topology

Compactification · One-point (Alexandroff), Stone-Čech, projective — embed a non-compact space densely into a compact one

A compactification of a topological space X is a compact space X̃ along with an embedding X ↪ X̃ as a dense subspace. Three canonical constructions: (

Topology

Completing the Square · x² + bx → (x + b/2)² − (b/2)²

A geometric trick turning any quadratic into a perfect square plus a constant. Derives the quadratic formula and powers conic-section analysis. From B

Algebra

Complex Numbers · Two real numbers, one orthogonal axis, and an algebra that closes every polynomial

A complex number a+bi pairs a real part with an imaginary part along an orthogonal axis. The Argand plane turns arithmetic into geometry — addition is

Complex Analysis

Complex Plane Arithmetic · Addition translates, multiplication rotates-and-scales, conjugation reflects

Arithmetic on the complex plane is geometry in disguise. Addition is vector translation, multiplication is rotation-and-scaling, conjugation is reflec

Complex Analysis

Conditional Expectation · E[X|Y] — the best mean-square predictor of X from Y, itself a random variable

E[X|Y] is the expected value of X given knowledge of Y — itself a random variable, a function of Y. Tower property: E[E[X|Y]] = E[X]. Foundation of re

Probability

Conformal Mapping · Holomorphic functions with f'(z) ≠ 0 preserve angles — bend regions while keeping local shape

A conformal mapping is a function f: U → ℂ (U ⊂ ℂ open) that preserves angles at every point — equivalently, f is holomorphic with f'(z) ≠ 0 throughou

Complex Analysis

Conic Sections · circle · ellipse · parabola · hyperbola

Slice a double cone with a plane at different angles to produce four curves. Horizontal cut → circle. Tilted cut → ellipse. Cut parallel to the cone's

Geometry

Conjugate Gradient · Iteratively solve Ax = b for symmetric positive-definite A in at most N steps using A-orthogonal search directions

The conjugate gradient (CG) method solves Ax = b for symmetric positive-definite A by walking a sequence of search directions that are A-conjugate. In

Optimization

Connectedness · "All one piece" — but the topologist's sine curve is one piece without a path between its ends

A topological space is connected if it can't be split into two disjoint open sets. Path-connected: any two points are joined by a continuous path. Pat

Topology

Continued Fractions · Express any real number as a tower of nested fractions — and the truncations are the best rational approximations

A continued fraction expresses a real number as a tower of nested reciprocals: x = a₀ + 1/(a₁ + 1/(a₂ + …)). Truncating gives a sequence of rational a

Number Theory

Continuity (ε-δ Definition) · The challenger picks any tolerance — you can always meet it

The ε-δ definition of continuity says a function f is continuous at a point c if for every tolerance ε > 0, there exists a closeness δ > 0 such that w

Analysis

Continuum Hypothesis (CH) · Cantor's question: no set has cardinality strictly between |ℕ| = ℵ₀ and |ℝ| = 2^ℵ₀

The Continuum Hypothesis (CH), posed by Georg Cantor in 1878, asserts that there is no set whose cardinality lies strictly between that of the natural

Set Theory

Convex Set and Convex Hull · The set that contains every segment between its points — and the smallest one wrapping a given cloud

A convex set contains every line segment between its members. The convex hull of a finite point set is the smallest convex set containing them, comput

Convex Analysis

Convolution Theorem · F(f ∗ g) = F(f) · F(g) — convolution in time becomes multiplication in frequency

The convolution theorem: the Fourier transform of a convolution is the pointwise product of Fourier transforms — F(f ∗ g) = F(f) · F(g). Turns O(n²) c

Fourier Analysis

Covering Spaces and the Galois Correspondence

Covering spaces and the Galois correspondence explained: the precise theorem matching subgroups of π₁(X) to connected coverings, with proof idea, exam

Algebraic Topology

Cramer's Rule · Solve Ax = b with one big determinant per unknown

Cramer's rule solves Ax = b for an invertible square matrix A by expressing each unknown as a ratio of determinants: xi = det(Ai) / det(A), where Ai i

Linear Algebra

Cross Product · A perpendicular vector whose length is the area between two others

The cross product takes two vectors in three-dimensional space and returns a third vector perpendicular to both, with magnitude equal to the area of t

Linear Algebra

Cross-Entropy · The bits you actually spend when your code is built for the wrong distribution

Cross-entropy H(P, Q) = −Σ P(x) log Q(x) is the average bits used to code samples from P with a code optimized for Q. Equal to entropy plus KL diverge

Information Theory

Cross-Ratio · (A, B; C, D) = ((C−A)(D−B))/((C−B)(D−A)) — preserved by every projective transformation

The cross-ratio of four collinear points A, B, C, D (or four concurrent lines) is the value (A, B; C, D) = ((C−A)(D−B))/((C−B)(D−A)) — the unique proj

Projective Geometry

Curl (Vector Calculus) · The local rotation of a vector field — circulation per unit area

Curl is the vector operator ∇×F that measures the local rotation of a vector field. At each point it tells you the axis and rate of swirl — the infini

Calculus

Cylindrical Coordinates · Polar plus a height — the natural coordinates for pipes, wires, and rotationally symmetric solids

Cylindrical coordinates (r, θ, z) extend polar coordinates by adding a vertical z axis: r is the perpendicular distance from the z-axis, θ is the azim

Geometry

De Moivre's Theorem · Powers of (cos θ + i sin θ) become a single rotation by nθ

De Moivre's theorem says (cos θ + i sin θ)^n = cos(nθ) + i sin(nθ). It turns powers of complex numbers into a single rotation, derives multiple-angle

Complex Analysis

Derivative Definition · lim h→0 [f(x+h)−f(x)] / h

As h → 0, the secant line through two points rotates into the tangent line. Its slope is the derivative — instantaneous rate of change. Calculus's fir

Calculus

Determinant · Signed Volume of a Transformation

|det(A)| is the scale factor for area (2D) or volume (3D) under the matrix transform. Sign tells orientation. Zero means the transform collapses dimen

Linear Algebra

Differential Equations (Overview) · Equations whose unknowns are functions, not numbers

A differential equation is an equation involving an unknown function and its derivatives. Solving one means finding the function. ODEs involve a singl

Differential Equations

Differential Forms · dx, dx∧dy, dx∧dy∧dz — the antisymmetric integrands that turn calculus into one theorem

Differential forms are antisymmetric covariant tensors — the integrands of higher-dimensional calculus. The wedge product, exterior derivative d, and

Differential Geometry

Dirac Delta Function · An infinite spike with integral one

The Dirac delta δ(x) is an idealised impulse — zero everywhere except a single point, with total integral one. Strictly it is a distribution rather th

Analysis

Directional Derivative · The slope in any direction — the gradient is the special vector that bundles them all

The directional derivative D_v(f) of a multivariable function f at a point gives the rate of change of f as you move in the direction of the unit vect

Calculus

Dirichlet Characters and Group Characters mod n

Dirichlet characters mod n explained: precise definition, orthogonality relations, the character group of (ℤ/nℤ)ˣ, L-functions, and Dirichlet's theore

Analytic Number Theory

Dirichlet Distribution · Multivariate Beta on the simplex — distribution over probability vectors

The Dirichlet distribution Dir(α₁, …, α_k) is the multivariate Beta — a distribution over probability vectors (p₁, …, p_k) summing to 1. Conjugate pri

Probability

Dirichlet Series · Σ aₙ/n^s — generating function for arithmetic data, generalizes ζ(s) and L-functions

A Dirichlet series is an infinite series of the form Σ_{n=1}^∞ aₙ/n^s, where {aₙ} is a sequence of complex numbers and s is a complex variable. Conver

Analytic Number Theory

Dirichlet's Theorem on Primes in Arithmetic Progressions

Dirichlet's theorem: every arithmetic progression a, a+q, a+2q,… with gcd(a,q)=1 contains infinitely many primes. Precise statement, proof idea via L-

Analytic Number Theory

Divergence (Vector Calculus) · The flux per unit volume — how much a field is spreading outwards from each point

The divergence of a vector field F = (F_x, F_y, F_z) is the scalar field ∇·F = ∂F_x/∂x + ∂F_y/∂y + ∂F_z/∂z. Geometrically, it is the limit of flux per

Calculus

Divergence Theorem (Gauss) · Outward flux through a closed surface = volume integral of divergence

The divergence theorem says outward flux through a closed surface equals the integral of divergence over the enclosed volume: ∯_∂V F·dS = ∭_V ∇·F dV.

Calculus

Dominated Convergence Theorem · Swap limit and integral whenever one fixed integrable function dominates the sequence

If fₙ → f pointwise and |fₙ| ≤ g with ∫g < ∞, then ∫fₙ → ∫f. Lebesgue's dominated convergence theorem justifies swapping limit and integral whenever a

Real Analysis

Dot Product · A single number that captures how much two vectors agree

The dot product takes two vectors and returns a single number that measures how much they point in the same direction. For vectors a and b in any dime

Linear Algebra

Dual Space · V = Hom(V, F) — the set of all linear functionals; same dimension as V in finite-dim*

The dual space V* of a vector space V over a field F is the set of all linear functionals — linear maps V → F. V* is itself a vector space; in finite

Linear Algebra

Egorov's Theorem · Almost Uniform Convergence

Egorov's theorem: on a finite measure space, a.e. pointwise convergence becomes uniform off a set of arbitrarily small measure. Precise statement, pro

Measure Theory

Eigenvalues & Eigenvectors · Av = λv

Eigenvectors are the directions a matrix stretches without rotating. Their scaling factors are eigenvalues. PageRank, PCA, quantum mechanics all rest

Linear Algebra

Eisenstein's Criterion · If p divides every coefficient except the leading one — irreducible over ℚ

Eisenstein's criterion (Gotthold Eisenstein, 1850) is a sufficient condition for a polynomial in ℤ[x] to be irreducible over ℚ. For f(x) = aₙxⁿ + aₙ₋₁

Abstract Algebra

Ergodic Theorem · For an ergodic system, time average equals space average — Birkhoff 1931

The ergodic theorem says that for an ergodic process, the time average of a function equals its space average — integrating against the stationary dis

Stochastic Processes

Euler Characteristic · χ(X) = V − E + F (vertices − edges + faces) — topological invariant for any cell decomposition

The Euler characteristic χ(X) is the alternating sum of cell counts in any CW-decomposition of a space X: χ = c₀ − c₁ + c₂ − …, where cₖ is the number

Topology

Euler's Identity · e^(iπ) + 1 = 0 · five constants · complex plane

Euler's identity e^(iπ) + 1 = 0 links five fundamental constants in one equation. Show the complex plane with the unit circle, Euler's formula e^(iθ)

Analysis

Euler's Totient Function · φ(n) counts integers from 1 to n coprime to n — the multiplicative function that powers RSA

Euler's totient function φ(n) counts how many positive integers up to n share no common factor with n. The closed form φ(n) = n × ∏(1 − 1/p) factors o

Number Theory

Expected Value · E[X] = Σ x

The long-run average of a random variable — weighted sum over outcomes and probabilities. Drives gambling, insurance, investment, and statistical esti

Probability

Exponential Growth · y = A

Constant doubling time makes exponential growth explode after seeming slow. Compound interest, pandemics, Moore's Law all follow this pattern. The bra

Algebra

Farkas' Lemma · The Theorem of the Alternative

Farkas' Lemma states that exactly one of two linear systems is solvable, giving a certificate of infeasibility. Precise statement, proof idea, geometr

Convex Analysis & Duality

Fast Fourier Transform · Cooley-Tukey 1965 cut DFT cost from O(n²) to O(n log n) — the algorithm that makes MP3, JPEG, and 5G possible

The Fast Fourier Transform computes the discrete Fourier transform of an N-sample signal in O(N log N) operations instead of the naïve O(N²). At N = 1

Numerical Analysis

Fermat's Last Theorem · aⁿ + bⁿ = cⁿ has no positive-integer solutions for n > 2 — and the margin was lying

No positive integers a, b, c satisfy aⁿ+bⁿ=cⁿ for n > 2. Stated 1637 by Fermat in a margin; proved 357 years later by Andrew Wiles in 1994 via modular

Number Theory

Fermat's Little Theorem · If p is prime and a is not divisible by p, then a^(p−1) ≡ 1 (mod p) — the bedrock of RSA primality testing

For any prime p and integer a coprime to p, raising a to the (p − 1) power leaves a remainder of 1 modulo p. The 1640 identity is the foundation of fa

Number Theory

Fibonacci Sequence · Golden Ratio

The Fibonacci sequence visualized — watch 1, 1, 2, 3, 5, 8, 13, 21 build into the golden spiral. See how the ratio converges to phi (1.618).

Number Theory

Field Extension · K ⊆ L makes L a vector space over K — degree [L:K] measures size

A field extension L/K is an inclusion K ⊆ L of fields. L becomes a vector space over K, with degree [L:K] = dim_K L (the dimension as a K-vector space

Abstract Algebra

Finite Difference Method · Approximate derivatives by discrete differences — the foundation of every PDE numerical solver and every numerical derivative your code has computed

The finite difference method approximates derivatives by differences of function values on a grid. Forward difference (f(x+h)−f(x))/h is O(h); central

Numerical Analysis

Finite Element Method · Mesh the PDE domain into elements, approximate the solution as a sum of basis coefficients, solve the Galerkin linear system — the algorithm behind every engineering simulation

The finite element method discretises a PDE domain into a mesh of elements (triangles or tetrahedra) and expresses the solution as a sum of coefficien

Numerical Methods

First-Order Differential Equations · Five types, five recipes — identify, then solve

A first-order ODE relates a function y(x) to its first derivative y'(x) but no higher. Five canonical types — separable, linear, exact, Bernoulli, and

Differential Equations

Fisher Information · The curvature of the log-likelihood at the truth — and the variance floor it sets on every estimator

Fisher information I(θ) = E[(∂ log f / ∂θ)²] = −E[∂² log f / ∂θ²] measures how much data tell you about a parameter. Sets the Cramér-Rao bound Var(θ̂)

Statistical Inference

Fixed-Point Iteration · Solve x = g(x) by repeatedly applying g — the foundation of nearly every iterative method in numerical analysis

Fixed-point iteration solves x = g(x) by repeatedly applying x_{n+1} = g(x_n). Converges if |g'(x*)| < 1 (contraction) at the fixed point; rate of con

Numerical Methods

Fourier Series · sine waves · any periodic function · frequency spectrum

Any periodic function can be built by adding sine and cosine waves of different frequencies. Show a square wave approximated by its first harmonic, th

Analysis

Fubini's Theorem · When iterated integrals equal the double integral — and when they don't

Fubini's theorem: if ∫|f| dxdy is finite, the double integral equals either iterated integral, and the order can be swapped. Counterexamples when ∫|f|

Measure Theory

Fundamental Group · π_1(X) classifies loops in a space up to continuous deformation — the algebraic invariant that distinguishes a sphere from a torus

The fundamental group of a topological space is the algebraic record of its loops: the set of inequivalent ways you can walk around in circles without

Topology

Fundamental Theorem of Calculus · Differentiation & Integration are Inverses

Area under a curve equals the antiderivative evaluated at endpoints: ∫ₐᵇ f(x) dx = F(b) − F(a). The theorem that unifies calculus.

Calculus

GCD & Euclidean Algorithm · Greatest Common Divisor

Euclid's 2300-year-old algorithm finds GCD by repeated remainders. Still the algorithm every modern cryptography library uses. A geometric interpretat

Number Theory

GMRES Method · Generalised Minimal Residual — solve Ax = b for non-symmetric A by minimising the residual over an Arnoldi Krylov subspace

GMRES solves Ax = b for non-symmetric A by building a Krylov subspace via Arnoldi iteration, then choosing the iterate that minimises ‖b − Ax‖ over th

Iterative Methods

Galois Theory · The duel-aged genius proved degree-5 polynomials have no general radical formula — by trading polynomial roots for groups

Galois theory translates polynomial equations into group theory. The roots of a polynomial are solvable in radicals exactly when the Galois group of i

Abstract Algebra

Gamma Distribution · The waiting time for the α-th event in a Poisson process

The Gamma distribution Gamma(α, β): waiting time for the α-th event in a Poisson process. Generalizes the exponential (α = 1) and reduces to chi-squar

Probability

Gamma Function · Γ(z) = ∫₀^∞ t^(z−1) e^(−t) dt — factorial, continued to every complex number

The gamma function Γ(z) = ∫₀^∞ t^(z−1) e^(−t) dt extends factorial to all complex numbers (except non-positive integers). Γ(n) = (n−1)! for positive i

Special Functions

Gauss Quadrature · Sample at orthogonal-polynomial roots — exact for polynomials up to degree 2n − 1

Gauss quadrature approximates ∫f(x)w(x)dx by Σ w_i f(x_i) using roots of orthogonal polynomials. Exact for polynomials up to degree 2n − 1 — the optim

Numerical Analysis

Gauss-Bonnet Theorem · ∫∫ K dA + ∫ kg ds = 2πχ(M) — local geometry constrained by global topology

The Gauss-Bonnet theorem is one of the most beautiful results in mathematics: for a compact 2-manifold M with (possibly empty) boundary, ∫∫_M K dA + ∮

Differential Geometry

Gauss-Seidel and Jacobi Iteration · Splitting a Matrix to Solve It

Jacobi and Gauss-Seidel iteration explained: matrix splitting A = M − N, the iteration matrix, the spectral-radius ρ(M) < 1 convergence theorem, diago

Iterative solvers

Gaussian Curvature · K = κ₁κ₂ — the intrinsic curvature of a surface, invariant under isometric deformation

Gaussian curvature K = κ₁κ₂ is the product of the two principal curvatures of a surface. Gauss's Theorema Egregium (1827) says K is intrinsic — comput

Differential Geometry

Gaussian Elimination · Sweep variables out of a linear system, one column at a time

Gaussian elimination reduces a system of linear equations to upper-triangular form by sweeping out variables column by column. With partial pivoting i

Linear Algebra

Geodesics · The Straightest Possible Paths on a Curved Surface

Geodesics explained: the geodesic equation γ̈ᵏ + Γᵏᵢⱼγ̇ⁱγ̇ʲ = 0, why they locally minimize length, the Hopf–Rinow theorem, worked sphere example, and

Differential Geometry

Gershgorin's Circle Theorem · Trapping Eigenvalues in Discs

Gershgorin's Circle Theorem localizes every eigenvalue of a matrix inside discs centered on the diagonal. Precise statement, proof idea, worked exampl

Spectral theory & eigenvalue localization

Girsanov's Theorem · Changing the Drift of Brownian Motion

Girsanov's theorem, stated precisely with the Novikov condition: how the stochastic exponential ℰ(−∫θ·dW) defines an equivalent measure that removes d

Probability & Statistics

Golden Ratio · φ ≈ 1.618 · Fibonacci · spirals in nature

The golden ratio φ ≈ 1.618 appears when a line is divided so the whole-to-large ratio equals the large-to-small ratio. Show the golden rectangle subdi

Number Theory

Gradient · Direction of Steepest Ascent

∇f is a vector pointing uphill on a multivariable function. Gradient descent — negative gradient direction — is the optimization engine of modern mach

Calculus

Gram-Schmidt Process · Turn any tilted basis into a perpendicular one, one projection at a time

The Gram-Schmidt process turns any linearly independent set of vectors into an orthonormal basis spanning the same subspace, by repeatedly projecting

Linear Algebra

Graph Coloring · Four-Color Theorem

Color a map so no two adjacent regions share a color. The Four-Color Theorem: four colors always suffice. Proved by computer in 1976 — the first major

Graph Theory

Green's Function · The inverse of a linear differential operator — point-source response, integrated

A Green's function G(x, x′) is the response of a linear differential operator L to a unit point source: L G = δ(x − x′). For any source f, the inhomog

PDEs

Green's Theorem · Boundary loop = sum of curl in the interior

Green's theorem ties a line integral around a closed plane curve to a double integral over the region it encloses: ∮(P dx + Q dy) = ∬(∂Q/∂x − ∂P/∂y) d

Calculus

Group Theory · Symmetry Made Precise

A set with a composition operation that obeys closure, identity, inverse, associativity. Rubik's cube, triangle rotations, and even the Standard Model

Abstract Algebra

Gröbner Bases · Solving Polynomial Systems by Reduction

Gröbner bases turn ideal membership and polynomial-system solving into a terminating computation. Precise definition, Buchberger's criterion and algor

Commutative Algebra

Gödel's Completeness Theorem · Provable Equals True in Every Model

Gödel's Completeness Theorem states T ⊨ φ iff T ⊢ φ for first-order logic. Precise statement, Henkin's proof, examples, and how it differs from incomp

Proof Theory

Gödel's Incompleteness Theorems · Any consistent formal system rich enough to do arithmetic contains true statements it cannot prove — and cannot prove its own consistency

In 1931, twenty-five-year-old Kurt Gödel constructed a sentence in the language of arithmetic that effectively said "I am not provable" — and proved i

Mathematical Logic

Hahn-Banach Theorem · Every bounded linear functional on a subspace extends to the whole space — without inflating the norm

The Hahn-Banach theorem says every continuous linear functional on a subspace extends to the whole space without enlarging its norm. The cornerstone o

Functional Analysis

Hall's Marriage Theorem · When a Perfect Matching Exists

Hall's Marriage Theorem states a bipartite graph has a perfect matching iff |N(S)| ≥ |S| for every subset S. Statement, proof idea, worked example, an

Graph Theory

Harnack's Inequality · Positive Solutions Can't Vary Too Wildly

Harnack's inequality states that a nonnegative harmonic (or elliptic/parabolic) solution satisfies sup_K u ≤ C·inf_K u with C independent of u. Precis

Partial Differential Equations

Heat Equation · ∂u/∂t = α∇²u — the canonical parabolic PDE behind every diffusion process

The heat equation ∂u/∂t = α∇²u is the canonical linear parabolic PDE — it governs heat diffusion, Brownian motion, image blur, and chemical mixing. Th

Partial Differential Equations

Heine-Borel Theorem · A subset of ℝⁿ is compact if and only if it is closed and bounded

The Heine-Borel theorem: a subset K of ℝⁿ (with the standard Euclidean metric) is compact if and only if K is both closed and bounded. Compact means e

Real Analysis

Hermite Polynomials · Orthogonal on ℝ with e^{−x²}; the quantum harmonic oscillator's eigenfunctions

Hermite polynomials H_n(x) are orthogonal on ℝ with weight e^{−x²}. They give the quantum harmonic oscillator's eigenstates ψ_n(x) ∝ H_n(x) e^{−x²/2},

Special Functions

Hermitian Matrix · A = A — self-adjoint, real-eigenvalued, the algebra of measurement*

A Hermitian matrix equals its own conjugate transpose: A = A*. All eigenvalues are real, and eigenvectors of distinct eigenvalues are orthogonal. The

Linear Algebra

Hilbert Space · A complete inner product space — ℓ², L², generalized Euclidean space for quantum mechanics

A Hilbert space H is a vector space (over ℝ or ℂ) equipped with an inner product ⟨·, ·⟩ such that the induced norm ‖x‖ = √⟨x, x⟩ makes H complete (eve

Functional Analysis

Hilbert's Nullstellensatz · The Dictionary Between Ideals and Varieties

Hilbert's Nullstellensatz precisely stated and explained: the weak and strong forms, I(V(I)) = √I over an algebraically closed field, the key proof id

Algebraic Geometry

Holonomy · How a Vector Rotates When Carried Around a Loop

Holonomy explained: how parallel transport around a loop rotates a vector, the holonomy group, the Ambrose-Singer theorem linking it to curvature, and

Riemannian Geometry

Homology Groups · Hₙ(X) measures n-dimensional holes — Betti numbers, Euler characteristic

Homology groups Hₙ(X) are abelian groups associated to a topological space X that algebraically capture its n-dimensional "holes" — connected componen

Algebraic Topology

Homotopy · Continuous deformation of maps — the equivalence relation algebraic topology was built around

A homotopy between continuous maps f, g : X → Y is a continuous H : X × [0,1] → Y with H(x, 0) = f(x), H(x, 1) = g(x). Homotopy is an equivalence rela

Topology

Hyperbolic Geometry · Negative-curvature world — Bolyai and Lobachevsky 1830, formalized by Beltrami's models

Hyperbolic geometry is the non-Euclidean geometry obtained by replacing Euclid's parallel postulate with: through any point not on a given line, there

Non-Euclidean Geometry

Hypergeometric Distribution · Counting successes when sampling without replacement from a finite population

The hypergeometric distribution counts successes when sampling n items WITHOUT replacement from a finite population of N items containing K successes.

Probability

Hypothesis Testing · Null vs Alternative

Collect data, compute a test statistic, calculate the p-value. If p < 0.05, reject the null hypothesis. The foundation of experimental science.

Statistics

Hölder's Inequality · ∫|fg| ≤ ‖f‖_p · ‖g‖_q whenever 1/p + 1/q = 1

Hölder's inequality: ∫|fg| ≤ (∫|f|^p)^(1/p) · (∫|g|^q)^(1/q) whenever 1/p + 1/q = 1. Generalizes Cauchy-Schwarz (p = q = 2). Foundation of L^p spaces

Inequalities

Ideals in Rings · I ⊆ R closed under addition and absorbs R-multiplication — kernels of homomorphisms

An ideal of a ring R is a subset I ⊆ R such that I is closed under addition (a − b ∈ I whenever a, b ∈ I) and "absorbs" multiplication by R (rI ⊆ I an

Abstract Algebra

Inclusion-Exclusion Principle · Add the parts, subtract the overlaps, add back the triple overlaps — counting unions of sets without double-counting

When you count a union of overlapping sets by adding their individual sizes, the overlaps get counted twice. The inclusion-exclusion principle correct

Combinatorics

Inner Product Space · Where geometry comes from algebra

An inner product space is a vector space equipped with an inner product ⟨·,·⟩ — a bilinear (or sesquilinear), symmetric, positive-definite pairing of

Linear Algebra

Integration by Parts · ∫u dv = uv − ∫v du

The product rule in reverse. Choose u and dv cleverly (LIATE order) to simplify the integral. Essential for products of functions — physics, signal an

Calculus

Interior-Point Method · Solve constrained optimization by adding a barrier — Karmarkar's 1984 polynomial-time LP algorithm

Interior-point methods add a logarithmic barrier to a constrained optimization problem, replacing inequality constraints with a penalty that diverges

Optimization

Intermediate Value Theorem · A continuous function never skips a value

The Intermediate Value Theorem says that a continuous function on [a,b] takes every value between f(a) and f(b). It guarantees roots, fixed points and

Analysis

Inversive Geometry · f(z) = r²/(z̄ − c̄) — inversion in a circle maps circles+lines to circles+lines

Inversion in a circle of center O and radius r is the map sending each point P (other than O) to the point P' on ray OP with OP · OP' = r². In complex

Geometry

Itô's Lemma · The Chain Rule for Random Motion

Itô's Lemma stated precisely with hypotheses and proof idea: the stochastic chain rule df = f′dX + ½f″σ²dt, why the second-order term appears, worked

Probability & Statistics

Jacobian Change of Variables · The local stretch factor of a coordinate map — and the missing piece of multivariable substitution

The Jacobian determinant |J| measures how a coordinate transformation stretches local area (in 2D) or volume (in 3D). The change-of-variables formula

Calculus

Jensen's Inequality · Convex function of a mean is at most the mean of the function — the chord-above-curve fact, applied to probability distributions

Jensen's inequality says f(E[X]) ≤ E[f(X)] for any convex function f. The concave version flips the sign. It is the engine behind every probability bo

Inequalities

Jordan Canonical Form · Every matrix over ℂ is similar to a block-diagonal matrix of Jordan blocks

The Jordan canonical form (JCF) is the unique (up to block ordering) representation of any square matrix A over an algebraically closed field as a sim

Linear Algebra

Kan Extensions · The Universal Way to Extend a Functor

Kan extensions explained: the precise universal property, the pointwise colimit/limit formula, the proof via comma categories and the coend, worked ex

Category Theory & Homological Algebra

Karush-Kuhn-Tucker Conditions · Four equations characterise every constrained optimum — necessary always, sufficient under convexity

The KKT conditions characterise the optimum of an inequality-constrained nonlinear program: stationarity of the Lagrangian, primal feasibility, dual f

Optimization

Kleene's Recursion Theorem · Programs That Read Their Own Source Code

Kleene's Recursion Theorem states that for any total computable f there is a fixed point e with φ_e = φ_{f(e)}. Precise statement, s-m-n proof, quines

Logic & Computability

Klein Bottle · A non-orientable closed surface with no inside and no outside — like a Möbius strip with the boundary glued to itself

The Klein bottle is the canonical non-orientable closed surface: a 2D world that wraps around so completely that the very notion of "inside" and "outs

Topology

Kronecker Product · Tile every entry of A with a scaled copy of B

The Kronecker product A⊗B replaces every entry a_ij of A by the scaled block a_ij·B. An m×p matrix kron a n×q matrix gives an mn×pq matrix. The mixed-

Linear Algebra

Kullback-Leibler Divergence · The extra bits you pay when you code samples from P using a code built for Q

KL divergence D(P‖Q) = Σ P(x) log(P(x)/Q(x)) measures how many extra bits you spend coding samples from P with a code optimized for Q. Zero iff P = Q,

Information Theory

Kőnig's Theorem · Matchings and Vertex Covers in Bipartite Graphs

Kőnig's Theorem states that in any finite bipartite graph the maximum matching equals the minimum vertex cover (ν = τ). Statement, proof idea, example

Graph Theory

L'Hôpital's Rule · Differentiating top and bottom to crack 0/0 limits

L'Hôpital's rule evaluates indeterminate limits of the form 0/0 or ∞/∞ by replacing f(x)/g(x) with f′(x)/g′(x). The other indeterminate forms — 0·∞, ∞

Calculus

LU Decomposition · Factor a matrix once, solve many systems cheaply

LU decomposition writes a square matrix as A = L·U, the product of a lower-triangular L and an upper-triangular U. Once factored, every subsequent sol

Linear Algebra

Lagrange Multipliers · Constrained Optimization

Find the max or min of f subject to constraint g = 0. At the optimum, gradients align: ∇f = λ∇g. Essential from economics to machine learning.

Calculus

Lagrange's Theorem · Why the Order of a Subgroup Always Divides the Order of the

Lagrange's Theorem states the order of a subgroup H divides the order of a finite group G. Precise statement, coset proof, worked example, and why it

Group theory

Lanczos Algorithm · Three-term recurrence that tridiagonalises symmetric A with O(N) per iteration

The Lanczos algorithm is Arnoldi specialised to symmetric (or Hermitian) matrices. The Hessenberg projection collapses to a tridiagonal T_k, and Gram–

Iterative Methods

Laplace Equation · ∇²φ = 0 — the equation of equilibrium and the harmonic functions that solve it

The Laplace equation ∇²φ = 0 governs every steady-state diffusion: equilibrium temperature, electrostatic potential in vacuum, gravity in empty space.

Partial Differential Equations

Laplace Transform · Trade calculus for algebra in the s-domain

The Laplace transform converts a function of time t into a function of complex frequency s, turning differential equations into algebraic ones. It's t

Differential Equations

Law of Cosines · c² = a² + b² − 2ab cos(C)

The Pythagorean theorem generalized to any triangle. When C = 90°, cos = 0 and the extra term vanishes. Non-right angles bend the formula predictably.

Trigonometry

Law of Large Numbers · (X̄ₙ) → μ as n → ∞: weak (in probability), strong (almost surely)

The Law of Large Numbers (LLN) states that the sample average X̄ₙ = (X₁ + … + Xₙ)/n of i.i.d. random variables with finite mean μ converges to μ as n

Probability

Lebesgue Differentiation Theorem · Averages Recover the Function

The Lebesgue Differentiation Theorem states that averaging an integrable function over shrinking balls recovers its value at almost every point. State

Measure Theory

Lebesgue Integral · Partition the codomain, not the domain — handles characteristic function of ℚ

The Lebesgue integral, introduced by Henri Lebesgue in his 1902 PhD thesis, generalizes the Riemann integral by partitioning the function's range

Measure Theory

Legendre Polynomials · Orthogonal polynomials on [−1, 1]; the m = 0 spherical harmonics; the Gaussian-quadrature nodes

Legendre polynomials P_n(x) are orthogonal polynomials on [−1, 1] with weight 1: ∫_{−1}^{1} P_n P_m dx = 0 for n ≠ m. P_n(cos θ) is the m = 0 spherica

Special Functions

Legendre Symbol · (a|p) = +1, -1, or 0 — tells you instantly if a is a square mod prime p

The Legendre symbol (a|p), introduced by Adrien-Marie Legendre in 1785, is defined for an odd prime p and any integer a as: +1 if a is a nonzero quadr

Number Theory

Limits (Formal Definition) · What "approaches" actually means, written down precisely

The formal ε-δ definition of a limit says limx→c f(x) = L if for every ε > 0 there exists δ > 0 such that 0 < |x − c| < δ implies |f(x) − L| < ε. The

Analysis

Line Integrals · Integrating along a curve — work, mass, circulation

A line integral integrates a function along a curve. For scalar fields it sums values weighted by arc length; for vector fields it sums F·dr — the wor

Calculus

Linear Programming Duality · Every primal LP has a dual LP — and at the optimum the two objectives coincide

LP duality pairs every primal linear program with a dual LP whose variables are shadow prices on the primal's constraints. Weak duality says the dual

Optimization

Linear Transformations · Matrices as Maps

Every linear transformation is represented by a matrix. Rotations, scalings, shears, reflections — all encoded as 2×2 or 3×3 matrices. Graphics engine

Linear Algebra

Liouville's Theorem · Why Bounded Entire Functions Are Constant

Liouville's Theorem states every bounded entire function is constant. Precise statement, the Cauchy-estimate proof, worked examples, why boundedness i

Complex Analysis

Lipschitz Continuity · |f(x) − f(y)| ≤ L|x − y| — a single slope bound holds everywhere

A function f is Lipschitz continuous with constant L if |f(x) − f(y)| ≤ L|x − y| for all x, y. Bounded-slope condition — stronger than continuous, wea

Analysis

Logarithms · Inverse of Exponentials

log asks 'what power gives this value?' log(ab) = log(a) + log(b) — multiplication becomes addition. Compresses wildly different scales: pH, decibels,

Algebra

Lyapunov Exponents · Measuring the Rate of Chaos

Lyapunov exponents measure the exponential rate at which nearby trajectories diverge in a dynamical system. Precise definition, the Oseledets Multipli

Dynamical Systems

Lyapunov Stability · An energy function that proves stability of a fixed point without solving the ODE

Lyapunov's direct method certifies stability of a fixed point without solving the ODE. Find V(x) with V(0) = 0, V > 0 elsewhere, and dV/dt ≤ 0 along t

Dynamical Systems

Mandelbrot Set · z² + c · fractal boundary · infinite complexity

The Mandelbrot set is the set of complex numbers c for which z_(n+1) = z_n² + c stays bounded when iterated from z_0 = 0. Points inside never escape;

Geometry

Manifold — Definition · Locally Euclidean, globally curved — the playground for calculus on arbitrary spaces

A manifold is a topological space that locally looks like Euclidean R^n. Smooth manifolds add compatible C^∞ charts and transition maps. Spheres, tori

Differential Geometry

Markov Chain · State Transitions

A memoryless random process — next state depends only on current. Converges to a steady-state distribution regardless of start. PageRank, weather mode

Probability

Markov's Inequality · For non-negative X, P(X ≥ a) is never more than E[X]/a

Markov's inequality: for a non-negative random variable X with finite mean, P(X ≥ a) ≤ E[X]/a. The simplest concentration bound and the building block

Probability

Martingale · E[Xₙ₊₁ | F_n] = Xₙ — your expected future given the past equals the present

A martingale is a stochastic process (Xₙ) on a filtered probability space (Ω, F, (Fₙ), P) satisfying E[Xₙ₊₁ | F_n] = Xₙ — given the entire history up

Probability

Matrix Inverse · The matrix that undoes A — when one exists

The inverse of a square matrix A is the unique matrix A⁻¹ satisfying A·A⁻¹ = A⁻¹·A = I. It exists if and only if det A ≠ 0 (equivalently, A's columns

Linear Algebra

Matrix Multiplication · Row × Column

The (i,j) entry of AB is the dot product of row i of A and column j of B. Represents composition of linear transformations. Graphics, quantum mechanic

Linear Algebra

Maximum Likelihood Estimation · Pick the parameter that makes your observed data most probable — Fisher 1922's keystone of modern statistics

You have data and a parametric model. Maximum likelihood estimation picks the single parameter value that makes your observed dataset the most probabl

Statistics

Maxwell's Equations (Mathematical Form) · Four coupled PDEs whose vacuum limit is a wave equation with c = 1/√(μ₀ε₀)

Maxwell's four PDEs — ∇·E = ρ/ε₀, ∇·B = 0, ∇×E = −∂B/∂t, ∇×B = μ₀J + μ₀ε₀ ∂E/∂t — are the foundation of classical electromagnetism. In vacuum they col

PDEs

McDiarmid's Bounded Differences Inequality

McDiarmid's bounded differences inequality: precise statement, martingale proof via Azuma-Hoeffding, worked examples, why independence and bounded dif

Concentration Inequalities

Mean Value Theorem · Somewhere on every smooth curve, the instantaneous matches the average

The Mean Value Theorem says that for a function continuous on [a,b] and differentiable on (a,b), there is a point c where the instantaneous slope f′(c

Analysis

Method of Characteristics · Turning a PDE Into ODEs Along Curves

A rigorous graduate-level explainer of the method of characteristics: how first-order PDEs reduce to ODEs along characteristic curves, the geometry of

Partial Differential Equations

Metric Space · Distance, axiomatized

A metric space is a set X equipped with a distance function d : X × X → ℝ≥0 satisfying three axioms: identity (d = 0 only at equal points), symmetry,

Topology

Metropolis-Hastings Algorithm · Sample from any distribution proportional to a known function — the MCMC workhorse

Metropolis-Hastings is the workhorse MCMC algorithm: sample from any distribution proportional to a known function by proposing moves and accepting wi

Stochastic Processes

Miller-Rabin Primality Test · Probabilistic primality with cosmic-ray-grade error in 40 rounds

Miller-Rabin is a probabilistic primality test: for a composite n, a random base catches it with probability at least 3/4. Forty rounds collapses the

Computational Number Theory

Minkowski's Inequality · ‖f + g‖_p ≤ ‖f‖_p + ‖g‖_p — the triangle inequality for L^p norms

Minkowski's inequality: ‖f + g‖_p ≤ ‖f‖_p + ‖g‖_p for any 1 ≤ p ≤ ∞. The triangle inequality for L^p norms — what makes L^p a normed space. Generalize

Inequalities

Mixing vs. Ergodicity · The Baker's Map and Decay of Correlations

A rigorous graduate-level explainer of mixing versus ergodicity, worked through the baker's map, with the spectral proof, exponential decay of correla

Ergodic Theory

Modular Arithmetic · Clock Math

Arithmetic that wraps around at n — like a clock. 7 + 8 ≡ 3 (mod 12). Powers RSA cryptography and explains calendars, music theory, and checksums.

Number Theory

Moment-Generating Function · M_X(t) = E[e^(tX)] — its k-th derivative at 0 gives E[X^k]; uniquely identifies distribution

The moment-generating function (MGF) of a random variable X is M_X(t) = E[e^(tX)] = Σ E[X^k] t^k / k!, when this expectation exists in some neighborho

Probability

Monads · Endofunctors With Unit and Multiplication

A rigorous, motivated explainer of monads in category theory: the precise definition (endofunctor T with unit η and multiplication μ), the monoid-in-e

Category Theory & Homological Algebra

Monty Hall Problem · 3 doors · always switch · 2/3 probability

Three doors: behind one is a car, behind two are goats. You pick door 1. The host (who knows what's behind each door) opens door 3 to reveal a goat. S

Probability

Moore–Penrose Pseudoinverse · The "inverse" that always exists — and always solves least squares

The Moore–Penrose pseudoinverse A⁺ generalises the inverse to non-square, rank-deficient matrices. The least-squares solution to Ax = b is x* = A⁺b —

Linear Algebra

Multigrid Methods · Killing Error at Every Scale

Multigrid methods explained: how coarse-grid correction plus smoothing solves elliptic PDEs in O(N) work with mesh-independent convergence. Statement,

Iterative solvers

Multinomial Distribution · Binomial generalized to k categories — joint count distribution

The multinomial distribution counts outcomes in n independent trials with k possible categories. PMF P(n₁, …, n_k) = n!/∏nᵢ! · ∏pᵢ^nᵢ. Generalizes the

Probability

Mutual Information · Bits of information one random variable carries about another — every kind of dependence, not just linear

Mutual information I(X; Y) = H(X) − H(X|Y) = D_KL(p(x, y)‖p(x)p(y)) is the bits one random variable reveals about another. Zero iff independent. Symme

Information Theory

Myers' Theorem · Positive Curvature Forces a Compact, Small Universe

Myers' Theorem: a complete Riemannian manifold with Ricci ≥ (n−1)k > 0 has diameter ≤ π/√k, is compact, and has finite fundamental group. Precise stat

Differential geometry

Möbius Function · μ(n) = (-1)^k if n is a product of k distinct primes, 0 if n has a squared factor

The Möbius function μ(n), introduced by August Möbius in 1832, is defined for positive integers as: μ(1) = 1; μ(n) = (-1)^k if n is a product of k dis

Number Theory

Möbius Strip · One-Sided Surface

A rectangle joined with a half-twist has only one side and one edge. Cutting it lengthwise doesn't split it — it stays as one longer loop. Topology's

Topology

Natural Transformations · Morphisms Between Functors

Natural transformations explained: the precise definition, the commuting-square (naturality) condition, worked examples like V ≅ V**, Yoneda, and why

Category Theory

Nesterov Acceleration · Why the Optimal First-Order Method Works

Nesterov's accelerated gradient achieves the optimal O(1/k²) convergence rate for smooth convex optimization. Precise statement, the estimate-sequence

Optimization

Newton's Method (Optimization) · Use 2nd-order information x_{k+1} = x_k − H⁻¹∇f for quadratic local convergence to a minimum

Newton's method for optimisation minimises a smooth function by fitting a local quadratic at each iterate and jumping to the minimum of that quadratic

Numerical Optimization

Newton-Raphson Method · Iterate x_{n+1} = x_n - f(x_n)/f'(x_n) and quadratically converge to a root — the method behind every calculator's square-root button

The Newton-Raphson method solves f(x) = 0 by drawing the tangent line at the current guess and using its x-intercept as the next guess. Near a simple

Numerical Analysis

Non-Euclidean Geometry · Triangles on Curved Surfaces

On flat space, triangle angles sum to 180°. On a sphere, more. On a saddle, less. Einstein's general relativity describes the universe as curved space

Geometry

Normal Distribution · bell curve · 68-95-99.7 rule · central limit theorem

The normal (Gaussian) distribution is the symmetric bell curve defined by mean μ and standard deviation σ. Show a Galton board where balls bounce thro

Statistics

Orthogonal Polynomials · Legendre, Hermite, Chebyshev, Laguerre — one idea, four weights

Orthogonal polynomial families satisfy ∫ p_n(x) p_m(x) w(x) dx = 0 for n ≠ m. Legendre, Hermite, Chebyshev, Laguerre — the basis of spectral methods.

Special Functions

Orthogonal Projection · The closest point in a subspace, found by dropping a perpendicular

An orthogonal projection drops a vector perpendicularly onto a line, plane, or higher-dimensional subspace, returning the closest point inside that su

Linear Algebra

Orthonormal Basis · A basis where every dot product is a coordinate read

An orthonormal basis is a set of mutually perpendicular unit vectors that span a vector space. Formally, {q₁, q₂, …, qₙ} satisfies qᵢ · qⱼ = δᵢⱼ — one

Linear Algebra

Parallel Transport and the Failure to Return Home

Parallel transport explained: the precise definition, the holonomy that measures curvature, the proof mechanism via the curvature tensor, worked examp

Riemannian Geometry

Parametric Curves · (x(t), y(t))

x and y each defined as functions of t trace out curves that can loop and cross themselves. Circle, cycloid, Lissajous — every trajectory in physics i

Geometry

Parseval's Identity · ∫|f(t)|² dt = (1/2π) ∫|F(ω)|² dω — energy preserved between time and frequency

Parseval's identity: ∫|f(t)|² dt = (1/2π) ∫|F(ω)|² dω. The Fourier transform is an isometry of L²(ℝ). Energy in the time domain equals energy in the f

Fourier Analysis

Partial Derivatives · Differentiate one variable at a time — the building block of every multivariable result

A partial derivative ∂f/∂x of a multivariable function f(x, y, ...) is the rate of change of f with respect to x while every other variable is held fi

Calculus

Partial Fractions Integration · Decompose P(x)/Q(x) into simpler fractions, integrate each piece

To integrate a rational function P(x)/Q(x), decompose it into a sum of simpler fractions — A/(x−a) terms for linear factors, (Bx+C)/(x²+bx+c) for irre

Integration

Partition Function p(n) · p(n) = number of ways to write n = a₁ + a₂ + … + aₖ with a_i ≥ a_{i+1} ≥ 1

The partition function p(n) counts the number of ways to write a positive integer n as an unordered sum of positive integers. p(0) = 1, p(1) = 1, p(2)

Combinatorics

Pascal's Triangle · binomial coefficients · C(n,k) · hidden patterns

Pascal's triangle: each number is the sum of the two directly above it. Row n gives the binomial coefficients C(n,k) — the number of ways to choose k

Combinatorics

Peano Axioms · Five axioms that uniquely characterize ℕ via successor + induction

The Peano axioms, formulated by Giuseppe Peano in 1889 (Arithmetices Principia), characterize the natural numbers ℕ from a single starting object 0 (o

Mathematical Logic

Permutations & Combinations · nPr and nCr

Permutations (ordered) count arrangements; combinations (unordered) count choices. P(n,r) = n!/(n−r)! and C(n,r) = P(n,r)/r!. Poker, lottery, and gene

Combinatorics

Pigeonhole Principle · More Items Than Boxes

If you put n+1 items into n boxes, some box has ≥ 2 items. Trivial statement, deep consequences. In any group of 13, two share a birth month. Always.

Combinatorics

Poincaré Recurrence · Why Systems Return to Where They Started

Poincaré's recurrence theorem, stated precisely for measure-preserving maps on finite measure spaces: full statement, the pigeonhole proof, worked exa

Dynamical Systems

Poisson Distribution · P(X=k) = λ^k e^(-λ)/k! — count of rare events in fixed time, from radioactive decay to Premier League goals

The Poisson distribution gives the probability of seeing exactly k events when events arrive independently at average rate λ in a fixed window. It is

Probability

Poisson Equation · ∇²φ = f — Laplace's equation with a source term, solved by Green's function

The Poisson equation ∇²φ = -ρ/ε₀ generalises Laplace by adding a source term. It governs electrostatic potential with charge density, gravity with mas

Partial Differential Equations

Poisson Summation · Where Sampling Meets Periodization

The Poisson summation formula ∑f(n) = ∑f̂(k): precise statement, hypotheses, proof via periodization and Fourier series, worked Gaussian/theta example

Harmonic Analysis

Polar Coordinates · Distance and direction instead of x and y — the right grid for anything circular

Polar coordinates locate a point by distance r from the origin and angle θ from the positive x-axis. Curves with rotational symmetry — circles, spiral

Geometry

Polynomial Roots · Fundamental Theorem of Algebra

Every polynomial of degree n has exactly n roots when complex numbers are counted. Real roots live on the real axis; complex roots come in conjugate p

Algebra

Power Iteration · Find the dominant eigenvector by repeated Av/‖Av‖ — the algorithm behind PageRank

Power iteration finds the dominant eigenvector of a matrix A by repeatedly multiplying a starting vector by A and renormalising. The iterate aligns wi

Iterative Methods

Power Series · Polynomials with infinitely many terms — the building blocks of analytic functions

A power series is an infinite polynomial ∑ aₙ(x−a)ⁿ. Inside its disk of convergence it defines an analytic function that can be differentiated, integr

Analysis

Prime Numbers · Sieve of Eratosthenes · fundamental theorem · infinitely many

Prime numbers are integers greater than 1 divisible only by 1 and themselves. Show the Sieve of Eratosthenes on a 10×10 grid: remove 1, then cross out

Number Theory

Principal Component Analysis · Find the directions a dataset stretches along, keep only the longest

Principal component analysis finds the directions of greatest variance in a dataset by computing the eigenvectors of its covariance matrix (or, equiva

Linear Algebra

Projective Geometry · Add 'points at infinity' to make every pair of lines meet

Projective geometry extends Euclidean geometry by adding points at infinity (one for each direction in the plane), so that every pair of lines — inclu

Projective Geometry

Proof by Induction · Base Case

Prove a statement for n = 1, then prove that if it holds for k it holds for k+1. Like dominoes falling, this proves the statement for every n. Backbon

Proofs

Pythagorean Theorem · a² + b² = c²

3D visualization of the Pythagorean theorem with squares built on each side of a right triangle.

Mathematics

QR Decomposition · An orthonormal basis for the columns, and a triangular record of how to get there

QR decomposition factors a matrix into an orthogonal Q and an upper-triangular R, exposing an orthonormal basis for the column space. It is the backbo

Linear Algebra

Quadratic Formula · (-b ± √(b²-4ac)) / 2a

The closed-form solution to any quadratic. Discriminant b²-4ac determines the number of real roots: positive → two, zero → one, negative → none. Every

Algebra

Quadratic Reciprocity · Gauss's Golden Theorem — predicts which numbers are squares mod prime

Quadratic reciprocity is one of the deepest theorems in elementary number theory. For distinct odd primes p and q, with the Legendre symbol (a|p) = ±1

Number Theory

Quasi-Newton (BFGS) · Approximate the Hessian from gradient differences for superlinear convergence without forming H

BFGS (Broyden–Fletcher–Goldfarb–Shanno, 1970) builds an approximate inverse Hessian directly from gradient differences via a rank-2 update. L-BFGS kee

Numerical Optimization

Quotient Group · G/N — equivalence classes (cosets) of a normal subgroup form a new group

A quotient group G/N (read "G mod N") is constructed by partitioning a group G into cosets of a normal subgroup N (a subgroup invariant under conjugat

Abstract Algebra

Radius of Convergence · How far from the centre does a power series still make sense?

The radius of convergence R of a power series ∑ aₙ(x−a)ⁿ is the half-width of the interval where the series converges absolutely. Compute it with the

Analysis

Ramsey Theory · Color any 6-vertex complete graph with 2 colors and you'll always find a monochromatic triangle

Ramsey theory is the branch of combinatorics that studies the conditions under which order must appear in sufficiently large structures. The classical

Combinatorics

Random Matrix · Symmetric matrices with random entries — eigenvalues fill a universal semicircle

A random matrix has entries drawn from a probability distribution. The empirical eigenvalue density of a symmetric N×N random matrix with i.i.d. zero-

Random Matrix Theory

Random Walk · Take a step left or right, fair coin each time — simple, but the math behind diffusion, finance, and Google's PageRank

A random walk is a path built from independent random steps. The simplest version moves +1 or −1 each second on a fair coin flip, but the same idea dr

Probability

Rank–Nullity Theorem · The conservation law every linear map obeys

For any linear map T : V → W, dim(ker T) + dim(im T) = dim V. For matrices, rank + nullity equals the number of columns. The conservation law of dimen

Linear Algebra

Ratio Test · If the terms shrink geometrically, the series converges

The ratio test decides convergence of ∑ aₙ from the asymptotic ratio L = lim |a_{n+1}/aₙ|. If L < 1 the series converges absolutely; if L > 1 it diver

Analysis

Rational Functions · Vertical & Horizontal Asymptotes

f(x) = P(x)/Q(x). Vertical asymptotes where denominator is zero; horizontal asymptotes determined by degree comparison. Essential in filter design, ra

Algebra

Related Rates · Implicit Differentiation of Time

Differentiate a relationship with respect to time to link rates of related quantities. A balloon's volume rate ties to its radius rate via dV/dt = 4πr

Calculus

Residue Theorem · A contour integral around a region equals 2πi times the sum of residues at the poles inside — turning hard real integrals into algebra

Cauchy's residue theorem says the integral of a meromorphic function around a closed contour is 2πi times the sum of the residues at the poles enclose

Complex Analysis

Rice's Theorem · Every Nontrivial Semantic Property of Programs Is Undecidable

Rice's Theorem states that every nontrivial semantic property of the partial function a program computes is undecidable. Precise statement, the reduct

Logic & Computability

Riemann Hypothesis · All non-trivial zeros of ζ(s) on the line Re(s) = 1/2 — Clay's $1M and 167 years of resistance

All non-trivial zeros of the Riemann zeta function lie on the critical line Re(s)=1/2. Conjectured by Riemann in 1859, verified for the first 10^13 ze

Number Theory

Riemann Sum · rectangles → integral · area under a curve · limit

A Riemann sum approximates the area under a curve by slicing it into rectangles. Show a smooth curve with 4 rectangles (rough fit), then 8, 32, 64 — t

Calculus

Riemann Zeta Function · ζ(s) = Σ 1/n^s — encodes the primes via Euler's product, hides them via the critical strip

The Riemann zeta function is defined for Re(s) > 1 by the Dirichlet series ζ(s) = Σₙ₌₁^∞ 1/n^s = 1 + 1/2^s + 1/3^s + …, and is extended by analytic

Analytic Number Theory

Riemannian Metric · g_p: TₚM × TₚM → ℝ — measures lengths, angles, and volumes on a manifold

A Riemannian metric on a smooth manifold M is a smoothly-varying inner product g_p on each tangent space TₚM. In local coordinates, g is represented b

Differential Geometry

Ring Theory · An abelian group under +, a monoid under ×, with × distributing over +

A ring is an algebraic structure (R, +, ×) where (R, +) is an abelian group, (R, ×) is a monoid (or semigroup, depending on convention), and multiplic

Abstract Algebra

Rodrigues' Formula · Building Orthogonal Polynomials by Repeated Differentiation

Rodrigues' formula generates classical orthogonal polynomials (Legendre, Hermite, Laguerre, Jacobi) by repeated differentiation of a weight. Precise s

Special Functions

Roots of Unity · The n complex solutions to z^n = 1 — vertices of a regular polygon, generators of cyclic symmetry, engine of the FFT

The n-th roots of unity are the n complex solutions to z^n = 1. They sit at the vertices of a regular n-gon inscribed in the unit circle, form a cycli

Complex Analysis

Rouché's Theorem and the Argument Principle · Counting Zeros by Winding

Rouché's Theorem and the Argument Principle explained: count zeros of analytic functions by winding number, with precise statements, proof idea, worke

Complex Analysis

Runge-Kutta Methods · RK4 evaluates the slope four times per step and ranks fourth-order — the workhorse for solving ODEs you can't solve in closed form

Most differential equations describing the real world have no closed-form solution — you have to integrate them numerically. Runge-Kutta methods sampl

Numerical Analysis

Russell's Paradox · Let R = {x : x ∉ x}. Is R ∈ R? Either answer leads to contradiction

Russell's Paradox, discovered by Bertrand Russell in 1901 (independently by Ernst Zermelo earlier), asks: consider R = {x : x ∉ x}, the set of all set

Mathematical Logic

Secant Method · Root finding by two-point linear approximation — superlinear convergence at golden-ratio order without needing a derivative

The secant method finds a root of f(x) = 0 by repeatedly drawing the secant line through the last two iterates and taking its x-intercept as the next

Numerical Analysis

Second-Order Differential Equations · From the harmonic oscillator to resonance — three cases of the characteristic equation

A second-order linear ODE has the form ay'' + by' + cy = f(x). The characteristic equation ar² + br + c = 0 splits into three cases — real distinct ro

Differential Equations

Separation of Variables · Move y to one side, x to the other, integrate — the first ODE technique

Solve a first-order ODE dy/dx = f(x)g(y) by moving y to one side and x to the other: ∫ dy/g(y) = ∫ f(x) dx. Works iff the right-hand side factors. Als

Differential Equations

Series Convergence Tests · Six tools to decide if an infinite sum is finite

A series ∑ aₙ converges when its partial sums approach a finite limit. Convergence tests — integral, comparison, ratio, root, alternating, p-series —

Analysis

Shannon Entropy · Average information per symbol — the 1948 number that built the digital age

Shannon entropy H(X) = −Σ p(x) log p(x) measures the average information content of a random variable. Maximum at uniform; zero at deterministic. Foun

Information Theory

Sherman–Morrison Formula · Update a matrix inverse after a rank-1 change in O(n²) instead of O(n³)

Sherman–Morrison rewrites (A + uvᵀ)⁻¹ in terms of A⁻¹ as A⁻¹ − (A⁻¹uvᵀA⁻¹)/(1 + vᵀA⁻¹u). Updating an inverse after a rank-1 change costs O(n²) instead

Linear Algebra

Sieve of Eratosthenes · Strike out the composites; survivors are prime — 2200-year-old still the fastest enumerator

Find every prime up to N by striking out composites. For each prime p ≤ √N, cross out multiples 2p, 3p, .... Survivors are primes. O(N log log N) — fa

Computational Number Theory

Sigma-Algebra · A collection of subsets closed under complement, countable union, and countable intersection

A sigma-algebra (σ-algebra) F on a set Ω is a collection of subsets of Ω satisfying: (1) Ω ∈ F, (2) if A ∈ F then Aᶜ ∈ F (closed under complement), (3

Measure Theory

Simplex Method · Dantzig 1947 — pivot from vertex to vertex of the feasible polytope, climbing toward the optimum

A linear program asks for the largest c·x over a region carved out by linear constraints. The simplex method walks the vertices of that region: at eac

Linear Programming

Simpson's Rule · Fit a parabola through three points, integrate it — fourth-order accurate in h

Simpson's rule approximates ∫f(x)dx via parabolas through 3 points: (h/3)(f(a) + 4f(c) + f(b)). Error O(h⁵·f⁽⁴⁾), fourth-order accurate.

Numerical Analysis

Singular Value Decomposition · Every matrix is a rotation, a stretch along axes, then another rotation

The singular value decomposition writes any matrix A as A = U·Σ·VT — an orthogonal rotation, a non-negative diagonal stretch, and another orthogonal r

Linear Algebra

Spectral Theorem · Every Hermitian (or real symmetric) matrix has an orthonormal eigenbasis with real eigenvalues

The spectral theorem: every Hermitian matrix H ∈ ℂ^(n×n) (i.e. H = H*) has an orthonormal basis of eigenvectors with real eigenvalues — equivalently,

Linear Algebra

Spherical Coordinates · Radius, azimuth, and polar angle — the natural coordinates for anything centred on a point

Spherical coordinates (ρ, θ, φ) describe a point in three-dimensional space by its distance from the origin, its azimuthal angle around the z-axis, an

Geometry

Spherical Harmonics · Yₗᵐ(θ, φ) — angular eigenfunctions of the Laplacian on the sphere

Spherical harmonics Yₗᵐ(θ, φ) are the angular eigenfunctions of the Laplacian on the sphere — an orthonormal basis for square-integrable functions on

Special Functions

Squeeze Theorem · Two converging bounds force the function between them to the same limit

The squeeze (sandwich) theorem: if g(x) ≤ f(x) ≤ h(x) near a and g→L, h→L, then f→L. The standard tool when direct substitution and L'Hôpital fail — u

Calculus

Steepest Descent · Move in the direction of −∇f at every step — the simplest first-order optimisation and the textbook starting point

Steepest descent (gradient descent with exact line search) iterates x_{k+1} = x_k − α_k ∇f(x_k) where α_k is the step that minimises f along the negat

Optimization

Stereographic Projection · Conformal map from sphere minus a point onto Euclidean space — circles go to circles or lines

Stereographic projection maps a sphere minus one point onto Euclidean space. The map is conformal (angle-preserving) and sends circles to circles or l

Differential Geometry

Stirling Numbers · S(n,k) — partitions of n into k blocks; c(n,k) — permutations of n with k cycles

Stirling numbers, named after James Stirling (1730), come in two kinds. Stirling numbers of the second kind S(n, k) count the number of ways to partit

Combinatorics

Stirling's Approximation · n! ≈ √(2πn)(n/e)ⁿ — relative error 0.83% at n = 10, 0.083% at n = 100

Stirling's approximation gives n! ≈ √(2πn)(n/e)ⁿ — an asymptotic formula whose relative error drops below 1% by n = 10 and below 0.1% by n = 100. Foun

Asymptotic Analysis

Stokes' Theorem · Surface integral of curl = line integral around the rim

Stokes' theorem says the surface integral of curl equals the line integral around the boundary: ∬_S (∇×F)·dS = ∮_∂S F·dr. It generalises Green's theor

Calculus

Stone-Weierstrass Theorem · Any continuous function on a compact space is uniformly approximated by a subalgebra that separates points and contains constants

The Stone-Weierstrass theorem says any continuous function on a compact metric space is uniformly approximated by elements of a subalgebra that separa

Functional Analysis

Student's t-Distribution · Small samples + unknown variance → heavier tails than Normal

Student's t-distribution: the distribution of (X̄ − μ)/(s/√n) when sample size is small and population variance is unknown. Heavier tails than Normal.

Statistics

Sturm-Liouville Theory · The Master Eigenvalue Problem Behind Special Functions

Sturm-Liouville theory explained: the self-adjoint eigenvalue problem −(pu′)′ + qu = λwu whose eigenfunctions form a complete orthonormal basis of L²(

Differential Equations

Successive Over-Relaxation · Tuning ω for Faster Convergence

Successive Over-Relaxation (SOR) explained: the iteration, the optimal ω = 2/(1+√(1−μ²)) formula, Ostrowski–Reich and Kahan convergence theorems, and

Iterative solvers

Surface Integrals · Sum a field over a curved sheet — or measure how much flows through it

A surface integral generalizes the line integral to two-dimensional surfaces in 3D space. The scalar version ∬_S f dS sums values like density or temp

Calculus

Szemerédi's Regularity Lemma · Every Graph Looks Random in Blocks

Szemerédi's Regularity Lemma states every graph partitions into boundedly many blocks that look random between them. Precise statement, ε-regularity,

Combinatorics

Tangent Space · TₚM is the n-dimensional vector space of velocity vectors of curves through p

The tangent space TₚM at a point p of an n-dimensional smooth manifold M is the n-dimensional vector space that "best approximates" M near p. Three eq

Differential Geometry

Taylor Series · Approximating Functions with Polynomials

Any smooth function can be expanded as a polynomial near a point. Each added term expands the accurate region. Calculators, simulations, and numerical

Calculus

The Adjoint Representation · A Lie Algebra Acting on Itself

The adjoint representation ad_X(Y)=[X,Y] and Ad_g of conjugation: precise definitions, the Jacobi identity as a representation, the Killing form, root

Representation Theory

The Azuma-Hoeffding Inequality · Concentration for Martingales

The Azuma-Hoeffding inequality gives Gaussian tail bounds for martingales with bounded increments. Precise statement, proof mechanism, worked example,

Concentration Inequalities

The Baker–Campbell–Hausdorff Formula · When exp(X)exp(Y) Isn't exp(X+Y)

The Baker–Campbell–Hausdorff formula expresses log(exp X exp Y) as a Lie series in nested commutators of X and Y. Precise statement, Dynkin's explicit

Lie Theory

The Bochner Technique · How Ricci Curvature Kills Harmonic Forms

Bochner's theorem: on a compact manifold with positive Ricci curvature there are no nonzero harmonic 1-forms, so b₁ = 0. Precise statement, the Bochne

Differential geometry

The Cartan–Hadamard Theorem · Nonpositive Curvature and the Exponential Map

The Cartan–Hadamard theorem: a complete, simply connected manifold of nonpositive sectional curvature is diffeomorphic to ℝⁿ via the exponential map.

Differential geometry

The Chebotarev Density Theorem · How Primes Split

The Chebotarev Density Theorem states that Frobenius elements of primes equidistribute over conjugacy classes of a Galois group with density |C|/|G|.

Number Theory

The Chernoff Bound · Exponential Tail Estimates

The Chernoff bound gives exponentially small tail estimates for sums of independent random variables via the moment generating function. Statement, pr

Concentration Inequalities

The Compactness Theorem · When Every Finite Piece Fits

The Compactness Theorem states a set of first-order sentences is satisfiable iff every finite subset is. Precise statement, proof idea, ultraproducts,

Model Theory

The Courant-Fischer Min-Max Theorem · Eigenvalues as Optimization Problems

The Courant-Fischer min-max theorem characterizes eigenvalues of symmetric matrices as min-max values of the Rayleigh quotient. Statement, proof idea,

Spectral theory & variational characterization

The Elliptic Curve Group Law · Adding Points on a Cubic

A rigorous, graduate-level explainer of the elliptic curve group law: the chord-and-tangent addition on a Weierstrass cubic, why it forms an abelian g

Algebraic Geometry

The Euler Product · Factoring the Zeta Function over Primes

The Euler product formula ζ(s) = ∏ₚ(1−p⁻ˢ)⁻¹ explained: precise statement, proof via unique factorization, worked example, why Re(s)>1 matters, and it

Analytic Number Theory

The Expander Mixing Lemma · Eigenvalues Force Pseudorandom Edges

The Expander Mixing Lemma states that a graph's second eigenvalue λ bounds edge deviation: |e(S,T) − (d/n)|S||T|| ≤ λ√(|S||T|). Precise statement, pro

Graph Theory

The Explicit Formula · Primes as a Sum Over Zeta Zeros

The Riemann–von Mangoldt explicit formula writes ψ(x), the prime counting function, as x minus a sum over the nontrivial zeros of the zeta function. P

Number Theory

The Exponential Map · From Lie Algebra to Lie Group

The Lie group exponential map exp: 𝔤 → G, defined precisely via one-parameter subgroups, its status as a local diffeomorphism, the matrix case, BCH,

Lie Theory

The Fenchel Conjugate · Legendre Duality for Convex Functions

The Fenchel conjugate f*(y)=sup(⟨y,x⟩−f(x)) explained: precise statement, the Fenchel–Moreau biconjugation theorem f**=f, proof idea, worked examples,

Convex Analysis & Duality

The Feynman-Kac Formula · PDEs Solved by Random Paths

The Feynman-Kac formula expresses solutions of parabolic PDEs as expectations over Brownian paths weighted by a potential. Precise statement, martinga

Probability & Statistics

The Five Lemma · When the Middle Map Must Be an Isomorphism

The Five Lemma states that in a commutative ladder of two exact sequences, if the four outer vertical maps are isomorphisms then the middle map is too

Category Theory & Homological Algebra

The Fourier Transform on L² · The Plancherel Theorem

The Plancherel theorem: the Fourier transform extends to a unitary operator on L²(ℝ) preserving the L² norm. Precise statement, proof mechanism, hypot

Harmonic Analysis

The Fredholm Alternative · Either Solve Always or Solve Almost Never

The Fredholm alternative for compact operators: precise statement, the Riesz–Schauder proof mechanism, worked integral-equation example, why compactne

Functional Analysis

The Halting Problem · Why No Program Can Predict Every Program

Turing's halting problem: the precise theorem that no algorithm decides whether an arbitrary program halts, the diagonal proof, worked example, and wh

Logic & Computability

The Heisenberg Uncertainty Principle for Functions

A rigorous graduate-level explainer of the Heisenberg uncertainty inequality for functions: precise L² statement, the integration-by-parts + Cauchy–Sc

Harmonic Analysis

The Hopf-Rinow Theorem · When Geodesics Reach Everywhere

The Hopf-Rinow theorem states that geodesic completeness, metric completeness, and the Heine-Borel property coincide on a connected Riemannian manifol

Differential geometry

The Killing Form · Detecting Semisimplicity with a Bilinear Trace

Cartan's Criterion via the Killing form B(x,y)=tr(ad x ad y): a Lie algebra is semisimple iff this trace form is nondegenerate. Precise statement, pro

Lie Theory

The Lax-Milgram Theorem · Existence From a Bounded, Coercive Form

The Lax-Milgram theorem gives existence, uniqueness, and stability for a(u,v)=f(v) whenever the bilinear form a is bounded and coercive on a Hilbert s

Partial Differential Equations

The Levi-Civita Connection · The One Compatible with the Metric

The Levi-Civita connection is the unique torsion-free, metric-compatible connection on a Riemannian manifold. Statement, Koszul-formula proof, Christo

Riemannian Geometry

The Logistic Map and the Feigenbaum Route to Chaos

The logistic map xₙ₊₁ = rxₙ(1−xₙ), its period-doubling cascade, the Feigenbaum constants δ=4.6692 and α=2.5029, universality, and Lanford's renormaliz

Chaos Theory

The Lorenz Attractor · Deterministic Chaos and the Butterfly Effect

The Lorenz attractor explained: the precise ODE system, the butterfly effect, sensitive dependence, Lyapunov exponents, the geometric Lorenz model, an

Chaos Theory

The Lovász Local Lemma · When Rare Bad Events Can All Be Avoided

The Lovász Local Lemma states that if bad events are individually rare and each depends on few others (ep(d+1)≤1), all can be simultaneously avoided.

Combinatorics

The Löwenheim-Skolem Theorem · Shrinking and Stretching Models

The Löwenheim-Skolem theorem explained: precise downward and upward statements, the Skolem hull proof, Skolem's paradox, and why first-order logic can

Model Theory

The Max-Flow Min-Cut Theorem

The Max-Flow Min-Cut Theorem: precise statement, proof via augmenting paths and residual graphs, worked example, why capacities and finiteness matter,

Graph Theory

The Maximum Modulus Principle · Peaks Live on the Boundary

The Maximum Modulus Principle explained: a non-constant holomorphic function's |f| attains its maximum only on the boundary. Precise statement, proof,

Complex Analysis

The Maximum Principle · Why Harmonic Functions Peak on the Boundary

A rigorous graduate-level explainer of the maximum principle for harmonic and subharmonic functions: precise weak and strong statements, the Hopf proo

Partial Differential Equations

The Mayer-Vietoris Sequence · Homology by Cutting and Pasting

The Mayer-Vietoris sequence explained: precise statement, proof via the snake lemma, worked computation of Hₙ(Sⁿ), why the interior-cover hypothesis m

Homology & Cohomology

The Optional Stopping Theorem · When You Can't Beat a Fair Game

The Optional Stopping Theorem states you cannot beat a fair game by clever stopping: for a martingale Xₙ and stopping time τ, 𝔼[X_τ]=𝔼[X₀] under bou

Probability & Statistics

The Orbit-Stabilizer Theorem · Counting Symmetries

The Orbit-Stabilizer Theorem states |orbit(x)| = [G:Stab(x)]. Precise statement, the bijection proof, worked cube example, hypotheses, and Burnside's

Group theory / group actions

The Oseledets Multiplicative Ergodic Theorem

The Oseledets Multiplicative Ergodic Theorem: precise statement, Lyapunov exponents, the Oseledets splitting, proof mechanism via Kingman's subadditiv

Ergodic Theory

The Perron Formula · Coefficients from Contour Integrals

Perron's formula recovers partial sums of Dirichlet-series coefficients as a contour integral (1/2πi)∫ F(s)xˢ/s ds. Precise statement, proof via the d

Number Theory

The Phase Plane · Geometry of 2D autonomous ODEs — fixed points, limit cycles, separatrices, without solving anything

The phase plane is the (x, ẋ) plot of a 2D autonomous system. Trajectories are integral curves of the vector field (f, g). Fixed-point types — node, s

Dynamical Systems

The Picard-Lindelöf Theorem · Existence and Uniqueness for ODEs

The Picard-Lindelöf theorem guarantees a unique solution to y′=f(t,y), y(t₀)=y₀ when f is Lipschitz in y. Precise statement, proof via Banach fixed po

Existence & Uniqueness

The Poincaré–Bendixson Theorem · Trapping Trajectories in the Plane

The Poincaré–Bendixson theorem explained: precise statement, proof mechanism via transversals and the Jordan curve theorem, why it forbids planar chao

Dynamical Systems

The Proximal Gradient Method · Splitting Smooth and Nonsmooth

The proximal gradient method minimizes f + g (smooth plus nonsmooth convex) by alternating a gradient step with a proximal operator. Rigorous statemen

Optimization

The Resolvent Set and Resolvent Operator · Where (T−λI)⁻¹ Lives

The resolvent set ρ(T) is where (T−λI)⁻¹ exists as a bounded operator; the resolvent R(λ,T) is analytic there. Rigorous definition, proof, and spectru

Functional Analysis

The Riesz Representation Theorem · Every Functional Is Secretly an Integral

The Riesz Representation Theorem explained: every bounded linear functional on a Hilbert space is an inner product, and every positive functional on C

Functional Analysis

The Schur Decomposition · Every Matrix Is Unitarily Triangular

The Schur decomposition A = UTU* factors any complex matrix into unitary times upper-triangular. Precise statement, proof, worked example, and why it

Matrix factorizations

The Schwarz Lemma and Automorphisms of the Disk

The Schwarz Lemma stated and proved precisely: |f(z)| ≤ |z| for self-maps of the disk fixing 0, the rigidity equality case, and how it classifies all

Complex Analysis

The Seifert-van Kampen Theorem · Gluing Fundamental Groups

The Seifert-van Kampen theorem computes the fundamental group of a space glued from two open pieces as an amalgamated free product (pushout). Statemen

Algebraic Topology

The Snake Lemma · Connecting Homology Across Exact Rows

The Snake Lemma states that three vertical maps between two short exact sequences induce a six-term exact sequence of kernels and cokernels, linked by

Category Theory & Homological Algebra

The Spectral Radius Formula · Gelfand's r(T) = lim ‖Tⁿ‖^{1/n}

Gelfand's spectral radius formula r(T) = lim ‖Tⁿ‖^{1/n} = sup|σ(T)|: precise statement, proof via the resolvent's analyticity and Hadamard, worked exa

Functional Analysis

The Subdifferential · Subgradients of Convex Functions

The subdifferential ∂f(x) of a convex function is the set of subgradients g with f(y) ≥ f(x) + ⟨g,y−x⟩. Definition, proof of existence, Fermat's rule,

Convex Analysis & Duality

The Sylow Theorems · Finding Prime-Power Subgroups

The Sylow theorems explained: precise statements, the conjugacy-and-counting proof via group actions, worked examples, and why n_p ≡ 1 (mod p) forces

Group theory

The Vitali Covering Lemma · Extracting Disjoint Balls

The Vitali Covering Lemma explained: extract a countable disjoint subfamily of balls whose 5-fold dilations still cover the union. Precise statement,

Measure Theory

The Wronskian · A 2×2 determinant that decides whether two ODE solutions span the whole solution space

The Wronskian W(y₁, y₂)(x) = y₁ y₂' − y₂ y₁' is the determinant that tests whether two solutions of a linear ODE are linearly independent. W ≠ 0 at on

Differential Equations

The Yoneda Lemma · How Objects Are Known by Their Relationships

The Yoneda Lemma explained: precise statement, proof mechanism, worked example, and why an object is determined by its morphisms. Rigorous category th

Category Theory

Topology · Donuts ≡ Coffee Cups

Topology cares about shapes up to smooth deformation. A coffee cup is a donut (one hole). A sphere is not (zero holes). The number of holes — genus —

Topology

Trace of a Matrix · Sum the diagonal — and get the sum of every eigenvalue for free

The trace of a square matrix is the sum of its diagonal entries. It equals the sum of eigenvalues, is invariant under similarity, and obeys the cyclic

Linear Algebra

Trapezoidal Rule · ∫f ≈ (h/2)(f(a) + f(b)) — the simplest Newton-Cotes quadrature, and the first numerical integrator every student meets

The trapezoidal rule approximates the integral ∫_a^b f(x) dx as the area of a trapezoid: (h/2)(f(a) + f(b)). The composite rule subdivides [a, b] into

Numerical Analysis

Triangle Inequality · d(x, z) ≤ d(x, y) + d(y, z) — detours never shorten the journey

The triangle inequality d(x,z) ≤ d(x,y) + d(y,z) is the defining axiom of metric spaces — detours never shorten a journey. Reverse form: |d(x,y) − d(y

Inequalities

Trig Identities · sin²+cos²=1

Equations true for every angle — powerful rewriting tools. Pythagorean, double-angle, sum formulas. Derive from the unit circle, apply to integrals, e

Trigonometry

Trig Substitution · Square roots in integrands collapse under the Pythagorean identity

For integrands containing √(a²−x²), √(a²+x²), or √(x²−a²), substitute x = a·sin θ, a·tan θ, or a·sec θ respectively. The Pythagorean identity collapse

Integration

Triple Integrals · Add up a function over a 3D region — volume, mass, charge, moment

A triple integral ∭_V f(x,y,z) dV adds up the values of a function across a 3D region. Cartesian coordinates use dV = dx dy dz; cylindrical coordinate

Calculus

Turán's Theorem · The Densest Triangle-Free Graph

Turán's theorem gives the maximum edges in a K_{r+1}-free graph: (1−1/r)n²/2, achieved uniquely by the balanced Turán graph. Statement, proof idea, ex

Extremal Combinatorics

U-Substitution · ∫f(g(x))g'(x)dx = ∫f(u)du — the chain rule, integrated

U-substitution is the integration counterpart to the chain rule: ∫f(g(x))g'(x)dx = ∫f(u)du with u = g(x). The first technique every calculus student l

Integration

Ultraproducts and Łoś's Theorem · Building Models from Averages

Łoś's theorem explained: how ultraproducts glue structures via an ultrafilter so that first-order truth equals truth on a "large" index set. Statement

Model Theory

Uniform Convergence · f_n → f uniformly iff sup_x |f_n(x) − f(x)| → 0 — preserves continuity, integrability

A sequence of functions f_n: D → ℝ converges uniformly to f if for every ε > 0 there exists N such that |f_n(x) − f(x)| < ε for all x ∈ D and al

Real Analysis

Unit Circle · sin θ

A point on a unit circle has coordinates (cos θ, sin θ). As θ sweeps around, sine and cosine trace their waveforms. Every trig identity is geometry on

Trigonometry

Unitary Matrix · UU = I — the complex isometry, the matrix model of a quantum gate*

A unitary matrix U satisfies U*U = I — its inverse is its conjugate transpose. Unitary maps preserve every inner product, every length, every angle. E

Linear Algebra

Variation of Parameters · The universal recipe for a particular solution to Ly = f, when f is anything you like

A general method for finding a particular solution to a non-homogeneous linear ODE Ly = f. Start with homogeneous solutions y₁, y₂. Replace constants

Differential Equations

Vector Calculus Identities · The algebraic glue of grad, div, and curl — and how they collapse Maxwell's equations

Vector calculus identities are the algebraic glue of E&M and fluid mechanics. The three signature identities — ∇·(∇×F) = 0, ∇×(∇φ) = 0, and ∇×(∇×F) =

Vector Calculus

Vector Spaces · Basis

An abstract structure where vectors can be added and scaled. A minimal spanning set is a basis; its size is the dimension. Generalizes beyond arrows t

Linear Algebra

Wasserstein Distance · Earth Mover's Distance — the geometric cost of morphing one distribution into another

The Wasserstein distance W_p(μ, ν) is the minimum cost to transport probability mass from distribution μ to distribution ν, where cost is distance rai

Optimal Transport

Wave Equation · ∂²u/∂t² = c²∇²u — the canonical hyperbolic PDE behind light, sound, and strings

The wave equation ∂²u/∂t² = c²∇²u is the canonical linear hyperbolic PDE — it governs light, sound, vibrating strings, and seismic waves. d'Alembert's

Partial Differential Equations

Weyl's Inequality · How Eigenvalues Move Under Perturbation

Weyl's inequality bounds how eigenvalues of a Hermitian matrix shift under perturbation: λₖ moves by at most ‖B‖₂. Precise statement, min-max proof, e

Matrix perturbation theory

Wilson's Theorem · n is prime if and only if (n−1)! ≡ −1 (mod n)

Wilson's theorem: an integer n > 1 is prime if and only if (n−1)! ≡ −1 (mod n). Beautifully clean characterization of primes — but factorials grow too

Number Theory

Wishart Distribution · Multivariate chi-squared — distribution of sample covariance matrices

The Wishart distribution W_p(n, Σ) is the distribution of n·S where S is the sample covariance matrix of n samples from a multivariate normal in p dim

Statistics

Young's Inequality · ab ≤ a^p/p + b^q/q for 1/p + 1/q = 1 — the scalar engine of Hölder

Young's inequality: for conjugate p, q with 1/p + 1/q = 1, ab ≤ a^p/p + b^q/q for non-negative a, b. Engine that drives Hölder's inequality. Young's c

Inequalities

Z-Transform · Algebra for sampled signals

The Z-transform converts a discrete-time sequence x[n] into a function X(z) of a complex variable z. It is the discrete-domain analogue of the Laplace

Discrete Mathematics

Zorn's Lemma · Every partially ordered set in which every chain has an upper bound contains a maximal element

Zorn's Lemma states: if every totally ordered subset (chain) of a partially ordered set P has an upper bound in P, then P contains at least one maxima

Set Theory

p-adic Numbers · ℚₚ — completion of ℚ where small means highly divisible by p

The p-adic numbers ℚₚ (for prime p) form an alternative completion of the rational numbers ℚ — one where two numbers are "close" if their difference i

Number Theory

Mathematics

ADMM · The Alternating Direction Method of Multipliers

AM-GM Inequality · (Σxᵢ)/n ≥ (Πxᵢ)^(1/n) — arithmetic mean ≥ geometric mean

Abel Summation · Turning Sums into Integrals

Adjoint Functors · The Free-Forgetful Duality

Analytic Continuation · If two analytic functions agree on a connected open set, they agree wherever both are defined

Arnoldi Iteration · Orthonormal basis of the Krylov subspace span{v, Av, A²v, ...} for non-symmetric A

Axiom of Choice · The axiom that decided ZFC

Baire Category Theorem · A complete metric space is not the countable union of nowhere-dense sets — completeness is fat

Banach Fixed-Point Theorem · Every k-contraction (k < 1) on a complete metric space has exactly one fixed point

Banach Space · A vector space (V, ‖·‖) where every Cauchy sequence converges — generalizes Euclidean ℝⁿ to infinite dimensions

Bayes' Theorem · P(A|B) = P(B|A)

Bessel Functions · The cylindrical sines and cosines — drumhead modes, FM sidebands, diffraction rings

Beta Distribution · The conjugate prior for proportions — Bayesian inference's two-parameter workhorse

Binomial Distribution · n independent yes/no trials with success probability p — the foundation of statistics, from coin flips to clinical trials

Bisection Method · Halve [a,b] where f(a)·f(b)<0 and squeeze a root with guaranteed linear convergence

Bolzano-Weierstrass Theorem · In ℝⁿ, every bounded sequence has at least one convergent subsequence

Brouwer Fixed-Point Theorem · Any continuous f: D^n → D^n on the closed unit ball has a point with f(x) = x

Brownian Motion · B(t) — continuous-time path with independent Gaussian increments, almost-surely nowhere differentiable

Burnside's Lemma · |X/G| = (1/|G|) Σ_{g∈G} |Fix(g)| — average over the group of fixed-point counts

Bézout's Theorem · Counting Intersections of Plane Curves

CW Complexes · Building Spaces by Attaching Cells

Cantor's Diagonalization · The proof technique that revealed uncountable infinity, the halting problem, and Gödel's incompleteness — all from one off-diagonal flip

Cardinality of Infinite Sets · Cantor's hierarchy of infinities

Catalan Numbers · C_n = (2n choose n)/(n+1) — the count that appears in 60+ disguises, from balanced parentheses to triangulated polygons

Cauchy Integral Formula · f(a) = (1/2πi) ∮ f(z)/(z−a) dz — a holomorphic function is determined by its boundary values

Cauchy Sequence · (aₙ) is Cauchy iff for every ε > 0, ∃N: |aₘ − aₙ| < ε for all m, n ≥ N

Cauchy-Riemann Equations · For f = u + iv, holomorphicity ⇔ ∂u/∂x = ∂v/∂y and ∂u/∂y = −∂v/∂x

Cauchy-Schwarz Inequality · |⟨u, v⟩| ≤ ‖u‖·‖v‖ — universal in every inner product space, equality iff u and v are collinear

Cayley-Hamilton Theorem · Every matrix satisfies its characteristic polynomial: p_A(A) = 0

Central Limit Theorem · Sums Become Normal

Chain Rule · (f(g(x)))' = f'(g(x))

Chebyshev Polynomials · Equioscillation, minimax approximation, and the death of the Runge phenomenon

Chebyshev's Inequality · At most 1/k² of any distribution sits more than k standard deviations from the mean

Cheeger's Inequality · The Spectral Gap Bounds Graph Expansion

Chi-Squared Distribution · Sum of k squared standard normals — mean k, variance 2k

Chinese Remainder Theorem · Solve a system of modular congruences with pairwise-coprime moduli — and recover a unique answer mod the product

Cholesky Decomposition · The square root of a positive-definite matrix — twice as fast as LU

Cohomology · The dual of homology — equipped with a cup product that makes H*(X) a graded ring

Compact Set · The topology that makes maxima exist

Compactification · One-point (Alexandroff), Stone-Čech, projective — embed a non-compact space densely into a compact one

Completing the Square · x² + bx → (x + b/2)² − (b/2)²

Complex Numbers · Two real numbers, one orthogonal axis, and an algebra that closes every polynomial

Complex Plane Arithmetic · Addition translates, multiplication rotates-and-scales, conjugation reflects

Conditional Expectation · E[X|Y] — the best mean-square predictor of X from Y, itself a random variable

Conformal Mapping · Holomorphic functions with f'(z) ≠ 0 preserve angles — bend regions while keeping local shape

Conic Sections · circle · ellipse · parabola · hyperbola

Conjugate Gradient · Iteratively solve Ax = b for symmetric positive-definite A in at most N steps using A-orthogonal search directions

Connectedness · "All one piece" — but the topologist's sine curve is one piece without a path between its ends

Continued Fractions · Express any real number as a tower of nested fractions — and the truncations are the best rational approximations

Continuity (ε-δ Definition) · The challenger picks any tolerance — you can always meet it

Continuum Hypothesis (CH) · Cantor's question: no set has cardinality strictly between |ℕ| = ℵ₀ and |ℝ| = 2^ℵ₀

Convex Set and Convex Hull · The set that contains every segment between its points — and the smallest one wrapping a given cloud

Convolution Theorem · F(f ∗ g) = F(f) · F(g) — convolution in time becomes multiplication in frequency

Covering Spaces and the Galois Correspondence

Cramer's Rule · Solve Ax = b with one big determinant per unknown

Cross Product · A perpendicular vector whose length is the area between two others

Cross-Entropy · The bits you actually spend when your code is built for the wrong distribution

Cross-Ratio · (A, B; C, D) = ((C−A)(D−B))/((C−B)(D−A)) — preserved by every projective transformation

Curl (Vector Calculus) · The local rotation of a vector field — circulation per unit area

Cylindrical Coordinates · Polar plus a height — the natural coordinates for pipes, wires, and rotationally symmetric solids

De Moivre's Theorem · Powers of (cos θ + i sin θ) become a single rotation by nθ

Derivative Definition · lim h→0 [f(x+h)−f(x)] / h

Determinant · Signed Volume of a Transformation

Differential Equations (Overview) · Equations whose unknowns are functions, not numbers

Differential Forms · dx, dx∧dy, dx∧dy∧dz — the antisymmetric integrands that turn calculus into one theorem

Dirac Delta Function · An infinite spike with integral one

Directional Derivative · The slope in any direction — the gradient is the special vector that bundles them all

Dirichlet Characters and Group Characters mod n

Dirichlet Distribution · Multivariate Beta on the simplex — distribution over probability vectors

Dirichlet Series · Σ aₙ/n^s — generating function for arithmetic data, generalizes ζ(s) and L-functions

Dirichlet's Theorem on Primes in Arithmetic Progressions

Divergence (Vector Calculus) · The flux per unit volume — how much a field is spreading outwards from each point

Divergence Theorem (Gauss) · Outward flux through a closed surface = volume integral of divergence

Dominated Convergence Theorem · Swap limit and integral whenever one fixed integrable function dominates the sequence

Dot Product · A single number that captures how much two vectors agree

Dual Space · V* = Hom(V, F) — the set of all linear functionals; same dimension as V in finite-dim

Egorov's Theorem · Almost Uniform Convergence

Eigenvalues & Eigenvectors · Av = λv

Eisenstein's Criterion · If p divides every coefficient except the leading one — irreducible over ℚ

Cohomology · The dual of homology — equipped with a cup product that makes H(X) a graded ring*

Dual Space · V = Hom(V, F) — the set of all linear functionals; same dimension as V in finite-dim*

Hermitian Matrix · A = A — self-adjoint, real-eigenvalued, the algebra of measurement*