Math Concept Library

The mathematics of symmetry — studying what remains unchanged under transformations, from polynomial solvability to crystal structure and particle classification.

A language centred on objects, morphisms, and composition, used to compare recurring structures across mathematical fields.

The mathematics of counting possible structures without duplication and proving what must occur even when exhaustive enumeration is impossible.

Vectors a matrix only stretches without rotating. The "principal axes" of high-dimensional data — PageRank, PCA, quantum mechanics.

The same numbers appear in counting parentheses, trees, and lattice paths.

A theory that reads solvability by radicals from the symmetry structure of a polynomial’s roots, originating in work Galois completed by age 20.

Coloring nodes so neighbors differ — from maps to compilers to scheduling.

Mathematics that tables relationships and transforms directions, spaces, and data together. Chinese equation tables, Japanese and European determinant traditions, and nineteenth-century vector spaces and matrix algebra grew from different problems before becoming important computational languages in graphics, quantum theory, statistics, search, and machine learning.

The mathematics of strategic decision-making — from poker to economics to evolutionary biology.

Continuous symmetries — the language of modern particle physics and gauge theory.

An algebra of true/false relations, later connected by Shannon to switching-circuit design and now central to digital logic.

Any map can be colored with 4 colors — first major theorem proved by computer.

A rule assigning exactly one output to each input. The notion grew from quantities attached to curves into a central language for mathematical relations and computation.

The mathematics of chance — from 17th-century gambling to modern AI.

The mathematics of change (derivatives) and accumulation (integrals).

Optimization where the unknown is itself a function — the foundation of physical "principle of least action".

The mathematics of minimizing or maximizing a stated objective subject to constraints. An optimum exists only after “better” and “feasible” are specified; local versus global, exact versus approximate, computable versus socially desirable remain distinct.

Can you add infinitely many things and get a finite sum? From Zeno to Euler's magical summations.

An extension of Fourier analysis to groups and general spaces, with decomposition and convergence depending on the domain and function space.

Equations relating functions to their derivatives — the language of *laws of change*.

Calculus extended to complex numbers — surprisingly elegant: once differentiable means infinitely differentiable.

Optimize a linear objective over a feasible region defined by linear constraints. Kantorovich’s 1939 production planning and Dantzig’s 1947 planning model and simplex method formed in different institutions; a model may be infeasible, unbounded, or have multiple optima.

Differential equations with a random noise term. Foundation of Black-Scholes option pricing (1973) and modern finance.

Solving hard problems by random simulation — from atomic bombs to modern AI.

A stochastic process where the next state depends only on the present. Foundation of PageRank, GPT, speech recognition.

Time average equals space average. Von Neumann and Birkhoff (1931) — foundational in statistical physics, chaos, and dynamical systems.

The rigorous formalization of "approaching" via ε-δ — the foundation of all of calculus.

Updating beliefs with evidence — from an 18th century reverend to AI today.

In real-world data, the leading digit is 1 about 30% of the time, 9 only 5%.

Exact deterministic rules can amplify tiny differences in initial conditions and frustrate long-range trajectory prediction. Chaos theory studies that sensitivity together with bifurcations, attractors, and periodic windows—the structure that remains inside apparently irregular motion. Chaos is not another word for randomness or lawlessness.

The mathematics of reasoning from observed cases to a wider population, process, or effect. Its central task is not merely gathering more numbers but exposing who was counted, what was measured, and which comparisons and assumptions support a conclusion.

Methods for analysing functions and signals by frequency components. Fourier’s heat work opened questions about which expansions converge, and in what sense.

A transform between time or spatial descriptions and frequency descriptions, with existence and inversion conditions depending on the function space.

Calculus on functions of several variables. Partial derivatives, gradients, Jacobians, multiple integrals — the language of ML loss, field theory, utility functions.

A rigorous foundation for "length" and "area" — Lebesgue's integral generalized Riemann's and underpins probability theory.

A neural network with one hidden layer can approximate any continuous function.

How to compute reliable answers on finite machines — the algorithms behind every simulation.

Calculus in space — the language of electromagnetism, fluid dynamics, and field theory.

Encoding entire sequences as functions — a unifying tool for combinatorics.

A bit-valued measure of average uncertainty in a probability distribution. Building on Nyquist and Hartley, Shannon’s 1948 theory treats source-coding and noisy-channel limits under distinct assumptions; it does not directly measure semantic depth or one file’s compressed size.

Long-term behavior of time-evolving systems. Started by Poincaré (3-body problem) — fixed points, periodic orbits, chaos.

Variables and their dependencies as a graph. Bayesian networks (directed) and Markov random fields (undirected) — the framework of AI uncertainty.

The simplest grouping algorithm — from 1957 Bell Labs to every data scientist's first tool.

How many numbers do you need? — From 0D to ∞D and even fractional dimensions in fractals.

Spaces that look flat locally but curve globally — from Earth's surface to spacetime.

Self-similar shapes at every scale — revealing nature's jagged geometry beyond smooth curves.

φ = (1+√5)/2 ≈ 1.618, a self-similar ratio with exact links to Fibonacci ratios, continued fractions, and the golden angle; many popular art and anatomy claims are retrospective.

The study of relations preserved by continuous transformations rather than particular lengths or angles. It distinguishes connectivity, dimension, orientation, boundary, cycles, and richer algebraic invariants, extending beyond rubber-sheet jokes to high-dimensional manifolds and multiscale data.

A way to record holes in a space with algebra. H₀ detects connected components, H₁ detects loops that do not bound a filled region, and H₂ detects higher-dimensional enclosed structure. Rather than merely counting holes, homology uses groups and boundary maps to distinguish topology that survives stretching and bending.

Geometry with "points at infinity" — discovered for Renaissance perspective, foundational for algebraic geometry.

In a right triangle, the square of the hypotenuse equals the sum of the other two squares. Related calculations and proofs appear in Babylonian, Indian, Chinese, and Greek sources.

Centuries of failed attempts to prove Euclid's fifth postulate revealed that changing it can produce consistent alternative geometries, opening hyperbolic, elliptic, and more general curved spaces.

The mathematics of connection — born from a Sunday walk puzzle, now the language of social networks and the brain.

Geometric shapes defined by polynomial zeros. Grothendieck's schemes (20th c.) revolutionized it. Fermat's Last Theorem fell to algebraic geometry in 1995.

When can two knots be deformed into each other without cutting? From Kelvin's atom theory to DNA biology.

Integers greater than 1 divisible only by 1 and themselves — the multiplicative *atoms of whole numbers*. Euclid proved there are infinitely many; primes also support several, but not all, modern cryptographic systems.

Representing reals as nested fractions. φ = [1;1,1,1,...], π = [3;7,15,1,292,...]. The most efficient rational approximations of irrationals.

Equations seeking only integer solutions. Fermat's Last Theorem is one. Hilbert's 10th Problem (Matiyasevich 1970) proved them generally undecidable.

Every even integer > 2 is the sum of two primes. Conjectured 1742; verified to 4×10¹⁸ but unproven for 280 years.

No integer solutions for x^n + y^n = z^n when n>2 — Fermat's 1637 margin note, finally proved by Wiles in 1994.

The arithmetic of clocks — wrapping around at a modulus. The basic language of number theory and cryptography.

The mathematics of secrets — from Caesar ciphers to quantum key distribution. *Information security* is fundamentally a question of which math is hard.

Take any number, halve it if even, 3n+1 if odd — does it always reach 1?

Are there infinitely many primes p with p+2 also prime? Formally conjectured by Polignac in 1846 — open for ~180 years. In 2013 Yitang Zhang proved the first finite-gap version.

The prime-counting function π(x) is asymptotic to x/ln(x). Gauss recalled an early table-based observation in an 1849 letter; Hadamard and de la Vallée Poussin proved the theorem independently in 1896.

A nonsingular cubic curve whose points carry a group law. Elliptic curves connect ancient questions about rational solutions, nineteenth-century elliptic functions, modern number theory, and selected cryptographic systems.

A family of mathematical ideas that distinguish potential processes, infinite sets, and different infinite cardinalities—developed through ancient, medieval, and modern debates.

A language for collections, membership, and size. Cantor’s work and later paradoxes led to axiomatic systems, with ZF and ZFC now widely used foundations.

A consistent, effectively axiomatized formal system strong enough for arithmetic has sentences it cannot prove and, under the theorem's conditions, cannot prove its own consistency.

If n+1 pigeons go into n holes, at least one hole has ≥2. A trivial-looking principle that powers Ramsey theory, finite combinatorics, and Dirichlet approximation.

Does the set of all sets that do not contain themselves contain itself? Russell's 1901 paradox exposed a contradiction in unrestricted set formation and prompted responses including type theory and axiomatic set theory.

Is there an infinity between the natural numbers and the reals? Gödel’s 1940 and Cohen’s 1963 complementary relative-consistency results established that CH is independent of ZFC.

When the proof is too long for humans — the new mathematical methodology.

From any collection of non-empty sets, one can pick one element from each — sounds trivial, but implies the Banach-Tarski paradox.

Logic of degrees between true and false — for rice cookers, washing machines, and AI.

The mathematical study of what fixed procedures can compute. In 1936 Church and Turing clarified its power and limits with different formal models.

Algorithms using quantum superposition — Shor 1994 cast a shadow on RSA security.

Generalization of |A∪B| = |A|+|B|−|A∩B|. Counting unions by alternating sums of intersections — the workhorse of combinatorics.

Seven problems with $1M each — only Poincaré conjecture solved (and prize refused).