Concept atlas
Search the concept atlas
Filter across point-set topology, algebraic topology, and graphics-facing concepts — the field this project grew out of. The detail panel ties each item back to a chapter or extension area.
The map
How the zoos connect
The clusters are not islands. The same theorem, identity, or structure surfaces in several of them at once — here are the bridges where one zoo's definition is another zoo's tool.
Compactness lab
The compactness equivalence web
Compactness has several faces: covers, closed-set intersections, nets, filters, sequences, total boundedness, completion, and the Euclidean closed-and-bounded theorem. The graph separates true equivalences from nearby compactness-like properties, so the proof routes show both the shortcuts and the traps.
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Greedy proof lab
Minimum spanning tree proof web
MST theory is a compact little universe: slice/cut and cycle/circle properties, exchange arguments, Prim and Kruskal invariants, minimax paths, threshold connectivity, lexicographic bottleneck optimality, matroid greedy, and nearby statements that look similar but are weaker or stronger. This panel treats those statements as proof faces and lets the bridges form a clique where that is valid.
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Fourier proof lab
Spatial and frequency operator zoo
Convolution, multiplication, differentiation, Laplacians, Green functions, heat flow, deconvolution, sampling, and windowing form a beautiful dictionary. The panel shows which identities are true dualities, which are recipe routes, and which are warnings about instability or discretization.
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Trig identity lab
Trigonometry, complex numbers, and hyperbolic functions
The same small engine appears as circle geometry, complex multiplication, power series, rotation matrices, and hyperbolic motion. This zoo keeps the definitions, proof routes, and caveats visible at once: Euler formula, cis, angle addition, De Moivre, sinh/cosh, boosts, analytic continuation, and the branch-cut warning for logarithms.
Fractal geometry lab
Dimension definitions and examples
Fractal geometry has its own compact zoo of dimensions: Hausdorff, box/Minkowski, packing, Assouad, similarity, Rényi, correlation, information, spectral, walk, and multifractal viewpoints. This panel separates equalities that need self-similar hypotheses from inequalities, examples, and warnings where two dimension ideas are answering different questions.
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Noise and roughness lab
fBM, spectra, and procedural noise
Noise is a bridge between probability, Fourier analysis, fractal geometry, and graphics. This zoo separates stochastic processes such as Brownian motion and fractional Brownian motion from colored-noise spectra such as red and blue noise, plus procedural recipes such as value, Perlin, simplex, octave, turbulence, and ridged noise.
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Spectral lab
Spectral & eigendecomposition
A matrix's spectrum is described by a web of definitions and theorems that prove one another — but only some links run both ways. Here we separate genuine equivalences from one-way implications, scope facts, and tempting false friends.
Curvature lab
Curvature & Gauss-Bonnet
The curvature cluster: how the two fundamental forms generate every local curvature notion, why Gauss's curvature is secretly intrinsic, and how local turning sums to a global topological invariant. Watch which links are genuine equivalences and which only run one way.
Choice lab
The Axiom of Choice and its equivalents
Over ZF, a surprising web of theorems from across mathematics all turn out to be the same statement in disguise. The point of this zoo is to separate the genuine full-strength equivalents from the strictly weaker fragments that merely follow from Choice.
Choice lab
Induction principles
A cluster of induction schemes that look different but mostly prove the same thing. Over the Peano axioms, weak, strong, and well-ordering on ℕ are the same principle in three costumes; the rest are either special cases of well-founded induction or its generalization to well-ordered domains.
Choice lab
The complexity class zoo
Most class relations are proven one-way inclusions; only a handful are true equivalences, and the most famous questions remain stubbornly open. The art is telling these apart.
Field theory · Galois theory
The Galois Correspondence
A cluster of definitions and theorems where equivalent characterizations prove one another. The keystone is the Fundamental Theorem of Galois Theory: for a finite normal-and-separable extension, subgroups of the Galois group correspond, inclusion-reversingly, to intermediate fields. Around it sit the genuine equivalences (Galois ⇔ normal+separable; solvable-by-radicals ⇔ solvable group; constructible ⇔ tower of quadratics), the one-way consequences (finite fields ⇒ cyclic Galois group), and the false friends (degree a power of 2 is necessary, not sufficient, for constructibility).
Min-max theorems and LP duality
Optimization Duality
Linear programming duality is the engine behind a family of combinatorial min-max theorems. The cluster's organizing claim: strong LP duality is the general fact, and max-flow/min-cut, Menger, Konig, and Hall are integral specializations that hold because their constraint matrices are totally unimodular. Within the bipartite layer, Konig, Hall, and bipartite max-flow/min-cut are mutually derivable equivalents. The recurring trap is conflating weak duality (always true, one direction) with strong duality (equality, only under feasibility and boundedness), and assuming an LP relaxation is integral without checking total unimodularity.
Metric spaces
Completeness & Fixed Points
Completeness is the promise that Cauchy sequences keep their word: if the terms crowd together, a limit is waiting for them inside the space. Several statements turn out to say exactly this — convergence of Cauchy sequences, Cantor's nested-set criterion, being your own completion — and these are genuine equivalences (↔). From completeness flow one-way consequences (→): the Baire category theorem and the Banach contraction principle. Watch the boundaries: Baire holds for spaces that are not complete, totally bounded spaces need not be complete, and Banach demands a strict contraction. Compactness joins the cluster as completeness plus total boundedness, bridging to the compactness zoo.
Category Theory
Universal Properties, Limits & Adjunctions
A guided cluster around the engine of category theory: the universal property. Objects defined by universal properties are pinned down up to unique isomorphism, limits and colimits are the master construction, adjunctions are the symmetric pairing that organizes them, and Yoneda is the lemma that makes "an object is what it maps to/from" precise. Edges separate genuine equivalences (definitions that prove each other) from one-way consequences, special cases, and seductive false friends.
Abstract algebra
Group theory core
The objects, maps, and counting theorems that organize symmetry. The spine is normality: a subgroup is normal exactly when it is a kernel, exactly when the quotient is a group — three faces of one idea. Around it sit the structural theorems (Lagrange, Sylow, Cayley, orbit-stabilizer) and the special classes (cyclic, abelian, simple). Watch the classic false friends: Lagrange has no converse, and normality is not transitive.
Measure theory
Measure-theory foundations
The scaffolding of integration: which sets you can measure (sigma-algebras, Borel, Lebesgue), how you assign size (measures, outer measures, the Caratheodory extension), which functions you can integrate (measurable functions, almost everywhere), and the structural theorems (Radon-Nikodym, Fubini-Tonelli). True equivalences are separated from one-way consequences, conditional scope (sigma-finiteness, completeness), and false friends (Borel vs Lebesgue, a.e.-equal vs equal).
Functional analysis
Hilbert spaces & bounded operators
The geometry of complete inner-product spaces and the structure theory of the operators on them. Equivalences (↔) are genuine two-way theorems — completeness, Parseval, Riesz, the spectral theorem; one-way consequences (→) flow from the strong operator classes to their spectral payoffs; scope edges (⊃) mark special cases and inclusions; warnings (≠) collect the infinite-dimensional surprises where finite-dimensional intuition breaks. Everything here stays with BOUNDED operators to avoid unbounded-domain subtleties.
Elementary & modular number theory
Number theory core
The arithmetic of Z and Z/n, organized around one central equivalence — the Chinese Remainder Theorem is a ring isomorphism — and the chain of group-theoretic consequences (Lagrange in (Z/n)* gives Fermat, then Euler). Watch the classic false friends: the converse of Fermat fails (Carmichael numbers), and not every modulus has a primitive root.
Convex analysis
Convexity & separation
Convex analysis runs on one engine: a convex set is the intersection of the halfspaces that support it, and separation theorems turn that geometry into algebra. The same duality appears as Legendre-Fenchel conjugation, subgradients, and Lagrangian saddle points. The recurring gotcha is hypotheses: separation needs an openness or compactness condition, biconjugation needs closedness, and strong duality needs a constraint qualification like Slater. Keep genuine equivalences (↔) apart from one-way consequences (→), special cases (⊃), and false friends (≠).
Convergence lab
Modes of convergence
A sequence of functions (or random variables) can converge in many inequivalent senses. The web below sorts strict one-way implications from the conditional bridges (finite measure, subsequences, uniform integrability) and the famous counterexamples that keep the arrows from reversing.
Integral zoo
Which functions become integrable?
Integral definitions form a small hierarchy of scopes: bounded interval sums, measure-theoretic integration, improper limits, Stieltjes weights, gauge refinements, and weak testing against smooth probes. The key question is not just “what is the value?” but “which functions are admitted, and under which hypotheses?”
Field integral lab
Static domains and moving-domain transport
Field integrals sit between multivariable calculus, vector calculus, PDEs, and physics. The static card organizes scalar, line, surface, volume, flux, and change-of-variables integrals; the moving card asks what changes when the domain itself moves, which is exactly where Leibniz and Reynolds transport belong. The theorem atlas then expands Green, Gauss, Stokes, differential forms, Green identities, multivariate Leibniz rules, Lagrangian/material views, and complex-plane shortcuts.
Separation ladder
From T0 to the full separation taxonomy
This expanded module follows the named axioms on Wikipedia's separation-axiom page: the main definitions, the summary-table additions, and the “other separation axioms” section. Switch between the connection map and focused axiom families, then pick any axiom to see the witness condition appear on an interactive diagram.
Counterexample lab
Spaces that separate similar definitions
Inspired by Steen and Seebach's Counterexamples in Topology and modern searchable counterexample databases, this panel collects small spaces whose job is to say exactly where a tempting implication breaks.
Invariant explorer
Classic spaces at a glance
Pick a space and compare orientability, boundary, Euler characteristic, fundamental group, and homology. This is the bridge between picture-level intuition and algebraic invariants.
Graphics lens
What topology actually does in geometry processing
Some ideas are pipeline staples. Others are research-heavy, niche, or just starting to cross over. This panel makes that split explicit.
Algorithms roadmap
Techniques as clusters of ideas
A LeetCode roadmap organized by the underlying technique, not the problem. Each cluster carries its core idea, when to reach for it, the pattern, canonical problems, and a link to the math zoo that explains why it works.
Number-systems roadmap
Complex, quaternion & geometric algebra
One ladder of number systems built for rotation and geometry: ℝ → ℂ → ℍ → 𝕆, and the Clifford/geometric algebra that unifies them. Each topic carries its core idea, the geometric intuition, the defining identities, sub-topics, applications, and links to the theory zoos.
Group theory roadmap
From the axioms to symmetry groups
A path through group theory — foundations, structure and homomorphisms, classification (Sylow, solvable, simple), the basics of representation theory, and Lie/symmetry groups — with links back to the Groups, Galois, and spectral zoos.
Algebraic topology roadmap
Turning spaces into algebra
A path through algebraic topology: the fundamental group and covering spaces, homology and cohomology, Poincaré duality, bundles and characteristic classes, and fixed-point theorems — with links back to the spaces, Gauss–Bonnet, and field-integral zoos.
Complex analysis roadmap
Holomorphy, residues & Riemann surfaces
A deep path through complex analysis: the Cauchy–Riemann equations and holomorphy, contour integration and the residue calculus, conformal maps and the Riemann mapping theorem, analytic continuation and Riemann surfaces, up to the Γ/ζ and elliptic functions.
Differential geometry roadmap
Manifolds, curvature & Gauss–Bonnet
A path through differential and Riemannian geometry: smooth manifolds and differential forms, bundles and connections, metrics and geodesics, the curvature tensors, up to Gauss–Bonnet in all dimensions, Hodge theory, and general relativity.
Functional analysis roadmap
Banach, Hilbert & the spectral theorem
A path through functional analysis: normed, Banach and Hilbert spaces, the Hahn–Banach and Baire-category theorems, bounded and compact operators, the spectral theorem, up to C*-algebras, semigroups, distributions, and Sobolev spaces.
Probability roadmap
Measure, martingales & Itô calculus
A path through measure-theoretic probability: probability spaces and expectation, independence and the limit theorems (LLN, CLT), conditional expectation and martingales, Markov chains and Brownian motion, up to the Itô integral and stochastic differential equations.
Lie groups roadmap
Continuous symmetry: groups, algebras & representations
A path through Lie theory: Lie groups and their Lie algebras, the exponential map, the structure theory of semisimple algebras (root systems, Dynkin diagrams, Weyl groups), highest-weight representation theory, and the geometry and physics of continuous symmetry — linking back to the Groups, spectral, and Gauss–Bonnet zoos.
PDE roadmap
Characteristics, the classical trio & weak solutions
A path through partial differential equations: first-order characteristics and the elliptic/parabolic/hyperbolic classification, the Laplace–heat–wave trio, the classical methods (separation, transforms, Green's functions, maximum and energy principles), then the modern weak-solution theory — Sobolev spaces, Lax–Milgram, regularity — and the nonlinear frontier.
Fourier analysis roadmap
Frequency: series, transforms & singular integrals
A path through Fourier and harmonic analysis: series and the delicate question of their convergence, the Fourier transform and convolution, the uncertainty principle, distributions and multipliers, the real-variable theory (maximal functions, interpolation, Calderón–Zygmund, BMO), and wavelets, the FFT and harmonic analysis on groups.
Dynamical systems roadmap
Flows, stability, bifurcation & chaos
A path through ODEs and dynamical systems: existence and uniqueness, linear systems and phase portraits, flows, Lyapunov stability and invariant manifolds, limit cycles, Hamiltonian dynamics and bifurcations, then chaos, strange attractors, ergodic theory and numerical integration.
Category theory roadmap
Functors, adjunctions & derived functors
A path through category theory and homological algebra: categories, functors, natural transformations and Yoneda; limits, universal properties, adjunctions and monads; then abelian categories, chain complexes and exact sequences, resolutions and derived functors (Ext/Tor), spectral sequences, sheaves and topoi.
Number theory roadmap
Primes, number fields & L-functions
A path through number theory: the elementary theory (divisibility, congruences, quadratic reciprocity, Diophantine equations), the algebraic theory (number fields, ideal factorization, class groups, units, p-adics), the analytic theory (Dirichlet series, the prime number theorem), L-functions and sieves, then class field theory and elliptic curves.
Logic & set theory roadmap
Proof, models, sets & the limits of computation
A path through mathematical logic: propositional and first-order logic, proof systems and cut elimination, completeness and compactness, model theory and quantifier elimination; then ZFC, ordinals and cardinals, choice, the continuum hypothesis and forcing; and finally computability, undecidability, Gödel incompleteness and the Turing degrees.
Optimization roadmap
Convexity, duality & the algorithms that exploit them
A path through optimization: convex sets, separation and convex functions, optimality conditions; the duality core — Lagrange multipliers and KKT, Lagrangian and Fenchel duality, linear programming; the algorithms — simplex, interior point, conic/SDP, gradient, accelerated, stochastic and proximal methods; and the discrete and nonconvex edges.
Algebraic geometry roadmap
From rings and ideals to schemes & Riemann–Roch
A path from commutative algebra into algebraic geometry: rings, modules, Noetherian and local theory, Krull dimension; the Nullstellensatz and the ideal–variety dictionary, affine and projective varieties, the Zariski topology, smoothness and singularities; then sheaves, Spec and schemes; and finally divisors, sheaf cohomology, Riemann–Roch and Serre duality.
Roadmap
The chapter-level map — where this started
This is the project's origin: the Munkres topology spine. It stays visible as the starting map even though the zoos now branch well beyond topology into analysis, optimization, probability, and geometry.