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The Lumenary

Lumenary Math

The knowledge graph

Every topic, every prerequisite, every optional discovery. Solid lines are required prerequisites; dashed gold lines are optional Wonder Trail connections.

201 topics · 390 required prerequisites · 318 optional discoveries

Required prerequisite Optional discovery Lesson available Planned
FoundationsNumberAlgebraGeometry & TopologyAnalysisDiscrete MathematicsProbability & StatisticsCategory TheoryComputationApplied MathematicsK–23–56–89–12UG 1UG 2GradResearchCollectionsCollectionsCountingCountingAddition, subtraction and place value to 100Addition, subtraction…PatternsPatternsShapesShapesMeasuring length and timeMeasuring length and …Data, tallies and picture graphsData, tallies and pic…Doing and undoingDoing and undoingFollowing and giving precise instructionsFollowing and giving …SetsSetsTrue, false, and, or, notTrue, false, and, or,…Whole-number arithmeticWhole-number arithmet…Fractions and decimalsFractions and decimalsFactors, multiples and primesFactors, multiples an…Expressions, unknowns and rulesExpressions, unknowns…Area, perimeter and anglesArea, perimeter and a…Symmetry: slides, turns and flipsSymmetry: slides, tur…Estimation and roundingEstimation and roundi…Systematic listing and countingSystematic listing an…Dots and lines: networks and puzzlesDots and lines: netwo…Data displays and the mean as a fair shareData displays and the…One thing after anotherOne thing after anoth…Repetition and decisionsRepetition and decisi…Secret codesSecret codesFunctionsFunctionsRelations, equivalence relations and partitionsRelations, equivalenc…Set-builder notation, quantifiers and infinite setsSet-builder notation,…IntegersIntegersClock arithmetic: remainders and modular arithmeticClock arithmetic: rem…Rational numbersRational numbersRatios, rates and proportional reasoningRatios, rates and pro…Variables and algebraic expressionsVariables and algebra…Linear equations and inequalitiesLinear equations and …Linear relationships and simultaneous equationsLinear relationships …The coordinate planeThe coordinate planeThe Pythagorean theoremThe Pythagorean theor…Rigid motions, congruence and similarityRigid motions, congru…Graphs of changeGraphs of changeCounting principlesCounting principlesStatistics: centre, spread and distributionsStatistics: centre, s…ProbabilityProbabilityFunction machines in sequenceFunction machines in …Diagrams where paths agreeDiagrams where paths …Programming for mathematicsProgramming for mathe…Check digits and error detectionCheck digits and erro…Propositional logic and truth tablesPropositional logic a…Proof: direct, contrapositive, contradiction and inductionProof: direct, contra…Countability and Cantor's diagonal argumentCountability and Cant…Russell's paradox and why sets need axiomsRussell's paradox and…Real numbers and irrationalityReal numbers and irra…Complex numbersComplex numbersDivisibility, Euclid's algorithm and unique factorisationDivisibility, Euclid'…PolynomialsPolynomialsQuadratic equations and functionsQuadratic equations a…Euclidean geometry and proofEuclidean geometry an…TrigonometryTrigonometryFunctions and their graphs (precalculus)Functions and their g…Exponential and logarithmic functionsExponential and logar…Sequences and seriesSequences and seriesIntroductory calculusIntroductory calculusCombinatoricsCombinatoricsGraphsGraphsConditional probability and independenceConditional probabili…Sampling and statistical reasoningSampling and statisti…CompositionCompositionStructure-preserving maps: homomorphisms and isomorphismsStructure-preserving …Products by maps: a first universal propertyProducts by maps: a f…Algorithms, correctness and efficiencyAlgorithms, correctne…First steps in LeanFirst steps in LeanMathematical modelling with functionsMathematical modellin…Vectors and coordinate geometryVectors and coordinat…Conic sectionsConic sectionsOrders, lattices and well-ordersOrders, lattices and …First-order logicFirst-order logicCongruencesCongruencesConstructing ℤ, ℚ and ℝConstructing ℤ, ℚ and…Linear algebraLinear algebraAlgebraic structures: monoids, groups and ringsAlgebraic structures:…Transformation geometry and the Erlangen programmeTransformation geomet…Calculus of one variableCalculus of one varia…Multivariable and vector calculusMultivariable and vec…Ordinary differential equationsOrdinary differential…Recurrences and generating functionsRecurrences and gener…Probability theoryProbability theoryStatistical inferenceStatistical inferenceCategoriesCategoriesFunctorsFunctorsUniversal propertiesUniversal propertiesNumerical methodsNumerical methodsAlgorithm design and analysisAlgorithm design and …Formalising mathematics in Lean and MathlibFormalising mathemati…Classical mechanicsClassical mechanicsCryptographyCryptographySurfaces, polyhedra and the Euler characteristicSurfaces, polyhedra a…Non-Euclidean geometry: spherical and hyperbolicNon-Euclidean geometr…ZFC: axiomatic set theoryZFC: axiomatic set th…Ordinals and cardinalsOrdinals and cardinalsThe axiom of choice and its equivalentsThe axiom of choice a…Completeness, compactness and Löwenheim–SkolemCompleteness, compact…Type theory and propositions as typesType theory and propo…Mathematical writingMathematical writingNumber theoryNumber theoryGroupsGroupsRings, ideals and fieldsRings, ideals and fie…Quotient groups and quotient ringsQuotient groups and q…Fields and Galois theoryFields and Galois the…Advanced linear algebraAdvanced linear algeb…Metric spaces and point-set topologyMetric spaces and poi…Differential geometry of curves and surfacesDifferential geometry…Homotopy and the fundamental groupHomotopy and the fund…Real analysisReal analysisComplex analysisComplex analysisPartial differential equations and Fourier seriesPartial differential …Graph theoryGraph theoryEnumerative combinatoricsEnumerative combinato…Markov chainsMarkov chainsNatural transformationsNatural transformatio…Products, coproducts, initial and terminal objectsProducts, coproducts,…Limits and colimitsLimits and colimitsAdjunctionsAdjunctionsRepresentable functorsRepresentable functorsThe Yoneda lemmaThe Yoneda lemmaComputabilityComputabilityThe λ-calculusThe λ-calculusComputational complexityComputational complex…Numerical analysisNumerical analysisOptimisationOptimisationInformation theoryInformation theoryMathematical biologyMathematical biologyProjective geometryProjective geometryDifferential forms and Stokes' theoremDifferential forms an…Knot theoryKnot theoryMatrix groups and continuous symmetryMatrix groups and con…Dynamical systems and chaosDynamical systems and…Model theoryModel theoryGödel's incompleteness theoremsGödel's incompletenes…Proof theoryProof theoryAxiomatic set theory: hierarchy, cardinal arithmetic and LAxiomatic set theory:…Forcing and independenceForcing and independe…Descriptive set theoryDescriptive set theoryReading the mathematical literatureReading the mathemati…Algebraic number theoryAlgebraic number theo…ModulesModulesCommutative algebraCommutative algebraRepresentation theory of finite groupsRepresentation theory…Homological algebraHomological algebraQuivers and path algebrasQuivers and path alge…Algebraic topologyAlgebraic topologySmooth manifoldsSmooth manifoldsAlgebraic geometryAlgebraic geometrySheavesSheavesMeasure theory and integrationMeasure theory and in…Functional analysisFunctional analysisThe probabilistic methodThe probabilistic met…Measure-theoretic probabilityMeasure-theoretic pro…Stochastic processesStochastic processesMonadsMonadsMonoidal categoriesMonoidal categoriesAbelian categoriesAbelian categoriesEnriched categoriesEnriched categoriesCategorical logicCategorical logicElementary topos theoryElementary topos theo…Kan extensionsKan extensionsFormal methodsFormal methodsComputational reproducibilityComputational reprodu…Experimental mathematics: conjectures and counterexamplesExperimental mathemat…Mathematical foundations of quantum mechanicsMathematical foundati…Riemannian geometryRiemannian geometryLie groups and Lie algebrasLie groups and Lie al…Differential topologyDifferential topologyHomotopy theoryHomotopy theoryVector bundles and characteristic classesVector bundles and ch…Riemann surfacesRiemann surfacesFourier and harmonic analysisFourier and harmonic …Partial differential equations: modern theoryPartial differential …Ergodic theoryErgodic theoryAnalytic number theoryAnalytic number theoryThe geometry of spacetime: special and general relativityThe geometry of space…Large cardinalsLarge cardinalsHomotopy type theoryHomotopy type theoryResearch ethics and attributionResearch ethics and a…Quiver representationsQuiver representationsHigher categoriesHigher categories∞-categories∞-categoriesDerived categoriesDerived categoriesTriangulated categoriesTriangulated categori…Stable ∞-categoriesStable ∞-categoriesCategorical representation theoryCategorical represent…Formalising published mathematicsFormalising published…AI-assisted mathematical researchAI-assisted mathemati…Low-dimensional topology: 3- and 4-manifoldsLow-dimensional topol…Geometric group theoryGeometric group theorySymplectic geometry and topologySymplectic geometry a…Complex and Kähler geometry; Hodge theoryComplex and Kähler ge…Stable homotopy theory and generalised cohomologyStable homotopy theor…Geometric analysisGeometric analysisArithmetic geometry and elliptic curvesArithmetic geometry a…The Langlands programme: an orientationThe Langlands program…Random structures: walks, percolation and random geometryRandom structures: wa…Gauge theory and the geometry of physicsGauge theory and the …Extremal and additive combinatoricsExtremal and additive…

Select a topic to see its prerequisites and connections.

A note for the curious: because the required edges form a directed acyclic graph, “is a (possibly indirect) prerequisite of” is a partial order on topics — and every partially ordered set can be viewed as a category with at most one arrow between any two objects.