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The Lumenary

Lumenary Math

知识图谱

每个主题、每项前置知识、每个可选发现。实线表示必需的前置知识；金色虚线表示可选的探奇小径联系。

169 个主题 · 306 项必需的前置知识 · 232 个可选发现

必需的前置知识 可选发现 课程已上线 计划中
FoundationsNumberAlgebraGeometry & TopologyAnalysisDiscrete MathematicsProbability & StatisticsCategory TheoryComputationApplied MathematicsK–23–56–89–12UG 1UG 2GradResearchCollectionsCollectionsCountingCountingAddition, subtraction and place value to 100Addition, subtraction an…PatternsPatternsShapesShapesMeasuring length and timeMeasuring length and timeData, tallies and picture graphsData, tallies and pictur…Doing and undoingDoing and undoingFollowing and giving precise instructionsFollowing and giving pre…SetsSetsTrue, false, and, or, notTrue, false, and, or, notWhole-number arithmeticWhole-number arithmeticFractions and decimalsFractions and decimalsFactors, multiples and primesFactors, multiples and p…Expressions, unknowns and rulesExpressions, unknowns an…Area, perimeter and anglesArea, perimeter and angl…Symmetry: slides, turns and flipsSymmetry: slides, turns …Estimation and roundingEstimation and roundingSystematic listing and countingSystematic listing and c…Dots and lines: networks and puzzlesDots and lines: networks…Data displays and the mean as a fair shareData displays and the me…One thing after anotherOne thing after anotherRepetition and decisionsRepetition and decisionsSecret codesSecret codesFunctionsFunctionsRelations, equivalence relations and partitionsRelations, equivalence r…Set-builder notation, quantifiers and infinite setsSet-builder notation, qu…IntegersIntegersClock arithmetic: remainders and modular arithmeticClock arithmetic: remain…Rational numbersRational numbersRatios, rates and proportional reasoningRatios, rates and propor…Variables and algebraic expressionsVariables and algebraic …Linear equations and inequalitiesLinear equations and ine…Linear relationships and simultaneous equationsLinear relationships and…The coordinate planeThe coordinate planeThe Pythagorean theoremThe Pythagorean theoremRigid motions, congruence and similarityRigid motions, congruenc…Graphs of changeGraphs of changeCounting principlesCounting principlesStatistics: centre, spread and distributionsStatistics: centre, spre…ProbabilityProbabilityFunction machines in sequenceFunction machines in seq…Diagrams where paths agreeDiagrams where paths agr…Programming for mathematicsProgramming for mathemat…Check digits and error detectionCheck digits and error d…Propositional logic and truth tablesPropositional logic and …Proof: direct, contrapositive, contradiction and inductionProof: direct, contrapos…Countability and Cantor's diagonal argumentCountability and Cantor'…Russell's paradox and why sets need axiomsRussell's paradox and wh…Real numbers and irrationalityReal numbers and irratio…Complex numbersComplex numbersDivisibility, Euclid's algorithm and unique factorisationDivisibility, Euclid's a…PolynomialsPolynomialsQuadratic equations and functionsQuadratic equations and …Euclidean geometry and proofEuclidean geometry and p…TrigonometryTrigonometryFunctions and their graphs (precalculus)Functions and their grap…Exponential and logarithmic functionsExponential and logarith…Sequences and seriesSequences and seriesIntroductory calculusIntroductory calculusCombinatoricsCombinatoricsGraphsGraphsConditional probability and independenceConditional probability …Sampling and statistical reasoningSampling and statistical…CompositionCompositionStructure-preserving maps: homomorphisms and isomorphismsStructure-preserving map…Products by maps: a first universal propertyProducts by maps: a firs…Algorithms, correctness and efficiencyAlgorithms, correctness …First steps in LeanFirst steps in LeanMathematical modelling with functionsMathematical modelling w…Orders, lattices and well-ordersOrders, lattices and wel…First-order logicFirst-order logicCongruencesCongruencesConstructing ℤ, ℚ and ℝConstructing ℤ, ℚ and ℝLinear algebraLinear algebraAlgebraic structures: monoids, groups and ringsAlgebraic structures: mo…Transformation geometry and the Erlangen programmeTransformation geometry …Calculus of one variableCalculus of one variableMultivariable and vector calculusMultivariable and vector…Ordinary differential equationsOrdinary differential eq…Recurrences and generating functionsRecurrences and generati…Probability theoryProbability theoryStatistical inferenceStatistical inferenceCategoriesCategoriesFunctorsFunctorsUniversal propertiesUniversal propertiesNumerical methodsNumerical methodsAlgorithm design and analysisAlgorithm design and ana…Formalising mathematics in Lean and MathlibFormalising mathematics …Classical mechanicsClassical mechanicsCryptographyCryptographyZFC: axiomatic set theoryZFC: axiomatic set theoryOrdinals and cardinalsOrdinals and cardinalsThe axiom of choice and its equivalentsThe axiom of choice and …Completeness, compactness and Löwenheim–SkolemCompleteness, compactnes…Type theory and propositions as typesType theory and proposit…Mathematical writingMathematical writingNumber theoryNumber theoryGroupsGroupsRings, ideals and fieldsRings, ideals and fieldsQuotient groups and quotient ringsQuotient groups and quot…Fields and Galois theoryFields and Galois theoryAdvanced linear algebraAdvanced linear algebraMetric spaces and point-set topologyMetric spaces and point-…Differential geometry of curves and surfacesDifferential geometry of…Homotopy and the fundamental groupHomotopy and the fundame…Real analysisReal analysisComplex analysisComplex analysisPartial differential equations and Fourier seriesPartial differential equ…Graph theoryGraph theoryEnumerative combinatoricsEnumerative combinatoricsMarkov chainsMarkov chainsNatural transformationsNatural transformationsProducts, coproducts, initial and terminal objectsProducts, coproducts, in…Limits and colimitsLimits and colimitsAdjunctionsAdjunctionsRepresentable functorsRepresentable functorsThe Yoneda lemmaThe Yoneda lemmaComputabilityComputabilityThe λ-calculusThe λ-calculusComputational complexityComputational complexityNumerical analysisNumerical analysisOptimisationOptimisationInformation theoryInformation theoryMathematical biologyMathematical biologyModel theoryModel theoryGödel's incompleteness theoremsGödel's incompleteness t…Proof theoryProof theoryAxiomatic set theory: hierarchy, cardinal arithmetic and LAxiomatic set theory: hi…Forcing and independenceForcing and independenceDescriptive set theoryDescriptive set theoryReading the mathematical literatureReading the mathematical…Algebraic number theoryAlgebraic number theoryModulesModulesCommutative algebraCommutative algebraRepresentation theory of finite groupsRepresentation theory of…Homological algebraHomological algebraQuivers and path algebrasQuivers and path algebrasAlgebraic topologyAlgebraic topologySmooth manifoldsSmooth manifoldsAlgebraic geometryAlgebraic geometrySheavesSheavesMeasure theory and integrationMeasure theory and integ…Functional analysisFunctional analysisThe probabilistic methodThe probabilistic methodMeasure-theoretic probabilityMeasure-theoretic probab…Stochastic processesStochastic processesMonadsMonadsMonoidal categoriesMonoidal categoriesAbelian categoriesAbelian categoriesEnriched categoriesEnriched categoriesCategorical logicCategorical logicElementary topos theoryElementary topos theoryKan extensionsKan extensionsFormal methodsFormal methodsComputational reproducibilityComputational reproducib…Experimental mathematics: conjectures and counterexamplesExperimental mathematics…Mathematical foundations of quantum mechanicsMathematical foundations…Large cardinalsLarge cardinalsHomotopy type theoryHomotopy type theoryResearch ethics and attributionResearch ethics and attr…Quiver representationsQuiver representationsHigher categoriesHigher categories∞-categories∞-categoriesDerived categoriesDerived categoriesTriangulated categoriesTriangulated categoriesStable ∞-categoriesStable ∞-categoriesCategorical representation theoryCategorical representati…Formalising published mathematicsFormalising published ma…

选择一个主题，查看它的前置知识和联系。

给好奇者的一点说明：由于必需的边构成一个有向无环图，“是……的（可能间接的）前置知识”是主题上的一个偏序——而每个偏序集都可以看作一个范畴，其中任意两个对象之间至多有一个箭头。