Modern Perspectives on Mock Theta Functions

Contents
1 Introduction to Mock Theta Functions 7
1.1 HistoricalBackground ............................ 7
1.2 BasicDefinitionsandExamples ....................... 8
1.3 Classicalvs.ModernPerspectives...................... 9
1.4 FurtherExamplesandProperties ...................... 11
1.5 ApplicationsandModernResearch ..................... 13
1.5.1 ApplicationsinNumberTheory................... 13 1.5.2 ApplicationsinCombinatorics.................... 14
1.5.3 ApplicationsinMathematicalPhysics. . . . . . . . . . . . . . . . 14
1.6 Conclusion................................... 16
2 Ramanujan’s Legacy 17
2.1 Ramanujan’s Contributions to Mock Theta Functions . . . . . . . . . . . 17
2.2 TheLostNotebook: RediscoveryandImpact. . . . . . . . . . . . . . . . 18
2.3 EarlyDevelopmentsPost-Ramanujan.................... 19
2.4 TheDevelopmentofHarmonicMaassForms . . . . . . . . . . . . . . . . 20
2.5 ModernResearchandOpenProblems.................... 21
3 Applications and Modern Developments 23
3.1 ApplicationsinModularForms ....................... 23
3.2 CombinatorialInterpretations ........................ 23
3.3 PhysicalApplications............................. 24
3.4 QuantumModularForms .......................... 24
3.5 FurtherDevelopmentsandOpenProblems................. 25
3.6 Conclusion................................... 25
4 Quantum Modular Forms 27
4.1 DefinitionandProperties........................... 27
4.2 RelationshipwithMockThetaFunctions.................. 28
4.3 RecentAdvancesinQuantumModularForms . . . . . . . . . . . . . . . 29
5 Mock Theta Functions and Quantum Invariants 33
5.1 QuantumInvariantsofKnotsand3-Manifolds . . . . . . . . . . . . . . . 33
5.2 ConnectionswithMockThetaFunctions .................. 34
5.3 RecentDevelopmentsandExamples..................... 35
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4 CONTENTS
6 Combinatorial Interpretations 37
6.1 PartitionTheoryandMockThetaFunctions . . . . . . . . . . . . . . . . 37
6.2 RecentCombinatorialResults ........................ 38
6.3 ApplicationsinCombinatorics........................ 39
7 Mock Theta Functions in Representation Theory 41
7.1 Role in the Theory of Lie Algebras and Vertex Operator Algebras . . . . 41
7.2 RecentResearchandFindings........................ 42
7.3 CaseStudiesandExamples ......................... 43
8 Harmonic Maass Forms and Mock Theta Functions 47
8.1 IntroductiontoHarmonicMaassForms................... 47
8.2 ConnectionsandApplications ........................ 48
8.3 RecentResearchDirections ......................... 51
9 Mock Modular Forms and Applications 53
9.1 BasicTheoryofMockModularForms ................... 53
9.2 ApplicationsinMathematicalPhysics.................... 54
9.3 CurrentResearchTrends........................... 55
9.4 Conclusion................................... 56
10 Analytical Approaches to Mock Theta Functions 57
10.1AsymptoticAnalysis ............................. 57 10.1.1 ClassicalAsymptoticMethods.................... 57 10.1.2 Modular Transformations and Asymptotics . . . . . . . . . . . . . 57 10.1.3 SaddlePointMethodandLargeDeviations. . . . . . . . . . . . . 58
10.2IntegralRepresentations ........................... 59 10.2.1 Mellin-BarnesIntegral ........................ 59 10.2.2 Residue Calculus and Contour Integration . . . . . . . . . . . . . 59 10.2.3 LaplaceandFourierTransformMethods . . . . . . . . . . . . . . 60
10.3RecentAnalyticalTechniques ........................ 60 10.3.1 HarmonicMaassForms........................ 60 10.3.2 QuantumModularForms ...................... 61 10.3.3 AlgebraicandTopologicalTechniques. . . . . . . . . . . . . . . . 61 10.3.4 ConnectionstoPhysicalTheories .................. 62
11 Computational Aspects 63
11.1 AlgorithmsforComputingMockThetaFunctions . . . . . . . . . . . . . 63 11.1.1 DirectSeriesSummation....................... 63 11.1.2 ModularTransformationMethods.................. 64 11.1.3 ThetaLiftandPoincar ́eSeries.................... 64
11.2SoftwareToolsandResources ........................ 65 11.2.1 MathematicaandMaple ....................... 65 11.2.2SageMath............................... 66 11.2.3PARI/GP............................... 66
11.3 RecentAdvancesinComputationalMethods . . . . . . . . . . . . . . . . 66 11.3.1AlgorithmicDevelopments...................... 67 11.3.2 MachineLearningTechniques .................... 67 11.3.3 High-PerformanceComputing .................... 67

CONTENTS
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12 The Future of Mock Theta Functions
12.1 Introduction . . . . . . . . . . . . . . . 12.2 SummaryofKeyConcepts . . . . . . . 12.3MajorDevelopmentsandApplications 12.4 Current Research Trends . . . . . . . . 12.5 Open Problems and Future Directions 12.6 Conclusion . . . . . . . . . . . . . . . .
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