Mathematics Going Forward: Collected Mathematical Brushstrokes

Description

This volume is an original collection of articles by 44 leading mathematicians on the theme of the future of the discipline. The contributions range from musings on the future of specific fields, to analyses of the history of the discipline, to discussions of open problems and conjectures, including first solutions of unresolved problems. Interestingly, the topics do not cover all of mathematics, but only those deemed most worthy to reflect on for future generations. These topics encompass the most active parts of pure and applied mathematics, including algebraic geometry, probability, logic, optimization, finance, topology, partial differential equations, category theory, number theory, differential geometry, dynamical systems, artificial intelligence, theory of groups, mathematical physics and statistics. Preface Contents Letter from Jean-Pierre Eckmann Part I Algebraic Geometry Une liste de problèmes Introduction 1. Variétés rationnelles et variétés proches 2. Principe de Hasse, approximation faible, obstruction de Brauer–Manin 3. Points rationnels des variétés rationnellement connexes sur un corps global 3.1. Espaces homogènes de groupes algébriques linéaires connexes 3.2. Surfaces de del Pezzo et variétés de Fano 3.3. Espaces totaux de fibrations en variétés rationnellement connexes au-dessus de la droite projective 3.4. Au-delà des variétés rationnellement connexes 4. Zéro-cycles des variétés sur un corps global 5. Rationalité des variétés et invariants birationnels 5.1. Unirationalité 5.2. R-équivalence 5.3. Rationalité des intersections de deux quadriques 5.4. Cohomologie non ramifiée 6. Points rationnels et indice des variétés algébriques. 7. Groupes algébriques linéaires Références Some Ideas in Need of Clarification in Resolution of Singularities and the Geometry of Discriminants 1 Problems Related to Resolution of Singularities 2 Problems Related to the Geometry of Discriminants of Miniversal Unfoldings References Shifted Sheaves for Space-Time References Lefschetz Fixed Point Theorems for Correspondences 1 Correspondences 2 Lefschetz Number of a Smooth Correspondence 3 Fixed Points of a Smooth Correspondence 4 The Trace of a Smooth Correspondence 5 The Lefschetz Fixed Point Theorem for a Smooth Correspondence 6 A Conjecture for a Holomorphic Correspondence 7 Extension to Holomorphic Vector Bundles Appendix References Part II Dynamical Systems Vous avez dit ``qualitatif'' ? 1 Introduction 2 En guise de hors-d'œuvre 3 Intégrer une équation différentielle ? 4 L'espace des phases et ses habitants 5 Intégrer ou décrire ? Le problème des trois corps 5.1 Sundman 5.2 Poincaré 5.3 Poser les bonnes questions 6 Stabilité dans le problème restreint 6.1 Hill 6.2 Poincaré 6.3 KAM 7 Généricité 7.1 Le général et le particulier : mesure et catégorie 7.1.1 Poincaré 7.1.2 Borel, Baire 7.2 De la stabilité structurelle aux figures universelles de bifurcations 7.2.1 Andronov, Pontryagin 7.2.2 Thom 8 Symétries 8.1 Symétries et intégrales premières : Noether 8.2 Invariants intégraux : Poincaré, Cartan 8.3 Minimiser l'action sous contrainte de symétrie 9 Images, formes, noms 9.1 La faune de la dynamique qualitative : Smale 9.2 L'importance des images 9.3 La convention du Nom 9.4 La variété infinie et joyeuse des formes 10 Conclusion Références Tale o' pi by pilota Reference Part III Finance What Next? Controlled Markov Processes Preliminaries Optimal Stopping Problems Unfinished Business Our Catriona References Some Remarks on Enlargement of Filtration and Finance 1 Introduction 2 Mathematical Facts 2.1 Problem of Enlargement of Filtration 2.2 Particular Cases 2.2.1 Discrete Time 2.2.2 Immersion 3 Initial Enlargement 3.1 Jacod's Conditions 3.2 Brownian Bridge 4 Progressive Enlargement 4.1 Before τ 4.2 Some Facts on the Predictable Representation Property 4.3 Immersion 4.4 Honest Times 5 Information Drift 6 Conclusion and Open Problems References Modern Extreme Value Theory at the Interface of Risk Management, Bayesian Networks and Heavy-Tailed Time Series 1 Introduction 1.1 An Extremal ode to Catriona Byrne 1.2 New Aspects of Extreme Value Theory 2 The Fundamental Theorems of Quantitative Risk Management (QRM) (by Paul Embrechts) 2.1 The 1st FTQRM 2.2 Discussion 3 Max-Linear Bayesian Networks (by Claudia Klüppelberg) 3.1 An Extremal Graphical Model 3.2 Conditional Independence 3.3 Discussion 4 A Light History of Heavy Tails (by Thomas Mikosch) References Limits of Limit-Order Books 1 Introduction 1.1 Background 1.2 Survey of the Literature 1.3 A Representative Model 2 Description of the Representative Model 3 Conclusion References Part IV Geometry Spinors in 2022 1 The Initial and Persistent Strangeness of Spinors 2 The Unabated Importance of Spinors in Physics 3 The Role of Natural Differential Operators on Spinor Fields 4 Towards a True Spinorial Geometry? References Cohomological Localizations and Set-Theoretical Reflection Introduction 1 Homology Theories and Cohomology Theories 1.1 Categories of Fractions 1.2 Solution-Set Conditions 2 Set-Theoretical Reflection 2.1 Cardinality and Rank 2.2 Structures 2.3 Reflection Principles 2.4 The Lévy Hierarchy 2.5 Existence of Localizations 3 Conclusions References Geometry of Data 1 Introduction 2 Preliminaries from Metric Geometry 3 Tripod Spaces 4 Hyperconvexity 5 Relation with Topological Data Analysis (TDA) 6 Curvature 7 Conclusions References A Chapter About Asymptotic Geometric Analysis: Isomorphic Position of Centrally Symmetric Convex Bodies A Few Words on Asymptotic Geometric Analysis Isomorphic Position of a Convex Body Isomorphic Version of Bourgain's Slicing Problem Isomorphic Version of Log-Brunn–Minkowski Inequality More Isomorphic Versions of Well-Known Problems of AGA References A Mysterious Tensor Product in Topology References Part V Groups Conjectures on Reductive Homogeneous Spaces 1 Introduction 2 Basic Setting 3 Problems on Discrete Series for G/H 4 Problems on Discontinuous Groups for G/H 5 Spectral Analysis for Pseudo-Riemannian Locally Homogeneous Spaces "026E30F G/H References Profinite Rigidity and Free Groups 1 Introduction 2 Full-Sized Groups that Are Profinitely Rigid 3 Restrictions on the Nature of Profinitely-Free Groups 4 Failure of Profinite Rigidity Close to Free Groups References On the Algebraic K-Theory of Hecke Algebras 1 Introduction 1.1 Conventions and Notations 2 Hecke Algebras 2.1 Basic Setup 2.2 The Construction of the Hecke Algebra 2.3 Functoriality in Q 2.4 Approximate Units 2.5 Discarding μ 3 Z-Categories, Additive Categories and Idempotent Completions 4 The Algebraic K-Theory of Z-Categories 5 Covirtually Z Groups 6 A Review of the Twisted Bass–Heller–Swan Decomposition for Unital Additive Categories 7 Hecke Algebras Over Compact td-Groups and Crossed Product Rings 7.1 Existence of Normal K P 7.2 Crossed Products Rings of Finite Groups and Regularity 7.3 The Hecke Algebra and Crossed Products 7.4 Filtering the Hecke Algebra of a Compact Group by Normal Compact Open Subgroups 7.5 Proof of Theorem 7.2 8 Negative K-Groups and the Projective Class Group of Hecke Algebras Over Compact td-Groups 9 On the Algebraic K-Theory of the Hecke Algebra of a Covirtually Z Totally Disconnected Group 10 Some Input for the Farrell–Jones Conjecture 11 Characteristic p References Groupes de Coxeter finis: centralisateurs d'involutions 1. Premiers énoncés 2. Les groupes Gu+, Gu-, G"0365Gu+, G"0365Gu- 3. Détermination des groupes G+u 4. Les types A1, I2(m), H3 et H4 5. Type An-1 6. Type Bn 7. Type Dn 8. Résultats auxiliaires sur les groupes Gu- et G"0365Gu- 9. Type E6 10. Type E7 11. Type E8 12. Type F4 Références Groups, Drift and Harmonic Measures 1 Introduction 2 Preliminaries 2.1 Hyperbolic Geometry 2.2 Geometric Group Theory 2.3 Random Walks and the Drift 3 Some Examples 3.1 Regular Octagon Tilings 3.2 Hyperbolic Triangle Groups 4 Two Problems 4.1 The Harmonic Measure on the Unit Circle 4.2 Singularity of the Harmonic Measure 4.3 Dimension of the Harmonic Measure 4.4 Relation to the Avez Entropy 4.5 Final Remarks References Part VI History of Mathematics Some Problems in the History of Modern Mathematics 1 A Brief Historiography 2 The Problems 2.1 Write a Social-Historical Book in the History of Mathematics 2.2 Write a Book on the History of Applied Mathematics 2.3 A History of Differential Equations 2.4 A Book on the History of Mathematics in the Eighteenth Century 2.5 The History of Mathematics from a New Philosophical Perspective 3 Concluding Remarks References Mathematics Going Backward? A Logological Encounter Between Mathematics and Archaeology 1 The Archaeologist and the Mathematician 2 The Mathematician and the Archaeologist 3 A Link That Everyone Recognizes as a Knot 4 The Brunnian Link and the Pomegranate References Max Dehn as Historian of Mathematics 1 Introduction: Biography and History 2 Dehn in Frankfurt 3 Dehn on the History of Geometry 4 Historical Studies on Leibniz and Newton References Part VII Information Theory Multiterminal Statistical Inference: An Unsolved Problem 1 Introduction 2 Formulation of the Problem Conclusions References Information in Probability: Another Information-Theoretic Proof of a Finite de Finetti Theorem 1 Entropy and Information in Probability 2 Information-Theoretic Proof of a Finite de Finetti Theorem 2.1 Proof References Part VIII Logic Between the Rings Z/pnZ and the Ring Zp: Issues of Axiomatizability, Definability and Decidability 1 Introduction 1.1 The Diagrams and Some Axioms Connected to Them 1.2 Tarski's Legacy 1.3 Metamathematics of Z 1.4 Metamathematics of PA 2 Classes C of L-Structures 2.1 Let p Vary 2.2 Ax 3 The Metamathematical Analysis of Zp 3.1 The Axioms for Zp 3.2 Definable Sets 3.3 Decidability 4 Ax on Finite Fields and p-Adic Uniformities 4.1 CFin 4.2 Pseudofinite Fields 4.3 How Does Pseudofinite Relate to Finite? 5 Towards Decidability Uniform in p 6 Uniformity for the Zp 7 What Do We Know About the Z/pnZ? 7.1 Some Uninformative Decidability Results 7.2 Generalizing ``Pseudofinite'' 7.3 Digression on PA 8 How to Get at Axioms for the Z/pn Z 8.1 Definability Theory for the Z/pn Zp 8.1.1 The Simplest Case 8.2 Quantifier Elimination 9 Axioms for the Class of All Z/pnZ 10 Conclusion References Part IX Mathematical Models Mathematical Biology: Looking Back and Going Forward 1 What Is Mathematical Biology? 2 How Has Mathematical Biology Changed? 3 Going Forward References Kermack and McKendrick Models on a Two-Scale Network and Connections to the Boltzmann Equations 1 Introduction 2 The Basic Kermack–McKendrick Equation 3 The Two-Scale s-i-Model 4 More General Models of McKendrick Type with Several Competing Infectious Agents 5 Boltzmann's Equation in McKendrick Form References The Pygmalion Syndrome, or How to Fall in Love with Your Model 1 A Missed Opportunity? 2 Testing Microeconomic Theory 3 The Model as an Icon 4 The Right Use of Mathematics References Can We Teach Functions to an Artificial Intelligence by Just Showing It Enough ``Ground Truth''? 1 Introduction 2 Depth Estimation and Its Impossible Ground Truth 3 Human Ground Truth and Line Segment Detection 3.1 Line Segments Detected by Statistical Testing 3.2 Learning by Examples What a Line Segment Is 3.3 Comparison of All Methods 4 Conclusion References Part X Mathematical Physics and PDEs Divergence-Free Tensors and Cofactors in Geometry and Fluid Dynamics Hommage Notations 1 Divergence-Free Symmetric Tensors 1.1 Special Div-Free Tensors 1.1.1 Multi-linearization 1.2 The Homogeneous Case: The Minkowski Problem 1.3 Mixed Convex Bodies 1.4 The Euclidian Case 2 Compensated Integrability 2.1 The Role of Div-Free Tensors in Mathematical Physics 3 Div-BV Tensors and the Curvature of Hypersurfaces Comments Appendix: Cofactors and Geometric Mean in SPDn References Hyperbolic Conservation Laws: Past, Present, Future 1 Introduction 2 The Past 3 The Present 4 The Future References Which Nuclear Shape Generates the Strongest Attraction on a Relativistic Electron? An Open Problem in Relativistic Quantum Mechanics 1 A Conjecture for Relativistic Electrons 2 Dirac Operator with External Charges 2.1 Self-adjointness 2.2 Dirac Eigenvalues in the Gap 3 Two Results from EstLewSer-21aCh12,EstLewSer-21bCh12 3.1 Existence of an Optimal Measure μ 3.2 The Potential Energy Surface References Strong Singularities of Solutions to Nonlinear Elliptic Equations 1 An Open Problem References Some Connections Between Stochastic Mechanics, Optimal Control, and Nonlinear Schrödinger Equations 1 Introduction 2 Quantum Mechanics and Bose–Einstein Condensation 3 Nelson's Stochastic Mechanics 4 Non-linear Stochastic Mechanics 5 Convergence of Markovian N-Particle Approximation 6 The Case of the Dirac Delta Potential 6.1 The Intermediate Scaling Limit 6.2 The Gross–Pitaevskii Scaling Limit 7 Future Research Lines References Part XI Number Theory Can We Dream of a 1-Adic Langlands Correspondence? 1 Combinatorics Around the Breuil–Mézard Conjecture 1.1 Review on the Breuil–Mézard Conjecture 1.2 Encoding Representations with Combinatorial Data 1.2.1 The Case of 2-Dimensional Representations 1.2.2 The Gene 1.2.3 Higher Dimension and Group-Theoretic Formulation 1.3 Explicit Computations of Rλ,tρ 1.3.1 Review on Kisin's Construction of Rλ,tρ 1.3.2 Examples in Dimension 2: The Generic Case 1.3.3 Examples in Dimension 2: The Nongeneric Case 1.3.4 Higher Dimension and Group-Theoretic Formulation 2 The Field with One Element 2.1 Clues in Favor of a 1-Adic Breuil–Mézard Conjecture 2.2 Major Challenges 2.3 Conclusion Appendix: Remarks on Galois Theory in Characteristic 1 Brief Review of Geometry over F1 Galois Theory over F1 Galois Theory over Q1 References Computational Number Theory, Past, Present, and Future 1 Introduction 2 The Development of Pari/GP and Computational Number Theory Books 3 Arithmetic Statistics 4 Automorphic Forms, L-Functions, and Pari/GP Implementations 4.1 Algebraic Number Fields 4.2 Elliptic and Hyperelliptic Curves 4.3 L-Functions and Automorphic Forms 4.4 Numerical Methods 4.5 Software Enhancements 5 Additional Available Software and Algorithms 6 The Future References The Four Exponentials Problem and Schanuel's Conjecture 1 Introduction 2 Leopoldt's Conjecture on the p-Adic Rank of the Group of Units of an Algebraic Number Field 3 Conjecture on the Algebraic Independence of Logarithmsof Algebraic Numbers 4 The Four Exponentials Problem and Six Exponentials Theorem 5 Rank of Matrices 6 The Strong Six Exponentials Theorem and the Strong Four Exponentials Problem 7 Schanuel's Conjecture 8 Roy's Conjecture References Part XII Probability and Applications Research Going Forward? Reference The Future of Probability References Selected Problems in Probability Theory Personal Remarks 1 Introduction 2 Self-Avoiding Walks 2.1 Origins 2.2 Asymptotics 2.3 Self-Avoiding Walks in a Random Environment 3 Lorentz Scatterers 3.1 Background 3.2 Ehrenfest Wind/Tree Model 3.3 Poisson Mirrors 3.4 Manhattan Pinball 4 Two Stochastic Inequalities 4.1 Bunkbed Inequality 4.2 Negative Correlation 5 Randomly Oriented Square Lattice 6 Dynamic Stochastic Epidemics References Optimal Matching of Random Samples and Rates of Convergence of Empirical Measures 1 Euclidean Random Optimal Matching and Rates of Convergence of Empirical Measures 2 The One-Dimensional Case 3 The Ajtai–Komlós–Tusnády Theorem in Dimension 2 and the Ultimate Matching Conjecture 4 General Distributions and Higher Dimension 5 Mass Transportation, PDE, and Exact Limits and AsymptoticExpansions References Space-Time Stochastic Calculus and White Noise 1 Introduction 2 Equations with Noise and White Noise Theory 2.1 A More Elaborate Model 3 Multi-Parameter White Noise Calculus 3.1 The d-Parameter White Noise Probability Space 3.2 Chaos Expansion in Terms of Hermite Polynomials 3.3 The Stochastic Test Function Spaces (S) and the Stochastic Distribution Space (S)* 3.4 The Wick Product 3.5 Wick Product and Hermite Polynomials 3.6 Wick Multiplication and Itô Integration 4 The Space-Time Hida–Malliavin Calculus 4.1 The General Hida–Malliavin Derivative 5 Solving the Population Growth Equation 6 Concluding Remarks References