Mathematics Going Forward: Collected Mathematical Brushstrokes

ISBN-13: 9783031122439 / Angielski / Miękka / 2023 / 603 str.

Jean-Michel Morel; Bernard Teissier

Opis

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.

Spis treści

1. Introduction.- Jean-Pierre Eckmann.-Part I Algebraic Geometry.- 2. Jean-Louis Colliot-Thélène: Une liste de problems.-3. Bernard Teissier: Some ideas in need of clarification in resolution of singularities and the geometry of discriminants.- 4. Pierre Schapira: Shifted sheaves for space-time.-5. Loring W. Tu: Lefschetz fixed point theorems for correspondences.- Part II Dynamical Systems.- 6. Alain Chenciner: Vous avez dit “qualitatif” ?.- 7. Ariel Amir and Tadashi Tokieda: A Tale o’ pi by pilota.- Part III Finance.- 8. L.C.G Rogers: What next ?.- 9: Monique Jeanblanc: Some remarks on enlargement of filtration and finance.- 10. Paul Embrechts, Claudia Klüppelberg and Thomas Mikosch: Modern extreme value theory at the interface of risk management, Bayesian networks and heavy-tailed time series.- 11. Christopher Almost, John Lehoczky, Steven Shreve and Xiaofeng Yu: Limits of limit-order books.- Part IV Geometry.- 12. Jean-Pierre Bourguignon: Spinors in 2022.- 13. Carles Casacuberta: Cohomological localizations and set-theoretical reflection.- 14. Parvaneh Joharinad and Jürgen Jost: Geometry of data.- 15. Vitali Milman:  A chapter about Asymptotic Geometric Analysis: Isomorphic position of centrally symmetric convex bodies.- 16. Ieke Moerdijk: A mysterious tensor product in topology.- Part V Groups.- 17. Toshiyuki Kobayashi: Conjectures on reductive homogeneous spaces.-  18. Martin R. Bridson: Profinite rigidity and free groups.-19. Arthur Bartels and Wolfgang Lück: On the algebraic -theory of Hecke algebras.- 20. Jean-Pierre Serre: Groupes de Coxeter finis: centralisateurs d’involutions .- 21: Mark Pollicott and Polina Vytnova: Groups, drift and harmonic measures.-Part VI History of Mathematics.- 22. Jeremy Gray: Some problems in the history of modern mathematics.-23. Norbert Schappacher: Mathematics going backward? A logological encounter between mathematics and archaeology.-24. David E. Rowe: Max Dehn as historian of mathematics.- Part VII Information Theory.- 25. Shun-ichi Amari: Multiterminal statistical inference: An unsolved problem.- 26. Lampros Gavalakis and Ioannis Kontoyiannis: Information in probability: Another information-theoretic proof of a finite de Finetti theorem.- Part VIII Logic.- 27. Angus J. Macintyre: Between the rings Z/Z and the ring Z: Issues of axiomatizability, definability and decidability.- Part IX Mathematical Models.-28. Philip K Maini: Mathematical biology: Looking back and going forward.- 29. Stephan Luckhaus and Angela Stevens: Kermack and McKendrick models on a two-scale network and connections to the Boltzmann equations.- 30. Ivar Ekeland: The Pygmalion syndrome, or how to fall in love with your model.- 31. Adrien Courtois, Thibaud Ehret, Pablo Arias and Jean-Michel Morel: Can we teach functions to an artificial intelligence by just showing it enough “ground truth”?.- Part X Mathematical Physics and PDEs.-32. Denis Serre: Divergence-free tensors and cofactors in geometry and fluid dynamics.-33. Constantine M Dafermos: Hyperbolic conservation laws: Past, present, future.- 34.Maria J. Esteban, Mathieu Lewin and Éric Séré: Which nuclear shape generates the strongest attraction on a relativistic electron? An open problem in relativistic quantum mechanics.- 35.Vladimir Maz’ya: Strong singularities of solutions to nonlinear elliptic equations.- 36. Sergio Albeverio, Francesco Carlo De Vecchi and Stefania Ugolini: Some connections between stochastic mechanics, optimal control, and nonlinear Schrödinger equations.- Part XI Number Theory.- 37. Xavier Caruso, Agnès David and Ariane Mézard: Can we dream of a 1-adic Langlands correspondence?- 38. Henri Cohen: Computational number theory, past, present, and future.- 39. Michel Waldschmidt: The four exponentials problem and Schanuel’s conjecture.- Part XII Probability and Applications.- 40. S.R.S. Varadhan: Research going forward?  41. Philip Protter: The future of probability.- 42. Geoffrey R. Grimmett: Selected problems in probability theory.- 43. Michel Ledoux: Optimal matching of random samples and rates of convergence of empirical measures.- 44. Bernt Øksendal: Space-time stochastic calculus and white noise.

O autorze

Jean-Michel Morel is professor of mathematics at Ecole Normale Supérieure Paris-Saclay.  He works on theories and algorithms for the restoration and automatic analysis of digital images and video.

Bernard Teissier is Emeritus Director of Research of CNRS at the Institut Mathématique de Jussieu-Paris Rive Gauche. His research interests center on the algebra and geometry of singularities in algebraic and analytic geometry.