This volume stems from the Linde Hall Inaugural Math Symposium, held from February 22–24, 2019, at California Institute of Technology, Pasadena, California. The content isolates and discusses nine mathematical problems, or sets of problems, in a deep way, but starting from scratch. Included among them are the well-known problems of the classification of finite groups, the Navier-Stokes equations, the Birch and Swinnerton-Dyer conjecture, and the continuum hypothesis. The other five problems, also of substantial importance, concern the Lieb–Thirring inequalities, the equidistribution problems in number theory, surface bundles, ramification in covers and curves, and the gap and type problems in Fourier analysis. The problems are explained succinctly, with a discussion of what is known and an elucidation of the outstanding issues. An attempt is made to appeal to a wide audience, both in terms of the field of expertise and the level of the reader.
Author(s): A. Kechris, N. Makarov, D. Ramakrishnan, X. Zhu
Series: Proceedings of Symposia in Pure Mathematics 104
Publisher: American Mathematical Society
Year: 2021
Language: English
Commentary: decrypted from AFC79CA346E232BFF0D02553B8350A91 source file
Pages: 221
Cover
Title page
Contents
Preface
The Linde Hall Inaugural Math Symposium at Caltech
Lectures
The finite simple groups and their classification
Motivation
Groups of prime order
Alternating groups
Groups of Lie type
Sporadic groups
The proof of CFSG
The local theory of finite groups
References
The Birch and Swinnerton-Dyer Conjecture: A brief survey
1. Introduction
2. The Birch and Swinnerton-Dyer conjecture
3. Results
4. Methods: an instructive example
5. \color{blue}Some open problems
Acknowledgments
References
Bounding ramification by covers and curves
1. Introduction
2. Elementary properties of \sS(?,?,?)
3. Reduction of Theorem 1.1 to the case ?=\A^{?}
4. Proof of Theorem 1.1 and Corollary 1.3
5. Rank one
6. Remarks
Acknowledgments
References
The Lieb–Thirring inequalities: Recent results and open problems
1. The Lieb–Thirring problem
2. Application: Stability of Matter
3. The Lieb–Thirring inequality for Schrödinger operators
4. Lieb–Thirring inequalities for Schrödinger operators. II
5. Further directions of study
6. Some proofs
References
Some topological properties of surface bundles
1. Introduction
2. Constructions
3. Flat circle bundles
4. Selfintersection numbers of sections
5. Cohomology of surface bundles
References
Some recents advances on Duke’s equidistribution theorems
1. Introduction
2. Duke’s Equidistribution Theorems: the original proof
3. ?-functions and Waldspurger’s formula
4. Ergodic methods
Acknowledgments
References
Gap and Type problems in Fourier analysis
1. Introduction
2. Forms of UP
3. The Gap problem
4. The Type problem
5. Pólya sequences and oscillations of Fourier Integrals
Acknowledgments
References
Quantitative bounds for critically bounded solutions to the Navier-Stokes equations
1. Introduction
2. Notation
3. Basic estimates
4. Carleman inequalities for backwards heat equations
5. Main estimate
6. Applications
References
The Continuum Hypothesis
1. Introduction
2. The Universe of Sets
3. The cumulative hierarchy
4. Cohen’s method
5. Beyond the \ZFC axioms
6. Perhaps \CH simply has no answer
7. Back to the problem of \CH
8. An unexpected entanglement
9. The effective cumulative hierarchy: Gödel’s universe ?
10. The axiom ?=? and large cardinals
11. The universally Baire sets
12. The universally Baire sets as the ultimate generalization of the projective sets
13. Gödel’s transitive class \HOD
14. \HOD^{?(?,\reals)} and large cardinals
15. The axiom ?=\UL
16. The language of large cardinals: elementary embeddings
17. The ?-cover and ?-approximation properties
18. The ?-genericity property and strong universality
19. The \UL Conjecture and the two futures of Set Theory
20. Concluding remarks
References
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