Nine Mathematical Challenges: An Elucidation

Description

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. 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(𝑗,𝑟,𝐷) 3. Reduction of Theorem 1.1 to the case 𝑋=\A^{𝑑} 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. 𝐿-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 𝐿 10. The axiom 𝑉=𝐿 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^{𝐿(𝐴,\reals)} and large cardinals 15. The axiom 𝑉=\UL 16. The language of large cardinals: elementary embeddings 17. The 𝛿-cover and 𝛿-approximation properties 18. The 𝛿-genericity property and strong universality 19. The \UL Conjecture and the two futures of Set Theory 20. Concluding remarks References Back Cover