**Proceedings of Symposia in Pure Mathematics**

Volume: 104;
2021;
221 pp;
Softcover

MSC: Primary 03; 14; 11; 20; 30; 35; 37; 57;

**Print ISBN: 978-1-4704-5490-6
Product Code: PSPUM/104**

List Price: $137.00

AMS Member Price: $109.60

MAA Member Price: $123.30

**Electronic ISBN: 978-1-4704-6746-3
Product Code: PSPUM/104.E**

List Price: $137.00

AMS Member Price: $109.60

MAA Member Price: $123.30

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# Nine Mathematical Challenges: An Elucidation

Share this page *Edited by *
*A. Kechris; N. Makarov; D. Ramakrishnan; X. Zhu*

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.

#### Readership

Graduate students and researchers interested in mathematics.

# Table of Contents

## Nine Mathematical Challenges: An Elucidation

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- The Linde Hall Inaugural Math Symposium at Caltech ix10
- The finite simple groups and their classification 112
- The Birch and Swinnerton-Dyer Conjecture: A brief survey 1122
- Bounding ramification by covers and curves 3142
- The Lieb–Thirring inequalities: Recent results and open problems 4556
- Some topological properties of surface bundles 8798
- Some recents advances on Duke’s equidistribution theorems 107118
- Gap and Type problems in Fourier analysis 133144
- Quantitative bounds for critically bounded solutions to the Navier-Stokes equations 149160
- The Continuum Hypothesis 195206
- 1. Introduction 195206
- 2. The Universe of Sets 196207
- 3. The cumulative hierarchy 199210
- 4. Cohen’s method 199210
- 5. Beyond the \ZFC axioms 201212
- 6. Perhaps \CH simply has no answer 202213
- 7. Back to the problem of \CH 203214
- 8. An unexpected entanglement 205216
- 9. The effective cumulative hierarchy: Gödel’s universe 𝐿 207218
- 10. The axiom 𝑉=𝐿 and large cardinals 208219
- 11. The universally Baire sets 209220
- 12. The universally Baire sets as the ultimate generalization of the projective sets 210221
- 13. Gödel’s transitive class \HOD 211222
- 14. \HOD^{𝐿(𝐴,\reals)} and large cardinals 212223
- 15. The axiom 𝑉=\UL 213224
- 16. The language of large cardinals: elementary embeddings 215226
- 17. The 𝛿-cover and 𝛿-approximation properties 216227
- 18. The 𝛿-genericity property and strong universality 217228
- 19. The \UL Conjecture and the two futures of Set Theory 218229
- 20. Concluding remarks 219230
- References 221232

- Back Cover Back Cover1234