**Proceedings of Symposia in Pure Mathematics**

Volume: 97;
2018;
635 pp;
Hardcover

MSC: Primary 14; 53;

**Print ISBN: 978-1-4704-3578-3
Product Code: PSPUM/97.2**

List Price: $133.00

AMS Member Price: $106.40

MAA Member Price: $119.70

**Electronic ISBN: 978-1-4704-4680-2
Product Code: PSPUM/97.2.E**

List Price: $133.00

AMS Member Price: $106.40

MAA Member Price: $119.70

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# Algebraic Geometry: Salt Lake City 2015

Share this page *Edited by *
*Tomasso de Fernex; Brendan Hassett; Mircea Mustaţă; Martin Olsson; Mihnea Popa; Richard Thomas*

A co-publication of the AMS and Clay Mathematics Institute

This is Part 2 of a two-volume set.

Since Oscar Zariski organized a meeting in 1954, there has
been a major algebraic geometry meeting every decade: Woods Hole
(1964), Arcata (1974), Bowdoin (1985), Santa Cruz (1995), and Seattle
(2005). The American Mathematical Society has supported these summer
institutes for over 50 years. Their proceedings volumes have been
extremely influential, summarizing the state of algebraic geometry at
the time and pointing to future developments.

The most recent Summer Institute in Algebraic Geometry was held
July 2015 at the University of Utah in Salt Lake City, sponsored by
the AMS with the collaboration of the Clay Mathematics Institute. This
volume includes surveys growing out of plenary lectures and seminar
talks during the meeting. Some present a broad overview of their
topics, while others develop a distinctive perspective on an emerging
topic.

Topics span both complex algebraic geometry and arithmetic
questions, specifically, analytic techniques, enumerative geometry,
moduli theory, derived categories, birational geometry, tropical
geometry, Diophantine questions, geometric representation theory,
characteristic \(p\) and \(p\)-adic tools, etc. The
resulting articles will be important references in these areas for
years to come.

#### Readership

Graduate students and researchers working in algebraic geometry and its applications.

# Table of Contents

## Algebraic Geometry: Salt Lake City 2015

Table of Contents pages: 1 2

- Cover Cover11
- Title page iii4
- Contents v6
- Preface vii8
- Scientific program ix10
- Part 2 118
- Betti Geometric Langlands 320
- Specializing varieties and their cohomology from characteristic 0 to characteristic 𝑝 4360
- How often does the Hasse principle hold? 89106
- Tropical methods in the moduli theory of algebraic curves 103120
- A graphical interface for the Gromov-Witten theory of curves 139156
- Some fundamental groups in arithmetic geometry 169186
- From local class field to the curve and vice versa 181198
- Introduction 181198
- 1. The curve 182199
- 2. Vector bundles 185202
- 3. The curve compared to ℙ¹ 187204
- 4. 𝐺-bundles on the curve ([4]) 188205
- 5. Archimedean/𝑝-adic twistors 190207
- 6. The fundamental class of the curve is the fundamental class of class field theory ([4]) 192209
- 7. Conjectures: ramified local systems and coverings 193210
- 8. Speculations: Fourier transform and 𝑝-adic local Langlands correspondence 196213
- References 197214

- Intrinsic mirror symmetry and punctured Gromov-Witten invariants 199216
- Diophantine and tropical geometry, and uniformity of rational points on curves 231248
- On categories of (𝜑,Γ)-modules 281298
- 1. The original category of (𝜑,Γ)-modules 283300
- 2. Interlude on perfectoid fields 286303
- 3. Slopes of 𝜑-modules 291308
- 4. From 𝜑-modules to (𝜑,Γ)-modules 293310
- 5. Cohomology of (𝜑,Γ)-modules 295312
- 6. The cyclotomic deformation 297314
- 7. Iwasawa cohomology and the cyclotomic deformation 298315
- 8. Coda: beyond the cyclotomic tower 302319
- References 302319

- Principal bundles and reciprocity laws in number theory 305322
- Bi-algebraic geometry and the André-Oort conjecture 319336
- 1. Introduction 319336
- 2. The André-Oort conjecture 322339
- 3. Special structures on algebraic varieties 329346
- 4. Bi-algebraic geometry 331348
- 5. O-minimal geometry and the Pila-Wilkie theorem 337354
- 6. O-minimality and Shimura varieties 340357
- 7. The hyperbolic Ax-Lindemann conjecture 341358
- 8. The two main steps in the proof of the André-Oort conjecture 345362
- 9. Lower bounds for Galois orbits of CM-points 350367
- 10. Further developments: the André-Pink conjecture 353370
- References 355372

- Moduli of sheaves: A modern primer 361378
- Part 1. Background 363380
- Part 2. A thought experiment 375392
- Geometric invariants for non-archimedean semialgebraic sets 389406
- Symplectic and Poisson derived geometry and deformation quantization 405422
- Varieties that are not stably rational, zero-cycles and unramified cohomology 459476
- On the proper push-forward of the characteristic cycle of a constructible sheaf 485502
- The 𝑝-adic Hodge decomposition according to Beilinson 495512
- 1. Introduction 496513
- 2. The cotangent complex and the derived de Rham algebra 503520
- 3. Differentials and the de Rham algebra for 𝑝-adic rings of integers 517534
- 4. Construction of period rings 532549
- 5. Beilinson’s comparison map 538555
- 6. The comparison theorem 549566
- A. Appendix: Methods from simplicial algebra 560577
- A.1. Simplicial methods 560577
- A.2. Associated chain complexes 562579
- A.3. Bisimplicial objects 563580
- A.4. Simplicial resolutions 564581
- A.5. Derived functors of non-additive functors 564581
- A.6. Application: derived exterior powers and divided powers 565582
- A.7. Cohomological descent 568585
- A.8. Hypercoverings 568585

- References 570587

- Specialization of ℓ-adic representations of arithmetic fundamental groups and applications to arithmetic of abelian varieties 573590

Table of Contents pages: 1 2