Note: Attendance is by invitation only.

Organizers:

Bhargav Bhatt, University of Michigan

Martin Olsson, UC Berkeley

Organized by Bhargav Bhatt (University of Michigan) and Martin Olsson (UC Berkeley), the symposium will bring together experts to explore recent developments in p-adic Hodge theory and understand the emerging relationship of p-adic Hodge theory with other subjects in mathematics.

- The relationship between p-adic Hodge theory, algebraic K-theory, and topological Hochschild homology.
- Recent developments in integral p-adic Hodge theory.
- The connection between p-adic Hodge theory and derived algebraic geometry.

## Agenda

SUNDAY | |
---|---|

7:30 -9:30 PM | Dinner at La Salle |

MONDAY | |

8:30 – 10:30 AM | Breakfast |

10:30 – 11:30 AM | Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series #1: Matthew Morrow An overview of “Integral p-adic Hodge Theory” |

11:30 -12:00 PM | Break |

12:00 – 1:00 PM | Laurent Fargues | Simple connectedness of the fibers of an Abel-Jacobi morphism and local class field theory |

1:00 – 2:00 PM | Lunch |

2:00 – 4:30 PM | Discussion & Recreation* |

5:00 – 6:00 PM | Bryden Cais | Breuil-Kisin modules and crystalline cohomology |

6:00 – 6:30 PM | Break |

6:30 – 7:30 PM | Minhyong Kim | Reciprocity laws and principal bundles |

8:00 – 9:30 PM | Dinner at La Salle |

TUESDAY | |

8:30 – 10:30 AM | Breakfast |

10:30 – 11:30 AM | Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series 2: Peter Scholze Hochschild homology, cyclic homology and relations to de Rahm cohomology |

11:30 -12:00 PM | Break |

12:00 – 1:00 PM | Michel Gros | Simpson correspondance in characteristic p>0 and splittings of the algebra of PD-differential operators |

1:00 – 2:00 PM | Lunch |

2:00 – 4:30 PM | Discussion & Recreation* |

5:00 – 6:00 PM | Ahmed Abbes | Lifting the Cartier transform of Ogus-Vologodsky modulo p^n |

6:00 – 6:30 PM | Break |

6:30 – 7:30 PM | Dmitry Kaledin | Co-periodic cyclic homology |

8:00 – 9:30 PM | Dinner at Wintergarden |

WEDNESDAY | |

7:30 – 9:30 AM | Breakfast |

9:45 – 2:00 PM | Guided Hike to Partnach Gorge |

2:00 – 3:00 PM | Lunch at Wintergarden |

3:00 – 5:00 PM | Recreation & Discussion |

5:00 – 5:30 PM | Tea |

5:30 – 6:30 PM | Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series 3: Jacob Lurie Topological Hochschild Homology |

6:30 – 7:30 PM | Wieslawa Niziol | Cohomology of p-adic Stein spaces |

8:00 – 9:30 PM | Dinner at La Salle |

THURSDAY | |

8:30 – 10:30 AM | Breakfast |

10:30 – 11:30 AM | Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series 4: Lars Hesselholt THH and cyclotomic spectra |

11:30 -12:00 PM | Break |

12:00 – 1:00 PM | Pierre Colmez | Cohomology of p-adic analytic curves |

1:00 – 2:00 PM | Lunch |

2:00 – 4:30 PM | Discussion & Recreation* |

5:00 – 6:00 PM | Ana Caraiani | Galois representations and torsion classes |

6:00 – 6:30 PM | Break |

6:30 – 7:30 PM | Kiran Kedlaya | Tautological local systems and (phi, gamma)-modules |

8:00 – 9:30 PM | Dinner at La Salle |

FRIDAY | |

8:30 – 10:30 AM | Breakfast |

10:30 – 11:30 AM | Integral p-adic Hodge Theory & Topological Cyclic Homology: A Five-talk Series 5: Peter Scholze A “weight” filtration on THH and its relation with crystalline cohomology and A\Omega |

11:30 -12:00 PM | Break |

12:00 – 1:00 PM | Ruochuan Liu | Logarithmic OB_dR |

1:00 – 2:00 PM | Lunch |

2:00 – 4:30 PM | Discussion & Recreation* |

5:00 – 6:00 PM | Takeshi Tsuji | The relative Fontaine-Laffaille theory and Ainf representations with Frobenius |

8:00 – 9:30 PM | Dinner at Kaminstüberl |

LOCATIONS | |

SESSIONS | Pavillion located at the Schloss Elmau Retreat |

MEALS | La Salle unless otherwise noted |

TEA & DISCUSSION | Pavillion located at the Schloss Elmau Retreat |

EXCURSION | Meet in Schloss Elmau Lobby |

SATURDAY DEPARTURE | Meet in Schloss Elmau Lobby |

*Participants may explore the hotel property and its surrounding areas as well as engage in informal discussion with other participants.

### Start Time

### Meeting Location

### AV

### Wednesday Excursion

### Dress Code

If you plan on taking part in the hike to Partnach Gorge we advise you to wear hiking boots, or even better, light mountain boots, warm clothing (e.g. a sweater), sun protection (e.g. light cap) and take waterproofs (e.g. raincoat or umbrella) with you. Bringing along a small backpack or satchel in which to carry your water, camera and other items may also be useful to you.

## Participants

Download participant list PDF here.

Ahmed Abbes | Institut des Hautes Études Scientifiques |

Bhargav Bhatt | University of Michigan |

Bryden Cais | University of Arizona |

Ana Caraiani | University of Bonn |

Pierre Colmez | Institut de Mathématiques de Jussieu |

Brian Conrad | Stanford University |

Aise Johan de Jong | Columbia University |

Gerd Faltings | Max-Planck-Institute for Mathematics |

Laurent Fargues | Institut de Mathématiques de Jussieu |

Jean-Marc Fontaine | Université Paris-Sud |

Ofer Gabber | Institut des Hautes Études Scientifiques |

Michel Gros | Université Rennes 1 |

Lars Hesselholt | University of Copenhagen |

Dmitry Kaledin | Steklov Math Institute |

Kiran Kedlaya | UC San Diego |

Minhyong Kim | University of Oxford |

Ruochuan Liu | Beijing International Center for Mathematical Research |

Jacob Lurie | Harvard University |

Matthew Morrow | Universität Bonn |

Wiesława Nizioł | École Normale Supérieure de Lyon |

Martin Olsson | UC Berkeley |

Peter Scholze | Universität Bonn |

Takeshi Tsuji | University of Tokyo |

## Abstracts

### Ahmed Abbes: Lifting the Cartier transform of Ogus-Vologodsky modulo p^n [following H. Oyama, A. Shiho and D. Xu]

Almost simultaneously, G. Faltings proposed a \(p\)-adic analogue of the Simpson correspondence. The relationship between these two correspondences remains mysterious, and the first challenge is to lift the Cartier transform modulo \(p^n\). A. Shiho did the first step by lifting the “local” correspondence modulo \(p^n\), given a lifting of the relative Frobinus modulo \(p^{n+1}\). Independently, in his thesis under the supervision of T. Tsuji, H. Oyama proposed a very beautiful interpretation of the Cartier transform (modulo \(p\)) as the pull-back by a morphism of ringed topoi. In his PhD thesis, my student D. Xu uses Oyama topos to “glue” Shiho’s local constructions and hence lift the Cartier transform modulo \(p^n\), under the (only) assumption that \(X\) lifts to a smooth formal scheme over the Witt vectors of k.

Abbes will report on the works of Shiho, Oyama and Xu.

### Bryden Cais: Breuil—Kisin modules and crystalline cohomology

In this talk, Cais will explain how to descend this result to obtain the Breuil—Kisin module over \(W(k)[[u]]\) when \(i < p-1\) and the crystalline cohomology of the special fiber of \(X\) is \(p\)-torsion-free in degrees \(i\) and \(i+1\).

This is joint work with Tong Liu.

### Ana Caraiani: Galois representations and torsion classes

### Pierre Colmez: Cohomology of \(p\)-adic analytic curves

### Laurent Fargues: Simple connectedness of the fibers of an Abel-Jacobi morphism and local class field theory

### Michel Gros: Simpson correspondance in characteristic \(p>0\) and splittings of the algebra of PD-differential operators

### Dmitry Kaledin: Co-periodic cyclic homology

### Kiran Kedlaya: Tautological local systems and (phi, Gamma)-modules

### Minhyong Kim: Reciprocity laws and principal bundles

### Ruochuan Liu: Logarithmic \(OB_dR\)

### Wiesia Niziol: Cohomology of \(p\)-adic Stein spaces

### Takeshi Tsuji: The relative Fontaine-Laffaille theory and Ainf representations with Frobenius.

### Integral \(p\)-adic Hodge theory and topological cyclic homology (series)

### 1) Matthew Morrow: An overview of “Integral \(p\)-adic Hodge theory”

### 2) Peter Scholze: Hochschild homology, cyclic homology, and relations to de Rham cohomology

### 3) Jacob Lurie: Topological Hochschild Homology

*topological Hochschild homology*, which is obtained by taking \(R\) to be the sphere spectrum.

In this talk, Lurie will give a brief introduction to the language of spectra, outline the construction of topological Hochschild homology, and give some sense of why it might be preferable to its algebraic cousin.

### 4) Lars Hesselholt: \(THH\) and cyclotomic spectra

### 5) Peter Scholze: A “weight” filtration on \(THH\), and its relation with (crystalline cohomology and) \(A\Omega\)

For \(p\)-complete commutative rings, we define a “weight” filtration on THH and related objects like \(TR^n\), \(TF\), \(THH^{hS^1}\) and \(TC\) (the latter two under mild hypothesis on the ring). For smooth \(F_p\)-algebras, the graded pieces recover objects known from crystalline cohomology, such as (truncations of) \(\Omega\) (for \(THH\)), \(W_n\Omega\) (for \(TR^n\)), \(W\Omega\) (truncated for \(TF\), untruncated for \(THH^{hS^1})\), along with the Nygaard filtration on it, which can be used to define Milne’s sheaves \(Z_p(r)=\nu_r[-r]\) (which are \(r\)-th graded piece of \(TC\)). Similarly, for smooth \(O_{C_p}\)-algebras, the graded pieces recover objects known from our paper on integral \(p\)-adic Hodge theory, such as (truncations of) \(\tilde{\Omega}\) (for \(THH\)), \(\tilde{W_n\Omega}\) (for \(TR^n\)) and \(A\Omega\) (truncated for \(TF\), untruncated for \(THH^{hS^1}\)), along with a Nygaard filtration on it, which can be used to construct certain syntomic complexes \(Z_p(r)\) (which are the \(r\)-th graded piece of \(TC\)).

Scholze will explain the construction of the filtration, and outline the proof of these comparison results. (Already the crystalline case is new.) An application is an extended definition of \(A\Omega\), which gives the descent of Breuil—Kisin—Fargues modules to Breuil—Kisin modules for proper smooth (formal) schemes defined over a discretely valued field.

## Travel

### Air

Participants arriving via plane will arrive via Munich International Airport (MUC).

### Train

All participants are required to arrive on Sunday prior to the meeting’s start and attend the entire week. No accommodations will be made for partial participation.

### Additional Night at Elmau

### Passports

### Local Ground Transportation

#### Arrival at Munich International Airport

Taxi Mathe, Mr. Mathe

taxi.mathe(replace this with the @ sign)googlemail.com

phone 08821 9663691

mobile 0170 8399 933

Travel time from Munich to Schloss Elmau is 90 minutes.

#### Arrival at Klais Train Station

Travel time from Kalis to Elmau is approximately 15 minutes.

#### Departure from Schloss Elmau

## Hotel

In Elmau 2, 82493 Krün

Germany

Tel: +49 (0) 8823 18-0

www.schloss-elmau.de

Check-in time is at 4:30 PM

Check-out time is 11:30 AM

## Meals

### While Traveling

### Day of Arrival (Sunday)

### Monday through Friday

**Breakfast** will be served daily starting at 7:30 AM at La Salle restaurant. Food at La Salle is served buffet-style and is open to all hotel guests and a table has been set aside so that symposia participants may dine together.

**Lunch** will be served daily at La Salle.

**Dinner** will be hosted around the hotel property at one of the restaurants. Locations are noted on the agenda.

### Day of Departure (Saturday)

## Contacts

### Registration and Travel Assistance

christophe.vergnol(replace this with the @ sign)protravelinc.com

(646) 747-9767

### Registration, Hotel and General Meeting Assistance

*Senior Executive Assistant, Simons Foundation*

mfazzi(replace this with the @ sign)simonsfoundation.org

(212) 524-6080

## Reimbursement and Travel Policy

Receipts are required for any expenses over $50 USD and are due within THIRTY DAYS (30) after the conclusion of the symposium. Should you have any questions, please contact Meghan Fazzi.