## Tentative Schedule

Monday Tuesday Wednesday Thursday Friday
9:00 Registration 9:00 Habegger 9:00 Zhu 9:30 Pappas 9:00 Esnault
10:00 Rapoport 10:30 Huber 10:30 Kerz 11:00 Tamme 10:30 Lin
11:30 Temkin 11:45 Pilloni 11:45 Nicaise 13:30 Scholze
15:15 Caraiani 15:30 Gee Excursion 15:00 Ziegler 15:00 Fargues
16:45 Hellmann 17:00 Schmidt Excursion 16:30 Zhang
18:00 Get together Conference dinner

The duration of the talks is 60 minutes.

## Speakers

Ana Caraiani
Hélène Esnault
Laurent Fargues
Toby Gee
Philipp Habegger
Eugen Hellmann
Annette Huber
Moritz Kerz
Jie Lin
Johannes Nicaise
George Pappas
Vincent Pilloni
Michael Rapoport
Alexander Schmidt
Peter Scholze
Georg Tamme
Michael Temkin
Wei Zhang
Xinwen Zhu
Paul Ziegler

Unfortunately, Ruochuan Liu had to cancel his participation.

## Abstracts

#### Toby Gee: Moduli stacks of $(\varphi, \Gamma)$-modules

I will review some properties of the moduli stacks of $(\varphi, \Gamma)$-modules which I recently constructed with Matthew Emerton.

#### Eugen Hellmann: On automorphy lifting for semi-stable representations

Automorphy lifting theorems aim to show that a $p$-adic global Galois representation that is unramified almost everywhere and de Rham at places dividing $p$ is associated to an automorphic representation, provided its reduction modulo $p$ is. In the past years there has been a lot of progress in the case of polarizable representations that are crystalline at $p$. In the semi-stable case much less is known (beyond the 2-dimensional case). I will explain how one can use results about $p$-adic automorphic forms in combination with the Taylor-Wiles method to deduce the semi-stable case from the crystalline case.

#### Annette Huber: Exponential periods and o-minimality

(work in progress with Johan Commelin and Philipp Habegger) Exponential periods arise from the period isomorphism for irregular connections established by Hien and Roucairol. The underlying motivic theory was recently developed and studied by Fresan and Jossen. We give an equivalent characterisation as $\int_Ge^{-f}\omega$ with $G\subset \mathbf{C}^n$ semi-algebraic and satisfying additional properties, algebraic $f$ and $\omega$. This can be used to show that the absolute values of real and imaginary part of all effective exponential periods are volumes of sets definable in a certain o-minimal structure. These results are in analogy to a similar statement for classical periods and volumes of semi-algebraic sets. We take this to indicate that there is a deeper connection between the theory of periods and o-minimiality.

#### Johannes Nicaise: Stable rationality of complete intersections

I will explain an ongoing project with John Christian Ottem to establish several new classes of stably irrational complete intersections. Our results are based on degeneration techniques and a birational version of the nearby cycles functor that was developed in collaboration with Evgeny Shinder.

#### Michael Rapoport: $p$-adic uniformization of unitary Shimura curves

Cherednik's theorem states that a Shimura curve attached to a quaternion algebra over a totally real field $F$ which is split at exactly one archimedean place and is ramified at some $p$-adic place admits $p$-adic uniformization by the Drinfeld $p$-adic upper half plane. This theorem is now almost 50 years old but is still not understood, one of the main difficulties being that outside the case $F=\mathbb Q$ the Shimura curve does not represent a moduli problem of abelian varieties. I will show that when the multiplicative group of the quaternion algebra is replaced by the group of unitary similitudes of a binary hermitian space, things become drastically better. This is joint work with S. Kudla and Th. Zink, and improves on results of P. Scholze (yes, this can be done!).

#### Michael Temkin: New algorithms to resolve singularities

The classical desingularization method of Hironaka was studied by many mathematicians for nearly 40 years. These efforts led to a number of relatively simple constructions/descriptions of essentially the same algorithm by Bierstone-Milman, Villamayor, Włodarczyk, Kollár and others. Very recently, logarithmic and stack-theoretic methods were used to construct new desingularization algorithms that possess better properties -- work faster, and resolve morphisms and log schemes or do not use divisors and history at all. In addition, the basic tools are more complicated (e.g. weighted stack-theoretic blow ups replace the usual blow ups), but the algorithms are simpler. In my talk I will outline the main ideas of the classical method and its recent successors. (Joint work with D. Abramovich and J. Włodarczyk.)