## Schedule

Below you can find the schedule for the joint activities and talks by the postdocs.

All times are in **Eastern Time**.

### Monday, August 3

- 12:00–13:00
- introduction
- 15:30–16:00
- On "completely decomposed" Grothendieck topologies and applications
- Elden Elmanto
- 16:00–16:30
- Upper ramification groups for arbitrary valuation rings
- Vaidehee Thatte

### Tuesday, August 4

- 12:00–12:30
- Constructing varieties with prescribed Hodge numbers modulo \(m\) in characteristic \(p\)
- Remy van Dobben de Bruyn
- 12:30–13:00
- Derived invariants of varieties in positive characteristic
- Daniel Bragg

### Wednesday, August 5

- 12:00–12:30
- The moduli of maps to an algebraic stack has a canonical obstruction theory
- Rachel Webb
- 12:30–13:00
- Period sheaves via derived de Rham cohomology
- Shizhang Li
- 15:30–?
- [
*optional*] treasure hunt in the Stacks Project

### Thursday, August 6

- 12:00–12:30
- Stable pairs with a twist
- Dori Bejleri
- 12:30–13:00
- Keel's theorem and quotients in mixed characteristic
- Jakub Witaszek
- 15:30–?
- [
*optional*] reference and bug hunt

### Friday, August 7

- 15:00–16:30
- group reports

## Abstracts

### Stable pairs with a twist

Dori Bejleri (Harvard University)

The notion of a stable log variety or stable pair \( (X,D) \) is the higher dimensional analogue of a stable pointed curve. The existence of a proper moduli space of stable pairs in any dimension has been established thanks to the last several decades of advancements in the minimal model program. However, the notion of a family of stable pairs remains quite subtle, and in particular a deformation-obstruction theory for this moduli problem is not known. Building on the work of Abramovich-Hassett, I will describe an approach to this question using a certain Deligne-Mumford stack canonically associated to the stable pair \( (X,D) \) and mention some applications of this approach. This is joint work with G. Inchiostro.

### Derived invariants of varieties in positive characteristic

Daniel Bragg (University of Californa, Berkeley)

A fundamental derived invariant associated to a smooth projective variety is its Hochschild homology. In positive characteristic, topological Hochschild homology gives another canonical derived invariant. We will explain how to compute with this object in practice by relating it to crystalline and de Rham-Witt cohomology. As a consequence, we obtain some new restrictions on the Hodge numbers of derived equivalent varieties in positive characteristic. We will also present an example of two derived equivalent 3-folds in characteristic 3 with different Hodge numbers. This is joint work with Benjamin Antieau and Nick Addington.

### Constructing varieties with prescribed Hodge numbers modulo \( m \) in characteristic \( p \)

Remy van Dobben de Bruyn (Princeton University)

The inverse Hodge problem asks which possible Hodge diamonds can occur for smooth projective varieties. While this is a very hard problem in general, Paulsen and Schreieder recently showed that in characteristic 0 there are no restrictions on the modulo \( m \) Hodge numbers, besides the usual symmetries. In joint work with Matthias Paulsen, we extend this to positive characteristic, where some new phenomena occur.

### On "completely decomposed" Grothendieck topologies and applications

Elden Elmanto (Harvard University)

At the time of writing, the Stacks project does not have a tag on "completely decomposed" topologies (like Nisnevich or cdh) in the sense of Voevodsky (though alluded in Tag 08GL). I will explain why I like them and introduce the cdarc topology which is a "completely decomposed" counterpart to Bhatt and Mathew's arc topology. This is based on joint work with Hoyois, Iwasa and Kelly.

### Period sheaves via derived de Rham cohomology

Shizhang Li (University of Michigan)

Fontaine's mysterious rings of periods have been understood via constructions related to differential (Colmez) and derived de Rham complex (Bhatt, Beilinson). Following this theme, we explain how to understand Scholze's period sheaf OBdR+ (which is one key ingredient in his proof of the de Rham comparison) in terms of the derived de Rham complex. This is a joint work with Haoyang Guo.

### Upper ramification groups for arbitrary valuation rings

Vaidehee Thatte (Binghamton University)

We define logarithmic and non-logarithmic ramification filtrations for arbitrary Henselian valuation rings. This joint work with K. Kato generalizes the ramification theory of complete discrete valuation rings of Abbes-Saito. We will discuss the notion of "defect", also known as ramification deficiency, that is specific to arbitrary valuation rings in positive residue characteristic.

### The moduli of maps to an algebraic stack has a canonical obstruction theory

Rachel Webb (University of Michigan)

Morphisms from nodal marked curves to a target \( X \) form an algebraic stack, called the moduli of maps to \( X \). These moduli have applications to string theory, where integrals over the moduli stack correspond to particle interactions. Making this notion of integration rigorous requires us to understand the deformation/obstruction theory of the moduli problem: one approach is the construction of the virtual fundamental class by Behrend-Fantechi. Their machinery requires a certain kind of obstruction theory on the moduli of maps. The goal of this talk is to explain why the moduli of stable maps carries this obstruction theory when \( X \) is an algebraic stack. Even when \( X \) is Deligne-Mumford (or a scheme), the explanation is more satisfying than the standard one in the literature.

### Keel's theorem and quotients in mixed characteristic

Jakub Witaszek (University of Michigan)

In trying to understand the geometry of characteristic zero varieties one can apply a wide range of techniques coming from analytic methods such as vanishing theorems. More complicated though they are, positive characteristic varieties come naturally with Frobenius action which sometimes allows for imitating analytic proofs or even showing results which are false over complex numbers. Of all the three classes, the mixed characteristic varieties are the most difficult to understand as they represent the worst of both worlds: one lacks the analytic methods as well the Frobenius action. What is key for many applications of Frobenius in positive characteristic (to birational geometry, moduli theory, constructing quotients, GIT, etc.) is the fact that every universal homeomorphism of algebraic varieties factors through a power of Frobenius. In this talk I will discuss a particular analogue of this fact (and applications thereof) in mixed characteristic.