## Schedule

• the coffee breaks are in the East Hall, Common Room (on the 2nd floor)
• the talks are in the East Hall, Room 1360
• the group work is in the basement of the Modern Languages Building,
Bhargav Bhatt
MLBB111
MLBB115A
Johan de Jong
MLBB122
Matthew Emerton
MLBB124
Max Lieblich
MLBB134
Davesh Maulik
MLBB135
• the (optional) evening sessions are in East Hall, 1360 and Common Room

### Monday, July 31

09:00–09:30
registration
09:30–10:30
welcome
history and future of the Stacks project (slides)
10:30–11:00
coffee break
11:00–12:30
group work
14:00–15:30
group work
15:30–16:00
coffee break
16:00–17:00
Nonconnective simplicial commutative rings
Akhil Mathew

### Tuesday, August 1

09:00–09:30
coffee break
09:30–10:30
Shimura stacks
Lenny Taelman
10:30–11:00
coffee break
11:00–12:30
group work
14:00–15:30
group work
15:30–16:00
coffee break
16:00–17:00
Geometric realizations of logarithmic schemes
Mattia Talpo
17:15–18:00
editing the Stacks project, and an introduction to Git (in EH 1360) (slides)
18:00–...
cross-referencing, typos and pizza

### Wednesday, August 2

09:00–09:30
coffee break
09:30–10:30
Associated forms and applications
Maksym Fedorchuk
10:30–11:00
coffee break
11:00–12:30
group work
14:00–15:30
group work
15:30–16:00
coffee break
16:00–17:00
Extremely indecomposable division algebras and algebraic cycles
Daniel Krashen
17:15–18:00
infrastructure of the Stacks project (slides)
18:00–...
cross-referencing, typos, hackathon and pizza

### Thursday, August 3

09:00–09:30
coffee break
09:30–10:30
Finiteness results for hypersurfaces over number fields
Ariyan Javanpeykar
10:30–11:00
coffee break
11:00–12:30
group work
14:00–15:30
group work
15:30–16:00
coffee break
16:00–17:00
brainstorming session

### Friday, August 4

09:00–09:30
coffee break
09:30–10:30
A gentle approach to crystalline cohomology
Jacob Lurie
10:30–11:00
coffee break
11:00–12:30
group work
14:00–15:30
group work
15:30–16:00
coffee break
16:00–17:00
group presentations

## Abstracts

### Associated forms and applications

Maksym Fedorchuk (Boston College)

I will describe the theory of associated forms (these are Macaulay inverse systems of certain balanced complete intersections) and two of its applications. The first is a proof of their GIT polystability and a resulting invariant-theoretic version of the Mather-Yau theorem for homogeneous hypersurface singularities, obtained in joint work with Alexander Isaev. The second is an if-and-only-if criterion for direct sum decomposability of a smooth homogeneous polynomial in terms of the factorization properties of its associated form. This criterion leads to an algorithm for computing direct sum decompositions over any field either of characteristic zero, or sufficiently large positive characteristic, for which polynomial factorization algorithms exist.

### Finiteness results for hypersurfaces over number fields

Ariyan Javanpeykar (University of Mainz)

I will explain why the stack of smooth hypersurfaces is uniformisable by a scheme, and use this "uniformisation" to deduce arithmetic consequences of the Lang-Vojta conjecture for smooth hypersurfaces over number fields. This is joint work with Daniel Loughran.

### Extremely indecomposable division algebras and algebraic cycles

Daniel Krashen (University of Georgia)

In this talk, I will explore some of the connections between the arithmetic of division algebras and the geometry of Brauer-Severi varieties. A division algebra is called decomposable if it may be written as a nontrivial tensor product of two other division algebras. One can strengthen this notion and consider whether or not a division algebra is similar to a tensor product of some number of division algebras of a fixed smaller index. This question is closely tied to the notion of symbol length in Galois cohomology. I will describe how to approach this problem by giving some extensions of results of Karpenko which connect indecomposability of division algebras with the Chow groups and K-theory of the corresponding Brauer-Severi varieties.

### A gentle approach to crystalline cohomology

Jacob Lurie (Harvard University)

Let $X$ be a smooth algebraic variety over a field $k$. The algebraic de Rham cohomology of $X$ is defined as the (hyper)cohomology of the de Rham complex $$\Omega^0_X \rightarrow \Omega^1_{X} \rightarrow \Omega^2_X \rightarrow \cdots$$ When $k$ is a perfect field of characteristic $p$, Deligne and Illusie introduced an analogue of the de Rham complex, called the de Rham-Witt complex, which instead computes the crystalline cohomology of $X$. In this talk, I'll describe an alternative construction of the de Rham-Witt complex, from which one can deduce some of the central properties of crystalline cohomology in an essentially calculation-free way. (Joint work with Bhargav Bhatt and Akhil Mathew.)

### Nonconnective simplicial commutative rings

Akhil Mathew (Harvard University)

Simplicial commutative rings are one of the first steps into the world of "derived" rings that one can take (e.g., which allows derived tensor products of ordinary rings). However, simplicial commutative rings are always connective, while many objects one wishes to consider (e.g., arising from cohomology theories) need not be. I will explain ongoing work with Bhargav Bhatt on an extended theory of "generalized rings" which extends this category to allow nonconnective objects. Many "equational" constructions which cannot work with $\mathrm{E}_\infty$-rings extend well to generalized rings.

### Shimura stacks

Lenny Taelman (University of Amsterdam)

Motivated by questions about complex multiplication, we describe how moduli stacks of abelian varieties and of K3 surfaces are 'stacky' Shimura varieties. This talk will be a mixture of exposition of well-known (folklore) mathematics, new results, and open questions.

### Geometric realizations of logarithmic schemes

Mattia Talpo (Simon Fraser University)

Logarithmic schemes are schemes equipped with a sheaf of monoids that records some additional information of interest - typically either a boundary divisor or some infinitesimal data about a degeneration, of which our scheme is a fiber. Initially introduced for arithmetic purposes, these objects have later found applications to different areas of algebraic geometry, in particular in moduli theory. I will talk about two different (but related) geometric realizations of logarithmic schemes, the "Kato-Nakayama space" and the "infinite root stack", and about past and upcoming applications of these constructions.