## Spring 2022, Ames, IA

We will hold a meeting in April 2022, at Iowa State in Ames, IA. Below you'll find a schedule, titles, and abstracts.

### Speakers

• Juliette Bruce (University of California, Berkeley)
• Eloísa Grifo (University of Nebraska, Lincoln)
• Jack Jeffries (University of Nebraska, Lincoln)
• Aida Maraj (University of Michigan)
• Thomas Polstra (MSRI/University of Virginia)
• Connor Simpson (University of Wisconsin)
• Aleksandra Sobieska (University of Wisconsin)

### Registration

Please register here for CA+ 2022. If you would like to be considered for funding (mileage for a carpool + two nights of lodging), please apply by April 1.

### Funding?

Stay tuned for a link to apply for travel funding. We gratefully acknowledge NSF funding from grant number DMS-2000390 and from the ISU Department of Mathematics and the ISU College of Liberal Arts and Sciences.

### Schedule (All talks are in Marston Hall 2155.)

• Saturday:
• 8:30-9:00: Registration
• 9:00-9:50: Juliette Bruce: Multigraded regularity on products of projective spaces
• 10:00-10:30: Coffee break
• 10:30-11:20: Aleksandra Sobieska: Subcomplexes of Certain Free Resolutions
• 11:30-1:30: Lunch and discussion
• 1:30-2:20: Connor Simpson: Chow rings for polymatroids via fans
• 2:30-3:20: Aida Maraj: Staged Tree Models with Toric Structure
• 3:30-4:00: Coffee break
• 4:00-5:00: Turbo talks by grad students
• Dinner is on your own.
• Sunday:
• 9:00-9:50: Jack Jeffries: Are determinantal rings direct summands of polynomial rings?
• 9:50-10:05: Break
• 10:05-10:55: Thomas Polstra: Powers of ideals and Frobenius powers of ideals: A comparison of Hilbert-Samuel and Hilbert-Kunz theory
• 10:55-11:10: Break
• 11:10-12:00: Eloísa Grifo: Cohomological support varieties

### Titles and Abstracts

Multigraded regularity on products of projective spaces
by Juliette Bruce (University of California, Berkeley)
On projective space the Castelnuovo-Mumford regularity of a module can be characterized in terms of the betti numbers, local cohomology, and truncations of the module. I will discuss recent work exploring the relationship between multigraded Castelnuovo-Mumford, truncations, Betti numbers, and virtual resolutions for modules on products of projective spaces. This is joint work with Lauren Cranton Heller and Mahrud Sayrafi.

Cohomological support varieties
by Eloísa Grifo (University of Nebraska, Lincoln)
Given a ring R, we can associate a variety to each R-module, or more generally each complex of R-modules, which encodes homological information. We will discuss what these varieties are, how they can detect ring theoretic properties of R, and other applications. This is joint work with Ben Briggs and Josh Pollitz.

Are determinantal rings direct summands of polynomial rings?
by Jack Jeffries (University of Nebraska, Lincoln)
Over any infinite field, the generic determinantal rings are known to be fixed subrings of the action of the general linear group on a polynomial ring. Since the general linear group is linearly reductive in characteristic zero, these generic determinantal rings are direct summands of polynomial rings, which explains many of their good properties in this case. In positive characteristic, these determinantal rings have many of the same good properties even though the general linear group is no longer linearly reductive. In this talk we investigate if these determinantal rings continue to be direct summands of polynomial rings in characteristic p>0. We will also encounter some interesting varieties related to linear algebra along the way.

Staged Tree Models with Toric Structure
by Aida Maraj (University of Michigan)
A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety. For certain trees, called balanced, the model is known to be exactly the intersection of the toric variety and the probability simplex. We show that in this case the defining toric ideal is Kosul, normal, and it has a Gröbner basis with binomials of degree one and two. The highlight of the talk will be that the class of staged tree models with a toric structure extends far outside of balanced trees, if we allow a change of coordinates. The talk is based on the preprint (https://arxiv.org/abs/2107.04516) with Christiane Görgen and Lisa Nicklasson.

Powers of ideals and Frobenius powers of ideals: A comparison of Hilbert-Samuel and Hilbert-Kunz theory
by Thomas Polstra (MSRI/University of Virginia)
Let $R$ be a Noetherian ring of prime characteristic $p>0$ and $I\subseteq R$ an ideal. For each integer $n$ let $I^n$ be the $n$th power of the ideal $I$ and for each integer $e$ let $I^{[p^e]}$ denote the expansion of the ideal $I$ along the $e$th iterate of the Frobenius endomorphism. The study of the asymptotic behavior of the sequences of ideals $\{I^n\}_{n\in\mathbb{N}}$ and $\{I^{[p^e]}\}_{e\in\mathbb{N}}$ has many similarities. For example, both theories can be used to study and relate the singularities of $R$, especially when $R$ is local and $I$ is the maximal ideal. There are sharp differences in the two studies as well. For example, a classical result of Brodmann asserts that $\bigcup \mbox{Ass}_R(R/I^n)$ is a finite set whereas examples of Katzman show that the set $\bigcup \mbox{Ass}_R(R/I^{[p^e]})$ can be infinite. In this talk we will survey recent developments that further relate and deviate Hilbert-Samuel theory and Hilbert-Kunz theory. Along the way we will pose open problems.

Chow rings for polymatroids via fans
by Connor Simpson (University of Wisconsin)
Matroids abstract the combinatorics of hyperplane arrangements. They are generalized by polymatroids, which abstract the combinatorics of subspace arrangements. Many problems on matroids have recently succumbed to algebraic geometry-inspired methods, often involving the "Chow ring" of a matroid. We will discuss Pagaria & Pezzoli's generalization of this ring to polymatroids. We show the Chow ring of a polymatroid can be realized as the Chow ring of a smooth fan. Applications include a new proof of Hodge theory for the Chow ring of a polymatroid. This is joint work with Colin Crowley, June Huh, Matt Larson, and Botong Wang.

Subcomplexes of Certain Free Resolutions
by Aleksandra (Ola) Sobieska (University of Wisconsin)
What are the subcomplexes of a free resolution? This question is simple to state, but the naive approach leads to a computational quagmire that is infeasible even in small cases. In this talk, I invoke the Bernstein--Gelfand--Gelfand (BGG) correspondence to address this question for free resolutions given by two well-known complexes, the Koszul and the Eagon--Northcott. This novel approach provides a complete characterization of the ranks of free modules in a subcomplex in the Koszul case and imposes numerical restrictions in the Eagon--Northcott case. This is joint work with Maya Banks.

### Lodging and Travel Info

Those participants who are granted funding will be housed at the Gateway Hotel in Ames. The conference organizers will arrange rooms for funded participants.

Most parking lots on campus allow free parking all day Saturday and Sunday. Recommended parking lots are: 4, 5, 8, 65, or the East Parking Deck. Here is a campus map.

Iowa State University is located in Ames, IA. If you are flying, your target airport is Des Moines (DSM). Ames is a 40 minute ride by car North from the airport. Executive Express runs a shuttle service to and from the airport. If you are driving, there are directions here. We expect most participants to arrive Friday evening April 29 and depart Sunday afternoon May 1.