### Algebra Seminar

## Algebra Seminar

## Discrete CATS & Algebra Seminar

Speaker: Ana Garcia Elsener, Universidad Nacional de Mar del Plata

Title: skew-Brauer graph algebras

Abstract:

Brauer graph algebras are defined by combinatorial data based on graphs:

Underlying every Brauer graph algebra is a finite graph, the Brauer graph, equipped with with a cyclic orientation of the edges at every vertex and a multiplicity function. This combinatorial data encodes much of the representation theory of Brauer graph algebras and is part of the reason for the ongoing interest in this class of algebras. A known result by Schroll states that Brauer graph algebras, with multiplicity function one, give rise to all possible trivial extensions for gentle algebras. On the other hand, Geiss and de la Peña studied a generalization of gentle algebras called skew-gentle algebras.

In our ongoing project we establish the right definition of skew-Brauer graph algebra in such a way that the result by Schroll can be enunciated in this context. That is,* A *is a skew-Brauer graph algebra with multiplicity function equal to one if and only if it is the trivial extension of a skew-gentle algebra. Moreover, the family of skew-Brauer graph algebras with arbitrary multiplicity function generalizes the family of Brauer graph algebras with arbitrary multiplicity function.

(Joint work with Victoria Guazzelli from Universidad Nacional de Mar del Plata, Argentina, and Yadira Valdivieso Diaz from Universidad de Puebla, México)

Ana Garcia Elsener is visiting Khrystyna Serhiyenko.

## Algebra Seminar

Title: Tropical Matroid Homology

Abstract: The intersection ring of matroids arises from the restriction of tropical intersection theory to Bergman fans. As an algebraic object, the intersection ring encodes many matroid invariants as homomorphisms and is naturally bigraded by the rank and ground set of the matroids which generate it. The matroid operations of deletion and contraction give rise to boundary maps, producing a tropical analog of the Kontsevich homology. We give an affirmative answer to the conjecture that these homology groups are trivial on the full intersection ring and show that these groups be used to measure whether a class of matroids is closed under certain extensions. We further extend this notion of homology to Chow rings of arbitrary matroids.

## Algebra Seminar

Title: The Birational Geometry of K-Moduli Spaces

Abstract: K-stability is a rapidly developing theory that allows one to construct moduli spaces for Fano varieties. In all known examples, K-moduli spaces are uniruled, so their Kodaira dimensions are negative infinity. In this talk we will describe components of K-Moduli spaces which are birational to M_g, in particular they have maximal Kodaira dimension when g is sufficiently large. This component parameterizes certain moduli spaces of vector bundles on smooth curves, and the main difficulty is to show that these moduli spaces are K-stable. To establish this we require good understanding of their toric degenerations.

## Algebra Seminar

Title: Computing Free Resolutions of OI-Modules

Abstract: Free resolutions are a powerful tool in commutative and homological algebra. Much of the structure of a module can be encoded in a free resolution. For example, in the case of graded modules, free resolutions can be used to study Betti numbers and Hilbert functions. Certain homological constructions such as the Ext and Tor functors can be computed with free resolutions as well. In this talk we show how to compute free resolutions in the case of OI-modules over a Noetherian polynomial OI-algebra, where OI denotes the category whose objects are totally ordered finite sets and whose morphisms are strictly increasing functions.

## Algebra Seminar

Title: Computing the Algebra of Conformal Blocks for sl_4

Abstract: Conformal blocks are finite-dimensional vector spaces that arise from the WZNW model of conformal field theory. These have applications in algebraic geometry, particularly in describing the moduli of principal bundles and the moduli of curves. In this talk, we will discuss recent progress on computing a presentation of the algebra of conformal blocks for sl_4. We also describe equations, the tropical variety, and a large family of toric degenerations for the case of a cone with genus 0 and 3 marked points.

## Algebra Seminar

__ Title:__ Mori Dream Spaces: What, Where, and Why You Would Care

__ Abstract:__ If you've asked me in the past two years about my research, I've likely launched into an explanation of Mori dream spaces, their properties, and/or my implementation of an algorithm in Macaulay2. However, my hallway explanation likely left you with even more questions: What are these objects? Where can I find them? Why would I care? Well, you're in luck! In this talk, I plan to answer all of these questions, give plenty of examples, and list some questions motivating future work.

## Algebra Seminar

Title: Pairs of quadratic forms over p-adic fields

Abstract: The Hasse-Minkowski theorem implies that if a quadratic form over a number field k has a nontrivial zero over every achimedean and non-achimedean completion of k, then the form will have a nontrivial zero over k. It's natural to ask whether this is true for common nontrivial zeros of pairs of quadratic forms. In an effort to answer this question, new results about pairs of quadratic forms over p-adic fields have been proven. One such result, by Heath-Brown, deals with finding forms in the pencil generated by a pair of quadratic forms over a p-adic field in 8 variables that split off 3 hyperbolic planes. In this talk, we will examine this result by Heath-Brown, and we will discuss the ongoing effort of generalizing Heath-Brown's hyperbolic plane result to pairs of quadratic forms over a p-adic field in an arbitrary number of variables.

## Algebra Seminar

Title: Local cohomology of thickenings

Abstract: Let $R$ be a standard graded polynomial ring that is finitely generated over a field, and let $I$ be a homogenous prime ideal of $R$. A recent paper of Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of the thickenings $R/I^t$ in characteristic $0$, and provided a stabilization result for these cohomology modules. I will explicitly construct isomorphisms between local cohomology modules of thickenings in several cases.