#### Lara BOSSINGER

Toric degenerations and maps to toric varieties.

## Abstract

I will report on recent joint work in progress with Takuya Murata. Motivated by work of Harada—Kaveh and the desire to construct moment map type maps to Newton-Okounkov polytopes we study maps to toric varieties. As algebro-geometric methods turn out to not be appropriate we use results of Mather involving Whitney stratifications to obtain a collapsing map. Although the result is more general our main application concerns toric degenerations where we construct a map from the general to the special fibre. Moreover, we generalize a result of Harada-Kaveh constructing an integrable system on a projective variety that admits a toric degeneration induced by the moment map on the toric variety. A first preprint of the work is available on arxiv:2210.13137.

#### Vyjayanthi CHARI

Higher order KR-modules, imaginary modules and monoidal categorification

## Abstract

We discuss a connection between cluster algebras, the representation theory of quantum affine algebras of type, the existence of imaginary modules and the relationship with the representation theory of the general linear group over a non-archimedean field. The talk is based on joint work with Matheus Brito.

#### Emily CLIFF

Quasi-universal sheaves and generic modules

## Abstract

This is based on joint work-in-progress with Colin Ingalls and Charles Paquette. Given a finite-dimensional algebra, we can associate a quiver and choose a dimension vector. Work of Alistair King shows that, if the dimension vector is unimodular, there is a moduli space of stable representations with a universal sheaf. Work of Reineke–Schröer and Hoskins–Schaffhauser shows that this does not hold for all dimension vectors; in general, we obtain a only “quasi-universal sheaf”. I will discuss this construction from several perspectives, and explain applications to the representation theory of finite-dimensional algebras.

#### Colleen DELANEY

HQFT interpretation of the zesting construction on ribbon categories

## Abstract

The zesting construction is a way to describe braided or ribbon categories whose fusion rings are related by central extension of a grading group by invertible objects. At the level of string diagrams this construction has a strong topological flavor that allows one to give a physical characterization of the relationship between the categories’ associated 3D Reshetikhin-Turaev TQFTs.

In this talk we will discuss recent work that defines a more general procedure for G-crossed braided fusion categories. This new point of view reproduces the original zesting construction, places it in the context of the well established extension theory of fusion categories by finite groups, and connects zesting to 3D homotopy quantum field theory (HQFT).

#### Tim HODGES

Belavin-Drinfeld quantum groups

## Abstract

Belavin-Drinfeld quantum groups are deformations of the algebra of functions of a semi-simple algebraic Poisson group whose Poisson structure is given by the classification of factorizable Lie bialgebra structures on (semi-)simple Lie algebras. This talk will survey the current state of knowledge of this subject together with a number of conjectures and future research directions.

#### Mindy HUERTA PEREZ

m-Periodic Gorenstein objects

## Abstract

In this talk we present the concept of m-periodic Gorenstein objects relative to a pair of classes of objects in an abelian category, as a generalization of m-strongly Gorenstein projective modules over associative rings. We mention some properties when the pair satisties certain homological conditions, like for example when the pair is a GP-admissible pair. Connections to Gorenstein objects and Gorenstein homological dimensions relate to these pairs are also established. This is a joint work with O. Mendoza and M. A. Pérez.

#### Alfredo NÁJERA CHÁVEZ

Newton-Okounkov bodies and minimal models for cluster varieties

## Abstract

I will explain a general procedure to construct (potentially infinitely many) Newton-Okounkov bodies for a given line bundle on a minimal model of a cluster variety. This construction applies for example to line bundles on Grassmannians and flag varieties, and recovers as particular examples distinguished polytopes arising in representation theory such as Gelfand-Tsetlin polytopes, Littelmann/Berenstein-Zelevisnky string polytopes and the flow valuation Newton-Okounkov polytopes constructed by Rietsch-Williams. Time permitting, I will provide further details of our construction for the Grassmannians as, in this case, one can use either the A or the X cluster structure to construct Newton-Okounkov polytopes. Interestingly, the Euler form of the dimer algebra used by Jensen-King-Su and Baur-King-Marsh to categorify the cluster structure on the homogeneous coordinate ring of a Grassmannian can be used to provide an equivalence between the two kinds of polytopes.

#### Daniel NAKANO

Monoidal triangular geometry with applications to representation theory

## Abstract

In this talk, I will provide some historical background on the subject of tensor triangular geometry and support theory for representations. The origins of this subject started with the work of Alperin and Carlson in the late 1970’s and 1980’s with a significant milestone being the introduction of tensor triangular geometry by Paul Balmer in the mid 2000’s. Since this time there have been many exciting new ideas in the subject with the modern development of tensor categories.

The general noncommutative version of Balmer’s tensor triangular geometry was recently developed by Nakano, Vashaw and Yakimov. Insights from noncommutative ring theory are used to obtain a framework for prime, semiprime, and completely prime (thick) ideals of a monoidal triangulated category, and then to associate to a topological space–the non-commutative Balmer spectrum. Our theory works well in finite tensor categories. In this talk, I will mainly focus on applications to studying the geometry around finite-dimensional Hopf algebras.

This talk represents joint work with Kent Vashaw and Milen Yakimov.

#### Emily PETERS

Categorifying connections

## Abstract

The road between fusion categories and subfactors (of von Neumann algebras) is well-trodden in both directions; subfactors provide examples of fusion categories not seen elsewhere, and fusion category techniques