Andrea Appel

University of Southern California

**Quantum affine algebras and monodromy of the Casimir connection**

**Abstract:** In 2005, V. Toledano Laredo proved that the monodromy of the Casimir connection of a simple Lie algebra is described by the quantum Weyl group operators of the corresponding Drinfeld-Jimbo quantum group. The proof relies upon the notion of quasi-Coxeter algebra, which is, informally speaking, a bialgebra carrying on its modules a joint action of the braid group and a given Artin-Tits group. In this talk, I will give a brief overview of the strategy to extend these results to affine Kac-Moody algebras, based upon a generalization of the notion of quasi-Coxeter algebra at a categorical level (joint work with V. Toledano Laredo).

Arkady Berenstein

University of Oregon

**Noncommutative marked surfaces.**

**Abstract**: The goal of my talk (based on joint work with Vladimir Retakh) is to attach a noncommutative cluster-like structure to

each marked surface. This is an algebra generated by "noncommutative geodesics" between marked points subject to certain triangular relations and noncommutative analogues of Ptolemy-Plucker relations. It turns out that the algebra exhibits a noncommutative Laurent Phenomenon

with respect to any triangulation of the surface, which confirms it cluster nature.

As a surprising byproduct, we obtain a new topological invariant of the surface, which is a free or a 1-relator group easily computable in terms of any triangulation of the surface.

Jonathan Brundan

University of Oregon

**Kac-Moody 2-categories and their cyclotomic quotients**

**Abstract:** I will review the definitions of Kac-Moody 2-categories given by Rouquier and by Khovanov and Lauda, and then discuss their generalized cyclotomic quotients introduced by Webster.

Ivon Dorado

Centro de Ciencias Matematicas, UNAM Campus Morelia**Algebraically equipped posets**

**Abstract: **We introduce partially ordered sets (posets) with an additional structure given by a collection of vector subspaces of an algebra $A$. We call them *algebraically equipped posets*. Some particular cases of these, are generalized equipped posets and p-equipped posets, for a prime number p.

We study their categories of representations and establish equivalences with some module categories, categories of morphisms and a subcategory of representations of a differential tensor algebra. Through this, we obtain matrix representations and its corresponding matrix classification problem.

Alexander Palen Ellis

University of Oregon

**Odd categorification of quantum sl(2) at a fourth root of unity**

**Abstract:** We will describe odd categorification of quantum sl(2) at a generic parameter (joint with Khovanov, Lauda) and at a fourth root of unity (joint with Qi).

Jacob Greenstein

UC Riverside

**Double canonical bases**

**Abstract: **The aim of this talk (based on a joint work with A. Berenstein) is to introduce a new class of bases for quantized universal enveloping algebra and other doubles attached to semisimple and Kac-Moody Lie algebras. These

bases contain dual canonical bases of upper and lower halves of the quantum group and are invariant under many symmetries including, in the semisimple case, all Lusztig's symmetries. It also turns out that a part of a double canonical basis of spans its center. We expect that double canonical bases carry a cluster-like structure extending that on the

upper and lower halves.

Jesús Jiménz /Efrén Pérez

**On the endomorphism ring of generically bounded modules for semigenerically tame algebras**

**Abstract:** Let k be a perfect field (perhaps finite), K the algebraic closure of k and $\Lambda$ a f.d. k –algebra. We say that the indecomposable \Lambda-module G is algebraically bounded if G^K \cong H_1 \oplus \ldots \oplus H_n, where n is a natural number and H_i is a generic \Lambda^K-module for each i.

If \Lambda^K is tame, G is an algebraically bounded \Lambda-module, and E_G is the endomorphism ring of G, then the center of E_G / rad (E_G) is a finite field extension of a field of rational functions of one variable, over a finite field extension of k.

We exhibit some consequences of the previous fact and other related theorems.

** **

Jonathan Kujawa

University of Oklahoma

**Tensor Triangular Geometry for Lie Superalgebras**

**Abstract**: Recently Paul Balmer defined the spectrum of a tensor triangulated category. His construction allows us to extract a natural geometry from the category. In particular, Balmer and others have shown that this spectrum provides a unifying generalization of previously known notions of spectrum and support varieties in widely different settings. However, its generality also makes Balmer's spectrum hard to compute in practice. The challenge, then, is to provide a concrete realization of Balmer's spectrum for other tensor triangulated categories of interest. Building on our earlier work, we show that this can be done in a very nice way for the stable category of finite dimensional representations of the complex Lie superalgebra gl(m|n). This is joint work with Brian Boe and Daniel Nakano.

Daniel Nakano

University of Georgia

**On good (p,r)-filtrations for rational G-modules**

**Abstract:** In this talk we investigate Donkin's (p,r)-Filtration Conjecture, and present two proofs of the ``if" direction of the statement when $p\geq 2h-2$. One proof involves the investigation of when the tensor product between the Steinberg module and a simple module has a good filtration. One of our main results shows that this holds under suitable requirements on the highest weight of the simple module. The second proof involves recasting Donkin's Conjecture in terms of the identifications of projective indecomposable G_{r}-modules with certain tilting G-modules, and establishing necessary cohomological criteria for the (p,r)-filtration conjecture to hold.

This is joint work with Tobias Kildetoft.

José-Antonio de la Peña

Centro de Investigación en Matemáticas

**On the trace of the Coxeter matrix of an algebra.**

**Abstract:** We review some (already) classical results by Happel on the relations of

traces of Coxeter matrices and the Hochschild cohomology of an algebra. We

present some recent results on algebras of cyclotomic type and

characterizations using traces of powers of the Coxeter matrix.

Daniel Labardini-Fragoso

Instituto de Matemáticas UNAM, Mexico City

**Species and potentials for surfaces with orbifold points**

**Abstract:**

The purpose of this talk is to show how to associate a species and a

potential to each triangulation of a surface with marked points and

orbifold points (of order 2), the aim of the association being that

triangulations related by a flip have species with potential related by

the corresponding mutation. The talk will be based on joint work in

progress with Jan Geuenich.

Vladimir Retakh

Rutgers University

**Noncommutative triangulations and the Laurent phenomenon**

**Abstract: **This is an introduction to the talk by Arkady Berenstein about our

joint work. I will explain the origin of noncommutative Ptolemy-Plucker

relations, introduce noncommutative angles as invariants of triangulations

of polygons, and discuss related examples of the noncommutative

Laurent phenomenon.

Dan Rogalski

UC San Diego

**Frobenius Ext-algebras arising from Artin-Schelter Gorenstein Algebras** (joint with Manuel Reyes, James Zhang)

**Abstract:** Artin-Schelter (AS) Gorenstein algebras are an important class of graded algebras in noncommutative geometry. Any AS Gorenstein algebra A has a graded automorphism associated to it called the Nakayama automorphism, which is a natural generalization to higher dimension of the usual Nakayama automorphism which is defined for a Frobenius algebra. In addition, we can associate to A the homological determinant, which is a multiplicative map from the group of graded automorphisms of A to the base field k that is important in noncommutative invariant theory.

We conjectured previously that for any AS Gorenstein algebra A, the Nakayama automorphism has homological determinant 1. This conjecture has numerous applications, which we explain in the talk. We then describe recent work which proves the conjecture in wide generality. The proof involves studying certain graded Frobenius Ext-algebras of objects in a suitable triangulated category associated to the Gorenstein algebra A. This method is of independent interest.

Valente Santiago Vargas

UNAM, Mexico-City

**Homologicla systems in triangulated categories.**

**Abstract:** We introduce the notion of homological systems for triangulated categories. Homological systems generalize,

on one hand, the notion of stratifying systems in module categories, and on the other hand, the notion of exceptional sequences in

triangulated categories. We will show that attached to a homological system, there are two standardly stratified algebras

A and B, which are derived equivalent.

Markus Schmidmeier

Florida Atlantic University

**Crossing and Noncrossing Partitions of the Disk**

**Abstract:** We categorify disk partitions as short exact sequences of nilpotent linear operators. Symmetry operations on the disk give insight about Littlewood-Richardson tableaux and the geometry of representationspaces.

The talk is a report on a joint project with Justyna Kosakowska from Torun.

Alexander Shapiro

UC Berkeley

**Quantization of Grothendieck-Springer resolution and quantum****coordinate rings**

**Abstract:** Let G be a complex semi-simple Poisson-Lie group with the standard Poisson structure, and G^* be its Poisson-Lie dual. Then the quantized algebra U_q(g) of functions on G^* is called a quantum group and

the quantized algebra O_q(G) of functions on G is called a quantum coordinate ring. I will outline a construction of a morphism of algebras from U_q(g) to (a localization of) O_q(G). This construction comes as a composition of morphisms from U_q(g) to a Heisenberg double H of U_q(b) and from H to O_q(G). I will explain how these morphisms appear naturally in a

theory of Hopf algebras, and how the second one is a quantization of the Grothendieck-Springer resolution. Motivation for this construction comes from the idea to use quantum cluster coordinates on O_q(G) to study the representation theory (e.g. principal series representations) of U_q(g).

This is joint work with Gus Schrader.

Peri Shereen

UC Riverside

**A Steinberg type decomposition theorem for higher level Demazure modules**

**Abstract**: We study Demazure modules which occur in a level $\ell$ irreducible integrable representation of an affine Lie algebra. We also assume that they are stable under the action of the standard maximal parabolic subalgebra of the affine Lie algebra. We prove that such a module is isomorphic to the fusion product of ``prime" Demazure modules, where the prime factors are indexed by dominant integral weights which are either a multiple of $\ell$ or take value less than $\ell$ on all simple coroots. Our proof depends on a technical result which we prove in all the classical cases and $G_2$. Calculations with mathematica show that this result is correct for small values of the level. Using our result, we show that there exist generalizations of $Q$--systems to pairs of weights where one of the weights is not necessarily rectangular and is of a different level. Our results also allow us to compare the multiplicities of an irreducible representation occurring in the tensor product of certain pairs of irreducible representations, i.e., we establish a version of Schur positivity for such pairs of irreducible modules for a simple Lie algebra.

Jose Simental Rodriguez

Northeastern University

**Harish-Chandra bimodules for Rational Cherednik algebras.**

**Abstract**: Rational Cherednik algebras are a certain class of associative algebras associated to complex reflection groups. They depend on a collection of complex numbers, and their representation theory is in some sense similar to that of universal enveloping algebras of semisimple Lie algebras. In particular, there is a notion of Harish-Chandra bimodules for these algebras. In my talk, I will describe irreducible maximally supported modules over rational Cherednik algebras associated to Coxeter groups.

Ben Webster

University of Virginia

**Uniqueness (or lack thereof) for categorical modules**

**Abstract:** Whenever one tries to explain categorification to a mathematician unfamiliar with it, there's one question that essentially unavoidable: "Is there a unique way of categorifying X, Y or Z? If there isn't how do you know you have the right one?"

This is a fair (if somewhat annoying) question. At this point we know many objects that don't have a unique categorification, but they do have ones which are in some sense "best." The examples we know seem to suggest a representation will have a "best class" of categorifications if it is the restriction of an irreducible representation under a good inclusion of subalgebras. While this is still preliminary work, I'll try to give some theoretical justification for this assertion.

Harold Williams

UC Berkeley

**Title: Toda Systems, Cluster Characters, and Spectral Networks**

**Abstract:** We discuss recent work relating the Hamiltonians of certain cluster integrable systems with the representation theory of an associated quiver with potential. We then reinterpret this as showing that traces of holonomies on certain simple wild SL_n-character varieties are cluster characters of nonrigid representations, generalizing known results in the SL_2 case. To make this precise we analyze in detail the spectral networks of the periodic Toda system.