## Research Leaders

The list of confirmed research group (co-)leaders are:

Martina Balagovic & Catharina Stroppel

Karin Baur & Gordana Todorov

Georgia Benkart & Rosa Orellana

Maria Gorelik & Vera Serganova

Susan Montgomery & Sonia Natale

María Julia Redondo & Andrea Solotar

Anne Shepler & Sarah Witherspoon

Susan Sierra & Michaela Vancliff

Brief descriptions of the leaders' research interests are given below.

**(Balagovic & Stroppel) Categorification and coideal subalgebras. **
Categorification is a process of replacing some set-theoretic object (e.g., a module over an algebra) by a category-theoretic
object (e.g., a category), so that the original object and information about it gets refined. Given a categorification,
the original object can be easily recovered (e.g., by taking the Grothendieck group of the category), but finding and
constructing a categorification is a difficult task. One of the first examples of categorification was the work of Khovanov,
who used it to obtain refined knot invariants by categorifying the Jones polynomial. This approach was generalized by Stroppel
and coauthors using categories arising in Lie theory (category O). Catharina Stoppel's work on categorification has
applications in knot homology, gives new results about multiplicity formulas and graded branching rules and establishes
unexpected equivalences (for instance between representations of Lie superalgebras and perverse sheaves on Grassmannians).
The general structure of quantum symmetric pairs is one of the research focuses of Martina Balagovic (joint with Stefan Kolb).
Stroppel and Balagovic are both interested in categorifying quantum symmetric pairs and their representations.

**(Baur & Todorov) Cluster categories.**
Cluster categories were introduced by Gordana Todorov in joint work with Aslak Buan, Robert Marsh, Marcus Reineke,
and Idun Reiten in the seminal paper "Tilting Theory and Cluster Combinatorics" to study the combinatorics of cluster algebras.
This work provided new and remarkable insights on the representation theory of hereditary Artin algebras. The study of cluster
categories also has a connection to the study of Calabi Yau categories. For instance, it was shown in the work of Claire Amiot,
Idun Reiten, and Gordana Todorov that large classes of Calabi-Yau categories are equivalent to generalized cluster categories
as triangulated categories. Karin Baur, in joint work with Robert Marsh, established a beautiful geometric description
of another generalization of a cluster category: m-cluster categories introduced by Bernhard Keller. Recent work of Baur and
Todorov provides several connections between cluster categories and other important objects arising in algebra and physics.
Todorov, in joint work with Kiyoshi Igusa and Jerzy Weyman, has investigated the relationship between cluster categories and
the study of periodic trees and semi-invariants and maximal green sequences. Baur, in joint work with Alastair King and Robert
Marsh, established a correspondence between cluster categories of Grassmannians and dimer models.

**(Benkart & Orellana) Diagram algebras, tensor invariants, representation theory.**
The centralizer algebra
of transformations, that commute with the action of a group or algebra on tensor powers of a representation,
can often be realized as an algebra of diagrams. Important examples include the
Temperley-Lieb algebras of statistical mechanics, the partition algebras, and the
Iwahori-Hecke algebras.
These centralizer (diagram) algebras have a rich combinatorics, as well as many connections with representation theory,
knot and link invariants,
categorification, and quantum computing. Georgia Benkart is an expert on centralizer algebras
and their connections with crystal bases, web bases, symmetric functions,
and representation theory. Her recent work determines, in a very general setting, an expression for the Poincare series
for the tensor invariants as a quotient of two determinants, and uses that to develop new connections between the
tensor invariants for the finite subgroups of *SU _{2}* and the polynomial invariants of Weyl groups via McKay's
celebrated correspondence. Orellana has tackled the longstanding problem of determining the Kronecker coefficients,
described by Richard Stanley as "one of the main problems in the combinatorial
representation theory of the symmetric group." Her recent work with Bowman and DeVisscher,
offers wonderful new insights to the problem and a brand new approach using partition algebras; this work began at the
2011 BIRS workshop "Algebraic Combinatorixx" .

**(Gorelik & Serganova) Representations of Lie superalgebras.**
Supersymmetry, an important tool in modern theoretical physics, has long motivated the theory of representations
of supergroups and superalgebras. Maria Gorelik and Vera Serganova are experts in the representation theory of Lie superalgebras;
each have authored over 20 articles on this subject alone. Gorelik's contributions include establishing the structure of the ghost centre of a
Lie superalgebra, and most recently, computing of characters of
relatively integrable irreducible highest weight modules over finite-dimensional basic Lie superalgebras and over
affine Lie superalgebras (joint with Victor Kac). Moreover, Vera Serganova has been a leader in study of Lie superalgebras for 30 years. Some of her many
contributions consists of establishing the character formula for atypical representations of the Lie superalgebras *gl(m|n)*,
computing the characters of finite-dimensional simple modules over a basic classical Lie superalgebra (joint with Caroline Gruson), and
providing formulae for Shapovalov's determinants for the Poisson Lie superalgebras (joint with Maria Gorelik). Her
recent interests includes study of the category * T * of tensor representations of *sl _{∞}*.

**(Montgomery & Natale) Fusion categories and finite-dimensional Hopf algebras.**
The program to classify
finite-dimensional complex Hopf algebras has made great advances over the last two decades,
especially for pointed Hopf algebras and semisimple Hopf algebras. A question receiving much attention of late is
whether all semisimple Hopf algebras can be obtained from group algebras by a handful of constructions: bosonizations,
extensions, Hopf duals, and twists. Sonia Natale answered this question affirmatively for semisimple Hopf algebras of
low dimension.
The categorical version of this program, the classification of fusion categories, is under active investigation by Sonia
Natale and others. Fusion categories are of independent interest in several areas of mathematics, including conformal
field theory and operator algebras. Moreover, Susan Montgomery has been a leading figure in the study of finite-dimensional
Hopf algebras and their representations for 30 years. She has authored over 40 papers on the subject
and wrote "Hopf algebras and their actions on rings" which is widely considered to be an influential
text of the field. Her current research interests include the study of Frobenius-Schur indicators for semisimple
Hopf algebras and fusion categories.

**(Redondo & Solotar) Hochschild (co)homology and linear categories.**
Hochschild cohomology of associative algebras is important in many areas of mathematics: such as ring theory, group theory,
representation theory, mathematical physics, homotopy theory and topology. It is well known that homological methods are useful
in order to study associative algebras and to understand their properties. Even though Hochschild cohomology is defined easily
in terms of very basic linear algebra, it has important invariance properties, for example it is invariant under derived,
stable, and Morita equivalences. The Hochschild cohomology space has a cup product and a graded Lie bracket. One wants to
understand its structure as a graded vector space, as a ring and as a graded Lie algebra. One might aim for a presentation
by generators and relations, though this appears to be a very difficult problem in general. In fact there are just a few
examples of computations of the ring structure of Hochschild cohomology completed. María Julia Redondo and Andrea Solotar
are both experts in Hochschild (co)homology, particularly in its connection to (Galois and universal) coverings of linear
categories. Further, their joint work with several authors investigates the Hochschild (co)homology of many important
classes of algebras, which includes cluster-tilted algebras, generalized Weyl algebras, incidence algebras, string algebras,
and Yang-Mills algebras.

**(Shepler & Witherspoon) Algebraic deformations and homological algebra.**
Many algebras of interest are deformations of simpler, better-understood
algebras, with Hochschild cohomology recording the possible deformations
and providing a tool for understanding them. Anne Shepler and Sarah
Witherspoon employ homological and combinatorial techniques to study
deformations of algebras formed from group actions on other algebras,
such as polynomial algebras, and generalizations. The resulting semidirect product algebras and their deformations arise in
combinatorics, geometry,
and representation theory, and go by various names such as symplectic
reflection algebras, graded Hecke algebras, and rational Cherednik algebras. Deformations of universal enveloping algebras,
Ore algebras,
and Sridharan algebras extended by groups give other examples. Many deformations
retain some of the structure of the semidirect product algebras from which they come, such as (PBW) bases of monomials
that harken back to the original Poincare-Birkhoff-Witt Theorem on universal enveloping algebras.
This fact helps in finding answers to questions about the algebraic structure, representations,
and cohomology of the deformed algebras.