WINART3 workshop

Women in Noncommutative Algebra and Representation Theory workshop 3.
Banff International Research Station (BIRS). April 3 – 8, 2022.

 

Organizers: Karin Baur, Andrea Solotar, Gordana Todorov , Chelsea Walton (contact)

Applications are now closed (Deadline: July 1, 2021).  Participant List
See the Research Group Descriptions below and Q & A page for more information.

 

Research Group Descriptions

Cluster Categories research group I, co-led by Karin Baur and Gordana Todorov

Cluster algebras were introduced by Fomin-Zelevinsky in 2002 in order to give a combinatorial framework for studying algebraic groups, and have since appeared in various fields including triangulations of surfaces, Teichmueller theory, Poisson geometry, and algebraic combinatorics. Their categories of representations, cluster categories, introduced by Buan-Marsh-Reineke-Reiten-Todorov have also had numerous applications throughout mathematics as described in Reiten’s ICM 2010 talk on this subject. Baur and Todorov are leaders in this area of research, particularly via their research work on the combinational aspects of cluster theory via friezes and triangulations in past WINART workshops. Their joint WINART works include “Mutation of friezes” (Bulletin des Sciences Math ́ematiques, 2018), “Conway-Coxeter friezes and mutation: a survey” (AWM Research Symposium proc., 2017), “Friezes satisfying higher SLk-determinants” (to appear in Algebra and Number Theory), and “Infinite friezes and triangulations of annuli” (arXiv/2007.09411).

Representation Theory of Hopf Algebras research group, co-led by Georgia Benkart and Ellen Kirkman

The representation theory of Hopf algebras has been a subject of intense investigation since the emergence of quantum groups in mathematical physics in the 1980s. This research group will focus on questions on this topic such as the following: When can the Drinfeld double of a finite-dimensional Hopf algebra be realized as the tensor product of a Hopf algebra and a commutative group algebra? This is true for the Drinfeld double of the Taft algebra. Having such a realization has significant implications for the representation theory and character theory of the double. The McKay matrix encodes the fusion rules for decomposing the tensor products of the simple modules with a given module. For such doubles, other questions to explore are What is the connection between the eigenvectors of the McKay matrix and what might be regarded as a character table for the double? When are the traces of the grouplike elements on the simple modules sufficient to give all the right eigenvectors? Computing some small examples should provide insight into how to approach these questions. Benkart and Kirkman’s recent WINART2 projects, “McKay matrices for finite-dimensional Hopf algebras” (arxiv/2007.05510) and “Tensor representations for the Drinfeld double of the Taft algebra” (arxiv/2012.1527), which are joint with R. Biswal, V.C. Nguyen, and J. Zhu, are the beginning of a long-term investigation of encoding representation-theoretic data of Hopf algebras through sophisticated combinatorial and linear-algebraic means.

Cluster Categories research group II, co-led by Ilke Canakci and Ana Garcia Elsener

Introduced by Fomin, Shapiro and Thurston in 2008, the cluster algebras associated with triangulated surfaces inspired much work on the geometric models for cluster categories. Such geometric models provide a practical way to interpret objects in cluster categories, representations over Jacobian algebras, and entries of frieze patterns as curves in marked surfaces. The leaders are prolific young researchers who have participated in previous editions of WinART and specialise in these interactions. See “Extensions in Jacobian algebras and cluster categories of marked surfaces” (Adv. Math. 2017), “Mutation of type D friezes” (J. Alg. Comb. Series A, 2020), and a former WinART cluster category project “Infinite friezes and triangulations of annuli” (arXiv/2007.09411). In this project we will incorporate geometric and combinatorial tools such as skein relations, surface triangulations and snake graphs, to investigate homological properties (and therein combinatorics) over cluster categories and related mathematical objects, ie. frieze patterns, Jacobian algebras, and cluster algebras.

Infinite-Dimensional Lie Theory research group, co-led by Lisa Carbone and Elizabeth Jurisich

Kac-Moody algebras are infinite-dimensional generalizations of finite-dimensional semisimple Lie algebras, and can be studied similarly to their finite-dimensional counterparts via analogues of their root system, irreducible representations, and connections to flag manifolds. One such Lie algebra of particular importance are the monster Lie algebras or “Borcherds algebras” due to their application to conformal field theory and string theory. In recent joint work with Scott Murray, Carbone and Jurisich proposed a Lie group analog for the monster Lie algebra (arxiv/2002.06658). The WINART3 workshop will be an excellent avenue to both follow-up on this exciting direction of research and to pursue research on Kac-Moody groups and on other aspects of infinite-dimensional Lie theory in which Carbone and Jurisich are leading experts.

Combinatorial Models in Representation Theory, co-led by Eleonore Faber and Bethany Marsh

Recently, many combinatorial models have arisen in the representation theory of algebras. Examples include the categorification of Coxeter-Conway friezes using cluster categories (first pointed out by Caldero-Chapoton), the description of module categories via Dyck paths (Moreno Cañadas-Bravo Riós) and the description of categories associated to gentle algebras via surface triangulations and ribbon graphs (Baur-Simoes, Lekili-Polishchuk, Opper-Plamondon-Schroll). In this group we will explore combinatorial models in the representation theory of algebras. Faber has extensive WINART experience. She has co-authored work on friezes with K. Baur, S. Gratz, K. Serhiyenko and G. Todorov (e.g. “Friezes satisfying higher SLk-determinants” (Algebra and Number Theory, 2021)) and has a recent preprint with J. August, M.-W. Cheung, S. Gratz and S. Schroll on “Grassmannian categories of infinite rank” ((arxiv/2007.14224). Combinatorial models have also appeared in work of Marsh, including a paper with K. Baur (“Categorification of a frieze pattern determinant,” Journal of Combinatorial Theory, Series A, 2012) and a paper with K. Baur and A. King (“Dimer models and cluster categories of Grassmannians,” Proc. LMS, 2016). She has mentored 9 postdoctoral researchers and supervised 5 PhD students.

Hochschild Cohomology research groups I and II, co-led by María Julia Redondo and Fiorela Rossi Bertone (I), and co-led by Sibylle Schroll and Andrea Solotar (II)

Hochschild cohomology is a theory in homological algebra that provides a framework for de- formations of associative algebras over rings, developed by Hochschild in the 1940s and further established by Cartan and Eilenberg in the 1950s. Nowadays, Hochschild cohomology plays a crucial role in algebraic geometry, category theory, functional analysis, and topology, and has vi- tal connections to cyclic homology and K-theory. Recent work of Redondo and Rossi Bertone contributed to this theory by establishing a careful correspondence between Morita equivalence and infinitesimal deformations of finite-dimensional associative algebras (arxiv/2003.10366), and by describing an explicit L∞-structure on an important complex used to compute Hochschild coho- mology (arxiv/2008.08122). Recent joint work of Schroll and Solotar includes the study of the first Hochschild cohomology of gentle algebras and Brauer graph algebras (J. Algebra, 2020), and the study of the first Hochschild cohomology of various finite-dimensional algebras (arxiv/1903.12145 and arxiv/1904.03565). The WINART3 workshop would provide an excellent avenue to push these new directions of research further, especially due the framework of the workshop. The authors of the article, arxiv/2003.10366, for instance, is a team of four women. Redondo and Solotar were also co-leaders at the WINART1 and WINART2 workshops, which resulted in the preprint “The Gerstenhaber structure on the Hochschild cohomology of a class of special biserial algebras” (arxiv/1803.10909) and works in preparation. Moreover, Schroll co-led a research team at the WINART2 workshop which resulted in the recent preprint, “Grassmannian categories of infinite rank” (arxiv/2007.14224). The participants of the WINART3 workshop will greatly benefit from these highly productive teams of research leaders.

Weak Quantum Symmetries research group, co-led by Chelsea Walton and Elizabeth Wicks.

Weak bialgebras and weak Hopf algebras first appeared in the study of symmetries of conformal field theories and partition functions in the early 1990s. They were axiomatized by Böhm-Nill-Szlach ́anyi in the late 1990s, and have since appeared in the study of dynamical quantum groups, fusion categories, and subfactor theory. Together with Robert Won, Chelsea Walton and Elizabeth Wicks have initiated the systematic study of symmetries of algebras using actions of weak bialgebras and weak Hopf algebras. This was achieved in the recent preprints “Algebraic structures in comodule categories over weak bialgebras” (arxiv/1911.12847) and “Universal quantum semigroupoids” (arxiv/2008.00606). Walton and Wicks have a range of potential projects in this area of research for the WINART3 workshop that are both computational and categorical in nature.

 

 

Testimonials from the WINART2 workshop

“Working the whole group together at the same time was a great experience. Having most of the time devoted to do research was the favorite for me. It helps to connect with the other members, to have time to think and overcome difficulties in the projects. I also enjoyed the talks during the afternoon, they were really good speakers and interesting results.”
– Julia Plavnik

“I enjoyed our team work during the workshop. Each group member had strengths and our strengths complimented each other.”
– Angela Tabiri

“This was a very productive week for our group, and that was because of as small number (very enjoyable) talks as possible, and as large number of hours for research as possible. I enjoyed working there a lot and hope to be a part of it again!”
– Maitreyee Kulkarni

“Our group was mathematically diverse, but cohesive in terms of the project. Everyone was able to make essential contributions and because of the different backgrounds, we all feel like we learned a lot. While we achieved a lot mathematically, the atmosphere in our group was very inclusive and we all were able to ask any question we had without feeling exposed.”
– Sibylle Schroll

“[I enjoyed] young and more senior mathematicians working together as equal – it was so much fun.”
– Gordana Todorov

“This workshop gave an opportunity to collaborate with people I knew from conferences and visits, but never had a chance to work with.
It was also wonderful and inspiring to talk to so many women in math.”
– Maitreyee Kulkarni

“I was always wanted to expand my research interests outside the subject of my PhD thesis. The teaching, the application process etc take so much time, that I usually focus on research areas I am already familiar with. This week not only I learnt and I worked on a new area, but I was doing this in an everyday basis, which helped a lot to start seeing progress in a short period of time. Moreover, I had the opportunity to collaborate with new people, learn about new techniques and expand my research interests. Being in a conference like that, seeing everyone around me working hard was really an inspiration for me.”
– Eirini Chavli

“The organizers did a wonderful job! The workshop went very smoothly and we had an awesome time! These research workshops like WINART are really beneficial to (especially young) women mathematicians to make connections and form collaborations. (THANK YOU!)^{\infinity} to the organizers and various funders to make such workshops as WINART happen!”
– Van Nguyen