Cluster Algebras and Categorification

Cluster algebras are a class of commutative rings with a rich combinatorial structure. There are nu- merous connections between these algebras and other areas of mathematics and theoretical physics, including mirror symmetry, representation theory, dynamical systems, Poisson and algebraic ge- ometry, which already produced signicant results. The goal of this proposal is to develop new combinatorial models for certain important classes of cluster algebras in order to get an explicit description of its generators and relations that are built recursively. The main tools rely on using additive categorication of cluster algebras in terms of representation theory of associative alge- bras. In particular, the PI will study the recently-found cluster structures coming from positroid and Richardson varieties in the Grassmannian, that are dened via categorication. Moreover, the PI proposes a new combinatorial description of Cohen-Macauley subcategories in the module category of Jacobian algebras which oers applications to various topics in cluster algebras and representation theory. Throughout this award, the PI will contribute to the advancement of education and research in the mathematical community by working with undergraduates, developing new courses, recruiting graduate students, and fostering international collaborations among women. Intellectual Merit: The proposed project is in a highly active research area aimed at answering fundamental questions about the particular structure of cluster algebras. It will investigate new connections between intricate combinatorial objects arising from cluster algebras and their various categorications to further their understanding and build parallels between the dierent areas. The PI will study a fascinating phenomenon of two seemingly dierent cluster structures dened on positroid varieties by comparing and contrasting their combinatorics in order to extract more information about them. In these cases, she will also apply another categorication model to compute generators of the cluster algebra that are not captured combinatorially. Furthermore, for a dierent class of cluster algebras the project will produce a novel description of elements in the cluster algebra via certain decorated polygons. It oers applications to studying periodicity properties of cluster algebras and Cohen-Macauley modules. The main ingredient behind solving these questions is the powerful machinery coming from the representation theory of algebras, which encode the structure of cluster algebras and constitute an invaluable tool in their study. In the past, the PI has made several important contributions to the eld such as the proof of the conjecture on the combinatorial cluster structure of (skew) Schubert varieties, the discovery of the connection between SLk friezes and Grassmannian cluster algebras, the development of new methods to study maximal green sequences, and the study of the representation theory of cluster-tilted algebras. Broader Impacts: The PI successfully mentored a group of undergraduates at an REU in 2015 at University of Connecticut, where the students solved an open research problem on lengths of maximal green sequences and published an article in a peer reviewed journal. Last semester the PI worked in Math Lab, a recently established program at University of Kentucky that enables the faculty to lead research projects with undergraduates throughout the academic year. She is expecting to continue participating in Math Lab in future semesters. She is also an active member of the AWM and a women's network in her research area that promotes international collaborations by organizing conferences and thematic workshops. Moreover, the PI will develop a new graduate course on cluster algebras and recruit graduate students to work on the problems outlined in the proposed 1 project. Currently, she is involved in the polytopes reading and research seminar at University of Kentucky, which includes many graduate students and will result in a publication. The seminar is expected to continue running next semester focusing on a dierent project in combinatorics and involving new students. The PI is constantly disseminating her results at domestic and international meetings, as well as organizing seminars and special sessions. 2