Cluster Algebras and Categorification

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, have an intricate structure that can appear in many other, seemingly unrelated, areas of mathematics and theoretical physics. In particular, they provide a rigid framework that captures various patterns formed by a given system of elements. Using cluster algebras, this system can then be translated into a different setting, which allows us to use new tools and techniques for its study.

Using this approach, the theory of cluster algebras has led to the establishment of many influential results in mathematics. This project is in a highly active research area aimed at answering fundamental questions about the structure of cluster algebras, as well as investigating new connections to other fields of mathematics. This project will focus both on obtaining explicit computational results and developing general properties.

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.

The project will explore new connections between cluster algebras and their various categorifications to further their understanding and build parallels between the different areas. Its goal is to develop novel 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 ingredient behind solving these questions is the powerful machinery coming from the representation theory of associative algebras, which encode the structure of cluster algebras and constitute an invaluable tool in their study. In particular, the PI will study the recently-found cluster structures coming from positroid and Richardson varieties in the Grassmannian.

Moreover, the PI proposes a new combinatorial description of Cohen-Macauley subcategories in the module category of certain Jacobian algebras which offers applications to various topics in cluster algebras and representation theory.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.