Friezes, Syzygies, and Connections to Cluster Algebras

Cluster algebras, introduced by Fomin and Zelevinsky in 2001, are a highly active and influential research area. These objects have a beautiful complex structure and are defined through certain specific patterns. Moreover, the same defining patterns appear in many other, seemingly unrelated, areas of mathematics and theoretical physics, enabling one to translate problems into different settings and apply new techniques to study them. In this way, cluster algebras have been instrumental in establishing numerous significant results in mathematics that reach across different fields. This project is aimed at answering fundamental questions about certain objects closely related to cluster algebras and appearing at the intersection of combinatorics and algebra, as well as investigating new connections to other areas of mathematics. The project will also contribute to the advancement of education and research in the mathematical community by working with graduate students and fostering international collaborations. The goal of this project is to study two classes of objects that lie at the intersection of algebra and combinatorics and are closely related to the theory of cluster algebras. The first one is SL_k friezes, certain arrays of numbers that are defined combinatorially and related to Grassmannian cluster algebras. The project aims to develop a thorough understanding of SL_k friezes and tilings, beyond the well-studied case k = 2. The second one is Cohen-Macauley subcategories of Jacobian algebras, which provide additive categorification of cluster algebras. The main objective is to classify algebras of finite Cohen-Macauley type and to derive a new combinatorial model for their Cohen-Macauley categories coming from dimer models. The main tools to answer these questions rely on utilizing existing connections between cluster algebras, representation theory, and combinatorics, as well as establishing new ones to strengthen the relations between the different topics. 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.