Combinatorial Structures in Cluster Algebras

Cluster varieties are certain highly symmetric and structured geometric spaces. Over the last twenty years, cluster varieties have turned up throughout geometry, representation theory and mathematical physics. Cluster algebras are algebraic structures which allow us to compute with cluster varieties and understand their structure. This research project pursues two lines of investigation in order to better understand cluster varieties. The project provides training opportunities for both undergraduate and graduate students. The first line of investigation is to construct cluster structures on braid varieties, Richardson varieties, and related spaces. These are geometric objects which originally arose in representation theory and have recently been discovered to play important roles in knot theory. Constructing cluster structures on these spaces will allow us to use all the techniques from cluster theory to explore these spaces. The second line of investigation is to develop relationships between cluster varieties and Coxeter groups, which describe the symmetries of collections of reflecting mirrors. This has been extensively done for cluster algebras "of finite type", which correspond to Coxeter groups with finitely many reflections, but an analogous story should exist for all cluster algebras, and this project proposes to develop it. 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.