Cluster Algebras, Combinatorics and Knot Theory

University Of Connecticut, Storrs CT

Investigators

Abstract

The theory of cluster algebras is a highly active research area in mathematics that was initiated in 2002. The original motivation came from representation theory, a branch of modern algebra which is concerned with studying the symmetries of an algebraic structure rather than studying the structure itself. Representation theory has numerous applications in physics and chemistry as well as in other mathematical fields. Cluster algebras capture fundamental underlying combinatorial patterns that occur throughout representation theory. Quite remarkably, these patterns turn out to be present as well in a number of other branches of mathematics and physics that had previously seemed mostly unrelated. This project will contribute to the development of cluster algebras and their relations to other areas, in particular to knot theory and representations of algebras. The project will involve graduate students in the proposed research. The project has several objectives. The principal investigator will develop a fundamental connection between cluster algebras and knot theory that realizes important knot invariants as specializations of cluster variables. The centerpiece of this project is the construction of a cluster algebra from an arbitrary knot or link, such that the cluster algebra contains a cluster in which each cluster variable specializes to the Alexander polynomial of the knot. The second objective is to study Cohen-Macaulay modules over 2-Calabi-Yau tilted algebras. These are non-commutative algebras that are associated to the clusters of a cluster algebra via categorification. One overarching goal is the classification of 2-Calabi-Yau tilted algebras that admit only finitely many Cohen-Macaulay modules. A third aim is to study maximal almost rigid modules, a new concept in representation theory inspired by the PI’s previous work on Catalan combinatorics. In this project, he PI will show that the triangulations of a surface with marked points and dissection correspond bijectively to the maximal almost rigid modules over an algebra associated to the surface dissection. 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.