Cluster Algebras, Combinatorics, and Knot Theory

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The theory of cluster algebras is a young research area in mathematics that was set in motion by Fomin and Zelevinsky in 2002. Their original motivation came from representation theory, a branch of modern algebra, which examines the symmetries of an algebraic structure rather than examining the structure directly. Representation theory has numerous applications in physics and chemistry as well as other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns that occur throughout representation theory. Surprisingly, these patterns are also observed in various other branches of science which, a priori, are not related to representation theory. This project will enhance the understanding of cluster algebras and provide new research directions in an already highly active research area. The investigator will establish and develop relations between cluster algebras and other areas of mathematics. These new connections will allow explicit computational results as well as structural development. The project will also investigate longstanding open questions. The project will involve graduate students in the proposed research.The research project focuses on cluster algebras and their relation to combinatorics, knot theory, number theory, and representation theory of finite-dimensional algebras. The investigator will pursue several objectives. He will work on establishing explicit formulas for the generators of a cluster algebra of arbitrary type. He will also develop a fundamental connection between cluster algebras and knot theory which will realize important knot invariants as specialized cluster variables. Furthermore, he will create a combinatorial model for the category of maximal Cohen-Macauley modules over a class of finite-dimensional algebras that arise naturally in additive categorifications of cluster algebras as the endomorphism algebras of clusters. He will also study Markov numbers from a novel point of view using the cluster algebra of the once-punctured torus.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.