Cluster Algebras, Combinatorics, and Knot Theory

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When cluster algebras were introduced by Fomin and Zelevinsky in 2002, their original motivation came from representation theory, which is a branch of modern algebra. Studying the symmetries of a model is often more fruitful than studying the model directly, and representation theory has found many applications in physics and chemistry as well as in other mathematical fields. The cluster algebras provide a mathematical framework for fundamental patterns which 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 provide new directions of research 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 allow explicit computational results as well as structural development. The project contains the instigation of new ideas as well as the investigation of longstanding open questions.The 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 investigations. He will apply his recent discovery of a relation between cluster algebras and continued fractions to study classical problems in number theory from a new point of view and to apply the machinery of continued fractions to develop new tools in cluster algebras. He will also establish and develop a new connection between cluster algebras and knot theory providing explicit combinatorial formulas for Jones polynomials and other knot invariants. Furthermore, he will continue his study of the representation theory of finite-dimensional non-commutative algebras that arise from cluster algebras.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.