The PI intends to study a model for cluster algebras in terms of certain tensor diagrams called webs. This model relates to higher Techmuller theory, specifically to representations of fundamental groups of Riemann surfaces inside special linear group of a three dimensional vector space. This class of cluster algebras allows one to have a conjectured purely combinatorial model for some cluster algebras of infinite mutation type. The PI also intends to study a generalization of cluster algebras called Laurent phenomenon algebras. In it, the exchange polynomials do not necessarily have to be binomial. The examples of such structures appear for example in the study of electrical networks, and in the study of hyperbolic metrics on non-orientable surfaces.
The proposed project is rooted in two areas of mathematics: representation theory and combinatorics. Representation theory is an important area that is fundamental for the study of quantum mechanics and particle physics. Cluster algebras provide a new approach to representation theory and related disciplines, and have found their application in string theory and quantum gravity. The PI's research focuses on cluster algebras that lie just beyond the current frontier of our understanding (the so called infinite mutation type). Another part of the proposed research focuses on a generalization of cluster algebras, called Laurent phenomenon algebras. The educational part of the proposal pertains to creating a summer undergraduate research program (URE) for international students which would run parallel to the already existing summer REU in combinatorics at University of Minnesota, Minneapolis.
