Collaborative Research: Cluster Algebras Approach to Poisson-Lie Groups and Higher Genus Directed Networks

University Of Notre Dame, Notre Dame IN

Investigators

Abstract

The proposed research builds upon PIs' previous collaboration on applications of Poisson geometry to cluster algebras. In the current project we will undertake a systematic study of multiple cluster structures in coordinate rings of a number of varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. Important examples include simple Lie groups, homogeneous spaces, configuration spaces of points, and are related to discrete and continuous integrable systems. The problems to be considered include cluster structures on simple Lie groups compatible with Poisson-Lie structures associated with the Belavin-Drinfeld classification, inverse problems for directed nets on surfaces of higher genus, continuous limits for directed networks with applications to moduli spaces of flat connections and growth rate classification of cluster algebras. The rapid development of the cluster algebra theory in recent years revealed relations between cluster algebras and a variety of areas including, among others, commutative and non-commutative algebraic geometry, quiver representations and Teichmuller theory. The proposed research is linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned with the goal to promote inter-institutional and inter-departmental cooperation, to attract graduate students from underrepresented groups and with diverse educational backgrounds, and, through a community outreach, at the early exposure of high school students to mathematical research.