Collaborative Research: Generalized Cluster Structures on Poisson Varieties and Applications

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This project lies in an area of algebra that is being developed with a view toward applications in Mathematical and Theoretical Physics. A particular focus is on the theory of Poisson-Lie groups and cluster algebras. The former has long served as a natural framework in which important exactly solvable models of classical and quantum mechanics can be studied. The latter, discovered by Fomin and Zelevinsky in 2001, have since been shown to have numerous exciting connections with a wide range of mathematical subjects, including combinatorics, representation theory, algebraic and Poisson geometry, as well as mirror symmetry and statistical and high energy physics. The PIs will build upon their previous collaborations to continue 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. This research will be linked to the development of undergraduate and graduate courses and research projects. Synergistic activities are planned to promote inter-institutional and inter-departmental cooperation, to attract graduate students from underrepresented groups and with diverse educational backgrounds, and, through community outreach, to expose high school students to mathematical research.The PIs will work on applications of Poisson geometry to the theory of cluster algebras. In more detail, the main goals of the project include: 1) construction and study of generalized cluster structures on Poisson-Lie groups and Poisson homogeneous varieties including Poisson-Lie groups equipped with Belavin-Drinfeld brackets, Drinfeld doubles and Poisson-Lie duals of simple Poisson-Lie groups,and K-theoretic Coulomb branches of 3d N = 4 SUSY gauge theories; and 2) applications of generalized cluster structures on Poisson varieties to classical and non-commutative discrete, integrable systems that arise as sequences of cluster transformations, representations of quantum groups at roots of unity, and higher Teichmuller theory.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.