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.
PI and collaborators continued their work on applications of Poisson Geometry to the theory of cluster algebras. Main topics of study included (i) exotic cluster structures on simple Lie groups compatible with Poisson-Lie brackets described by the Belavin-Drinfeld classification; (ii) generalized cluster structures on the Drinfeld double and the Poisson-Lie dual of a simple Poisson-Lie group; (iii) discrete integrable systems arising as sequences of cluster transformations and elementary transformations of higher genus networks; Coexistence of diverse mathematical structures supported on the same variety often leads to deeper understanding of its features. If the manifold is a Lie group, endowing it with a Poisson structure that respects group multiplication (Poisson-Lie structure) is instrumental in a study of classical and quantum mechanical systems with symmetries. In turn, a Poisson structure on a variety can be compatible with a cluster structure - a useful combinatorial tool that organizes generators of the ring of regular functions into a collection of overlapping clusters connected via rational transformations. Since the invention of cluster algebras by Fomin and Zelevinsky in 2001, the mathematical community witnessed an explosion of interest to the subject due to the deep connections that were revealed between and variety of branches of mathematics and theoretical physics ranging from quiver representations and algebraic geometry to string theory and statistical physics. PI and collaborators built upon their previous research to continue a systematic study of multiple cluster structures in coordinate rings of Poisson-Lie groups and a number of other varieties of importance in algebraic geometry, representation theory and mathematical physics and study an interaction between corresponding cluster algebras. The obtained results were utilized in the development of undergraduate and graduate courses and research projects. PI took part in synergistic activities that were aimed at promoting inter-institutional and inter-departmental cooperation, and strived to attract graduate students from underrepresented groupsand with diverse educational backgrounds, and, through a community outreach, at the early exposure of high school students to mathematical research.
