Collaborative Research: Generalized Cluster Structures of Geometric Type

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This project lies in an area of algebra that is being developed with a view to applications in mathematical and theoretical physics. The emphasis is on a type of structure called "cluster algebras" invented in 2001 that have since proven to be useful in a wide and still expanding range of mathematical subjects including combinatorics, geometry, and high energy physics. This project will naturally lead to the development of courses and research projects on both undergraduate and graduate levels. Planned activities will develop new inter-institutional and cross-disciplinary collaborations include conferences, workshops and seminar series. Particular attention will be paid to recruitment of graduate students from underrepresented groups and with diverse educational backgrounds, and, through community outreach, to attracting high school students to mathematical research.The PIs will continue their work on applications of Poisson Geometry to the theory of cluster algebras. The main goals of the project include construction and study of (i) exotic generalized 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 nets; and (iv) quivers of finite mutation type.