Research Seminar "Cluster Poisson Varieties"

Cluster varieties introduced by Fomin and Zelevinsky are commutative rings with unit andno zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlappingsubsets (clusters) of the same cardinality connected by exchange relations. Originally they were introduced inan attempt to create an algebraic and combinatorial framework for the study total positivity in semisimplegroups. In the case of GL𝑛 the notion of total positivity coincides with the classical one, first introducedby Gantmakher and Krein. Since then, the theory of cluster algebras has witnessed a spectacular growthdue to the many links that have been discovered with a wide range of subjects including representationtheory of quivers and finite-dimensional algebras and categorification; discrete dynamical systems based onrational recurrences; Teichmuller and higher Teichmuller spaces; combinatorics and the study of combinatorialpolyhedra; commutative and non-commutative algebraic geometry, projective configurations and their tropicalanalogues, the study of stability conditions in the sense of Bridgeland, Donaldson – Thomas invariants; Poissongeometry and theory of integrable systems. The purpose of the course is to give an introduction to the theoryof cluster poisson varieties.