2022/2023

Research Seminar "Cluster Poisson Varieties"

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

When: 1, 2 module

Open to: students of all HSE University campuses

Instructors: Vasily Gorbounov

Language: English

ECTS credits: 3

Contact hours: 32

Course Syllabus

Abstract

Cluster varieties introduced by Fomin and Zelevinsky are commutative rings with unit and no zero divisors equipped with a distinguished family of generators (cluster variables) grouped in overlapping subsets (clusters) of the same cardinality connected by exchange relations. Originally they were introduced in an attempt to create an algebraic and combinatorial framework for the study total positivity in semisimple groups. In the case of GL𝑛 the notion of total positivity coincides with the classical one, first introduced by Gantmakher and Krein. Since then, the theory of cluster algebras has witnessed a spectacular growth due to the many links that have been discovered with a wide range of subjects including representation theory of quivers and finite-dimensional algebras and categorification; discrete dynamical systems based on rational recurrences; Teichmuller and higher Teichmuller spaces; combinatorics and the study of combinatorial polyhedra; commutative and non-commutative algebraic geometry, projective configurations and their tropical analogues, the study of stability conditions in the sense of Bridgeland, Donaldson – Thomas invariants; Poisson geometry and theory of integrable systems. The purpose of the course is to give an introduction to the theory of cluster poisson varieties.