Cluster Algebras and Integrable Systems
The theory of cluster algebras was introduced in 2001 by Sergey Fomin and Andrei Zelevinsky in the effort to understand a famous representation theory construction of Lusztig-Kashiwara dual canonical bases. The theory of cluster algebras offers a fresh approach to some of the natural problems in mathematics and mathematical physics, more elementary and "down-to-earth" than many other existing approaches.
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. Important examples of cluster structures include simple Lie groups, homogeneous spaces, configuration spaces of points, and are related to discrete and continuous integrable systems.
The Department of Mathematical Sciences hosts a vibrant community of mathematicians, some of whom are world experts in Cluster Algebras. They have attracted the LMS Invited Lectures 2015 to Durham, delivered by Michael Shapiro between March 16 and March 20, 2015.
This minicourse, organised by Pavel Turmarkin, has consisted of ten lectures devoted to the fast growing area on the intersection of cluster algebras theory and integrable systems, and interactions of these with other areas of mathematics and theoretical physics. The course was designed to be fully accessible to postgraduate students and non-specialists interested in the topic.
The following topics were discussed:
- Introduction of cluster algebra
- Compatible Poisson structures and their applications
- Cluster approach to Teichmuller spaces
- Generalized cluster algebras associated with orbifold surfaces
- Finite type and finite mutation type classification
- Planar networks, inverse problems
- Cluster transformations as Backlund Darboux transformations of Toda lattices
- Pentagram maps and cluster integrability
- Dimers and cluster integrable systems
- Scattering Amplitudes and the Positive Grassmannian