150th Anniversary Invited Lecture Series

Monday 16th March, 2015 to Friday 20th March, 2015
Durham University, Stockton Road, Durham, DH1 3LE

Professor Michael Shapiro (MSU) will be giving a series of lectures on Cluster Algebras and Integrable Systems aimed at graduate students. 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 minicourse consisting of ten lectures will be 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 will be fully accessible to postgraduate students and non-specialists interested in the topic.

The preliminary content of the course:

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

Supplementary lectures will be given by Robert Marsh (Leeds), Andy Hone (Kent), Sebastian Franco (CCNY) and Alexander Veselov (Loughborough).

Deadline for registration 8 February 2015. Limited financial support is available with preference given to UK research students, please contact the organizers for further details.

Organiser: Pavel Tumarkin (Durham)