Program

Cluster algebras were conceived in the Spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. They are constructively defined commutative algebras with a distinguished set of generators (cluster variables) grouped into overlapping subsets (clusters) of fixed cardinality.  Both the generators and the relations among them  are not given from the outset, but are produced by an iterative process of successive mutations. Although this procedure appears counter-intuitive at first, it turns out to encode a surprisingly widespread range of phenomena, which might explain the explosive development of the subject in recent years. Cluster algebras provide a unifying algebraic/combinatorial framework for a wide variety of phenomena in diverse settings ranging from tropical calculus to Lie theory and from Poisson geometry  to invariant theory. Possibly due to the fact that cluster-like structures were for quite some time implicit in many areas of mathematics, the field has exploded in recent years. The program will focus on links between cluster algebras and other areas, such as: polyhedral combinatorics;  triangulations of surfaces; Y, Q, and T-systems;  additive categorification via quiver representations;  quivers with potentials and Donaldson-Thomas invariants; Lie theory and monoidal categorification;  Poisson geometry and Teichmueller theory. To apply please visit http://www.msri.org/web/msri/scientific/member-application (Deadlines: October 1, 2011 and December 1, 2011). Bibliography (PDF) Special Opportunity: RTG Mini-Course and Workshop on "Tensors and their Geometry in High Dimensions" From September 26-29, 2012, the Berkeley RTG on Representation Theory, Geometry, and Combinatorics will host a workshop on "Tensors and their Geometry in High Dimensions," featuring Andrew Snowden, Jan Draisma, and Giorgio Ottaviani giving mini-courses. There will also be opportunities for some of the other participants to give related talks. All the details are available here: http://math.berkeley.edu/~oeding/RTG/index.html. There are no registration fees, however participants are kindly asked to register online. There may be some funding for young researchers.