Research Focus: Andrei Zelevinsky's Cluster Algebras
Every once in a while, a truly innovative idea in mathematics comes along that seems to catch fire and spread rapidly, generating new theories in many different fields. One such innovative idea is that of a "cluster algebra", introduced in 2000 by Professor Andrei Zelevinsky of our Department, with his collaborator Professor Sergey Fomin of the University of Michigan. The idea of a cluster algebra originated in representation theory, but it provides a unifying algebraic/combinatoric foundation for a wide variety of mathematical theories.
Cluster algebras are commutative algebras that are defined using subsets or "clusters" of fixed cardinality ("rank") chosen from a set of a distinguished generators ("cluster variables"). A "seed" is a cluster together with an "exchange matrix" with integer entries that are indexed by elements of the cluster. Neither the generators nor the seeds are given at the outset, but both are produced by an iterative process of successive mutations. For a more detailed description and examples, see the Wikipedia article and Professor Zelevinsky's column "What is ... a Cluster Algebra?" in the series of columns on "hot" mathematical notions in the Notices of American Mathematical Society. The image above is from the Wikipedia article and represents a mutation between triangulations of a heptagon.
Once introduced, cluster algebras were recognized as being implicit in a wide range of mathematical theories; in this sense, they might be described more as a discovery than an invention. Some of the exciting connections and applications of cluster algebras include quiver representations, preprojective algebras, Calabi-Yau algebras and categories, Teichmüller theory, discrete integrable systems, and Poisson geometry. The idea has inspired several influential papers such as those by Bakke Buan-Marsh-Reineke-Reiten-Todorov (2006), Iyama (2007), Keller-Reiten (2008), Marsh (2009), and Amiot (2009). For more references to papers, courses, and conferences related to cluster algebras, see the Portal maintained by Professor Fomin.
As a further testament to their impact on mathematics, the Mathematical Sciences Research Institute (MSRI) in Berkeley, California has scheduled a four-month program in Fall 2012 dedicated to cluster algebras. Professor Zelevinsky will spend the Fall 2012 semester in residence at MSRI, holding a prestigious Eisenbud Professorship, delivering lectures, and carrying over joint research projects with other participants. We can expect that cluster algebras will continue to propagate and increase their impact on mathematics as a result of the MSRI program.