Special Meeting: Recent Advances in Combinatorics, CRM Thematic Semester 2007

University Of California-Berkeley, Berkeley CA

Investigators

Abstract

The 2007 special semester "Recent Advances in Combinatorics" will take place at the Centre de Recherche en Mathematiques in Montreal, Canada from January to July 2007. Combinatorics, a subject often motivated by the simplest and most basic of questions, is nevertheless breathtaking in its scope of techniques and breadth of applications. Many combinatorial topics are very exciting at the moment, following extraordinary recent breakthroughs in the understanding of some fundamental and difficult questions. Of particular emphasis at present is the development of a new "Combinatorics of Macdonald Polynomials," which ties together far-flung topics in a surprising way. The semester is organized as a series of four research workshops, each with an associated preparatory school for graduate and post-doctoral level participants. Each workshop/school combination concentrates on an area in which current research advances are occurring. This grant will support attendance at the workshops and schools by graduate students and junior researchers from the United States, giving them an unusual opportunity to interact with a large number of the world's leading researchers in these areas, particularly those from Europe and Asia. Three main themes guide the topics of the workshops. (1) Relations between Enumerative Combinatorics and Statistical Physics; more specifically, the uses of enumerative techniques for the study of gas models, Ising and Potts models, etc., for the purpose of computing phase transitions or thermodynamic limits. (2) Algebraic Aspects of Combinatorics on Words, underlining natural links with the study of free groups, free fields, free Lie algebras, and of special properties of continued fractions in relation with the study of Artin's billiards and real quadratic fields. (3) Interactions between Algebraic Combinatorics, Algebraic Geometry and Representation Theory, a theme of two of the workshops. The emphasis here is on the study of subjects such as Schubert varieties, Hilbert schemes, Gromov-Witten invariants, and their ties with symmetric functions such as Macdonald polynomials, and with classical combinatorial enumeration.