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

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The 2007 special semester "Recent Advances in Combinatorics" will takeplace at the Centre de Recherche en Mathematiques in Montreal, Canadafrom January to July 2007. Combinatorics, a subject often motivatedby the simplest and most basic of questions, is neverthelessbreathtaking in its scope of techniques and breadth of applications.Many combinatorial topics are very exciting at the moment, followingextraordinary recent breakthroughs in the understanding of somefundamental and difficult questions. Of particular emphasis atpresent is the development of a new "Combinatorics of MacdonaldPolynomials," which ties together far-flung topics in a surprisingway. The semester is organized as a series of four researchworkshops, each with an associated preparatory school for graduate andpost-doctoral level participants. Each workshop/school combinationconcentrates on an area in which current research advances areoccurring. This grant will support attendance at the workshops andschools by graduate students and junior researchers from the UnitedStates, giving them an unusual opportunity to interact with a largenumber of the world's leading researchers in these areas, particularlythose from Europe and Asia.Three main themes guide the topics of the workshops. (1) Relationsbetween Enumerative Combinatorics and Statistical Physics; morespecifically, the uses of enumerative techniques for the study of gasmodels, Ising and Potts models, etc., for the purpose of computingphase transitions or thermodynamic limits. (2) Algebraic Aspects ofCombinatorics on Words, underlining natural links with the study offree groups, free fields, free Lie algebras, and of special propertiesof continued fractions in relation with the study of Artin's billiardsand real quadratic fields. (3) Interactions between AlgebraicCombinatorics, Algebraic Geometry and Representation Theory, a themeof two of the workshops. The emphasis here is on the study ofsubjects such as Schubert varieties, Hilbert schemes, Gromov-Witteninvariants, and their ties with symmetric functions such as Macdonaldpolynomials, and with classical combinatorial enumeration.