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

2007年特别学期“组合学的最新进展”将于2007年1月至7月在加拿大蒙特利尔数学中心举行。组合学是一门经常由最简单和最基本的问题所激发的学科，它的技术范围和应用范围从来都是令人惊叹的。许多组合主题目前非常令人兴奋，因为最近在一些基本和困难问题的理解上取得了非凡的突破。目前特别强调的是开发一种新的“麦克唐纳多项式组合”，它以一种令人惊讶的方式将广泛的话题联系在一起。本学期由四个研究班组成，每个研究班都有一个相关的预科学校，供研究生和博士后参加。每个研讨会/学校的结合都集中在当前研究进展正在发生的一个领域。这笔赠款将支持来自美国的研究生和初级研究人员参加讲习班和学校，使他们有一个难得的机会与这些领域的大量世界顶尖研究人员，特别是来自欧洲和亚洲的研究人员交流。(1)计数组合学和统计物理学之间的关系；更具体地说，是为了计算相变或热力学极限而使用计数技术研究气体模型、Ising和Potts模型等。(2)关于词的组合学的代数方面，强调了与三元群、自由场、自由李代数的研究的自然联系，以及与Artin台球和实二次域的研究有关的连分式的特殊性质。(3)代数组合学、代数几何和表示论之间的相互作用，这是其中两个工作坊的主题之一。这里的重点是研究诸如Schubert变分、Hilbert格式、Gromov-Wittenant不变量，以及它们与对称函数如Macdonald多项式的联系，以及与经典组合计数的关系。

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.