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

2007年特别学期“组合数学的最新进展”将于2007年1月至7月在加拿大蒙特利尔的数学研究中心举行。 组合数学，一个通常由最简单和最基本的问题所激发的学科，在其技术范围和应用广度上无疑是令人惊叹的。许多组合主题在当前是非常令人兴奋的，因为最近在理解一些基本和困难的问题上取得了非凡的突破。 目前特别强调的是一个新的“麦克唐纳多项式组合学”的发展，它以一种简洁的方式将广泛的主题联系在一起。 本学期是由四个研究工作坊组成，每个工作坊都有一个为研究生和博士后水平的参与者提供的相关预备学校。 每个研讨会/学校组合集中在一个领域，其中目前的研究进展正在发生。 这笔赠款将资助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领先研究人员，特别是来自欧洲和亚洲的研究人员进行互动。 (1)计数组合学与统计物理学之间的交叉;更具体地说，计数技术在气体模型、伊辛和波茨模型等研究中的应用，for the purpose目的of computing计算phase相transitions转变or thermodynamic热力学limits极限. (2)代数方面的组合的话，强调自然的联系与研究ofree集团，自由领域，自由李代数，和特殊性质的连续分数与研究阿廷的台球和真实的二次领域。 (3)代数组合学、代数几何学和表示论之间的相互作用，是两个工作坊的主题。 这里的重点是研究的科目，如舒伯特品种，希尔伯特计划，格罗莫夫-维特不变量，以及他们的关系与对称函数，如麦克唐纳多项式，并与经典的组合计数。

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.