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

特别会议：组合学的最新进展，2007 年 CRM 主题学期

• 批准号: 0603479
• 负责人: Mark Haiman
• 金额: $ 6.85万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0603479&HistoricalAwards=false
• 关键词: Special; Meeting; Recent; Advances; Combinatorics

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.

2007 年特殊学期“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研究中心举行。组合学是一门经常由最简单和最基本的问题激发的学科，但其技术和范围的范围却令人惊叹。应用的广度。随着最近在一些基本和困难的理解上取得了非凡的突破，许多组合主题目前非常令人兴奋 问题。 目前特别强调的是新的“麦克唐纳多项式组合学”的发展，它以令人惊讶的方式将广泛的主题联系在一起。 本学期由四个研究研讨会组成，每个研讨会都有一个为研究生和博士后级别参与者提供的相关预科学校。 每个研讨会/学校组合都集中在当前研究进展正在发生的领域。 这笔赠款将支持来自美国的研究生和初级研究人员参加研讨会和学校，为他们提供一个难得的机会，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互动。三个主题引导这些主题的研讨会。 (1) 枚举组合学与统计物理的关系；更具体地说，使用枚举技术研究气体模型、伊辛模型和波茨模型等，以计算相变或热力学极限。 (2) 单词组合学的代数方面，强调与自由群、自由域、自由李代数研究的自然联系，以及与 Artin 台球和实二次域研究相关的连分数特殊性质的自然联系。 (3) 代数组合学、代数几何和表示论之间的相互作用，这是两个研讨会的主题。 这里的重点是研究舒伯特簇、希尔伯特方案、格罗莫夫-维滕不变量等主题，以及它们与对称函数（如麦克唐纳多项式）以及经典组合枚举的联系。