Conference Proposal: Talbot Workshops 2005 - 2007: Geometric Langlands

摘要奖：DMS-0512714主要研究者：Haynes R.米勒，迈克尔J霍普金斯塔尔博特计划是一系列的年度研讨会务虚会汇集研究生，最近的博士，和教师导师的积极数学研究的一个主题的激烈探索。塔尔博特2004年研讨会致力于椭圆上同调的Stolz-Teichner模型，由Stephan Stolz指导。塔尔博特2005年研讨会将集中在几何朗兰兹计划和指导由大卫本兹维。几何朗兰兹纲领是数学中最丰富的领域之一，它综合了表示论、拓扑学和代数几何学。朗兰兹的原始理论将无限维表示理论与伽罗瓦群的结构联系起来，并在数论中有着深远的应用。从著名的Borel-Weil-Bott定理导出的几何观点出发，人们希望通过参数化空间上的等变层对这种表示进行分类。Drinfeld和Laumon，除其他外，适应朗兰兹原来的想法forfunction领域制定了深远的geometricgeneralization，现在被称为几何朗兰兹计划。在这方面的工作在过去的二十年里已经改变了现代表征理论的面貌。最新的进展，如经典佐竹同构的新几何推广，使这成为一个理想的时机，聚集年轻的数学家来研究这些结果，用于导出它们的技术，以及未来研究的新途径。塔尔博特研讨会旨在向有抱负的数学家介绍数学研究的活跃领域，促进跨学科和机构的社区和合作，并在数学家和年轻的研究人员之间建立教学和研究联系。塔尔博特2004年的主题涉及到一个与周边科学领域密切相关的数学中心问题。例如，蛋白质的晶体学研究、粒子的量子态以及多项式方程的整数解的结构都是化学、物理学和密码学/数论的核心;对它们的深入研究需要探索支配其性质的代数结构。表示论正是致力于这一研究的数学领域。 此外，几何朗兰兹计划提供了一个统一的视野，为特定的代数结构，适合于每一个这些领域，这是所谓的反射群，李群，伽罗瓦群。这个塔尔博特研讨会将汇集不同专业的研究生和来自许多大学花一个星期的时间集中在这个重要的主题。参与者和他们的导师，大卫本-兹维，将分享一个住所以及讲座，讨论和膳食。这种非正式的气氛，混合奖学金和智力兴趣，将促进集中研究的材料，并奠定了基础，为未来的合作和研究。

AbstractAward: DMS-0512714Principal Investigator: Haynes R. Miller, Michael J. HopkinsThe Talbot program is a series of yearly workshop retreatsbringing together graduate students, recent PhDs, and facultymentors for an intense exploration of a topic of activemathematical research. The Talbot 2004 workshop was devoted tothe Stolz-Teichner model of elliptic cohomology and was mentoredby Stephan Stolz. The Talbot 2005 workshop will focus on thegeometric Langlands program and be mentored by David Ben-Zvi. Thegeometric Langlands program is one of the most fertile areas ofmathematics, integrating representation theory, topology, andalgebraic geometry. The original conjectures of Langlands tieinfinite-dimensional representation theory to the structure ofGalois groups, with profound applications to number theory. Fromthe geometric perspective deriving from the renowned theorem ofBorel-Weil-Bott, one would like to classify such representationsby equivariant sheaves on a parametrizing space. Drinfeld andLaumon, among others, adapted Langlands' original ideas forfunction fields to formulate a far reaching geometricgeneralization, now called the geometric Langlands program. Workin this area over the past twenty years has changed the face ofmodern representation theory. Recent advances, such as newgeometric generalizations of the classical Satake isomorphism,make this an ideal time to gather young mathematicians to studythese results, the techniques used to derive them, and newlyexposed avenues for future research.The Talbot workshops seek to introduce aspiring mathematicians toactive areas of mathematical research, foster community andcollaboration across subdisciplinary and institutional lines, andform pedagogical and research ties between establishedmathematicians and young researchers. The topic for Talbot 2004concerns a central problem in mathematics deeply related toneighboring areas of science. For instance, the crystallographicstudy of proteins, quantum states of particles, and the structureof integer solutions to polynomial equations are all central tochemistry, physics, and cryptography/number theory; their deeperstudy requires probing the algebraic structures that governaspects of their nature. Representation theory is precisely thefield of mathematics devoted to this study. Further, thegeometric Langlands program offers a unifying vision for thespecific algebraic structures suited to each of these areas,which are called reflection groups, Lie groups, and Galoisgroups. This Talbot workshop will bring together graduatestudents with different specialties and from many universities tospend a week focused on this important subject. The participantsand their mentor, David Ben-Zvi, will share a residence as wellas lectures, discussions, and meals. This informal atmosphere,mixing fellowship and intellectual interest, will promote aconcentrated study of the material and lay the foundation forfuture collaboration and research.