Conference Proposal: Talbot Workshops 2005 - 2007: Geometric Langlands

摘要奖：DMS-0512714首席研究员：海恩斯·R·米勒，迈克尔·J·霍普金斯塔尔伯特项目是一系列年度研讨会，将研究生、新近的博士和教师聚集在一起，对活跃的数学研究主题进行紧张的探索。2004年的Talbot研讨会专门讨论了椭圆上同调的Stolz-Teichner模型，并由Stephan Stolz指导。塔尔博特2005年的研讨会将重点讨论几何朗兰兹计划，并由大卫·本-兹维指导。几何朗兰兹程序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。朗兰兹最初的猜想将无限维表示理论与伽罗华群的结构联系在一起，并在数论中有了深刻的应用。从著名的Borel-Weil-Bott定理引出的几何观点来看，人们希望通过参数化空间上的等变薄片来分类这种表示。德恩菲尔德和劳蒙等人采用了朗兰兹关于函数域的原始思想，形成了一个影响深远的几何推广，现在被称为几何朗兰兹程序。在过去的二十年里，在这个领域的工作已经改变了现代表征理论的面貌。最近的进展，如经典的佐竹同构的新的几何推广，使现在成为聚集年轻数学家研究这些结果的理想时间，用于推导这些结果的技术，以及新暴露的未来研究途径。Talbot研讨会寻求将有抱负的数学家介绍到数学研究的活跃领域，促进跨分支学科和机构线的社区和合作，并在知名数学家和年轻研究人员之间形成教学和研究联系。Talbot 2004的主题涉及数学中的一个中心问题，这个问题与八个枯燥的科学领域密切相关。例如，蛋白质的结晶学研究，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；它们的深入研究需要探索支配其性质方面的代数结构。表示论正是致力于这项研究的数学领域。此外，几何朗兰兹程序为适合于每个区域的特定代数结构提供了统一的愿景，这些代数结构被称为反射群、李群和伽罗群。这个塔尔博特研讨会将汇集来自多所大学的不同专业的毕业生，用一周的时间专注于这一重要主题。参与者和他们的导师大卫·本-兹维将共享一处住所，以及演讲、讨论和餐饮。这种非正式的氛围，混合了友谊和智力兴趣，将促进对材料的集中学习，并为未来的合作和研究奠定基础。

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.