Conference Proposal: Talbot Workshops 2008-2010

该塔尔博特研讨会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，教师导师的积极数学研究的一个主题的激烈探索。过去的研讨会一直致力于拓扑模形式（由迈克尔霍普金斯指导），流形的自同构（由迈克尔韦斯指导），几何朗兰兹（由大卫Ben-Zvi指导）和椭圆上同调的几何模型（由斯蒂芬Stolz指导）。丹尼斯Gaitsgory将指导2008年塔尔博特研讨会，仿射李代数和手征结构，这将侧重于几何方法的表示理论的仿射李代数和量子群使用贝林森和德林费尔德的手征代数理论。手征代数是曲线上的某些结构化D-模，它们编码共形场论的算子乘积展开的数据;它们提供了顶点代数理论的灵活几何模拟，用于描述共形场论的对称性。在Gaitsgory和Jacob Lurie最近的工作中，手征代数理论与同伦理论和高等范畴理论的思想相结合，在几何表示论中产生了重要的新结果。研讨会的一个特别重点将是其中的一些工作，特别是Gaitsgory-Lurie的几何Satake等价的衍生版本和量子群的几何Langlands对偶的公式。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，并形成教学和研究之间的联系建立数学家和年轻的研究人员。塔尔博特2008年的主题是关于存在于基本物理理论中的对称性的数学研究。理解这些对称性的结构，就能理解定义物理理论的守恒原理，比如能量守恒。从最近的量子物理学的对称性的研究导致了新的代数结构定义这些对称性。2008年的研讨会将向与会者介绍这些代数结构的最新工作。该研讨会将汇集来自许多大学的不同专业的研究生和初级教师，花一周时间专注于这一重要主题。参与者和他们的导师，丹尼斯Gaitsgory，将共享一个住所，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，并为未来的合作和研究奠定基础。

The Talbot workshop program is a series of yearly workshop retreats bringing together graduate students, recent PhDs, and faculty mentors for an intense exploration of a topic of active mathematical research. Past workshops have been devoted to Topological Modular Forms (mentored by Michael Hopkins), Automorphisms of Manifolds (mentored by Michael Weiss), Geometric Langlands (mentored by David Ben-Zvi), and Geometric Models of Elliptic Cohomology (mentored by Stephan Stolz). Dennis Gaitsgory will mentor the 2008 Talbot workshop, Affine Lie Algebras and Chiral Structures, which will focus on geometric approaches to the representation theory of affine Lie algebras and quantum groups using Beilinson and Drinfeld's theory of chiral algebras. Chiral algebras are certain structured D-modules on curves that encode the data of the operator product expansion of a conformal field theory; they provide a flexible geometric analogue of the theory of vertex algebras for describing the symmetries of conformal field theories. In recent work of Gaitsgory and Jacob Lurie, the theory of chiral algebras has been combined with ideas of homotopy theory and higher category theory to produce important new results in geometric representation theory. A particular focus of the workshop will be some of this work, in particular Gaitsgory-Lurie's derived version of the geometric Satake equivalence and a formulation of geometric Langlands duality for quantum groups.The Talbot workshops seek to introduce aspiring mathematicians to active areas of mathematical research, foster community and collaboration across subdisciplinary and institutional lines, and form pedagogical and research ties between established mathematicians and young researchers. The topic for Talbot 2008 concerns the mathematical study of the symmetries that exist in fundamental physical theories. Understanding the structure of these symmetries lead to understanding the conservation principles, such as the conservation of energy, that define physical theories. The study of symmetries from recent quantum physics have led to new algebraic structures defining these symmetries. The 2008 workshop will introduce participants to recent work on these algebraic structures. The workshop will bring together graduate students and junior faculty with different specialties and from many universities to spend a week focused on this important subject. The participants and their mentor, Dennis Gaitsgory, will share a residence as well as lectures, discussions, and meals. This informal atmosphere, mixing fellowship and intellectual interest, will promote a concentrated study of the material and lay the foundation for future collaboration and research.