Representation Theory and Mathematical Physics

主要研究者：Kapranov，Mikhail 提案编号：DMS - 0925341机构：耶鲁大学标题：表示论和数学物理李群及其表示是数学的一个基本领域，与几何，拓扑，数论，物理，组合学和许多其他领域有联系。表示论是数论中朗兰兹纲领的基石之一，可以追溯到20世纪70年代。朱克曼的工作派生函子，翻译原则，并连贯的延续在于核心的现代理论表示李群。表示论中一个未解决的主要问题是酉对偶。事实上，在原则上，有一个有限的算法来计算酉对偶在很大程度上依赖于朱克曼的工作。近年来，数学和物理之间的相互作用，在几何表示论，弦理论，和其他领域。手征代数，仿射卡茨-穆迪代数的表示理论和几何朗兰兹对应的新发展将是会议的重点。几何朗兰兹计划的最新发展指出了某些自守表示和几何镜像对称中的SYZ型纤维化之间令人兴奋的联系。关于这些表示的问题现在可能会受到新的几何技术和镜像对称的发展。

ABSTRACTPrincipal Investigator: Kapranov, Mikhail Proposal Number: DMS - 0925341 Institution: Yale University Title: Representation Theory and Mathematical PhysicsLie groups and their representations are a fundamental area of mathematics, with connections to geometry, topology, number theory, physics, combinatorics, and many other areas. Representation theory is one of the cornerstones of the Langlands program in number theory, dating to the 1970s. Zuckerman's work on derived functors, the translation principle, and coherent continuation lie at the heart of the modern theory of representations of Lie groups. One of the major unsolved problems in representation theory is that of the unitary dual. The fact that there is, in principle, a finite algorithm for computing the unitary dual relies heavily on Zuckerman's work.In recent years there has been a fruitful interplay between mathematics and physics, in geometric representation theory, string theory, and other areas. New developments on chiral algebras, representation theory of affine Kac-Moody algebras, and the geometric Langlands correspondence will be some of the focal points of the conference. Recent developments in the geometric Langlands program point to exciting connections between certain automorphic representations and SYZ type fibrations in geometric mirror symmetry. Problems about these representations may now be amenable to new geometric techniques and insights developed for mirror symmetry.