Workshop on "Geometry and Representation Theory"

摘要奖：DMS-0400785主要研究人员：菲利普·A·福斯、保罗·布雷斯勒、柯蒂·N·乔希亚。几何方法和思想的强大涌入使几何表示理论取得了重大突破，并促使几何表示理论成为现代数学的一个重要研究领域。几何方法被用来探索表示论中的许多重要问题。朗兰兹计划提供了一个综合了表象理论、算术和几何的例子。在其算术化身中，朗兰兹通信设想用自同构表示来描述数域或函数域的伽罗瓦群的某些类型的表示。由Drinfel‘d和Laumon开创的朗兰兹通信的几何化身也引起了许多人的注意，并被证明是数学的几个领域的融合：表示理论、D-模、Kac-Moody和顶点代数以及可积系统、Hitchin映射等。表象理论在理论物理中无处不在，相反，物理学中的问题和发展推动了表象理论的发展，而表象理论又是对其他数学分支的重大输入。共形场理论导致了对Virasoro代数、Kac-Moody代数和顶点算子代数的表示理论的深入研究。表象理论的结果对理解模空间的结构产生了重要影响。在可积系统理论中，很早以前就观察到，物理上有意义的完全可积系统以及运动积分的显式公式与几何表示理论密切相关。所有这些发展都是我们组织几何表示理论会议的动力，这个论坛的主要参与者将是年轻的研究人员和高级研究生，向领先的专家学习，并进一步推进他们的研究项目。表示理论是现代数学的一个典型分支，研究各种代数系统的对称性。表象理论的结果和思想在纯数学和应用数学以及量子物理、生物学、经济学等领域都有许多重要的应用。最近，几何学思想的强大融合对该学科产生了重大影响，并导致了重大突破。我们会议的主要目标是聚集代表理论的顶尖专家以及初级研究人员和高级研究生，创建一个论坛，与会者可以在其中交流新思想，交流最新进展，并帮助年轻参与者制定成功的研究策略。特别强调吸引妇女和代表不足的少数群体参与者，特别是那些处于职业生涯初期的人。

AbstractAward: DMS-0400785Principal Investigator: Philip A. Foth, Paul Bressler, Kirti N. JoshiA powerful influx of methods and ideas from geometry into therepresentation theory led to significant breakthroughs andstimulated the emergence of geometric representation theory as animportant area of research in modern mathematics. Geometricmethods have been utilized to explore many significant problemsin representation theory. The Langlands program provides anexample of the synthesis of representation theory, arithmetic andgeometry. In its arithmetic avatar the Langlands correspondenceenvisages a description of certain kinds of representations ofthe Galois group of a number field or a function field in termsof automorphic representations. The geometric avatar of theLanglands correspondence pioneered by Drinfel'd and Laumon hasalso attracted a lot of attention and has turned out to be aconfluence of several areas of mathematics: representationtheory, D-modules, Kac-Moody and vertex algebras and integrablesystems, Hitchin maps to name a few. Representation theory hasbeen omnipresent in theoretical physics and conversely, problemsand developments in physics motivated much progress inrepresentation theory which, in turn, was a significant inputinto other branches of mathematics. Conformal Field Theory ledto intensive study of the representation theory of the Virasoroalgebra, Kac-Moody algebras and vertex operator algebras. Resultsin representation theory have had a significant impact on theunderstanding of the structure of moduli spaces. In the theoryof integrable systems it has been observed long ago in numerousexamples that physically meaningful completely integrable systemsas well as explicit formulas for the integrals of motion areintimately related with the geometric representation theory. Allthese developments serve as our motivation to organize aconference on geometric representation theory, a forum where themajority of participants will be young researchers and advancedgraduate students, learning from leading specialists and furtheradvancing their research projects.Representation theory is a quintessential branch of modernmathematics which studies symmetries of various algebraicsystems. The results and ideas from representation theory foundmany important applications in pure and applied mathematics aswell as quantum physics, biology, economics, just to name afew. More recently a powerful merge of ideas from geometry made asignificant impact on the discipline and led to importantbreakthroughs. The main goal of our conference is to gatherleading specialists in representation theory as well as beginningresearchers and advanced graduate students, to create a forumwhere participants can exchange new ideas, communicate recentadvances and assist younger participants in developing successfulresearch strategies. A special emphasis is made on attractingwomen and underrepresented minority participants, especiallythose at the dawn of their careers.