Shimura Varieties, the Trace Formula, Congruences and Galois Representations

下村变种、迹公式、同余和伽罗瓦表示斯蒂芬·S·库德拉(马里兰大学)弗雷登·沙希迪(普渡大学)这个项目将提供支持，让美国数学界的年轻研究人员受益于2000年春季学期在巴黎的亨利·庞加莱研究所(IHP)的特别计划。本课程主要讨论两个主题：(I)下村簇与迹公式；(Ii)同余与伽罗瓦表示。在21世纪的头几十年，这些主题，特别是它们的相互作用，肯定会成为自同构形式和数论的大部分研究活动的中心。国际水文计划的活动将把这些领域的世界领导人聚集在一起。本课程将围绕一系列讲座“课程”展开，内容涉及迹公式、内窥镜、基本引理、下村簇的L函数、全局和局部朗兰兹函数性、伽罗瓦表示、p-进赫克代数、p-进模形式、刚性分析、局部朗兰兹对应和几何朗兰兹对应。该计划的范围鼓励在这两个主要领域的交界处进行新的研究方向，参与将为年轻研究人员提供一个独特的机会，在他们职业生涯的早期阶段发展这一重要领域的专业知识。20世纪后半叶数学的两大发展是自同构形式/表示理论中的朗兰兹程序、费马大定理的Wiles和Taylor-Wiles证明以及谷山-下村猜想。这些把数论和几何联系在一起的进展实际上是非常紧密地联系在一起的，而这两个领域的技术的结合正在蓬勃发展。由此产生的领域将是21世纪头几十年数学研究活动的主要领域之一。这项研究计划于2000年春季学期在巴黎亨利·庞加莱研究所举行，以世界领导人的讲座为中心，提供了无与伦比的视野和洞察力。这项NSF资助奖将为来自美国数学界的年轻研究人员参与国际水文计划项目提供资金，因此将有助于确保美国在这些数论的新发展方面拥有强大的专业水平。该奖项由数学科学部(代数和数论计划)、国际方案司(西欧计划)以及数学和物理科学局多学科活动办公室提供支持。

Shimura varieties, the trace formula, congruences and Galois representationsStephen S. Kudla (University of Maryland)Freydoon Shahidi (Purdue University)This project will provide support allowing young researchers from the US mathematical community to benefit from participation in the special program at the Institute Henri Poincare (IHP) in Paris in the spring semester 2000. This program focuses on two topics: (i) Shimura varieties and the trace formula and (ii) congruences and Galois representations. These topics, and particularly their interaction, will certainly be at the center of much of the research activity in automorphic forms and number theory in the opening decades of the 21st century. The activity at IHP will bring together the world leaders in these areas. The program will center around a series of lecture `courses' covering the latest developments concering the trace formula, endoscopy, the fundamental lemma, L functions for Shimura varieties, global and local Langlands functoriality, Galois representations, p-adic Hecke algebras, p-adic modular forms, rigid analysis, the local Langlands correspondence and the geometric Langlands correspondence. The scope of the program encourages new directions for research at the interface of the two major fields and participation will provide young researchers a unique opportunity to develop expertise in this important area at an early stage in their careers. Two major developments in mathematics in the later part of the 20th century are the Langlands program in automorphic forms/representation theory and the Wiles and Taylor-Wiles proof of Fermat's Last Theorem and the Taniyama-Shimura conjecture. These advances, relating number theory and geometry, are in fact very closely linked, and a vigorous development of the union of the techniques from the two areas is currently taking place. The resulting field will be one of the main arenas of research activity in mathematics in the first decades of the 21st century. The research program taking place at the Institute Henri Poincare in Paris in the spring semester 2000 and centered around lecture courses by the world leaders provides an unparalleled level of vision and insight. This NSF Grant award will provide funding for young researchers from the US mathematical commmunity to participate in the IHP program, and hence will help to ensure a strong level of US expertise in these new developments in number theory. This award is being supported by the Division of Mathematical Sciences (Algebra and Number Theort program), the Divison of International Programs (Western Europe Program), and the Office of Multidisciplinary Activities of the Mathematical and Physical Sciences Directorate .