Shimura Varieties, the Trace Formula, Congruences and Galois Representations

Shimura簇，迹公式，同余和Galois表示. Kudla（马里兰州大学）Freydoon Shahidi（普渡大学）该项目将提供支持，使来自美国数学界的年轻研究人员能够从参与 2000年春季学期在巴黎的亨利·庞加莱研究所（IHP）的特别方案。该计划侧重于两个主题：（i）志村品种和迹公式和（ii）同余和伽罗瓦表示。 这些主题，特别是它们之间的相互作用，肯定会成为世纪开始的几十年里自守形式和数论研究活动的中心。 国际水文计划的活动将汇集这些领域的世界领导人。该计划将围绕一系列讲座'课程'涵盖的最新发展concerning迹公式，内窥镜，基本引理，L功能的志村品种，全球和当地朗兰兹函，伽罗瓦表示，p-adic Hecke代数，p-adic模形式，刚性分析，当地朗兰兹对应和几何朗兰兹对应。 该计划的范围鼓励在两个主要领域的接口和参与研究的新方向将为年轻的研究人员提供一个独特的机会，在他们的职业生涯的早期阶段发展在这一重要领域的专业知识。 世纪后期数学的两个主要发展是自守形式/表示论中的朗兰兹纲领和费马大定理的怀尔斯和泰勒-怀尔斯证明以及谷山-志村猜想。这些进展，有关数论和几何，事实上是非常密切相关的，并蓬勃发展的工会的技术，从这两个领域目前正在发生。由此产生的领域将是一个主要领域的研究活动在数学在第一个几十年的世纪。该研究计划发生在研究所亨利庞加莱在巴黎在2000年春季学期和围绕讲座课程的世界领导人 提供了无与伦比的视野和洞察力。NSF的这一资助将为美国数学界的年轻研究人员提供资金，以参与国际水文计划，从而有助于确保美国在数论新发展方面的专业知识水平。 该奖项由数学科学部（代数和数论计划），国际计划部（西欧计划）以及数学和物理科学理事会多学科活动办公室支持。

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 .