Mathematical Sciences: Topological Trace Formula

9303550亚瑟的Lefschetz迹公式是局部对称空间Y的L2上同调上Hecke对应诱导的映射的迹的交错和的一个表达式。这个表达式是用轨道积分、稳定基团的体积和离散级数表示的特征给出的。阿瑟的公式将被重新推导(在与麻省理工学院的R·麦克弗森的这个联合项目中)。通过将其解释为Hecke对应在Y的约化Borel-Serre紧化的加权上同调上的作用的Lefschetz不动点公式。这种解释将被用来(A)将Arthur的公式推广到包括普通上同调的情况，(B)得到上同调的Hodge分量上的迹和的一个类似的表达式，以及(C)猜想一个对于定义在特征p0的域上的Shimura簇应该有效的相关表达式。在20世纪70年代的S时期，朗兰兹(普林斯顿高等研究院)提出了一系列广度和深度都很大的猜想和想法，这些猜想和想法一旦得到充分探索和验证，将导致数论、李群表示理论和调和分析等几个数学分支的“大一统”。在过去的二十年里，这一方案取得了巨大的进展，克服了非常困难的障碍。然而，人们普遍认为，朗兰兹的思想可能还需要50年(或更长时间)才能得到充分的探索。本项目涉及一种新发现的几何和拓扑解释和“迹公式”的证明，“迹公式”是本学科的基本工具之一。利用这种解释，迹公式被推广和改进到可以应用于感兴趣的中心对象(即，Shimura簇上的Hecke对应)的程度。虽然这些几何技术主要是为了在朗兰兹的程序中实现这一步，但它们已经被应用于数学的其他领域的问题。***

9303550 Goresky The Lefschetz trace formula of Arthur is an expression for the alternating sum of traces of the map induced on the L2 cohomology of a locally symmetric space Y by a Hecke Correspondence. This expression is given in terms of orbital integrals, volumes of stabilizer groups, and characters of discrete series representations. Arthur's formula will be rederived (in this joint project with R. MacPherson of M.I.T.) by interpreting it as the Lefschetz fixed point formula for the action of the Hecke correspondence on the weighted cohomology of the reductive Borel-Serre compactification of Y. This interpretation will be used (a) to extend Arthur's formula to include the case of the ordinary cohomology, (b) to derive a similar expression for the alternating sum of traces on the Hodge components of the cohomology, and (c) to conjecture a related expression which should be valid for Shimura varieties defined over fields of characteristic p 0. During the 1970's, R. Langlands (of the Institute for Advanced Study in Princeton) outlined a series of conjectures and ideas of enormous scope and depth which, when fully explored and verified, will result in a "grand unification" of several branches of mathematics, including number theory, representation theory of Lie groups, and harmonic analysis. During the last twenty years enormous progress has been made on this program and very difficult obstacles have been overcome. Nevertheless, it is commonly believed that it may take another fifty years (or more) before Langlands' ideas are fully explored. The present project concerns a newly discovered geometric and topological interpretation and proof of the "trace formula," one of the basic tools in the subject. Using this interpretation, the trace formula has been generalized and refined to the point where it may be applied to the central objects of interest (namely, Hecke correspondences on Shimura varieties). Although these geometric techniques have b een developed primarily to carry out this step in Langlands' program, they have already been applied to problems in other areas of mathematics. ***