Mathematical Sciences: Topological Trace Formula

小行星9303550 亚瑟的莱夫谢茨迹公式是由赫克对应在局部对称空间Y的L2上同调上诱导的映射的迹的交替和的表达式。 这个表达式是根据轨道积分、稳定子群的体积和离散级数表示的特征给出的。 亚瑟的公式将被重新推导（在这个与R。（MacPherson of MIT）通过将其解释为Hecke对应作用于Y的约化Borel-Serre紧化的加权上同调的Lefschetz不动点公式。 这种解释将被用来（a）扩展亚瑟的公式，包括普通的上同调的情况下，（B）推导出一个类似的表达式交替和迹上的霍奇分量的上同调，和（c）猜想一个相关的表达式，这应该是有效的志村品种定义在外地的特征p 0。 20世纪70年代，R.朗兰兹（普林斯顿高等研究院）概述了一系列具有巨大广度和深度的理论和思想，当这些理论和思想得到充分探索和验证时，将导致数学的几个分支的“大统一”，包括数论、李群表示论和调和分析。 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。 尽管如此，人们普遍认为，可能还需要50年（或更长时间）才能充分探索朗兰兹的思想。 本项目涉及一个新发现的几何和拓扑的解释和证明的“迹公式”，在这个主题的基本工具之一。 使用这种解释，迹公式已被推广和完善的点，它可以应用于中心对象的利益（即，Hecke对应志村品种）。 虽然这些几何技巧B甚至主要是为了实现朗兰兹纲领中的这一步而发展起来的，但它们已经被应用于数学的其他领域。 ***

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. ***