Mathematical Sciences: Topological Trace Formula

Arthur的Lefschetz迹公式是用Hecke对应表示局部对称空间Y的L2上同调上映射的迹交替和的表达式。用轨道积分、稳定群的体积和离散级数表示的性质给出了这个表达式。亚瑟将重新推导出的公式与r·麦克弗森(在本合作项目的麻省理工学院)通过解释它为Lefschetz定点公式的作用Hecke对应的加权上同调的紧化还原Borel-Serre y将使用这种解读(一)延长亚瑟的公式,包括普通的上同调的情况下,(b)获得一个类似的表达式的交错和霍奇组件上同调的痕迹,(c)推测一个相关表达式，该表达式对特征为p 0的域上定义的志村变量有效。在20世纪70年代，r·朗兰兹（普林斯顿高等研究所的）概述了一系列范围和深度都很大的猜想和想法，如果这些猜想和想法得到充分的探索和验证，将导致数学几个分支的“大统一”，包括数论、李群的表示理论和谐波分析。在过去的二十年里，这个计划取得了巨大的进展，克服了非常困难的障碍。然而，人们普遍认为，要充分探索朗兰兹的思想可能还需要50年（或更长时间）。本项目涉及新发现的“迹公式”的几何和拓扑解释和证明，这是本学科的基本工具之一。利用这种解释，迹公式已经被推广和改进到可以应用于感兴趣的中心对象（即志村变量上的赫克对应）的程度。虽然这些几何技术的发展主要是为了执行朗兰兹计划中的这一步，但它们已经被应用于其他数学领域的问题。＊＊＊

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. ***