Mathematical Sciences: Topics in the Cohomology of Arithmetic Groups

调查员打算从事三个项目。第一个，与Avner Ash （Ohio State）联合，处理SL（3，Z）的算术子群G。G的逆上同调类是赫克特征类，它产生一个自同构表示，该自同构表示应附属于伽罗瓦表示族。他们将在两种特殊情况下研究这些伽罗瓦表示。第二个项目涉及理解G(N)在SL（3，Z）中对素数N的上同调的一般方法。第三个项目是与麻省理工学院的罗伯特·麦克弗森（Robert MacPherson）合作。对于Sp（4，Z）中的任意无扭转有限指标G，他们在Sp（4，R）的对称空间内建立了G等变变形缩回（“脊”）；它具有类似于SL(n)的全圆缩回的性质。他们打算尝试将他们的构造扩展到非无扭G，并为Sp(6)和更高阶群建立一个类似的对象。这里所设想的复杂计算旨在阐明在几个不同数学领域中以重要方式出现的结构。所谓的算术局部对称空间不仅引起了数论家的兴趣，也引起了几何学家的兴趣，而关于它们的一些问题已经被研究者转化为纯粹的拓扑学。没有任何变换可以消除它们的困难，但是当以正确的方式观察时，可以看到一些困难已经被早期的几何学者解决了。17和19世纪的几何学家研究了一类美丽的物体，被称为投影构型，并证明了数学上的格言：美丽的物体值得研究，因为它们以不同的形式反复出现，这些构型的性质与某些算术群的上同调有关，通常被认为纯粹是二十世纪的发明。

The investigator intends to work on three projects. The first, joint with Avner Ash (Ohio State), deals with arithmetic subgroups G of SL(3,Z). A cuspidal cohomology class for G which is a Hecke eigenclass generates an automorphic representation which should be attached to a family of Galois representations. They will study these Galois representations in two special cases. The second project involves a general approach to understanding the cohomology of G(N) in SL(3,Z) for prime N. The third project is joint with Robert MacPherson (M.I.T.). For any torsion-free finite-index G in Sp(4,Z), they have built a G-equivariant deformation retract (a "spine") inside the symmetric space for Sp(4,R); it has properties like those of the well-rounded retract for SL(n). They intend to attempt to extend their construction to non-torsion-free G, and to build a similar object for Sp(6) and higher-rank groups. The intricate computations envisioned here are intended to shed light on structures which arise in important ways in several different mathematical areas. The so-called arithmetic locally symmetric spaces interest number theorists as well as geometers, while some of the questions about them have been transformed by the investigator into pure topology. No transformation removes their difficulty, but when looked at in the right way, some difficulties can be seen to have been addressed by earlier generations of geometers. Seventeenth and nineteenth century geometers studied a beautiful class of objects known as projective configurations, and bearing out the mathematical dictum that beautiful objects repay study because they recur again and again in different guises, properties of these configurations have turned out to bear on the cohomology of certain arithmetic groups, normally considered a purely twentieth century invention.