Mathematical Sciences: Topics in the Cohomology of Arithmetic Groups

调查员打算从事三个项目。 的 首先，与阿夫纳阿什（俄亥俄州）联合，处理算术 SL（3，Z）的子群G. G的尖点上同调类， Hecke特征类生成自守表示， 应该附加到伽罗瓦表示族上。 他们 将在两种特殊情况下研究这些伽罗瓦表示。 的 第二个项目涉及一个一般性的方法来了解 G（N）在SL（3，Z）中的上同调。 第三个项目是 罗伯特·麦克弗森（Robert MacPherson）（MIT） 对于任何无扭转的 有限指标G在Sp（4，Z）中，他们建立了一个G-等变 变形收缩（“脊柱”）内的对称空间， Sp（4，R）;它具有与良好圆化收缩核相似的性质 对于SL（n）。 他们打算把他们的建筑扩展到 非无挠G，并为Sp（6）建立类似的对象， 更高级别的团体。 这里设想的复杂计算旨在 揭示了在几个重要方面出现的结构， 不同的数学领域。 所谓的局部算术 对称空间引起数论家和几何学家的兴趣， 虽然关于它们的一些问题已经被 纯粹拓扑学的研究者。 任何转换都不会移除它们的 困难，但当以正确的方式看待时，一些困难 可以看出， 几何学家 17和19世纪的几何学家研究了 美丽的一类物体被称为投影配置， 证明了数学格言美丽的物体 因为它们以不同的形式一次又一次地出现， 这些配置的属性已经被证明对 某些算术群的上同调，通常被认为是 纯粹是世纪的发明。

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.