Cohomology of locally symmetric spaces and applications to number theory

局部对称变异的上同调在数论特别是朗兰兹规划中起着重要的作用。一个原因是它们是空间的例子，展示了赫克代数和伽罗瓦群的作用。一个基本目标是将这些动作和相应的l函数联系起来。在之前的工作中，主要研究者创造了一个新的工具，l模，来研究这种上同调。在当前的项目中，首席研究员建议将Hecke和galois动作合并到l模块中。他还将证明l模的“分解定理”，并将其应用于构造与次局部对称变体相关的环。提出的研究研究几何和数论中的对称性。几何学和数论的应用比比皆是，例如密码学和晶体学。几何中的对称是不改变距离的空间变换，例如旋转。在数论中，对称是一种数字系统的变换，它将两个数的和变换成变换的和，乘积也是如此，例如复共轭。在这两门学科中，最有趣的是那些在许多对称情况下保持不变的物体。例如，在几何学中，球面在所有旋转下都是不变的，而在数论中，整系数多项式方程的根集在所有对称下都是不变的。研究者的研究涉及具有几何和数论对称性的高维物体，以及这些对称性之间的关系。

The cohomology of locally symmetric varieties plays an important role innumber theory and in particular Langlands's program. One reason is thatthey are examples of spaces exhibiting an action of both a Hecke algebra]and a Galois group. A fundamental goal is to relate these actions and thecorresponding L-functions. In previous work, the principal investigatorcreated a new tool, L-modules, to study such cohomology. In the currentproject, the principal investigator proposes to incorporate Hecke andGalois actions into L-modules. He will also prove a "decompositiontheorem" for L-modules and apply it to construct cycles associated tosub-locally symmetric varieties.The proposed research studies symmetry in geometry and number theory.Applications of geometry and number theory abound, for example tocryptography and crystallography. A symmetry in geometry is atransformation of space which doesn't change distance, for example arotation. In number theory a symmetry is a transformation of a numbersystem that transforms the sum of two numbers into the sum of thetransforms and likewise for the product, for example complex conjugation.In both subjects, the most interesting objects are those which remainunchanged under many symmetries. For example, in geometry the sphere isinvariant under all rotations, while in number theory the set of roots of apolynomial equation with integer coefficients are invariant under allsymmetries. The investigator's research concern higher-dimensional objectswhich possess both geometric and number theoretic symmetries, and therelations between these symmetries.