Shimura Varieties and Galois Modules

主要研究以下两个问题：(A)他试图描述非光滑约简素数下Shimura变量的积分模型。他特别研究了PEL Shimura变体的“局部模型”及其与无限维群、对称空间和伽罗瓦表示的变形空间的仿射标志变体的关系。其动机是获得可用于计算这些变量的Hasse-Weil zeta函数和其他算术应用的信息。(B)他正在研究算术变异与群作用的上同调中出现的表示。特别是，他继续他的工作，发展不动点公式计算不变量的（积分）表示使用立方结构的理论。这位研究者的研究领域是算术代数几何，这是一个融合了数学中两个最古老领域的学科：可以用最简单的方程（即多项式）定义的图形几何，以及对数字的研究。事实证明，这种结合非常富有成效——解决了几代人经受住考验的问题（比如“费马大定理”）。研究者的工作主要集中在研究某些具有许多对称性的多项式方程。它与物理学、纠错码的构造和密码学都有联系。

The principal investigator is working on the following two problems:(A) He is attempting to describe integral models for Shimura varieties at primes of non-smooth reduction. In particular, he studies ``local models" for PEL Shimura varieties and their relation with affine flag varieties for infinite dimensional groups, with symmetric spaces and with deformation spaces of Galois representations.The motivation is to obtain information that can be used in the calculation of the Hasse-Weil zeta function of these varieties and in other arithmetic applications. (B) He is studying the representations that appear in the cohomology of arithmetic varieties with a group action.In particular, he continues his work on developing fixed point formulas for calculating invariants of such (integral) representations usingthe theory of cubic structures.The investigator's research is in the field of arithmetic algebraic geometry, a subject that blends two of the oldest areas of mathematics:the geometry of figures that can be defined by the simplest equations, namely polynomials, and the study of numbers. This combination has proved extraordinarily fruitful - having solved problems that withstood generations (such as ``Fermat's last theorem"). The investigator's work mainly concentrates on the study of certain polynomial equations that have many symmetries. There are connections with physics, the construction of error correcting codes and cryptography.