Shimura varieties, Galois representations and Riemann-Roch theorems

主要研究人员正在研究以下三个问题：(A)他试图描述非光滑约化素数下的Shimura品种的积分模型。特别是，他研究了Shimura簇的“局部模型”及其与无限维群的仿射旗簇以及Galois表示的形变空间之间的关系。其动机是获得可用于计算这些簇的Hasse-Weil Zeta函数以及在其他数论应用中使用的信息。(B)他正在开发允许计算挠率信息的Grothendieck-Riemann-Roch定理的精化和函数式版本。(C)他正在研究出现在具有有限群作用的算术簇的上同调中的表示。特别是，他继续利用两个相互关联的主题：立方结构理论和代数循环群的中心扩张理论，开发用于计算这种(积分)表示的不变量的不动点公式。研究人员的研究是在算术代数几何领域，这是一门融合了两个最古老的数学领域的学科：可以由最简单的方程定义的图形几何，即多项式，以及数字研究。事实证明，这种结合非常有成效--解决了几代人都存在的问题(如费马最后定理)。研究者的工作主要集中在研究某些具有多种对称性的多项式方程。这与物理学、纠错码的构造和密码学有关。

The principal investigator is working on the following three 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 Shimura varieties and their relation with affine flag varieties for infinite dimensional groups 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 number theoretic applications.(B) He is developing refined and functorial versions of the Grothendieck-Riemann-Roch theorem that would allow for the calculation of torsion information.(C) He is studying the representations that appear in the cohomology of arithmetic varieties with a finite group action.In particular, he continues his work on developing fixed point formulas for calculating invariants of such (integral) representations using two interconnected themes: the theory of cubic structures and the theory of central extensions of algebraic loop groups.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.