Shimura Varieties, p-Adic Shtukas, and Local Systems

该协会将在算术代数几何领域进行研究。这是一个融合了两个最古老的数学领域的学科：可以用最简单的方程(即多项式)描述的形状的几何，以及对数字的研究。这种学科的结合被证明是非常有成效的--解决了几代人都能解决的问题(如“费马大定理”)。一般领域与物理有关，并在构造纠错码和密码学方面有重要的应用。PI的工作主要集中在研究描述具有多种对称性的形状的特定方程，以及学科与数学物理中某些结构的联系。PI计划让研究生参与一些项目。PI致力于描述非光滑约化素数的Shimura变种的积分模型，并研究相关空间。特别是，他将继续研究下村阿贝尔型变种在这样的素数下的奇异性。他计划用p-进shtukas的新理论来刻画这些积分模型，并在正交Shimura簇的情况下，显式地研究它们约化的局部结构。他还将Shimura簇解释为更一般的模空间的特例，并将Shimura簇的特殊点的概念推广到这种模空间。最后，由于类似于数学物理中出现的黎曼曲面上丛的模理论，PI将调查局部系统和伽罗瓦表示的变形空间的辛属性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The PI will conduct research in the field of arithmetic algebraic geometry. This is a subject that blends two of the oldest areas of mathematics: The geometry of shapes that can be described by the simplest equations, namely polynomials, and the study of numbers. This combination of disciplines has proved extraordinarily fruitful - having solved problems that withstood generations (such as "Fermat's last theorem"). The general field has connections with physics, and has found important applications to the construction of error correcting codes and cryptography. The PI's work mainly concentrates on the study of specific equations which describe shapes with many symmetries and on connections of the subject with certain constructions in mathematical physics. The PI plans to involve graduate students in some of the projects.The PI is working to describe integral models for Shimura varieties at primes of non-smooth reduction and study related spaces. In particular, he will continue to investigate the singularities of Shimura varieties of abelian type at such primes. He plans to characterize these integral models by using the novel theory of p-adic shtukas and, in the case of orthogonal Shimura varieties, explicitly study the local structure of their reductions. He would also like to interpret Shimura varieties as special cases of more general moduli spaces of "arithmetic shtukas" and to generalize the concept of special points of Shimura varieties to such moduli spaces. Finally, motivated by an analogy with the theory of moduli of bundles over Riemann surfaces as it appears in mathematical physics, the PI will investigate symplectic properties of deformation spaces of local systems and Galois representations.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.