ABSTRACT

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.

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