ABSTRACT

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.

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