This award supports the research in arithmetic algebraic geometry of Professor Ching-Li Chai of the University of Pennsylvania. Dr. Chai has proposed to work on two projects: the first is to investigate level structures over extraordinary primes, and the second is to construct rigid homogeneous spaces. Both of these projects are related to the arithmetic of automorphic forms and Shimura varieties. This is research in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.
