This award supports the research in algebraic geometry and number theory of Professor David Harbater of the University of Pennsylvania. Dr. Harbater has proposed to continue his study of Galois covers of arithmetic varieties and of arithmetic intersection theory. He will work to construct Galois covers over the rational line having special type of Galois group and ramification, and in his intersection-theoretic work, he plans to use a horizontal pairing that he has recently defined in order to study arithmetic surfaces and to enlarge the Arakelov divisor group and study the fine structure at infinity. 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.
