This award supports the research in arithmetic algebraic geometry of Professor Gerd Faltings of Princeton University. Dr. Faltings's research will be centered on questions about p-adic analysis and Diophantine geometry. He has recently discovered p- adic symmetric domains, and plans to study their cohomology and construct their Satake compactifications. In Diophantine geometry, he plans to use new methods, in particular the Product Theorem that he has recently proved, to reformulate the classical theory of Diophantine approximations. 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.