9401321 Kolyvagin This award funds the research of Prof. Victor Kolyvagin. Prof. Kolyvagin works on the theory of Euler systems and their application to number theory. For a given Euler system, he plans to extend the set of cohomology classes explicitly controlled locally, and connect them to the arithmetic of number fields and elliptic curves. This research falls under the general heading of number theory. Number theory is the study of the properties of the whole numbers and is the oldest branch of mathematics. From the beginning problems in number theory have furnished the driving force to creation of new mathematics in almost all parts of the discipline. Prof. Kolyvagin is a leader in the use of geometric ideas to study whole numbers. This kind of number theory is very technical and deep, but it has had astonishing applications in areas like theoretical computer science and coding theory.