Mathematical Sciences: "L-Adic Representations and Iwasawa Theory"

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This award supports the research in arithmetic algebraic geometry of Professor Andrew Wiles of Princeton University. Dr. Wiles plans to continue his researches in Iwasawa Theory by examining the symmetric square of a representation associated to a certain class of Hecke eigenforms. In this way, he hopes to attack the Greenberg conjectures, which concern the newly defined Selmer groups of compatible families of L-adic representations. 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.