Mathematical Sciences: Integral Hecke Structures, Crystalline Cohomology, & GL(2) Over Totally Real Fields

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Jordan This award funds the research of Professor Bruce Jordan into the Hecke structure of Shimura varieties. Recent advances in crystalline cohomology make it possible to control the action of inertia on p-adic cohomology groups of varieties over unramified extensions with good reduction over the rational p-adic field. Prof. Jordan expects to apply these techniques to 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 result has been an increased understanding of both areas of mathematics. ***