Topics in Arithmetic Geometry and K-Theory

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Henri GilletDMS-98 01219Professor Gillet will work on problems involving the interaction between K-theory and arithmetic geometry. He will study algebraic analogs to Arakelov theory for varieties over discretely valued fields. He hopes to establish an arithmetic Riemann-Roch theorem. It is hoped that these researches will lead to improved insights into the structure of sets of rational points on arithmetic varieties. In addition to these algebraic problems, Professor Gillet will study intersections of analytic cycles from a purely analytic perspective.Analysis, geometry and algebra are three of the major attacks used to investigate modern mathematical problems. All three have been used repeatedly on the ancient problem of solving equations using fractions alone. Geometrically this problem amounts to finding rational points (points described by fractions) on curves (equations viewed as graphs.) Giving such solutions a geometric interpretation makes them more concrete. But having done this, the modern approach to geometry constructs abstract algebraic objects that behave like familiar geometric objects. At the same time analytic techniques that use Calculus can help understand the same geometric objects in a different way. In this project Professor Gillet will study certain collections of rational solutions of equations, varieties, using geometry, algebra, and analysis in tandem. The combination of several modern ideas will lead to new insights into the structure of the rational solutions of rational equations.