Galois Structure and Arithmetic Geometry

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The principal investigator will apply tools from arithmetic geometry to study invariants attached to the actions of finite groups on schemes. These invariants are Euler characteristicsassociated to different kinds of cohomology. Methodsof computing these Euler characteristics will be studied,along with their connection to L-series, Arakelov theoryand the theory of motives. The principal investigatorwill also work on two other projects. These are tocomplete earlier research relating capacity theoryto arithmetic intersections theory, and to studythe universal deformation rings of representationsof finite groups and their applications to Galois theory. This proposal concerns using geometric ideas to studythe solutions of systems of algebraic equations. In geometry,it is often useful to consider the symmetries of an object,which are the ways the object can be rotated or flippedback onto itself. Another geometric idea, which goes backto the mathematician Euler, is that one can attach certainnumbers ("Euler characteristics") to objects. Thesenumbers provide a measure of complexity, and in simplecases they count various natural features, suchas the number of holes in the object. The current proposalhas to do with studying the symmetries of systemsof equations, and how one can assign natural Euler characteristicnumbers to them. The goal of this is to carry over to algebrasome of the insights gained in geometry from consideringsymmetries and Euler characteristics. Systems of equationsarise from many different mathematical applications.Euler numbers and symmetries can in many cases be usedto either rule out or count the number of particularkinds of solutions to such equations.