Professor Gillet will be studying a number of problems related to Arakelov Theory, Motives, and K-theory. He intends to use motivic cohomology to give a purely sheaf theoretic construction of the arithmetic Chow groups. This will include studying the action of Adams' operations on Grayson's model of the motivic cohomology complexes, with the goal of showing that the motivic cohomology of an arithmetic variety can be used to compute its Chow groups (with rational coefficients). For semi-stable arithmetic varieties, he intends to use deformation to the normal cone, techniques to construct an intersection product on the Chow groups with integer coefficients. (Previously it was only known how to do this when the variety is smooth over the base.) He will also study the possibility of defining arithmetic Chow groups with coefficients in a local system. This would allow the construction of new modular forms via generating series with coefficients that are arithmetic cycles. Two other topics that he plans to pursue are first to study the relationship between the proof, using Brauer lifting, of Gersten's conjecture for discrete valuation rings with finite residue fields and the K-theory of finite sets, and secondly, giving an explicit proof of the Riemann-Roch theorem, which will lead to improved information about secondary classes.
The overall goal of these areas of research is to improve our understanding of Diophantine equations, that is understanding the properties of the set of integer solutions of equations with integer coefficients. Diophantine equations have applications in particular to problems in cryptography and coding theory.
