The investigator is working on two projects. On the one hand, he tries to extend his results on Weil-etale cohomology and special values of zeta functions from smooth projective varieties over finite fields to general varieties over finite fields, and to varieties over local fields. The other project is the examination of properties of the de Rham-Witt complex and topological cyclic homology of smooth varieties over complete discrete valuation rings. Topological trace homology TR has a Frobenius operator, a Galois action and a filtration analog to Fontaine's functor. The investigator wants to exploit this structure to construct etale and crystalline cohomology, and to apply this to arithmetic problems.
In arithmetic algebraic geometry, solutions of polynomial equations are studied. Even though this field is more than two thousand years old, it turned out recently that there is a variety of applications to cryptography. One method to study a solution set of polynomials is to associate invariants called cohomology and zeta functions to it, and then study those invariants instead. Since the invariants are defined in very different ways, finding relationships between them allows to translate knowledge on one into knowledge on the other. The investigator studies the relationship between zeta functions and a new cohomology, called Weil-etale cohomology, on the one hand, and a new invariant called topological cyclic homology on the other hand.
