The proposer will study the algebraic K-theory and motivic theories of singular schemes. The first project is to prove Weibel's vanishing conjecture and Vorst's regularity conjecture for negative K-theory in characteristic p; the characteristic 0 case has been previously handled by Cortinas, Haesemeyer, Schlichting and Weibel. The second project deals with Suslin's singular homology. With coefficients prime to the characteristic over an algebraically closed field these groups are dual to etale cohomology by work of Suslin and Voevoedsky. Geisser proposes to study the behavior of its rational and p-primary part in characteristic p (especially over finite fields), as well as the relationship to etale cohomolgy over non-algebraically closed fields.
In this project, Geisser will study algebraic K-theory and motivic cohomology theories. These theories form invariants which help to understand the structure of the set of solutions to a system of polynomial equations with coefficients and solutions in fields, like the rational numbers, real numbers or complex numbers. These invariants are fairly well understood if the solution set is smooth (for example, curves are smooth if they have no intersections or cusps). The proposer will try to advance the understanding in the general case, using methods which have been developed recently. This could lead to applications in cryptography, because several cryptosystems rely on (the difficulty in) finding solutions to systems of polynomial equations.
Arithmetic geometry examines the solution of simultanuous polynomial equations in several variables, called varieties. Here the coefficients and solutions are elements of a field like, for example, the rational numbers or finite fields. The way to obtain information on such a variety is by associating to it invariants, which can then be related to other invarients, like the number of solutions. One class of important invariants are algebraic K-groups and motitic cohomology theories, invented in the 1970's-1980's. By work of many people, these invariants are reasonably well understood if the variety is smooth (for example, does not contain cusps or nodes), and if the characteristic of the field is zero. During the project I studied, in joint work with Lars Hesselholt, the properties of algebraic K-theory of fields of characteristic p (this means that p=0 in such a field, or equivalently that the integers reduced modulo p are contained in the field). In one paper we showed that the K-groups vanish in degrees smaller than minus the dimension of the variety, which had been conjectured in the 1980's by Charles Weibel. In another paper we proved a weaker version of a conjecture of Vorst, which related the vanishing of certain K-groups to the degree of singularity of the variety. In another article, I establised properties of Suslin homology (one of the motivic theories mentioned above) for varieties (singular or not) over fields of characteristic p.
