Research Group: Algebraic Geometry

We are particularly interested in the following topics:

• (Operations on) Higher Algebraic K-Theory: Exterior power operations furnish the Grothendieck group and higher K-groups of a ring or a scheme with additional structure which is at the heart of Grothendieck’s Riemann-Roch Theory. While higher K-groups have originally been defined and studied using homotopy theory, there are now purely algebraic definitions of both higher K-groups and exterior power operations available that invite to be studied from the purely algebraic point of view.
• Equivariant Riemann-Roch theory: In the equivariant context, Riemann-Roch spaces are studied as group representations when a group acts on the vector bundle compatible with a given action of the group on the underlying space. Many versions of equivariant Riemann-Roch theorems have been proved over the past three decades or so. Current research in this area concerns refinements of the equivariant Adams-Riemann-Roch theorem and equivariant Riemann-Roch theorems for inertial K-theory or for orbifold varieties.
• Geometric Galois Module Theory: Relating global invariants to invariants that are created from local data is a fundamental and widely studied topic in number theory. In Geometric Galois Theory, an equivariant version of this local-global principle is studied in the situation when a (finite) group acts on a variety over a finite field. More precisely, a goal is to relate epsilon constants appearing in functional equations of Artin L-functions to global invariants that are extracted from the equivariant structure of certain Riemann-Roch spaces. Particularly explicit and deep theorems exist when the space is a curve and the action of the group is tame. Interesting and sought-after generalisations concern wild actions on curves and actions on surfaces.