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The University of Southampton
Mathematical Sciences

Research Group: Algebraic Geometry

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The research here falls into the broadly understood area of Riemann-Roch theory. Riemann-Roch theory originates from the fundamental problem of computing the dimension of so-called Riemann-Roch spaces such as the vector space of global sections of a vector bundle or, more classically, the vector space of global meromorphic functions on a compact Riemann surface that satisfy certain pole and zero order conditions. The classical and celebrated Riemann-Roch theorem from the 19th century which solves the latter problem has seen vast generalisations over the past century which are central to Algebraic K-Theory, Algebraic Geometry, Arithmetic Geometry and Differential 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.

The group welcomes applications for postgraduate studies. Please contact group members for more information; and when ready, please apply through the Graduate School application page.

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