This research focuses on the concept of invariants in a number of different guises: moduli and parameter spaces, abelian and non-abelian Galois cohomology, K-cohomology of schemes, Clifford algebras, and the period and index of a genus 1 curve. In particular, it will investigate operad structures on certain configuration and moduli spaces, local-global principles for Galois cohomology via field patching, relations between the K-cohomology of homogeneous varieties and the K-theory of separable algebras, and Clifford algebra invariants of finite morphisms of schemes to obtain new results concerning the period-index problem.
A standard model encapsulating a good deal of current mathematics is that one begins by specifying a class of objects of study, be they `algebraic' or `geometric' in nature, and then one attempts to construct invariants in order to distinguish or classify them. One of the most fascinating aspects of this approach is the interplay that often arises between the invariants and the objects which they measure, and the fact that the invariants often achieve a life of their own, often becoming objects of study in themselves.
The main research goal of this paper was to gain insigts into invariants of algebraic and geometric objects, such as the period-index bounds for Galois cohomology of fields, u-invarinats, K-cohomology groups, period-index bounds for elliptic curves, clifford algebras of quadratic forms, and cohomology of confiuration spaces, by looking for new instrinsic algebro-geometric structures in the sets of invariants themselves. This strategic approach has indeed led to a number of new results. These included: Three new completed manuscripts joint with David Harbater and Julia Hartmann, one to appear in Commentarii Mathematici Helvetici, one to appear in Mathematiche Annalen, and the other in submission. We also have a fourth joint manuscript almost ready for submission. A new manuscript in submission "Period and index, symbol length, and generic splittings in Galois cohomlogy," which develops new tools to study Galois cohomology using "presentable functors," geometric objects which parametrize certain aspects of Galois cohomology. These are then used to get new results relating symbol length to the period index problem and the existence of generic splitting fields for cohomology classes. The work on Clifford algebras is still in progress and has branched into two separate papers, joint with both Max Lieblich and Adam Chapman. We now have been able to obtain new results on the existence of irreducible Ulrich bundles for certain families hypersurfaces, as well as obtained new results concerning the period-index problem for elliptic curves. The outreach activities have included both a mentoring program for high school students paired with graduate students, as well as a very successful day camp for high school students this past summer involving 5 faculty members, 5 graduate students, 5 math major undergraduates and about 17 local high school students working on projects in topology, graph theory, number theory and mathematical biology. We plan to try to run this program on a regular basis.
