The structure of invariants in algebra and geometry

University Of Georgia Research Foundation Inc, Athens GA

Investigators

Abstract

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.