This award supports the continued study of certain invariants, known as local Chern characters, defined for modules over Noetherian rings. While numerous properties of these invariants have recently been discovered, there are still many open questions. The main line of study concerns invariants defined by matrices in a free resolution of a module over a local ring, where relations between these invariants and classical results in commutative algebra have led to new interpretations of them, which, although they are still conjectural in many cases, give much easier methods for their computation. Another line of research concerns the details of the construction itself, and gives alternative definitions for intersection multiplicities, leading, among other things, to a new approach to Serre's positivity conjecture. This research is concerned with a number of questions in commutative algebra and algebraic geometry. Algebraic geometry studies solutions of families of polynomial equations. One can either study the geometry of the solution set or approach problems algebraically by investigating certain functions on the solution set that form what is called a commutative ring. This dual perspective creates a close connection between commutative algebra and algebraic geometry.
