Combinatorial and nonarchimedean methods in algebraic geometry

Yale University, New Haven CT

Investigators

Abstract

The investigator will pursue several lines of research in algebraic geometry involving the application of combinatorial and nonarchimedean methods to study algebraic curves and their moduli, plus intersection theory. In particular, this proposal deals with nonarchimedean approaches to the Gieseker-Petri Theorem and Maximal Rank Conjectures, the weight filtration on cohomology of moduli of curves, metric properties of tropicalizations and analytifications, and the development of a functorial tropicalization of intersection theory. Algebraic geometry studies solution sets of systems of polynomial equations. Over a nonarchimedean field, one can split the problem of understanding such a solution set into two parts. What are the possible valuations of solutions? And what are the solutions with a given valuation? The set of valuations of solutions has a rich combinatorial and polyhedral structure, and is the primary object of study in tropical geometry. Recent developments in this field make it possible to resolve subtle questions about the geometry of the actual solution set using the geometry of these sets of valuations. The current proposal aims to refine these new methods and explore deeper applications to open problems in algebraic geometry.