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.
This award supported the development of new combinatorial methods at the interface between algebraic and nonarchimedean analytic geometry, with significant applications to the classical theory of algebraic curves. In particular, in joint work with David Jensen, the PI introduced the notion of tropical independence and have applied this to give a new proof of the Gieseker-Petri Theorem and to make substantial progress toward the maximal rank conjecture. In joint work with Dan Abramovich and Lucia Caporaso, the PI also proved that the skeleton of the Deligne-Mumford compactification of the moduli space of curves is naturally identified with the moduli space of stable tropical curves, and that this identification is functorial for toroidal morphisms. In particular, it is compatible with the tautological forgetful, clutching, and gluing maps. These developments allow the computation of invariants of the algebraic moduli space via tropical methods, and this approach is pursued in forthcoming joint work with Melody Chan and Soren Galatius. In conjunction with these research developments, this award supported the training and development of six PhD students and five postdoctoral fellows working with the PI on the projects described above and other closely related work. One of these students Yoav Len completed his PhD in December 2014 and is now a postdoctoral research with Hannah Markwig in Saarbruecken. The other five have expected completion dates in 2016-2019. Three of the postdoctoral fellows have accepted tenure track positions at major research universities in the US, and one moved on to a research position in industry. In particular, this award contributed significantly to supporting a rising generation of young researchers working at the interface between algebraic and nonarchimedean geometry.
