Combinatorial and nonarchimedean methods in algebraic geometry

研究人员将从事代数几何的几条研究路线，涉及组合和nonarchimedean方法的应用，研究代数曲线及其模数，加上交叉理论。 特别是，该建议涉及Gieseker-Petri定理和最大秩猜想的非阿基米德方法，曲线模的上同调的权过滤，热带化和分析的度量性质，以及相交理论的函子热带化的发展。 在一个非阿基米德域上，我们可以把理解这样一个解集的问题分成两部分。 解决方案的可能估值是什么？ 在给定估值的情况下，解是什么？ 解的赋值集具有丰富的组合和多面体结构，是热带几何的主要研究对象。 最近在这一领域的发展使人们有可能解决微妙的问题，几何的实际解决方案集使用的几何这些套估值。 目前的建议旨在完善这些新方法，并探索更深层次的应用，以解决代数几何中的开放问题。

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.