CAREER: Tropical and Nonarchimedean Analytic Methods in Algebraic Geomoetry

PI将追求代数几何的几条研究路线，涉及组合和非阿基米德方法的应用，以研究代数曲线及其模数，加上相交理论。 特别是，他将调查nonarchimedean方法的Gieseker-Petri定理和最大秩猜想，重量过滤上同调的模的曲线，和发展的函热带化的交叉理论。在这个研究计划中，PI将整合一个教育计划，其中包括支持本科生作为研究助理，参与精心挑选的热带几何项目，并致力于增加美国机构的研究生和最近的博士生参加盖尔为早期职业代数几何学家举办的年度会议。代数几何研究多项式方程组的解集。 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。该奖项支持努力完善这些新方法，并探索更深层次的应用，以解决代数几何中的开放问题。

The PI 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, he will investigate nonarchimedean approaches to the Gieseker-Petri Theorem and Maximal Rank Conjectures, the weight filtration on cohomology of moduli of curves, and the development of a functorial tropicalization of intersection theory. Into this research program, the PI will integrate an educational program that will include supporting undergraduates as research assistants on carefully selected projects in tropical geometry and working to increase the participation of graduate students and recent PhDs from US institutions in the annual GAeL conference for early career algebraic geometers.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. This award supports efforts to refine these new methods and explore deeper applications to open problems in algebraic geometry.