CAREER: Tropical and Nonarchimedean Analytic Methods in Algebraic Geomoetry

PI将在代数几何中从事几个方面的研究，包括应用组合和非阿基米德方法来研究代数曲线及其模，以及交集理论。特别是，他将研究非阿基米德方法的吉赛克-佩里定理和最大秩猜想，关于曲线模的上同调的权过滤，以及交的函数热带化理论的发展。在这项研究计划中，PI将整合一项教育计划，包括支持本科生在精心挑选的热带几何项目中担任研究助理，并努力增加来自美国机构的研究生和新近的博士参加一年一度的职业生涯早期代数几何学年度GEL会议。代数几何研究多项式方程组的解集。在非阿基米德域上，人们可以将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，解决方案是什么？解的赋值集具有丰富的组合多面体结构，是热带几何的主要研究对象。这一领域的最新发展使得使用这些估值集的几何来解决关于实际解集的几何的微妙问题成为可能。该奖项支持改进这些新方法的努力，并探索代数几何中公开问题的更深层次应用。

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.