CAREER: Tropical and Nonarchimedean Analytic Methods in Algebraic Geomoetry

职业：代数几何中的热带和非阿基米德分析方法

• 批准号: 1149054
• 负责人: Sam Payne
• 金额: $ 48.05万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1149054&HistoricalAwards=false
• 关键词: CAREER; Tropical; Nonarchimedean; Analytic; Methods

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.

PI 将在代数几何领域开展多项研究，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 Gieseker-Petri 定理和最大等级猜想的非阿基米德方法、曲线模上同调的权重过滤，以及交集理论函子热带化的发展。在该研究计划中，PI 将整合一项教育计划，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，并努力增加美国机构的研究生和新近博士参加年度 GAeL 会议，以促进早期职业生涯代数几何。代数几何研究多项式方程组的解集。 在非阿基米德域上，我们可以将理解这样一个解决方案集的问题分为两部分。 解决方案的可能估值是多少？ 在给定估值的情况下，有哪些解决方案？ 解的评估集具有丰富的组合和多面体结构，是热带几何研究的主要对象。 该领域的最新发展使得使用这些评估集的几何形状来解决有关实际解决方案集的几何形状的微妙问题成为可能。该奖项支持改进这些新方法并探索更深入的应用来解决代数几何中的开放问题。