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

基本信息

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

项目摘要

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将在代数几何形状中追求几项研究线，涉及组合和非建筑方法的应用来研究代数曲线及其模量以及相交理论。特别是，他将研究对Gieseker-petri定理和最大秩构的非一切集方法，曲线模量共同体的权重过滤以及相交理论的功能性热带化的发展。进入该研究计划，PI将整合一个教育计划，其中包括在热带几何学上精心挑选的项目的研究助理，并努力增加研究生的参与以及美国机构的最新博士学位在年度GAEL年度职业代数会议上，参加早期职业代数几何学生。集中质量研究解决方案综合型系统组成的系统。在一个非架构的字段上，可以将理解该解决方案设置的问题分为两个部分。解决方案的可能估值是什么？给定估值的解决方案是什么？解决方案的一组估值具有丰富的组合和多面体结构，是热带几何学研究的主要对象。该领域的最新发展使得使用这些估值集的几何形状解决有关实际解决方案集的几何形状的细微问题。该奖项支持为完善这些新方法的努力，并探索更深层次的应用程序，以解决代数几何形状的开放问题。