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Tropical and nonarchimedean analytic methods in algebraic geometry

代数几何中的热带和非阿基米德解析方法

基本信息

批准号：
2001502
负责人：
Sam Payne
金额：
$ 35.97万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2001502&HistoricalAwards=false
关键词：
Tropical nonarchimedean analytic methods algebraic

项目摘要

Algebraic geometry studies solution sets of systems of polynomial equations. For instance, lines are solution sets of linear polynomial equations, while circles and hyperbolas are solution sets to quadratic polynomial equations, and their study goes back to the ancient Greeks. The solution sets of systems of many polynomial equations in many variables often have beautiful and complicated geometry. The PI will apply new and modern techniques to answer questions of classical interest in the field of algebraic geometry, and to address long standing open problems about the geometry of curves defined by polynomial equations. This project provides research training opportunities for graduate students.Over a nonarchimedean field, one can split the problem of understanding such solution sets 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; the solutions with a given valuation can be studied via nonarchimedean analytic geometry. Recent developments in these fields make it possible to resolve subtle questions about the geometry of the actual solution set using new and innovative tools and techniques. This project will refine, abstract, and generalize these new methods and explore deeper applications to open problems in algebraic geometry.The PI will use tropical and nonarchimedean methods to continue his work on refined curve counting, and on unstable cohomology of moduli spaces of curves, proceeding beyond the top graded piece of the weight filtration to examine the full weight-graded cohomology ring. He will also initiate an analogous study of the weight graded cohomology of the moduli space of abelian varieties, using the topology of moduli spaces of tropical abelian varieties and the combinatorial algebra of complexes built out of unimodular matroids and perfect quadratic forms. Additional projects will attack the motivic, p-adic, and topological monodromy conjectures for Newton nondegenerate singularities, using the combinatorics of relative local h-polynomials and Stapledon’s formulas for monodromy, and attempt to resolve outstanding (and mutually contradictory) conjectures of Kontsevich and Morita in the homology of commutative graph complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

代数几何研究多项式方程组的解集。例如，直线是线性多项式方程的解集，而圆和双曲线是二次多项式方程的解集，它们的研究可以追溯到古希腊人。多元多项式方程组的解集往往具有优美而复杂的几何形状。PI将应用新的和现代的技术来回答代数几何领域的经典问题，并解决长期存在的关于多项式方程定义的曲线几何的开放问题。这个计画提供研究训练的机会给研究生。在一个非阿基米德的领域中，我们可以将了解这类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;具有给定赋值的解可以通过非阿基米德解析几何来研究。这些领域的最新发展使人们有可能使用新的和创新的工具和技术来解决实际解决方案集的几何形状的微妙问题。这个项目将提炼、抽象和推广这些新方法，并探索更深层次的应用，以解决代数几何中的开放问题。PI将使用热带和nonarchimedean方法继续他在精细曲线计数和曲线模空间的不稳定上同调方面的工作，超越权过滤的顶级部分，研究全权分级上同调环。他还将启动一个类似的研究权重分级上同调的模空间的阿贝尔品种，使用拓扑结构的模空间的热带阿贝尔品种和组合代数的复合物建立了unimodular拟阵和完美的二次形式。其他项目将攻击牛顿非退化奇点的motivic，p-adic和拓扑monodromy代数，使用相对局部h-多项式的组合学和Stapledon的monodromy公式，并试图解决悬而未决的（相互矛盾）Kontsevich和Morita在交换图复形的同源性方面的贡献。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估。