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

• 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

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组的解集，它们的研究可以追溯到古希腊人的解集。许多变量的多项式方程通常具有美丽而复杂的几何形状，PI 将应用新的现代技术来回答代数几何领域中经典的问题，并解决长期存在的开放性问题。由多项式方程定义的曲线的几何形状。该项目为研究生提供了研究培训机会。在非阿基米德领域，人们可以将理解此类解决方案集的问题分为两部分：解决方案的可能评估是什么？具有给定估值的解？ 解的估值集合具有丰富的组合和多面体结构，并且是热带几何的主要研究对象，可以通过以下方式研究具有给定估值的解；这些领域的最新发展使得使用新的创新工具和技术解决有关实际解决方案集的几何的微妙问题成为可能，该项目将完善、抽象和概括这些新方法，并探索更深入的应用。 PI将使用热带和非阿基米德方法继续他的精细曲线计数和曲线模空间不稳定上同调的工作，超越权重过滤的顶级部分来检查他还将利用热带阿贝尔簇的模空间拓扑以及由单模拟阵和完美矩阵构建的复形的组合代数，发起对阿贝尔簇模空间的权分级上同调的类似研究。其他项目将使用牛顿非简并奇点来攻击动机、p-adic 和拓扑单向猜想。相对局部 h 多项式和 Stapledon 的一元性公式的组合，并试图解决 Kontsevich 和 Morita 在交换图复形同源性方面的突出（且相互矛盾的）猜想。该奖项反映了 NSF 的法定使命，并被认为值得通过以下方式获得支持：使用基金会的智力价值和更广泛的影响审查标准进行评估。

项目成果

Tropical curves, graph complexes, and top weight cohomology of $\mathcal {M}_g$

$mathcal {M}_g$ 的热带曲线、复合图和顶重上同调

• DOI: 10.1090/jams/965
• 发表时间: 2021-01
• 期刊: Journal of the American Mathematical Society
• 影响因子: 3.9
• 作者: Chan, Melody;Galatius, Søren;Payne, Sam
• 通讯作者: Payne, Sam

Tropical moduli spaces as symmetric Δ$\Delta$‐complexes

作为对称 Î$Delta$âcomplex 的热带模空间

• DOI: 10.1112/blms.12570
• 发表时间: 2022-03
• 期刊: Bulletin of the London Mathematical Society
• 影响因子: 0.9
• 作者: Allcock, Daniel;Corey, Daniel;Payne, Sam
• 通讯作者: Payne, Sam

Bitangents to plane quartics via tropical geometry: rationality, $$\mathbb {A}^1$$-enumeration, and real signed count

通过热带几何到平面四次曲线的双切线：理性、$$mathbb {A}^1$$-枚举和实数有符号数

• DOI: 10.1007/s40687-023-00383-1
• 发表时间: 2023-06
• 期刊: Research in the Mathematical Sciences
• 影响因子: 1.2
• 作者: Markwig, Hannah;Payne, Sam;Shaw, Kris
• 通讯作者: Shaw, Kris

Triangulations of simplices with vanishing local h-polynomial

具有消失局部 h 多项式的单纯形三角剖分

• DOI: 10.5802/alco
• 发表时间: 2020-01
• 期刊: Algebraic combinatorics
• 作者: de Moura, André;Gunther, Elijah;Payne, Sam;Schuchardt, Jason;Stapledon, Alan
• 通讯作者: Stapledon, Alan

Compactified Jacobians as Mumford models

压缩雅可比行列式作为 Mumford 模型

• DOI: 10.1090/tran/8875
• 发表时间: 2023-07
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Christ, Karl;Payne, Sam;Shen, Tif
• 通讯作者: Shen, Tif