New Bridges to Gromov-Witten Theory

通往格罗莫夫-维滕理论的新桥梁

• 批准号: 2302116
• 负责人: Nuromur Arguz
• 金额: $ 16万
• 依托单位: University of Georgia Research Foundation Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2302116&HistoricalAwards=false
• 关键词: New; Bridges; Gromov; Witten; Theory

In this project, the PI will work on Gromov--Witten theory, an area in mathematics lying in the intersection of algebraic geometry and symplectic topology, with implications to string theory in physics. Gromov--Witten theory is concerned with counting curves of a given type which intersect in special configurations. Such counts, in good situations, give rise to enumerative invariants of the space, called Gromov--Witten invariants. These invariants play a key role in modern enumerative algebraic geometry, as well as in theoretical physics, in the context of string theory. In this project, the PI will apply techniques to calculate Gromov-Witten invariants to several other questions coming from different areas of mathematics, to obtain major advances in mirror symmetry, in representation theory of quivers, and in the study of compactifications of moduli spaces of algebraic varieties. This award will also support graduate students working with the PI in these areas.The project is built around five main strands. In the first strand the project intends to develop new degeneration tools for calculating Gromov--Witten invariants with arbitrary insertions of a large class of spaces. The second strand is concerned with constructions of explicit examples of mirrors to log Calabi--Yau varieties and orbifolds using tropical geometric tools. In the third strand, the project intends to provide concrete descriptions of functorial compactifications of the moduli space of log Calabi--Yau pairs using mirror symmetry. The fourth strand of the project will provide a correspondence between Donaldson--Thomas theory of quivers and Gromov--Witten theory for cluster varieties. Finally, the project intends to implement tropical algebro-geometric tools to study the topology of near symplectic manifolds with a toric structure. The key new ingredient used throughout these strands is the recent advance in our understanding of Gromov--Witten theory using logarithmic geometry. This is a modern variant of algebraic geometry, developed in particular to understand solutions to degeneration problems, which naturally appear in the context of enumerative and tropical algebraic geometry.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.

在这个项目中，PI将研究Gromov- Witten理论，这是一个数学领域，位于代数几何和辛拓扑的交叉点，对物理学中的弦理论有影响。Gromov- Witten理论关注的是对以特殊构型相交的给定类型曲线进行计数。在好的情况下，这样的计数会产生空间的枚举不变量，称为Gromov—Witten不变量。这些不变量在现代枚举代数几何中以及在弦理论背景下的理论物理中发挥着关键作用。在这个项目中，PI将应用计算Gromov-Witten不变量的技术来解决来自不同数学领域的其他几个问题，以获得镜像对称，颤振的表示理论和代数变体模空间的紧化研究方面的重大进展。该奖项还将支持与PI在这些领域合作的研究生。该项目围绕五条主线建造。在第一部分中，该项目打算开发新的退化工具，用于计算具有大量空间任意插入的Gromov—Witten不变量。第二部分是关于使用热带几何工具来记录卡拉比-丘品种和轨道的镜子的明确例子的构建。在第三部分，该项目打算使用镜像对称提供对数Calabi—Yau对模空间的功能紧化的具体描述。该项目的第四部分将提供关于颤抖的Donaldson- Thomas理论和Gromov- Witten理论之间的对应关系。最后，本计画拟使用热带代数几何工具来研究具有环形结构的近辛流形的拓扑。贯穿这些分支的关键新成分是我们对使用对数几何的Gromov- Witten理论的理解的最新进展。这是代数几何的一种现代变体，特别是为了理解退化问题的解而发展起来的，退化问题自然出现在枚举和热带代数几何的背景下。该奖项反映了美国国家科学基金会的法定使命，并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，被认为值得支持。