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将致力于Gromov-维滕理论，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。格罗莫夫-维滕理论研究的是在特殊构形中相交的给定类型的计数曲线。这样的计数，在良好的情况下，引起枚举不变量的空间，称为格罗莫夫-维滕不变量。这些不变量在现代枚举代数几何中起着关键作用，在弦理论的背景下，也在理论物理中起着关键作用。在这个项目中，PI将应用技术来计算Gromov-Witten不变量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，箭图表示理论和代数簇模空间紧化研究的重大进展。该奖项也将支持研究生与PI在这些领域的工作。该项目是围绕五个主要方面。在第一链中，该项目打算开发新的退化工具，用于计算Gromov-维滕不变量与任意插入的一大类空间。第二条链是有关建筑的明确的例子镜子日志卡拉比-丘品种和orbifolds使用热带几何工具。在第三股，该项目打算提供具体的描述函紧的模空间的日志卡拉比-丘对使用镜像对称。该项目的第四链将提供唐纳森-托马斯理论的箭袋和格罗莫夫-维滕理论集群品种之间的对应关系。最后，该项目打算实现热带代数几何工具来研究具有环面结构的近辛流形的拓扑。关键的新成分使用整个这些股是最近的进展，我们的理解格罗莫夫-维滕理论使用对数几何。这是代数几何的现代变体，特别是为了理解退化问题的解决方案而开发的，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NSF的法定使命，并通过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，被认为值得支持。