Lefschetz Fibrations, Their Noncommutative Counterparts, and Formal Groups

Lefschetz 纤维、它们的非交换对应物以及形式群

• 批准号: 1904997
• 负责人: Paul Seidel
• 金额: $ 31.19万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1904997&HistoricalAwards=false
• 关键词: Lefschetz; Fibrations; Their; Noncommutative; Counterparts

The goal of this project is to integrate recent insights, originating from theoretical physics, into classical areas of geometry. One can consider these purely mathematical questions as a "theoretical laboratory", which allows one to quickly explore the structure of cutting-edge ideas, abstracting away the full complexity of their physical origin. Besides the expected scientific benefit, the project contains specific parts designed for graduate and undergraduate research. Undergraduate research is an increasingly important part of the training of next-generation scientists and mathematicians. A particular effort has been made in this project to find issues of current relevance which allow students to take charge, under suitable mentorship.In Kontsevich's formulation of the string theory notion of mirror symmetry, this becomes a relation between symplectic geometry and algebraic geometry, formulated in a common algebraic language of noncommutative geometry. The project intends to further develop noncommutative geometry thinking in symplectic geometry. This is useful as an organizing principle for the information arising from pseudo-holomorphic curve methods. In the framework of the project, it will lead to new methods for understanding and computing that information. One key question under consideration is the dependence of categorical structures on the Novikov parameter. Mirror symmetry also has an arithmetic aspect. That motivates another part of the project, which is to bring structures common in number theory, such as formal groups, to bear on symplectic 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.

这个项目的目标是将最近的见解，从理论物理学，到几何的经典领域。人们可以把这些纯粹的数学问题看作是一个“理论实验室”，它允许人们快速探索前沿思想的结构，抽象出它们物理起源的全部复杂性。除了预期的科学效益外，该项目还包含为研究生和本科生研究设计的具体部分。本科研究是培养下一代科学家和数学家的一个越来越重要的组成部分。 在这个项目中，我们特别努力寻找与当前相关的问题，让学生在适当的指导下掌握主动权。在孔采维奇对弦论镜像对称概念的表述中，这成为辛几何和代数几何之间的关系，用非对易几何的通用代数语言表述。该项目旨在进一步发展辛几何中的非对易几何思想。这是有用的组织原则所产生的信息从伪全纯曲线方法。在该项目的框架内，它将导致理解和计算这些信息的新方法。一个关键的问题正在考虑的是依赖的诺维科夫参数的范畴结构。镜像对称也有一个算术方面。这激发了该项目的另一部分，即将数论中常见的结构（如形式群）应用于辛几何。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。

项目成果

Covariant constancy of quantum Steenrold operations

量子 Steenrold 运算的协变常数

• DOI: 10.1007/s11784-022-00967-4
• 发表时间: 2022
• 期刊: JP Journal of Fixed Point Theory and Applications
• 作者: Seidel, P.;Wilkins, N.
• 通讯作者: Wilkins, N.