Lefschetz Fibrations, Their Noncommutative Counterparts, and Formal Groups

这个项目的目标是将源自理论物理的最新见解整合到几何的经典领域。人们可以把这些纯粹的数学问题视为“理论实验室”，它允许人们快速探索尖端思想的结构，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，该项目还包含专为研究生和本科生研究而设计的具体部分。本科生研究是培养下一代科学家和数学家的一个越来越重要的部分。在这个项目中，一个特别的努力是寻找与当前相关的问题，允许学生在适当的指导下掌握。在康采维奇关于镜像对称性的弦论概念的表述中，这成为辛几何和代数几何之间的关系，用非对易几何的公共代数语言来表述。该项目旨在进一步发展辛几何中的非对易几何思维。对于伪全纯曲线方法产生的信息，这是一个有用的组织原则。在该项目的框架内，它将产生理解和计算这些信息的新方法。正在考虑的一个关键问题是范畴结构对Novikov参数的依赖。镜面对称也有算术方面的问题。这激发了该项目的另一个部分，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优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.