Floer Theory, Arc Spaces, and Singularities

弗洛尔理论、弧空间和奇点

• 批准号: 2203308
• 负责人: Mark McLean
• 金额: $ 35万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-08-01 至 2025-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2203308&HistoricalAwards=false
• 关键词: Floer; Theory; Arc; Spaces; Singularities

This project is concerned with the development of certain tools for use in two areas of mathematics: algebraic geometry and symplectic geometry. Algebraic geometry studies mathematical objects called varieties that are solutions of equations built from addition and multiplication. Symplectic geometry is the natural geometry that emerges when one studies Hamilton's equations, which are equations describing the motion of classical physical systems. A very important collection of tools, called Floer theory, has been utilized to solve many problems in both of the subject areas above. However, these tools are hard to use since the computations involved can be quite difficult. A part of this project involves finding better computational techniques via an object called the arc space. Another part of this project is concerned with finding more refined ways of counting curves inside varieties as well as establishing efficient foundations for such counts using ideas from symplectic geometry. Additionally, the PI will help mentor a summer workshop for graduate students as well as engage with the Stony Brook math summer camp for high school students. In his role as the graduate director at Stony Brook, the PI will be involved in many student-centered activities.The broad aim of this project is to better understand Floer theory and its relationship with symplectic and algebraic geometry. To this end, the PI will utilize arc spaces to compute various Floer algebras. The arc space of a variety is the space of holomorphic maps from a disk to that variety. The PI will show that Floer cohomology of iterates of the monodromy of an isolated hypersurface singularity is compactly supported cohomology of jets of certain arcs. Another project is to compute the full contact homology of any isolated singularity in terms of its arc space. The PI aims to prove that there is a spectral sequence computing symplectic cohomology of affine varieties whose pages are also compactly supported cohomology groups of jets of certain arcs at infinity. The PI, along with collaborators Abouzaid and Smith, has a project defining Morava K-theoretic Gromov-Witten invariants in a more efficient way as well as over some other generalized cohomology theories. This project will also use the new idea of constructing a global Kuranishi chart from an enlarged moduli space of curves. Finally in joint work with Ritter, the PI will investigate the crepant resolution conjecture using Hamiltonian Floer cohomology.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.

该项目涉及在两个数学领域使用某些工具的开发：代数几何和符号几何形状。代数几何研究数学对象称为品种，是通过添加和乘法构建的方程解决方案。符号几何形状是当研究汉密尔顿方程的自然几何形状，它是描述经典物理系统运动的方程式。在上述两个主题领域中，已利用一个非常重要的工具集合，称为浮子理论。但是，这些工具很难使用，因为涉及的计算可能非常困难。该项目的一部分涉及通过称为ARC空间的对象找到更好的计算技术。该项目的另一部分涉及找到更精致的方法来计算品种内部的曲线，并使用符合性几何形状中的思想为这些计数建立有效的基础。此外，PI将帮助指导研究生的夏季研讨会，并与Stony Brook数学夏令营互动。 PI在Stony Brook担任研究生主任的角色中将参与许多以学生为中心的活动。该项目的广泛目的是更好地了解浮动理论及其与符号和代数几何形状的关系。为此，PI将利用弧形空间计算各种浮动代数。一种品种的弧形空间是从磁盘到该品种的全态图的空间。 PI将表明，孤立的超表面奇异性的迭代术的漂浮同谋是某些弧线的喷气机的共同支持。另一个项目是根据其弧空空间计算任何孤立的奇点的完整接触同源性。 PI的目的是证明仿射品种的频谱序列计算符号共同体，其页面也被紧凑地支持了无穷大的某些弧线的Jets组。 PI与合作者Abouzaid和Smith一起制定了一个项目，以更有效的方式以及其他一些广义的共同体理论来定义Morava K Theoretic Gromov-Witten不变性。该项目还将使用新的想法，即从扩大的模量曲线空间构建全球库兰尼图表。最终，在与Ritter的联合合作中，PI将使用汉密尔顿浮子同居研究来调查Crepant决议的猜想。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估评估标准来通过评估来获得支持的。