Floer Theory, Arc Spaces, and Singularities

这个项目是关于某些工具的发展，用于数学的两个领域：代数几何和辛几何。代数几何研究的数学对象称为变量，它是由加法和乘法建立的方程的解。辛几何是研究汉密尔顿方程时产生的自然几何，汉密尔顿方程是描述经典物理系统运动的方程。一个非常重要的工具集合，称为Floer理论，已被用于解决上述两个主题领域的许多问题。然而，这些工具很难使用，因为所涉及的计算可能相当困难。这个项目的一部分涉及到通过一个叫做弧空间的对象来寻找更好的计算技术。这个项目的另一部分是寻找更精细的方法来计算曲线内的变化，并利用辛几何的思想为这种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，并参与石溪高中学生数学夏令营。作为石溪大学的研究生主任，PI将参与许多以学生为中心的活动。这个项目的主要目的是更好地理解花理论及其与辛几何和代数几何的关系。为此，PI将利用弧空间来计算各种花代数。一个变种的弧空间是从一个圆盘到该变种的全纯映射的空间。证明了孤立超曲面奇点的单多项式的迭代的Floer上同调是某些弧的射流的紧支持上同调。另一个方案是计算任何孤立奇点在其弧空间中的全接触同调。PI的目的是证明在无穷远处存在一个谱序列计算仿射变体的辛上同调，这些仿射变体的页也是紧支持的某些弧的射流上同调群。PI，与合作者Abouzaid和Smith一起，有一个项目，以更有效的方式定义Morava k -理论Gromov-Witten不变量，以及其他一些广义上同调理论。该项目还将使用从扩大的曲线模空间构建全局Kuranishi图的新思想。最后，在与Ritter的联合工作中，PI将使用哈密顿弗洛尔上同调来研究蠕变分辨率猜想。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。

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.