Symplectic cohomology and quantum cohomology of Fano manifolds

辛几何是研究由经典力学中的相空间局部建模的空间。虽然辛流形没有局部不变量，但它们有有趣的全局不变量。这个项目的主要部分是关于“Gromov-Witten”不变量，其目的是通过研究从二维曲面到满足适当偏微分方程的流形的映射来探测辛流形。这些不变量非常强大，但由于它们的非局部性质，很难很好地控制它们。该项目的一个关键目标是证明由多项式方程定义的某些辛流形上关于Gromov-Witten不变量的新基本事实。PI将使用的一般策略是沿着除数“切开”流形。相应的除数补的不变量是非常容易处理的，也为理解原始空间的Gromov-Witten不变量提供了一个垫脚石。PI将继续担任麻省理工学院研究科学研究所高中生的导师，并将为研究生组织一个关于同调镜像对称的研讨会。他还将继续在马萨诸塞大学波士顿分校开设数学硕士课程。具体来说，在项目的主要部分，PI将证明具有光滑反正则除数的Fano流形上的量子连接具有非分支指数型奇点。该策略是将范诺流形的量子上同调看作是除数补的辛上同调的变形。补的辛上同调可以反过来通过包裹的Fukaya范畴来研究，允许人们从非交换几何中引入工具来承担。在另一个方向上，PI将在他之前的工作的基础上，将仿射对数Calabi-Yau变体的辛上同性与最近由代数几何家构造的某些本征镜像代数联系起来。然后，PI将继续使用先前工作中开发的辛和分类技术的组合来研究本征镜像对的同调镜像对称性。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。

Symplectic geometry is the study of spaces that are locally modeled on phase spaces from classical mechanics. While symplectic manifolds have no local invariants, they do have interesting global ones. A major part of this project is concerned with “Gromov-Witten” invariants, which aim to probe a symplectic manifold by studying maps from two-dimensional surfaces into the manifold that satisfy an appropriate partial differential equation. These invariants are very powerful, but it can be quite difficult to get good control over them because of their non-local nature. One key goal of the project is to prove new fundamental facts about Gromov-Witten invariants on certain symplectic manifolds that are defined by polynomial equations. The general strategy the PI will use is to “cut open” the manifold along a divisor. The corresponding invariants of the divisor complement turn out to be very tractable and also provide a stepping stone towards understanding the Gromov-Witten invariants of the original space. The PI will continue to serve as a mentor to high school students at MIT's Research Science Institute, and will organize a workshop on homological mirror symmetry for graduate students. He will also continue to work on developing a Master’s program in mathematics at the University of Massachusetts Boston. Specifically, in the main strand of the project, the PI will prove that the quantum connection on a Fano manifold with a smooth anti-canonical divisor has a singularity of unramified exponential type. The strategy is to view the quantum cohomology of the Fano manifold as a deformation of the symplectic cohomology of the complement of the divisor. The symplectic cohomology of the complement can in turn be studied via the wrapped Fukaya category, allowing one to bring tools from noncommutative geometry to bear. In a different direction, the PI will build on his previous work to relate the symplectic cohomology of an affine log Calabi-Yau variety to certain intrinsic mirror algebras recently constructed by algebraic geometers. The PI will then go on to study homological mirror symmetry for intrinsic mirror pairs using a combination of symplectic and categorical techniques also developed in previous work.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.