Symplectic cohomology and quantum cohomology of Fano manifolds

Fano流形的辛上同调和量子上同调

• 批准号: 2306204
• 负责人: Daniel Pomerleano
• 金额: $ 12.34万
• 依托单位: University of Massachusetts Boston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2306204&HistoricalAwards=false
• 关键词: Symplectic; cohomology; quantum; Fano; manifolds

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.

Symple Timely几何形状是对从经典力学的相位空间进行局部建模的空间的研究。虽然对称歧管没有局部不变性，但它们确实具有有趣的全球。该项目的主要部分与“ gromov-witten”不变性有关，该项目的目的是通过将二维表面的地图研究到满足适当的部分微分方程的歧管中来探测对称的歧管。这些不变的人非常强大，但是由于它们的非本地性质，可以很好地控制它们。该项目的一个关键目标是证明有关由多项式方程定义的某些符号歧管上的Gromov-witten不变性的新基本事实。 PI将使用的一般策略是“削减”沿划分的歧管。该部门的相应不变性事实非常容易，还为理解原始空间的Gromov编织的不变性提供了垫脚石。 PI将继续为麻省理工学院研究科学研究所的高中生提供心理，并将组织一个针对研究生的同源镜像对称性的研讨会。他还将继续在马萨诸塞州波士顿大学开发数学硕士课程。具体而言，在项目的主要链中，PI将证明具有光滑的反典型分隔线的Fano歧管上的量子连接具有未造成的指数类型的奇异性。该策略是将FANO歧管的量子共同体视为分隔物补体的对称共同体的变形。可以通过包装的福卡亚类别研究完成的对称共同体，从而使一个人从非交通几何形状中携带工具。在不同的方向上，PI将基于他以前的工作，以将仿射日志的同型共同体学Calabi-yau品种与最近由代数几何构建的某些内在的镜像代数相关联。然后，PI将继续研究以固有镜对的同源镜子对称性，同时在先前的工作中也开发了符号和分类技术的组合。该奖项反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的审查标准通过评估来通过评估来支持的。