Heegaard Floer Homology and Low-Dimensional Topology

Heegaard Floer 同调和低维拓扑

基本信息

批准号：
1811900
负责人：
Yi Ni
金额：
$ 20.3万
依托单位：
California Institute of Technology
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-09-01 至 2024-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811900&HistoricalAwards=false
关键词：
Heegaard Floer Homology Low Dimensional

项目摘要

Topology is a fundamental discipline of mathematics which studies the shape of spaces. The basic question is whether two spaces have the same shape. Moreover, if two spaces have different shapes, how different are the two shapes? Can we transform one space to another by certain simple processes? How is a space contained in higher dimensional spaces? Such questions are fundamental in modern mathematics, and are important to other disciplines such as physics, chemistry, and biology. This National Science Foundation funded project will study and develop techniques that help extract information about spaces, and thus answer such questions for some cases. This project also aims to make geometry and topology accessible to a broad audience, including undergraduate mathematics majors and scholars from other disciplines. The project will incorporate Caltech's SURF program, which facilitates undergraduate research. The PI will help broaden the influence of this field by mentoring graduate students and postdocs, organizing seminars and conferences, teaching short courses in summer schools and workshops.This project focuses on Heegaard Floer homology and its applications to low-dimensional topology. Heegaard Floer homology is a package of invariants defined via methods in gauge theory and symplectic geometry. The PI will investigate the relationship between Heegaard Floer homology and many other aspects of low-dimensional topology. In particular, the PI will study the generalization of Gabai's Property R theorem to null-homotopic knots in 3-manifolds. By doing so, the PI hopes to explore the relationship between Heegaard Floer homology and fundamental groups of 3-manifolds. Another problem the PI plans to attack is the classification of finite surgeries on knots in the 3-sphere. The PI will also address the applications of Floer homology and virtual techniques to 4-dimensional topology, for example, the topology of knot surgeries on the K3 surface.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.

拓扑是研究空间形状的数学基本学科。基本问题是两个空间是否具有相同的形状。而且，如果两个空间的形状不同，那么两个形状有何不同？我们可以通过某些简单的过程将一个空间转换为另一个空间吗？如何在更高维空间中包含一个空间？这些问题在现代数学中至关重要，对其他学科（例如物理，化学和生物学）很重要。该国家科学基金会资助的项目将研究和开发有助于提取空间信息的技术，从而在某些情况下回答此类问题。该项目还旨在使广泛受众访问几何学和拓扑，包括来自其他学科的本科数学专业和学者。该项目将纳入加州理工学院的冲浪计划，该计划促进了本科研究。 PI将通过指导研究生和博士学位，组织研讨会和会议，在暑期学校和研讨会上教短期课程来帮助扩大该领域的影响。该项目侧重于Heegaard Floer同源性及其在低维拓扑学上的应用。 Heegaard Floer同源性是通过尺度理论和符号几何形状中的方法定义的不变式包装。 PI将研究Heegaard Floer同源性与低维拓扑的许多其他方面之间的关系。特别是，PI将研究Gabai属性R定理对3个manifolds中无效结的概括。通过这样做，PI希望探索Heegaard Floer同源性与3个Manifolds的基本组之间的关系。 PI计划攻击的另一个问题是将有限的手术分类为三个球的结中。例如，PI还将解决浮子同源性和虚拟技术在4维拓扑上的应用，例如，K3 Surface上结的拓扑结构。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优点和更广泛影响的审查标准来通过评估来通过评估来支持的。

项目成果

期刊论文数量（5）

专著数量（0）

科研奖励数量（0）

会议论文数量（0）

专利数量（0）

Null-homotopic knots have Property R

空同伦结具有属性 R

DOI：
10.1017/s0305004123000129
发表时间：
2023
期刊：
Mathematical Proceedings of the Cambridge Philosophical Society
影响因子：
0.8
作者：
NI, YI
通讯作者：
NI, YI

The next-to-top term in knot Floer homology

结弗洛尔同源性中的次高项

DOI：
10.4171/qt/174
发表时间：
2022
期刊：
Quantum Topology
影响因子：
1.1
作者：
Ni, Yi
通讯作者：
Ni, Yi

A Characterization of T2g+1,2 among Alternating Knots

交替结中 T2g 1,2 的表征

DOI：
10.1007/s10114-021-0408-4
发表时间：
2021
期刊：
English Series
影响因子：
0
作者：
Ni, Yi
通讯作者：
Ni, Yi

The prism manifold realization problem

DOI：
10.2140/agt.2020.20.757
发表时间：
2016-12
期刊：
Algebraic & Geometric Topology
影响因子：
0.7
作者：
William Ballinger;Chloe Ching-Yun Hsu;Wyatt Mackey;Yi Ni;Tynan Ochse;F. Vafaee
通讯作者：
William Ballinger;Chloe Ching-Yun Hsu;Wyatt Mackey;Yi Ni;Tynan Ochse;F. Vafaee

The prism manifold realization problem III

棱镜流形实现问题三