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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.

拓扑学是研究空间形状的一门基本数学学科。基本问题是两个空间是否具有相同的形状。此外，如果两个空间具有不同的形状，那么这两个形状有多大的不同？我们能否通过某些简单的过程将一个空间转换成另一个空间？一个空间如何包含在高维空间中？这些问题是现代数学的基础，对其他学科如物理、化学和生物学也很重要。这个由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这些问题。该项目还旨在使几何和拓扑学可供广大受众使用，包括本科数学专业和其他学科的学者。该项目将纳入加州理工学院的SURF计划，这有利于本科生的研究。PI将通过指导研究生和博士后，组织研讨会和会议，在暑期学校和讲习班中教授短期课程来帮助扩大这一领域的影响力。该项目侧重于Heegaard Floer同调及其在低维拓扑中的应用。Heegaard Floer同调是通过规范理论和辛几何中的方法定义的一组不变量。PI将研究Heegaard Floer同调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的性质R定理推广到3-流形中的零同伦结。通过这样做，PI希望探索Heegaard Floer同调和3-流形的基本群之间的关系。PI计划解决的另一个问题是3球中结的有限手术分类。PI还将讨论Floer同源性和虚拟技术在四维拓扑中的应用，例如K3表面上结手术的拓扑。该奖项反映了NSF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。

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