Understanding Smooth Structures via Regular Homotopy of Surfaces in 4-Manifolds

通过 4 流形中曲面的正同伦了解光滑结构

基本信息

  • 批准号:
    2204367
  • 负责人:
  • 金额:
    $ 16.9万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2022
  • 资助国家:
    美国
  • 起止时间:
    2022-09-01 至 2025-08-31
  • 项目状态:
    未结题

项目摘要

The classification of smooth structures on 4-dimensional topological spaces is surprisingly subtle and complex, and far from understood. Lower dimensions (1, 2, and 3) do not have enough room for interesting problems to arise, while there is ample space to resolve them in higher dimensions (above 4). A consequence of this is that many well-known questions remain unanswered only smoothly in dimension 4, such as the Poincaré and Schönflies conjectures first posed in 1904 and 1908, respectively. The primary goal of this project is to advance the mathematical techniques and machinery necessary for the eventual resolution of these outstanding problems. This will be achieved by studying relatively "simple" smooth 4-manifolds and their submanifolds up to various notions of equivalence, through manipulating surfaces within these manifolds and understanding limitations on how these surfaces intersect and embed. As a broader impact, the PI is passionately involved with the Prison Teaching Initiative (PTI) at Princeton University, a program recruiting volunteer graduate students, postdocs, and faculty to teach college courses to incarcerated students in New Jersey Department of Corrections institutions. The PI is actively working with the PTI to co-develop a new math course for non-math majors to be offered as part of the BA curriculum, examining legal cases in which mathematics has been used (both correctly and incorrectly) in the courtroom. The PI is also designing and co-teaching a course in which students learn basic knot theory by using it to model circus arts such as aerial acrobatics, juggling, and tightrope walking. The PI is currently working with undergraduate students to compile their insights and observations from the first iteration of this course, with the goal of publishing these results in an undergraduate journal.The classification of closed, simply-connected 4-manifolds up to homeomorphism is well understood, due to groundbreaking work of Freedman from the 80's. The goal of this research project is to further understand the difference between the smooth and topological categories in dimension 4. Examples of compact topological 4-manifolds admitting infinitely many distinct smooth structures were first produced by Friedman and Morgan, using the work of Donaldson. In contrast, compact topological manifolds of dimension other than four admit at most finitely many smooth structures. The PI is interested in developing concrete and useful methods of relating pairs of smooth 4-manifolds that are homeomorphic but not diffeomorphic. In particular, the project will focus on (1) smooth 4-manifolds up to "stable" diffeomorphism, i.e. modulo connected summing with copies of the product S^2 × S^2, (2) the diffeomorphism types of topological 4-balls that embed in the standard 4-sphere, and (3) embeddings of contractible manifolds called corks up to regular homotopy and topological isotopy.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.
4维拓扑空间上光滑结构的分类是令人惊讶的微妙和复杂的,并且远未被理解。较低的维度(1、2和3)没有足够的空间来产生有趣的问题,而较高的维度(高于4)有足够的空间来解决这些问题。其结果是,许多众所周知的问题在4维空间中仍然没有得到顺利的回答,例如分别于1904年和1908年首次提出的庞卡莱和舍恩费尔斯猜想。这个项目的主要目标是推进最终解决这些悬而未决的问题所需的数学技术和机制。这将通过研究相对简单的光滑4-流形及其子流形直到各种等价概念来实现,通过操纵这些流形中的曲面并理解这些曲面如何相交和嵌入的限制来实现。作为更广泛的影响,PI热情地参与了普林斯顿大学的监狱教学倡议(PTI),这是一个招募志愿者研究生、博士后和教职员工的计划,为新泽西州惩教机构的被监禁学生教授大学课程。PI正在积极与PTI合作,共同为非数学专业的学生开发一门新的数学课程,作为BA课程的一部分,审查在法庭上(正确和错误地)使用数学的法律案例。PI还设计并共同教授了一门课程,学生们通过使用它来模拟空中杂技、杂耍和走钢丝等马戏团艺术,学习基本的结理论。PI目前正在与本科生合作,从本课程的第一次迭代开始汇编他们的见解和观察结果,目标是将这些结果发表在本科生期刊上。由于弗里德曼从80年代的S开始的开创性工作,闭的、单连通的直到同态的4-流形的分类得到了很好的理解。这个研究项目的目标是进一步理解4维光滑和拓扑范畴之间的区别。可以容纳无穷多不同光滑结构的紧致拓扑4-流形的例子首先是由弗里德曼和摩根利用唐纳森的工作产生的。相反,非四维的紧致拓扑流形至多允许有限多个光滑结构。PI感兴趣的是发展具体和有用的方法来联系光滑的4-流形对,这些流形对是同胚的而不是微分同胚的。具体而言,该项目将关注(1)光滑的4-流形直至“稳定”的微分同态,即与S^2×S^2积的副本的模连通求和,(2)嵌入标准4-球面的拓扑4-球的微分同态类型,以及(3)称为软木塞的可压缩流形的嵌入至正则同伦和拓扑同伦。该奖项反映了美国国家科学基金会的法定使命,并已通过使用基金会的智力优势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

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Hannah Schwartz其他文献

Perinatal Mental Healthcare Needs Among Women at a Community Hospital.
社区医院妇女的围产期心理保健需求。
The Potential Role of Glucagon-Like Peptide-1 (GLP-1) Receptor Agonists and Glucose-Dependent Insulinotropic Polypeptide (GIP) Receptor Agonists in Obstructive Sleep Apnea and Obesity
  • DOI:
    10.1007/s13665-025-00384-1
  • 发表时间:
    2025-07-25
  • 期刊:
  • 影响因子:
    1.000
  • 作者:
    Charlotte C. Ellberg;Hannah Schwartz;Annika Witt;Karen C. McCowen;Ana Lucia Fuentes;Atul Malhotra
  • 通讯作者:
    Atul Malhotra
Translating developmental origins of health and disease in practice: health care providers’ perspectives
将健康和疾病的发展起源转化为实践:医疗保健提供者的观点

Hannah Schwartz的其他文献

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