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

• 批准号: 2204367
• 负责人: Hannah Schwartz
• 金额: $ 16.9万
• 依托单位: Princeton University
• 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
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2204367&HistoricalAwards=false
• 关键词: Understanding; Smooth; Structures; via; Regular

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 维拓扑空间上的平滑结构的分类令人惊讶地微妙和复杂，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，但有足够的空间来解决。其结果是，许多众所周知的问题在 4 维中仍然没有得到顺利解答，例如 1904 年首次提出的庞加莱猜想和舍恩弗利斯猜想。该项目的主要目标是推进最终解决这些突出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 流形及其子流形到各种概念来实现。通过操纵这些流形内的表面并了解这些表面如何相交和嵌入的局限性，PI 热衷于参与普林斯顿大学的监狱教学计划 (PTI)。大学，一个招募志愿者研究生、博士后和教师的项目，为新泽西州惩教署机构中的被监禁学生教授大学课程。PI 正在积极与 PTI 合作，为非数学专业共同开发一门新的数学课程。作为文学学士课程的一部分，PI 还设计和共同教授一门课程，学生通过使用数学来学习基本的结理论。模型马戏艺术，如空中杂技、杂耍和走钢丝，PI 目前正在与本科生合作，汇编他们对本课程第一次迭代的见解和观察，目标是在本科生期刊上发表这些结果。由于 Freedman 在 80 年代的开创性工作，封闭、单连通 4 流形直至同胚已得到很好的理解。该研究项目的目标是进一步了解光滑拓扑和拓扑之间的差异。维度 4 中的类别。允许无限多个不同光滑结构的紧致拓扑 4 流形的示例首先由弗里德曼和摩根利用唐纳森的工作提出，相比之下，除了四维之外的紧致拓扑流形最多允许有限多个光滑结构。 PI 有兴趣开发具体且有用的方法来关联同胚但非微分同胚的平滑 4 流形对，特别是，该项目将重点关注 (1) 平滑。 4-流形达到“稳定”微分同胚，即与乘积 S^2 × S^2 的副本进行模连接求和，(2) 嵌入标准 4-球中的拓扑 4-球的微分同胚类型，以及 (3 ）称为软木塞的可收缩流形的嵌入达到正则同伦和拓扑同位素。该奖项反映了 NSF 的法定使命，并被视为值得通过使用基金会的智力优点和更广泛的影响审查标准进行评估来支持。