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维拓扑空间上平滑结构的分类是令人惊讶的微妙和复杂的，远非理解。较低的维度（1、2和3）没有足够的空间来出现有趣的问题，而有足够的空间可以在更高的维度（4）中解决它们。结果是，许多著名的问题仅在维度4中保持平稳，例如分别于1904年和1908年首次提出的庞加莱和Schenflies猜想。该项目的主要目标是推进事件解决这些杰出问题所需的数学技术和机械。这将通过研究相对“简单”平滑的4个manifolds及其子序列，直到各种等效性，通过操纵这些流形中的表面并理解这些表面如何相交和嵌入的局限性来实现。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计划（PTI），该计划是一项招募志愿研究生，博士后和教职员工，向新泽西州惩教局的被监禁学生教大学课程。 PI正在与PTI积极合作，共同开发了新的数学课程，以作为BA课程的一部分提供非记忆专业的新课程，研究法律学院在法庭中使用数学（正确和错误地）的法律案例。 PI还在设计和共同讲授一门课程，在该课程中，学生通过使用它来模拟马戏团艺术（例如空中杂技，杂耍和绳索步行）来学习基本结理论。 PI目前正在与本科生合作，从本课程的首次迭代开始汇编其见解和观察结果，目的是在本科期刊上发布这些结果。由于80年代的封面，封闭的，与同构的封闭式4个manifolds的分类非常令人沮丧。该研究项目的目的是进一步了解维度4中的平滑和拓扑类别之间的差异。紧凑型拓扑4 manifolds的示例是，弗里德曼（Friedman）和摩根（Morgan）首先使用唐纳森（Donaldson）的作品生产了无限的许多独特的平滑结构。相比之下，除四个平滑结构以外的尺寸以外的紧凑型拓扑歧管。 PI有兴趣开发混凝土和有用的方法，用于将同构但不是差异的平滑4个manifolds对成对。特别是，该项目将重点介绍（1）流畅的4个序列，直至“稳定”的差异性，即与产品S^2×S^2的副本相关的求和，（2）拓扑4球的差异类型，这些拓扑类型类型嵌入了标准的4-phere and（3）嵌入了（3）嵌入corcorts and Corcord sofds and Corcords and Cork cork cork cork cork，同位奖，该奖项反映了NSF的法定任务，并通过使用基金会的知识分子优点和更广泛的影响审查标准来评估，被认为是宝贵的支持。