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

四维拓扑空间上光滑结构的分类令人惊讶地微妙和复杂，而且远未被理解。较低的维度（1、2和3）没有足够的空间来出现有趣的问题，而在较高的维度（4以上）中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只有在第四维空间中才能顺利地得到解答，比如分别于1904年和1908年首次提出的庞加莱猜想和Schönflies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器。这将通过研究相对“简单”的光滑4流形及其子流形达到各种等效概念，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限制来实现。作为一个更广泛的影响，PI积极参与普林斯顿大学的监狱教学倡议（PTI），这是一个招募志愿研究生、博士后和教职员工的项目，为新泽西州惩教机构的在押学生教授大学课程。PI正积极与PTI合作，共同为非数学专业的学生开发一门新的数学课程，作为学士课程的一部分，研究在法庭上使用数学（无论正确与否）的法律案件。PI还设计并联合教授一门课程，让学生学习基本的结理论，并用它来模拟马戏艺术，如空中杂技、杂耍和走钢丝。项目负责人目前正在与本科生合作，收集他们在本课程第一次迭代中的见解和观察结果，目标是将这些结果发表在本科生期刊上。由于弗里德曼在80年代的开创性工作，闭合单连通4流形直到同胚的分类得到了很好的理解。本研究项目的目标是进一步理解光滑类别和拓扑类别在维度4上的区别。Friedman和Morgan利用Donaldson的工作，首先给出了包含无穷多个不同光滑结构的紧致拓扑4流形的例子。相反，非四维的紧致拓扑流形最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关联对光滑4流形是同胚的，但不是微分同胚的。特别是，该项目将重点关注(1)光滑4流形直至“稳定”的微分同构，即与乘积S^2 × S^2的副本的模连通求和，(2)嵌入标准4球的拓扑4球的微分同构类型，以及(3)嵌入称为corks的可收缩流形直至正则同伦和拓扑异构。该奖项反映了美国国家科学基金会的法定使命，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。

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.