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

四维拓扑空间上的光滑结构的分类是令人惊讶的微妙和复杂的，而且还远未被理解。较低的维度（1、2和3）没有足够的空间让有趣的问题出现，而在更高的维度（4以上）有足够的空间来解决它们。这样做的一个后果是，许多著名的问题仍然没有得到回答，只有顺利地在第4维，如庞加莱和Schönflies命题首次提出于1904年和1908年分别。这个项目的主要目标是推进数学技术和必要的机械最终解决这些悬而未决的问题。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价概念，通过操纵这些流形内的曲面并理解这些曲面如何相交和嵌入的限制来实现。作为一个更广泛的影响，PI热情地参与了普林斯顿大学的监狱教学计划（PTI），该计划招募志愿者研究生，博士后和教师在新泽西惩教机构部门向被监禁的学生教授大学课程。PI正在积极与PTI合作，共同开发一个新的数学课程，为非数学专业的学生提供作为BA课程的一部分，检查法律的案件中，数学已被使用（正确和不正确）在法庭上。PI还设计和共同教授一门课程，让学生学习基本的结理论，用它来模拟马戏艺术，如空中杂技，杂耍和走钢丝。PI目前正在与本科生合作，从本课程的第一次迭代中汇编他们的见解和观察，目标是将这些结果发表在本科期刊上。由于Freedman在80年代的开创性工作，封闭的单连通4-流形的分类直到同胚都得到了很好的理解。这个研究项目的目标是进一步理解四维空间中光滑和拓扑范畴之间的区别。允许无穷多个不同光滑结构的紧致拓扑4-流形的例子首先由弗里德曼和摩根利用唐纳森的工作提出。相比之下，四维以外的紧致拓扑流形最多只能容纳1000个光滑结构。PI感兴趣的是发展具体的和有用的方法，涉及对光滑的4-流形是同胚的，但不同胚。特别是，该项目将集中在（1）光滑4-流形到“稳定”的复同态，即与乘积S^2 × S^2的副本的模连接求和，（2）嵌入标准4-球面的拓扑4-球的复同态类型，以及（3）嵌入的收缩流形称为软木塞到定期同伦和拓扑合痕。这一奖项反映了NSF的法定使命，并已被视为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，

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.