Applications of Gauge Theory and Floer Homology to Low-Dimensional Topology

了解我们所处的四维宇宙的结构是现代数学和物理学研究的一个重要课题。几何学家和拓扑学家提出的许多问题都与四维空间中的三维空间的性质有关，以及与大空间的整体特性对这些子空间的限制有关。这项由美国国家科学基金会资助的研究项目使用现代分析和几何工具来揭示这些子空间的局部性质，包括展示这些空间中的奇点不能被平滑的新方法。相关的分析技术将用于探索四维空间的全局拓扑，包括对其对称性的研究。Daniel Ruberman将使用Seiberg-Witten规范理论、Heegaard-Floer同调和更传统的拓扑技术进行几何拓扑研究。该项目的第一部分，与林剑锋和Nikolai Saveliev合作，关注四流形的光滑拓扑，它们在同构上类似于三维流形与圆的乘积。中心问题涉及规范理论对经典罗林不变量和多特征不变量的解释；这些主要问题的解决将决定高维外科理论所预测的流形是否存在。PI将与Adam Levine合作，利用Heegaard flower理论中的精细技术，在四个空间中嵌入被刺穿的三流形。David Auckly， Hee Jung Kim和Paul Melvin正在进行的一个项目是关注四维流形的微分同构群的拓扑结构以及它如何受到流形稳定化的影响。最后，PI将与Saveliev和Demetre Kazaras合作研究一种新技术，以阻止偶数维流形之间正标量曲率协数的存在。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 understanding of the structure of the four-dimensional universe in which we live is a key topic of investigation in modern mathematics and physics. Many of the questions posed by geometers and topologists have to do with the nature of three-dimensional spaces sitting in a four-dimensional space, and with the restrictions that the global properties of larger space places on such sub-spaces. The research in this National Science Foundation funded project uses modern tools of analysis and geometry to shed light on the local nature of such subspaces, including new methods for showing that singularities in such spaces cannot be smoothed. Related analytical techniques will be used to explore the global topology of four-dimensional spaces, including an investigation of their symmetries.Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological techniques. The first parts of the project, joint with Jianfeng Lin and Nikolai Saveliev, are concerned with the smooth topology of four-manifolds that homologically resemble a product of a three-dimensional manifold with a circle. Central questions concern the interpretation of the classical Rohlin invariant and multi-signature invariants in terms of gauge theory; solutions of the main problems will decide the existence of manifolds predicted by high dimensional surgery theory. The PI will work with Adam Levine on embeddings of punctured three-manifolds in four-space, using refined techniques from Heegaard Floer theory to find obstructions. An ongoing project with David Auckly, Hee Jung Kim, and Paul Melvin is concerned with the topology of the diffeomorphism group of a four-dimensional manifold and how it is affected by stabilization of the manifold. Finally, the PI will work with Saveliev and Demetre Kazaras on a new technique to obstruct the existence of positive scalar curvature cobordisms between even-dimensional manifolds.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.