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

对我们所生活的四维宇宙结构的理解是现代数学和物理学研究的一个关键课题。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，也与较大空间的整体性质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项目的研究使用了现代分析和几何工具来揭示这种子空间的局部性质，包括显示这种空间中的奇点无法平滑的新方法。相关的分析技术将用于探索四维空间的整体拓扑，包括它们的对称性的调查。丹尼尔鲁伯曼将进行几何拓扑的研究，使用塞伯格-威滕规范理论，Heegaard-Floer同调，和更传统的拓扑技术。该项目的第一部分，与Jianfeng Lin和Nikolai Saveliev联合，关注四流形的光滑拓扑，这些流形同调类似于三维流形与圆的乘积。中心问题涉及经典的Rohlin不变量和多签名不变量的规范理论的解释;解决的主要问题将决定存在的流形预测高维手术理论。PI将与Adam Levine合作，使用Heegaard Floer理论的精细技术来寻找障碍物，在四维空间中嵌入穿孔的三维流形。 一个正在进行的项目与大卫奥克兰，喜贞金，和保罗梅尔文是关注的拓扑结构的同胚群的四维流形，以及它是如何影响稳定的流形。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存在正标量曲率协边。该奖项反映了NSF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.