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

理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涉及位于四维空间中的三维空间的性质，以及较大空间的全局性质对这些子空间施加的限制。国家科学基金会资助的这个项目的研究使用了现代分析和几何工具来阐明这种子空间的局部性质，包括证明这种空间中的奇点不能被平滑的新方法。丹尼尔·鲁伯曼将使用Seiberg-Witten规范理论、Heegaard-Floer同调和更传统的拓扑技术来研究几何拓扑。该项目的第一部分，与林剑锋和尼古拉·萨维列夫合作，涉及四维流形的光滑拓扑，这些流形在同源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Rohlin不变量和多重签名不变量的解释；主要问题的解决将决定高维外科理论所预测的流形的存在。PI将与Adam Levine合作，使用Heegaard Floer理论中的改进技术，在四空间中嵌入穿孔的三维流形，以找到障碍。David Auckly，Hee Jung Kim和Paul Melvin正在进行的一个项目涉及四维流形的微分同胚群的拓扑以及它如何受到流形的稳定化的影响。最后，PI将与Saveliev和Demetre Kazaras合作开发一种新技术，以阻止偶数维流形之间存在正标量曲率密码学。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势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.