Knot concordance, periodic ends, and Rohlin's invariant

Daniel Ruberman将利用Seiberg-Witten规范理论、Heegaard-Floer同调和更传统的拓扑思想进行几何拓扑学的研究。该项目的第一部分，与Nikolai Saveliev和Tomasz Mrowka合作，研究Seiberg-Witten理论和周期流形上的Dirac算子分析之间的关系。第二个项目是与Saso Strle和Elisenda Grigsby合作，对我们在协调性研究中引入的新的Heegaard-Floer纽结不变量进行进一步的计算。与Matthew Hedden和Hee Jung Kim的相关工作将探索4维流形中纽结理论和嵌入曲面的其他方面。理解我们所生活的4维宇宙的结构是现代数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位于4维空间中的2维表面的性质以及这些表面上存在的奇点有关。这项提案中的研究使用了现代分析和几何工具来阐明这种奇点的局部性质，包括证明这种奇点不能被平滑的新方法。相关的分析技术将被用来探索4维空间的全局拓扑。

Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and more traditional topological ideas. The first part of the project, in collaboration with Nikolai Saveliev and Tomasz Mrowka, studies the relation between Seiberg-Witten theory and the analysis of the Dirac operator on periodic manifolds. A second project, joint with Saso Strle and Elisenda Grigsby, is to carry out further calculations of new Heegaard-Floer knot invariants that were introduced in our study of concordance. Related work with Matthew Hedden and Hee Jung Kim will explore other aspects of knot theory and embedded surfaces in 4-dimensional manifolds.The understanding of the structure of the 4-dimensional universe in which we live is a key topic of investigation in modern mathematics. Many of the questions posed by geometers and topologists have to do with the nature of 2-dimensional surfaces sitting in a 4-dimensional space, and with the singularities present on such surfaces. The research in this proposal uses modern tools of analysis and geometry to shed light on the local nature of such singularities, including new methods for showing that such singularities cannot be smoothed. Related analytical techniques will be used to explore the global topology of 4-dimensional spaces.