Analytical and geometric methods in low-dimensional topology

Daniel Ruberman将利用Seiberg-Witten规范理论、Heegaard-Floer同调和纽结理论技术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，研究同调类似于三维流形与圆的乘积的4-流形的光滑拓扑。中心问题围绕着规范理论对经典罗林不变量的解释。技巧涉及Seiberg-Witten理论和具有周期末端的流形的一个新的指标定理。该项目的第二部分提出了PI与Saso Strle和Jae Choon Cha在纽结理论中的联合工作，将Heegaard-Floer理论中的新不变量应用于关于纽结和链接一致性的经典问题。这一部分的结果将有助于衡量平滑位于4维流形中的曲面的奇点的难度。最后一部分，与Paul Melvin和David Auckly合作，研究了4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。理解我们所处的4维宇宙的结构是现代数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，包括对其对称性的调查。

Daniel Ruberman will carry out research in geometric topology, using Seiberg-Witten gauge theory, Heegaard-Floer homology, and techniques of knot theory. The first part of the project, joint with Nikolai Saveliev and Tomasz Mrowka, studies the smooth topology of 4-manifolds that homologically resemble a product of a 3-dimensional manifold with a circle. The central questions center around the interpretation of the classical Rohlin invariant in terms of gauge theory. Techniques involve Seiberg-Witten theory and a new index theorem for manifolds with periodic ends. A second part of the project proposes joint work of the PI with Saso Strle and Jae Choon Cha in knot theory, applying new invariants from Heegaard-Floer theory to classical problems about knot and link concordance. The results from this part will help measure the difficulty of smoothing a singularity of a surface sitting in a 4-dimensional manifold. A final portion, joint with Paul Melvin and David Auckly, is concerned with the topology of the diffeomorphism group of a 4-dimensional manifold and how it is affected by stabilization of the manifold. 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, including an investigation of their symmetries.