Gauge theory and Floer homology in 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位于四维空间中的二维曲面的性质以及这些曲面上存在的奇点有关。该项目的研究使用现代分析和几何工具来揭示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Nikolai Saveliev和Tomasz Mrowka合作，关注的是4-流形的光滑拓扑，它在同调上类似于三维流形与圆的乘积。核心问题围绕着规范理论对经典罗林不变量的解释；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和Saso Strle合作研究Heegaard flower理论中的新不变量在4流形中的扭转同调类中的应用，Levine和Matthew Hedden将研究4流形的端点，Selman Akbulut将研究新的光滑4流形的构造。最后一部分，与David Auckly， Hee Jung Kim和Paul Melvin合作，关注四维流形的微分同构群的拓扑结构及其如何受到流形稳定化的影响。

The understanding of the structure of the 4-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 2-dimensional surfaces sitting in a 4-dimensional space, and with the singularities present on such surfaces. The research in this project 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.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 Nikolai Saveliev and Tomasz Mrowka, are concerned with 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; solutions of the main problems will decide the existence of manifolds predicted by high dimensional surgery theory. The PI will work with Adam Levine and Saso Strle on applications of new invariants from Heegaard Floer theory to torsion homology classes in 4-manifolds, with Levine and Matthew Hedden on ends of 4-manifolds, and with Selman Akbulut on the construction of new smooth 4-manifolds. A final portion, joint with David Auckly, Hee Jung Kim, and Paul Melvin, is concerned with the topology of the diffeomorphism group of a 4-dimensional manifold and how it is affected by stabilization of the manifold.