Analytical and geometric methods in low-dimensional topology

低维拓扑中的解析和几何方法

• 批准号: 1105234
• 负责人: Daniel Ruberman
• 金额: $ 11.52万
• 依托单位: Brandeis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2016-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1105234&HistoricalAwards=false
• 关键词: Analytical; geometric; methods; low; dimensional

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.

丹尼尔鲁伯曼将开展研究的几何拓扑结构，使用塞伯格-威滕规范理论，Heegaard-弗洛尔同源，和技术的结理论。该项目的第一部分，与Nikolai Saveliev和Tomasz Mrowka联合，研究4-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。 中心问题围绕着规范理论中经典Rohlin不变量的解释。 技术涉及塞伯格-威滕理论和一个新的指标定理流形周期结束。该项目的第二部分提出了PI与Saso Strle和Jae Choon Cha在结理论方面的联合工作，将Heegaard Floer理论的新不变量应用于关于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光滑四维流形中曲面奇点的难度。 最后一部分，联合保罗梅尔文和大卫奥克兰，是关注的拓扑结构的同胚群的4维流形，以及它是如何影响稳定的流形。对我们生活的四维宇宙结构的理解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四维空间中二维曲面的性质有关，并且与这些曲面上的奇点有关。本提案中的研究使用现代分析和几何工具来阐明这种奇点的局部性质，包括显示这种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全局拓扑，包括对它们的对称性的研究。