Topology, Geometry and Physics

该项目的近期目标是开发研究三维和四维空间的新工具和方法。关于这些空间的问题是当前许多几何学和拓扑学研究的核心。已经有了研究这些空间的强大工具，但还需要更多工具，特别是研究四维空间，那里的无知是广泛和深刻的。这种无知促使这里的焦点开发新工具并改进旧工具，同时牢记对所涉及的空间有更多了解的长期希望。这项研究可以在拓扑学、微分几何、辛几何和数学物理领域之间架起一座桥梁。这些领域不仅是数学的核心，也是理论物理的核心。例如，这些领域的概念被用于凝聚态物理、高能物理、弦理论和引力理论。在可预见的未来，陶布斯的研究将集中在微分拓扑学问题上，主要集中在以下三个具体领域。第一个集中领域将针对3流形上的平坦连接和4-流形上的自对偶连接的方程的某些推广的解的序列的行为，目的是确定发散序列是否注定了微分拓扑不变量的解的计数构造。第二个集中的领域将研究二维全纯二次微分概念的维度3和4的某些自然推广的微分拓扑。第三个集中区将使用Seiberg-Witten方程来分析三维和更高维度流形上的矢量场的动力学。特别令人感兴趣的是保证闭合轨道和其他类型的小维度不变集存在的条件。在所有这些领域，本研究将使用微分几何和分析的方法来构建研究低维空间的微分拓扑的新工具。尽管有特定的主题和问题，但长期目标是理解光滑、低维流形的结构。

The immediate goal for the project is to develop new tools and methods to study three and four dimensional spaces. Questions about these spaces are central to much of current research in both geometry and topology. There exist powerful tools for studying these spaces, but more is needed, especially to study four dimensional spaces where ignorance is broad and deep. This ignorance motivates the focus here to develop new tools and sharpen the old ones keeping in mind the long term hope of understanding more about the spaces involved. The research described in this proposal could construct bridges between the fields of topology, differential geometry, symplectic geometry and mathematical physics. These fields are central not just to mathematics, but to theoretical physics as well. For example, notions from these fields are used in condensed matter physics, high energy physics, string theory and theories of gravity. The foreseeable future research of C. H. Taubes will focus on differential topology questions with the following three specific areas of concentration. The first area of concentration will be directed towards understanding the behavior of sequences of solutions to certain generalizations of the equations for flat connections on 3 manifolds and for self dual connections on 4-manifolds with a goal to determine if divergent sequences doom solution counting constructions of differential topology invariants. The second area of concentration will investigate the differential topology of certain natural generalizations to dimensions 3 and 4 of the 2-dimensional notion of a holomorphic, quadratic differential. The third area of concentration will use the Seiberg-Witten equations to analyze the dynamics of vector fields on manifolds of dimension three and higher. Of particular interest are conditions that guarantee the existence of closed orbits and other sorts of small dimensional invariant sets. In all of these areas, the research will use methods from differential geometry and analysis to construct new tools for studying the differential topology of low dimensional spaces. Specific topics and questions notwithstanding, the long term goal is to understand the structure of smooth, low dimensional manifolds.