Topology, Geometry and Physics

该项目的近期目标是开发新的工具和方法来研究三维和四维空间。 关于这些空间的问题是当前几何学和拓扑学研究的核心。有强大的工具来研究这些空间，但需要更多，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重点开发新的工具和锐化旧的铭记长期希望了解更多的空间涉及。 该方案所描述的研究可以在拓扑学、微分几何、辛几何和数学物理之间建立桥梁。 这些领域不仅是数学的核心，也是理论物理的核心。 例如，来自这些领域的概念被用于凝聚态物理学、高能物理学、弦理论和引力理论。 展望了C. H. Taubes将专注于微分拓扑问题与以下三个具体领域的浓度。 集中的第一个领域将是针对理解的行为序列的解决方案，以某些推广的方程的平坦连接3流形和自对偶连接4流形的目标，以确定是否发散序列的厄运解决方案计数结构的微分拓扑不变量。 第二个领域的浓度将调查的微分拓扑的某些自然推广到三维和四维的二维概念的全纯，二次微分。 第三个集中的领域将使用塞伯格-威滕方程来分析三维及更高维流形上的向量场的动力学。 特别令人感兴趣的是条件，保证存在的封闭轨道和其他种类的小维不变集。 在所有这些领域中，研究将使用微分几何和分析的方法来构建研究低维空间微分拓扑的新工具。 尽管有特定的主题和问题，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.