Seiberg-Witten Invariants in Dimension three and four

提案：DMS-9971950首席研究员：Peter Ozsvath该提案涉及三维和四维规范理论。具体地说，在从Zoltan Szabo开始的工作中，我们打算将证明辛Thom猜想的技术扩展到寻找其他规范理论上的障碍，从而将三维流形嵌入到四维流形中。一个相关的目标是阐明规范理论不变量的性质--闭四流形的Seiberg-Witten不变量，三流形的Seiberg-Witten Floer同调，以及带边界的四流形的相对不变量--以期获得计算它们的组合技术。西蒙·唐纳森的工作将三维和四维空间的研究直接置于许多不同数学分支的交汇点上，包括几何、分析和拓扑学。事实上，他的工作可以被视为与现代理论物理学正在进行的对话的一部分，这使得物理学家塞伯格和维滕引入了一个具有深远数学和物理后果的新方程。拟议研究的目标是研究这个方程的基本数学性质，以及它与定义它的空间的基本几何之间的相互作用。

Proposal: DMS-9971950Principal Investigator: Peter OzsvathThe proposal deals with gauge theory in dimensions three and four. Specifically, in work begun with Zoltan Szabo, we intend to extend techniques used in proving the symplectic Thom conjecture to finding other gauge-theoretic obstructions to embedding three-manifolds in four-manifolds. A related goal is to shed light on the nature of gauge-theoretic invariants -- Seiberg-Witten invariants of closed four-manifolds, Seiberg-Witten Floer homology for three-manifolds, and relative invariants for four-manifolds with boundary -- with a view towards obtaining combinatorial techniques for calculating them. Work of Simon Donaldson has placed the study of three- and four-dimensional spaces squarely at a juncture between many diverse branches of mathematics, including geometry, analysis, and topology. And indeed, his work can be seen as part of the on-going dialogue with modern theoretical physics, which lead the physicists Seiberg and Witten to introduce a new equation with far-reaching mathematical and physical ramifications. The goal of the proposed research is to study both the fundamental mathematical properties of this equation, and its interaction with the underlying geometry of the spaces on which it is defined.