Floer theory in gauge theory and symplectic geometry

摘要/ abstract摘要：dms -1007846项目负责人：leyi - jen PI提出在Seiberg-Witten = Gromov的Taubess证明框架下，继续研究定子理论不变量与Gromov型不变量之间的关系；特别是它的各种花理论的延伸。她还打算探讨双方各种理论对等的丰富含义。辛几何起源于经典力学。然而，尽管取得了很大进展，许多基本问题仍未得到解答。一个例子是长期存在的温斯坦猜想，它指出在任何紧密接触流形上都有一个周期轨道。在物理学中，“接触流形”提供了描述粒子运动的框架。另一方面，规范理论起源于现代物理学中的基本粒子理论。令人惊讶的是，许多在两边定义的看似不相关的不变量实际上是相等的，因此，在一边更容易看到的事实在另一边意味着相似的结果，否则这可能很难建立。Taubes最近对三维温斯坦猜想的著名证明就是这种等价的许多可能应用的一个例子。

AbstractAward: DMS-1007846Principal Investigator: Yi-Jen LeeThe PI proposes to continue her study on the relation betweengauge theoretic invariants and Gromov-type invariants under theframework of Taubess proof of Seiberg-Witten equals Gromov;especially, its various Floer theoretic extensions. She alsointends to explore the rich implications of the equivalence ofvarious theories on both sides.Symplectic Geometry has its origin from classicalmechanics. However, in spite of much progress made, many basicquestions remain unanswered. An example is the long standingWeinstein conjecture, which states that there is a periodic orbiton any compact contact manifold. In physics, "contact manifolds"provide the framework to describe the motion of particles. On theother hand, gauge theory arises from the theory of elementaryparticles in modern physics. Surprisingly, many seeminglyunrelated invariants defined on both sides turn out to beequivalent, and therefore facts that are easier to see on onesides imply similar results on the other side, which mightotherwise be very difficult to establish. Taubes's celebratedrecent proof of the 3-dimensional Weinstein conjecture is anexample of many possible applications of such equivalence.