Floer theory in gauge theory and symplectic geometry

Purdue University, West Lafayette IN

Investigators

Abstract

Abstract Award: DMS-1007846 Principal Investigator: Yi-Jen Lee The PI proposes to continue her study on the relation between gauge theoretic invariants and Gromov-type invariants under the framework of Taubess proof of Seiberg-Witten equals Gromov; especially, its various Floer theoretic extensions. She also intends to explore the rich implications of the equivalence of various theories on both sides. Symplectic Geometry has its origin from classical mechanics. However, in spite of much progress made, many basic questions remain unanswered. An example is the long standing Weinstein conjecture, which states that there is a periodic orbit on any compact contact manifold. In physics, "contact manifolds" provide the framework to describe the motion of particles. On the other hand, gauge theory arises from the theory of elementary particles in modern physics. Surprisingly, many seemingly unrelated invariants defined on both sides turn out to be equivalent, and therefore facts that are easier to see on one sides imply similar results on the other side, which might otherwise be very difficult to establish. Taubes's celebrated recent proof of the 3-dimensional Weinstein conjecture is an example of many possible applications of such equivalence.