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.
The goal of the project is to analyze different filed theories and their role in geometry. In recent decades there has been a fruitful interaction between mathematics and physics. On one hand physicists are using more and more mathematical techniques to describe the objects they are studying on the other hand, mathematicians have taken ideas from physics and converted them into bona fide mathematical theories, many times with the aim to understand geometry of spaces. This goes back to classical mechanics. The mathematical version of this are symplectic manifolds. Their properties reflect the fact that in the Hamiltonian formalism everything can be written in terms of positions and momenta. The advent of field theory and string theory has brought new tools and correspondences. A major hindrance for this is that some of the techniques like the Feynman integral are not mathematically rigorous. This is overcome by introducing substitute theories that behave like physics would predict, but at mathematically well defined. These have added tremendously to the understanding of the geometry and topology of spaces. Since there are several ways to mathematically encode the ideas presented in physics, it is then an interesting problem to compare the theories and the information they carry. Part of the activity was to show that two mathematical incarnations of a theory acturally are isomorphic. That is Floer homology is equivalent to Seiberg-Witten Floer cohomology. Both are used as interesting extra structures for symplectic manifolds. The project has furthermore used field theory inspired techniques to study important spaces, such as Teichmueller space, which is a special space used to understand the geometry of surfaces. Lastly, the project has provided a general framework in which further theories can be addressed as being different or the same. This is the theory of Feynman categories. It allows to compare the different theories by analyzing their DNA in a metaphorical sense. One of these fingerprints are Hopf algebras, which arose in the project.
