The PI will continue his investigations into Floer homology invariants for three manifolds and knots in them. One particular focus will be joint work with Peter Kronheimer concerning a Floer homology theory built from connections with a prescribed singularity along a link in a three manifold. When specialized to links in the three sphere this will invariant appears to be closely related to the Khovanov homology of the link. Kronheimer and the PI intend to further explore this theory and understand more clearly its relation to Khovanov homology. With a student, Maksim Lypyanskiy, the PI intends to further explore a new foundation for Floer homology based on a theory of semi-infinite dimensional cycles. This appears to lead to a drastic simplification of theory. With another student, Ben Mares, the PI hopes to begin to put into place the mathematical theory of the N=4 supersymmetric Yang-Mills equations. Finally with third student, Timothy Nguyen, he hopes to understand the hyperbolic Yang-Mills equations especially in dimensions 3+1.
The models physicists have constructed for understanding particles in high energy physics, like that Yang-Mills and Seiberg-Witten equations, have proved to be a source for many exciting developments in mathematics as well. Most notably these models figure crucially in understanding phenomena in dimensions three and four that still seem out of reach by other methods. The PI has been a leader in the mathematical developments of these models and will continue research in a number of different directions on these models. One project with Peter Kronheimer seeks to relate two quite different seeming models. Another seeks to development new mathematical foundations for exploration of these models and will hopefully lead to a great simplification in the rigorous mathematical construction of these models. Finally with some students Mrowka will explore new models in hopes that they too will have interesting mathematical consequences.
The PI's project with Kronheimer combines many branches of mathematics, physics and has some impact on questions about the topological properties of DNA. The most important result coming from this grant is a proof that Khovanov homology reliably can distinguish an honest knot from a very complicated and yet unknotted loop. Khovanov homology is a subtle knot invariant coming from representation theory, but one that is computable in a rather direct manner. This project required the developing a deep understanding of solutions to the Yang-Mills equations with singularities along the loop under study. The Yang-Mills equations are believed by high energy physicists to govern the behavior of quarks. Potentially, the new understanding gained from this project will lead to new insights applicable to the study of quarks. Thanks to an offshot of this work, a popular attack on finding a counterexample to the four-dimensional smooth Poincare conjecture was shown to be doomed to failure. The four-dimensional smooth Poincare conjecture is one of the major un solved problems in mathematics. It can be viewed a questions about wherther the simplest possible models of space time are unique. This project originally proposed a study of the Yang-Mills equations with singularities along knotted loops. Kronheimer and the PI realized toward the end of the award that the Yang-Mills equations behave well even if they are allowed to have singularities along (knotted) trivalent graphs. This allows exploring a connection with Khovanov-Rozansky homology, a subtle invariant of knots or more generally knotted trivalents graphs having its origins in Representation theory.
