Low Dimensional Topology and Gauge Theory

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

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.