Instantons, Low Dimensional Topology and Knotted Graphs

High energy physicists believe that the Yang-Mills Equations model the behavior of quarks. The Yang-Mills Equations turn out to have a remarkably rich mathematical structure. These equations enable the study of models of space-time inaccessible by other means and in addition give tools for the study of the topological structure of DNA. The PI will continue his research on the Yang-Mills equations and their connections to many different parts of mathematics, the study of nonlinear partial differential equations, low dimensional topology, algebraic geometry, representation theory and graph theory.

The PI with study Floer homology invariants for three manifolds and knotted graphs in them. In particular with Peter Kronheimer, the PI is studying a family of Floer homology theories built from connections with a prescribed singularity along knotted graphs in three manifolds. Certain versions of these theories appear to be related to Khovanov-Rozansky SL(N)-homology for foams and the relations will be explored and elucidated. Other versions point to theories more general than Khovanov-Rozansky homology. Yet another rather delicate version appears to have bearing on questions of tri-colorability of spacial graphs, and in particular appears likely to provide novel insight to the question of tri-colorability of planar graphs.