New tools for gauge theory in dimensions 3 and 4

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High energy physicists believe that the Yang-Mills Equations model the behavior of quarks - the fundamental constituents of matter. The Yang-Mills Equations turn out to have a remarkably rich mathematical structure which, outside of direct applications to physics, enable the study of models of space-time inaccessible by other means. They even give tools for the study of the topological structure of DNA. The PI will continue his research on the Yang-Mills equations, and related ideas focusing on its 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 award provides support for students engaged in related research. The PI will 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. These theories have many applications to questions in low dimensional topology. The PI plans to put a key missing feature for Instanton Floer homology into place-- having a full “equivariant” version available for all three manifolds. This should enable new computations and give rise to new application. The PI will develop tools for a topological attack on some basic questions in Algebraic Geometry and Number theory related to conjectures of Land and Caporaso-Harris-Mazur. Yet another rather delicate version appears to have bearing on questions of tri-colorability of spacial graphs in particular appears likely to provide novel insight to the question of tri-colorability of planar graphs.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.