Grant: Discovery Grants Program - Individual

I work in the fields of geometry and topology. This roughly means that I study properties of shapes, though mathematicians prefer the term ""spaces"". The spaces of most interest to me are drawn from the world of mathematical physics. A rich source interesting spaces is gauge theory. Gauge theory is used in physics to model the fundamental forces of nature, like the electromagnetic and nuclear forces. The dynamics of these fields are controlled by the Yang-Mills action and fields which for the action is critical are called Yang-Mills fields. These critical fields combine to form a space, called the moduli space of Yang-Mills fields. These spaces tend to have rich geometric and topological structure, and properties of this structure often carry physical meaning. My research applies mathematical methods, like Morse theory and moment maps, to study the geometry and topology of Yang-Mills moduli spaces. I am particularly interested in the case when ""space-time"" is a two dimensional surface, which is of special relevance in the mathematics fields of representation theory and knot theory and in the physics of string theory.