Perspectives on families of solutions to geometric partial differential equations.
Monday October 15, 2018 at 16:00, SERC 110A
The notion of a moduli space (of geometric objects, or solutions to a problem) goes back (at least) to the 19th century and captures the idea of "fully describing" the family of all solutions to a given equation. This plays a central role in algebraic geometry. However, moduli spaces of "geometric structures", originating in the work of Riemann, have come to play an increasingly important role in many other parts of mathematics and physics. While this subject is far too large to do much justice to in an hour, I will describe some interesting settings, both old and new, concerning geometric features of the totality of solutions to a given partial differential equation This will be a light-hearted survey with a lot of pictures.

Older and newer gauge theories on surfaces and four-manifolds. I
Tuesday October 16, 2018 at 16:00, Wachman 617
In this and the next lecture I will describe a collection of related results by several authors, obtained over the past several years, concerning the geometry of the so-called Hitchin moduli space, and its relationship with the emerging Kapustin-Witten theory. The Hitchin moduli space is a rather fundamental object, with deep connections to several fields, and I will describe some of its basic features, assuming you have never seen it before and have no idea why it might be interesting. This will lead to newer work on its asymptotic geometry. I will then connect this to Kapustin-Witten theory, conjectured by Witten to lead to a new analytic way to study certain knot invariants. The connection between these different fields is that solutions to the Hitchin equations serve as boundary conditions for this higher dimensional theory.

Older and newer gauge theories on surfaces and four-manifolds. II
Wednesday October 17, 2018 at 16:00, Wachman 617
In this lecture I continue with the description of a collection of related results by several authors, obtained over the past several years, concerning the geometry of the so-called Hitchin moduli space, and its relationship with the emerging Kapustin-Witten theory. The Hitchin moduli space is a rather fundamental object, with deep connections to several fields, and I will describe some of its basic features, assuming you have never seen it before and have no idea why it might be interesting. This will lead to newer work on its asymptotic geometry. I will then connect this to Kapustin-Witten theory, conjectured by Witten to lead to a new analytic way to study certain knot invariants. The connection between these different fields is that solutions to the Hitchin equations serve as boundary conditions for this higher dimensional theory.