I am going to introduce you to the idea of studying manifolds (in this talk gravitational instantons) by looking at the level sets of functions. You would have seen something like this in your first year topology/geometry course as well as a question on every geometry qualifying exam. Except now instead of the ambient manifolds being R^n, they are infinite dimensional, and instead of polynomials, the functions are differential equations. 

 

The functions we will be looking at, the Self-Duality Equations, come from considering the minimum of the Yang-Mills Functional. Yang-Mills Theory is a physical theory seeking to describe the electroweak and strong forces. 

 

This talk would be of interest to students considering research in mathematical physics and/or differential geometry.