# Matthew Gursky

## University of Notre Dame

### Singular solutions to some PDEs arising in conformal geometry

I will describe some aspects of the k-Yamabe problem, a fully nonlinear version of the Yamabe problem, emphasizing the negative case. I will begin with a quick overview of the regularity theory in the ‘positive’ case, and contrast this with how little is known in the negative case. (This is a remarkable difference with the classical Yamabe problem, in which the negative case is easy). At the end of the talk I will discuss some ongoing work with M. Musso on constructing solutions with isolated singularities.