Title: Geometric analysis of (singular) 4-manifolds
Abstract: It's been known since the work of Freedman and Donaldson that not every simply connected topological 4-manifold admits a smooth structure and that in fact the intersection form of such manifolds must be special. Donaldson's (and Freedman's) work, through the work of Gompf's and Taubes, culminated in the proof of the existence of uncountably many inequivalent (exotic) smooth structure on R^4.
Since then, a lot of work has been done on studying the existence of exotic smith structure on compact 4-manifolds (finding one on the sphere would disprove the smooth Poincaré conjecture).
I will talk about the interplay between (exotic) smooth structures and the existence of solutions of geometric PDE's, such as non-singular solutions to the normalized Ricci flow. Time permitting, I will also discuss connections with quantum gravity.