Ancient Solutions to geometric flows are solutions which have existed for an infinite amount of time in the past. They arise naturally in the blow-up analysis of singularity formation, and typically have nice rigidity properties. In this talk, I will present a survey of results relating to the classification of ancient solutions to different geometric flows with a specific focus on the Ricci flow and the mean curvature flow.

Bourgain's slicing problem, also known as the hyperplane conjecture, ask the following question. Is there a universal constant C such that for any n and any centrally symmetric convex body in R^n of volume one there exists a hyperplane whose cross-section with the convex body has area at least C. This conjecture has been a powerful source of motivation for a lot of progress in convex geometry. He will discuss the slicing problem and its relation to the isotropic constant of a convex body.