Yair Shenfeld, MIT

Abstract: Water droplets are round because the ball is the unique shape which minimizes surface area for a given fixed volume. But what if we want to minimize surface area  where in addition to fixing the volume we also fix the average width of the shape? This question turns out to be a very special case of long-standing conjectures at the heart of convex geometry. In the first part of the talk I will explain these conjectures and the significant progress that we made towards their resolutions. In the second part of the talk I will explain how these  geometric problems can provide information on problems in combinatorics, which led us to the discovery of some surprising results in the theory of partially ordered sets. Along the way we will encounter objects and ideas from spectral theory and the theory of log-concave sequences.