Course information

The starting point of this course is the classical isoperimetric inequality, dating back to antiquity: in the plane, of all shapes of a given perimeter, the disc has the largest area. Isoperimetric inequalities are prevalent in many familiar settings e.g. Euclidean and Gaussian spaces and the sphere, and they give rise to the surprising concentration of measure phenomenon: a Lipschitz function on a high dimensional sphere or Gaussian space concentrates around its mean. These are part of a large family of important geometric and functional inequalities including Brunn-Minkowski, Prekopa-Leindler and Sobolev, with applications to other fields such as PDEs. An active research topic is the investigation of equality or near equality cases of these inequalities.  We will apply several techniques to study these inequalities: symmetrization processes, convex localization and optimal transport.  

 

The second part of the course focuses on high dimensional phenomena, in particular, the distribution of mass in high dimensional convex bodies. We will mention two recent breakthroughs: the resolution of Bourgain's slicing conjecture - every convex body of unit volume has a slice with constant volume, and of the thin shell conjecture. We will derive a  central limit theorem for convex bodies: in most directions the distribution of mass is approximately Gaussian. We will prove Dvoretzky's celebrated theorem: all Banach spaces have an almost Euclidean subspace of high dimension; equivalently, all convex bodies have almost ellipsoidal sections of quite high dimension. We will explore the notion of convex duality culminating with Blaschke-Santalo and Bourgain-Milman inequalities. Throughout, we will develop probabilistic techniques.