The third meeting took place on December 19th, 2018 and was coupled with the workshop Geometric analysis.
Expository talks:
Andrea Mondino (University of Warwick)
Some smooth applications of non-smooth Ricci curvature lower bounds
After a brief introduction to the synthetic notions of Ricci curvature lower bounds in terms of optimal transportation, due to Lott-Sturm-Villani, I will discuss some applications to smooth Riemannian manifolds. In particular I will discuss a quantitative Levy-Gromov inequality and an almost euclidean isoperimetric inequality motivated by the celebrated Perelman’s Pseudo-Locality Theorem for Ricci flow.
The slides of the two lectures: slides Lecture1, slides Lecture2.
Romain Petrides (Paris 7)
Maximisation of Steklov eigenvalues on a surface
Given a compact surface with a non-empty boundary, we ask the following question: is there a smooth Riemannian metric wich maximizes the k-th Steklov eigenvalue on this surface? We will explain the link between this problem and the existence question of minimal surfaces with free boundary in a ball. After stating the first eigenvalue results by Fraser and Schoen, we will address the harder question for higher eigenvalues and we will give an existence result of maximal metrics for metrics lying in a fixed conformal class.
Research talk:
Eleonora Di Nezza (Sorbonne Université)
Log-concavity of volume
In this talk we present a proof of the log-concavity property of total masses of positive currents on a given compact Kähler manifold, that was conjectured by Boucksom, Eyssidieux, Guedj and Zeriahi. The proof relies on the resolution of complex Monge-Ampère equations with prescribed singularities. As corollary we give an alternative proof of the Brunn-Minkowsky inequality for convex bodies. This is based on a joint work with Tamas Darvas and Chinh Lu.
