V. Tosatti - $C^{1,1}$ estimates for complex Monge-Ampère equations

I will describe work in progress, parts of which are joint with Marcelo Alves. Let L be a knot or link in the 3-sphere. I will explain how one can count minimal surfaces in hyperbolic 4-space

I will present a joint work with G. Carron and I. Mondello where we study Kato limit spaces. These are metric measure spaces obtained as Gromov-Hausdorff limits of smooth n-dimensional Riemannian

Over the last fifty years, the problem of finding sharp upper bounds for area-normalized Laplacian eigenvalues on closed surfaces has attracted the attention of many geometers, due in part to

The study of positive scalar curvature on noncompact manifolds has seen significant progress in the last few years. A major role has been played by Gromov's results and conjectures, and in

In this talk, we shall describe some topological rigidity and its relationship with positive scalar curvature. Precisely, we will present a proof that a complete contractible 3-manifold with

We propose a class of local definitions of weak lower scalar curvature bounds that is well defined for C0 metrics. We show the following: that our definitions are stable under greater-than-second

In this talk, I will discuss some recent developments on the topology of closed manifolds admitting Riemannian metrics of positive scalar curvature. In particular, we will prove if a closed PSC

The theory of non smooth spaces with lower Ricci Curvature bounds has undergone huge developments in the last thirty years. On the one hand the impetus came from Gromov’s precompactness theorem

The goal of the talk is to present a recent work in collaboration with Cavalletti (SISSA) on optimal transport in Lorentzian synthetic spaces. The aim is to set up a “Lorentzian analog” of the

We find a family of 3d steady gradient Ricci solitons that are flying wings. This verifies a conjecture by Hamilton. For a 3d flying wing, we show that the scalar curvature does not vanish at

We present a new compactness theory of Ricci flows. This theory states that any sequence of Ricci flows that is pointed in an appropriate sense, subsequentially converges to a synthetic flow.

Given a closed and oriented manifold M and Riemannian tensors g0, g1, ... on M that satisfy g0 < gj, vol(M, gj)→vol (M, g0) and diam(M, gj)≤D we will see that (M, gj) converges to (M, g0) in the