2021

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

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

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 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

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 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

We study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result

Mean curvature flow is the gradient flow of the area functional and constitutes a natural geometric heat equation on the space of hypersurfaces in an ambient Riemannian manifold. It is believed,

Given a positive epsilon, a closed Einstein 4-manifold admits a natural thick-thin decomposition. I will explain how, for any delta, one can modify the Einstein metric to a bounded sectional

In the talk I would like to discuss some  statements and questions about convex subsets and convex hulls in generic Riemannian manifolds of dimension at least 3. The statements, obtained jointly

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

We introduce various notions of convergence of Riemannian manifolds and metric spaces.  We then survey results and open questions concerning the limits of sequences of Riemannian manifolds with

I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between

I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between

I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between

I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief

It has become clear in recent years that to understand mean curvature flow through singularities it is essential to work with weak solutions to mean curvature flow. We will give a brief

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the

We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who

The idea of compactifying the space of Riemannian manifolds satisfying Ricci curvature lower bounds goes back to Gromov in the '80ies and was pushed by Cheeger-Colding in the ‘90ies, who

This is a series of lectures on Bishop--Gromov's type inequalities adapted to metric spaces. We consider the case of Gromov-hyperbolic spaces and draw consequences of these inequalities such as

This is a series of lectures on Bishop--Gromov's type inequalities adapted to metric spaces. We consider the case of Gromov-hyperbolic spaces and draw consequences of these inequalities such as

This is a series of lectures on Bishop--Gromov's type inequalities adapted to metric spaces. We consider the case of Gromov-hyperbolic spaces and draw consequences of these inequalities such as

This is a series of lectures on Bishop--Gromov's type inequalities adapted to metric spaces. We consider the case of Gromov-hyperbolic spaces and draw consequences of these inequalities such as

The purpose of this course is to introduce the synthetic treatment of sectional curvature upper-bound on metric spaces. The basic idea of A.D. Alexandrov was to characterize the curvature bounds

The purpose of this course is to introduce the synthetic treatment of sectional curvature upper-bound on metric spaces. The basic idea of A.D. Alexandrov was to characterize the curvature bounds

The purpose of this course is to introduce the synthetic treatment of sectional curvature upper-bound on metric spaces. The basic idea of A.D. Alexandrov was to characterize the curvature bounds

The purpose of this course is to introduce the synthetic treatment of sectional curvature upper-bound on metric spaces. The basic idea of A.D. Alexandrov was to characterize the curvature bounds

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This

I will present recent work with Kleiner in which we verify two topological conjectures using Ricci flow. First, we classify the homotopy type of every 3-dimensional spherical space form. This