2016

Plateau's problem asks whether there exists a minimal surface with a given boundary in Euclidean space. In this course, we will study related problems in broader classes of spaces and ask what the

Plateau's problem asks whether there exists a minimal surface with a given boundary in Euclidean space. In this course, we will study related problems in broader classes of spaces and ask what the

Plateau's problem asks whether there exists a minimal surface with a given boundary in Euclidean space. In this course, we will study related problems in broader classes of spaces and ask what the

Plateau's problem asks whether there exists a minimal surface with a given boundary in Euclidean space. In this course, we will study related problems in broader classes of spaces and ask what the

Plateau's problem asks whether there exists a minimal surface with a given boundary in Euclidean space. In this course, we will study related problems in broader classes of spaces and ask what the

W. Thurston's geometrization program has lead to manyoutstanding results in 3-manifold theory. Thanks to worksof G. Perelman, J. Kahn and V. Markovic, D. Wise, and I. Agol among others, compact 3

W. Thurston's geometrization program has lead to manyoutstanding results in 3-manifold theory. Thanks to worksof G. Perelman, J. Kahn and V. Markovic, D. Wise, and I. Agol among others, compact 3

W. Thurston's geometrization program has lead to manyoutstanding results in 3-manifold theory. Thanks to worksof G. Perelman, J. Kahn and V. Markovic, D. Wise, and I. Agol among others, compact 3

W. Thurston's geometrization program has lead to manyoutstanding results in 3-manifold theory. Thanks to worksof G. Perelman, J. Kahn and V. Markovic, D. Wise, and I. Agol among others, compact 3

The goal of the course is to introduce some of the recent developments on discrete conformal geometry of polyhedral surfaces. We plan to cover the following topics. - The Andreev-Koebe-Thurston

The goal of the course is to introduce some of the recent developments on discrete conformal geometry of polyhedral surfaces. We plan to cover the following topics. - The Andreev-Koebe-Thurston

The goal of the course is to introduce some of the recent developments on discrete conformal geometry of polyhedral surfaces. We plan to cover the following topics. - The Andreev-Koebe-Thurston

The goal of the course is to introduce some of the recent developments on discrete conformal geometry of polyhedral surfaces. We plan to cover the following topics. - The Andreev-Koebe-Thurston

The goal of the course is to introduce some of the recent developments on discrete conformal geometry of polyhedral surfaces. We plan to cover the following topics. - The Andreev-Koebe-Thurston

The Margulis lemma describes the structure of the group generated by small loops in the fundamental group of a Riemannian manifold, thus giving a picture of its local topology. Originally stated for

The Margulis lemma describes the structure of the group generated by small loops in the fundamental group of a Riemannian manifold, thus giving a picture of its local topology. Originally stated for

The Margulis lemma describes the structure of the group generated by small loops in the fundamental group of a Riemannian manifold, thus giving a picture of its local topology. Originally stated for

The Margulis lemma describes the structure of the group generated by small loops in the fundamental group of a Riemannian manifold, thus giving a picture of its local topology. Originally stated for

The Margulis lemma describes the structure of the group generated by small loops in the fundamental group of a Riemannian manifold, thus giving a picture of its local topology. Originally stated for

Schlenker and Krasnov have established a remarkable Schlaffli-type formula for the (renormalized) volume of a quasi-Fuchsian manifold. Using this, some classical results in complex analysis and Gromov

In this talk, I will present a joint work with H. Rosenberg where we give a characterization of the minimal hypersurface of least area in any Riemannian manifold. As a consequence, we give a lower

In 2012, Colding-Ilmanen-Minicozzi-White conjectured that the entropy of any closed surface in R^3 is at least that of the self-shrinking two-sphere. I will explain joint work with X. Zhou where we

The Riemann Mapping Theorem asserts that any simply connected planar domain which is not the whole of it, can be mapped by a conformal homeomorphism onto the open unit disk. After normalization, this

With Robert Haraway, Robert Meyerhoff, Nathaniel Thurston and Andrew Yarmola. We address the following question. What are all the 1-cusped hyperbolic 3-manifolds whose maximal cusps have low volume? 

We study sequences of closed minimal hypersurfaces (in closed Riemannian manifolds) that have uniformly bounded index and area. In particular, we develop a bubbling result which yields a bound on the

(joint work with Assaf Naor) The Heisenberg group $\mathbb{H}$ is a sub-Riemannian manifold that is unusually difficult to embed in $\mathbb{R}^n$. Cheeger and Kleiner introduced a new notion of

Manifolds with almost nonnegative curvature operator

We study the problem of classifying Kleinian groups via the topology of their limit sets. In particular, we are interested in one-ended convex-cocompact Kleinian groups where each piece in the JSJ

I will discuss a Kuranishi-type theorem for deformations of complex structure on ALE Kahler surfaces, which will be used to prove that for any scalar-flat Kahler ALE surface, all small deformations of

An old theorem of Huber asserts that the number of closed geodesics of length at most L on a hyperbolic surface is asymptotic to $\frac{e^L}L$. However, things are less clear if one either fixes the

Anti-de Sitter geometry is a Lorentzian analog of hyperbolic geometry. In the last 25 years a number of connections have emerged between 3-dimensional anti-de Sitter geometry and the geometry of

Sweep-out techniques in geometry and topology have recently received a great deal of attention, leading to major breakthroughs. In this talk, we will present several width estimates relying on min-max

On closed surfaces there are three basic ways to evolve a metric, by conformal change, by pull-back with diffeomorphisms and by horizontal curves, moving orthogonally to the first two types of

Very recently, Markovic, Lemm-Markovic and Benoist-Hulin, established the existence of a harmonic mapping in the homotopy class of an arbitrary quasi-isometry between rank 1 symmetric spaces. I will

Our recent joint work with D. Gu established a discrete version of the uniformization theorem for compact polyhedral surfaces.   In this talk, we prove that discrete uniformizaton maps converge to

We investigate the topology of the space of smoothly embedded n-spheres in R^{n+1}, i.e. the quotient space M_n:=Emb(S^n,R^{n+1})/Diff(S^n). By Hatcher’s proof of the Smale conjecture, M_2 is

I will discuss whether the Minkowski sum of two compact convex bodies can be made smoother by a generic rotation of one of them.  Here "generic" is understood in the sense of Baire category. The main

Percolation of random nodal lines

The goal of these lectures is to introduce some fundamental tools in the study of manifolds with a lower bound on Ricci curvature. We will first state and prove the laplacian comparison theorem

The goal of these lectures is to introduce some fundamental tools in the study of manifolds with a lower bound on Ricci curvature. We will first state and prove the laplacian comparison theorem

The goal of these lectures is to introduce some fundamental tools in the study of manifolds with a lower bound on Ricci curvature. We will first state and prove the laplacian comparison theorem

The goal of these lectures is to introduce some fundamental tools in the study of manifolds with a lower bound on Ricci curvature. We will first state and prove the laplacian comparison theorem

The goal of these lectures is to introduce some fundamental tools in the study of manifolds with a lower bound on Ricci curvature. We will first state and prove the laplacian comparison theorem

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of

In a Riemannian 3-manifold, minimal surfaces are critical points of the area functional and can be a useful tool to understand the geometry and the topology of the ambient manifold. The aim of