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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 for manifolds with a lower bound on the Ricci curvature, and derive some important consequences : Bishop-Gromov inequality, Myers theorem, Cheeger-Gromoll splitting theorem. Then we will define the Gromov-Hausdorff distance between metric spaces which will allow us to consider limits of sequences of Riemannian manifolds, along the way we will prove Gromov’s precompactness theorem for sequences of manifolds with a Ricci lower bound. We will also see on examples what type of degeneration can occur when considering these « Ricci limit spaces », we will in particular encounter curvature blow up and volume collapsing. One of the major point in the study of these limit spaces is to understand which results on smooth manifolds with a Ricci lower bound carry on to the limit spaces, we will give an introduction to this topic by outlining the proof by Cheeger.

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 total curvature along the sequence. As a consequence, we obtain qualitative control on the topology of minimal hypersurfaces in terms of index and area. This is joint work with Ben Sharp.

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 contractible. This is a highly nontrivial theorem generalizing in particular the Schoenflies theorem and Cerf’s theorem.In this talk, I will explain how geometric analysis can be used to study the topology of M_n respectively some of its variants.I will start by sketching a proof of Smale’s theorem that M_1 is contractible. By a beautiful theorem of Grayson, the curve shortening flow deforms any closed embedded curve in the plane to a round circle, and thus gives a geometric analytic proof of the fact that M_1 is path-connected. By flowing, roughly speaking, all curves simultaneously, one can improve path-connectedness to contractibility.In the second half of my talk, I’ll describe recent work on space of smoothly embedded spheres in the 2-convex case, i.e. when the sum of the two smallest principal curvatures is positive. Our main theorem (joint with Buzano and Hershkovits) proves that this space is path-connected, for every n. The proof uses mean curvature flow with surgery.

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 hyperbolic sufaces. We will explain how the study of ideal polyhedra in anti-de Sitter space leads to an answer to a question of Steiner (1832) on the combinatorics of polyhedra that can be inscribed in a quadric. Joint work with Jeff Danciger and Sara Maloni.

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 discuss these results and the more general conjecture which states that this result holds for quasi-isometries between negatively curved manifolds and metric spaces.

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 bound for the area of a minimal surface in a hyperbolic 3-manifold.

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? Among other things we will outline a proof that the figure-8 knot complement and its sister are the 1-cusped manifolds with minimal maximal cusp volume.

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 asymptotics of filling problems tell us about the geometry of surfaces in groups and spaces. What do minimal and nearly minimal surfaces look like in different spaces, and how is the geometry of surfaces related to the geometry of the ambient space? Our main examples will arise from geometric group theory, including nilpotent groups and symmetric spaces.

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 for manifolds with a lower bound on the Ricci curvature, and derive some important consequences : Bishop-Gromov inequality, Myers theorem, Cheeger-Gromoll splitting theorem. Then we will define the Gromov-Hausdorff distance between metric spaces which will allow us to consider limits of sequences of Riemannian manifolds, along the way we will prove Gromov’s precompactness theorem for sequences of manifolds with a Ricci lower bound. We will also see on examples what type of degeneration can occur when considering these « Ricci limit spaces », we will in particular encounter curvature blow up and volume collapsing. One of the major point in the study of these limit spaces is to understand which results on smooth manifolds with a Ricci lower bound carry on to the limit spaces, we will give an introduction to this topic by outlining the proof by Cheeger and Colding of the splitting theorem for limit spaces.

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 for manifolds with a lower bound on the Ricci curvature, and derive some important consequences : Bishop-Gromov inequality, Myers theorem, Cheeger-Gromoll splitting theorem. Then we will define the Gromov-Hausdorff distance between metric spaces which will allow us to consider limits of sequences of Riemannian manifolds, along the way we will prove Gromov’s precompactness theorem for sequences of manifolds with a Ricci lower bound. We will also see on examples what type of degeneration can occur when considering these « Ricci limit spaces », we will in particular encounter curvature blow up and volume collapsing. One of the major point in the study of these limit spaces is to understand which results on smooth manifolds with a Ricci lower bound carry on to the limit spaces, we will give an introduction to this topic by outlining the proof by Cheeger and Colding of the splitting theorem for limit spaces.

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 decomposition is a free group, and we describe interesting examples in this situation. In certain cases we show that the type of Kleinian group is determined by the topology of its group boundary. We conjecture that this is not the case in general. We also determine the homeomorphism types of planar boundaries that can occur. This is joint work in progress with Peter Haissinsky and Luisa Paoluzzi.

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 for manifolds with a lower bound on the Ricci curvature, and derive some important consequences : Bishop-Gromov inequality, Myers theorem, Cheeger-Gromoll splitting theorem. Then we will define the Gromov-Hausdorff distance between metric spaces which will allow us to consider limits of sequences of Riemannian manifolds, along the way we will prove Gromov’s precompactness theorem for sequences of manifolds with a Ricci lower bound. We will also see on examples what type of degeneration can occur when considering these « Ricci limit spaces », we will in particular encounter curvature blow up and volume collapsing. One of the major point in the study of these limit spaces is to understand which results on smooth manifolds with a Ricci lower bound carry on to the limit spaces, we will give an introduction to this topic by outlining the proof by Cheeger.

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 these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.

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 these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.

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 homogeneous spaces by C. Jordan, L. Bieberbach, H. J. Zassenhaus, D. Kazhdan-G. Margulis, it has been extended to the Riemannian setting by G. Margulis for manifolds of non positive curvature. The goal of these lectures is to present the recent work of V. Kapovitch and B. Wilking who gave a sharp version of the Margulis lemma under the assumption that the Ricci curvature is bounded below. Their method uses the structure of « Ricci limit spaces » explained by T. Richard during his lectures.

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 theorem on circle packing polyhedral metrics and Marden-Rodin’s proof- Thurston’s conjecture on the convergence of circle packings to the Riemann mapping and its solution by Rodin-Sullivan- Finite dimensional variational principles associated to polyhedral surfaces- A discrete conformal equivalence of polyhedral surfaces and its relationship to convex polyhedra in hyperbolic 3-space- A discrete uniformization theorem for compact polyhedral surfaces- Convergence of discrete conformality and some open problems

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 these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.

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 asymptotics of filling problems tell us about the geometry of surfaces in groups and spaces. What do minimal and nearly minimal surfaces look like in different spaces, and how is the geometry of surfaces related to the geometry of the ambient space? Our main examples will arise from geometric group theory, including nilpotent groups and symmetric spaces.

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 theorem on circle packing polyhedral metrics and Marden-Rodin’s proof- Thurston’s conjecture on the convergence of circle packings to the Riemann mapping and its solution by Rodin-Sullivan- Finite dimensional variational principles associated to polyhedral surfaces- A discrete conformal equivalence of polyhedral surfaces and its relationship to convex polyhedra in hyperbolic 3-space- A discrete uniformization theorem for compact polyhedral surfaces- Convergence of discrete conformality and some open problems

(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 differentiation that they used to show that it does not embed nicely into $L_1$. This notion is based on surfaces in $\mathbb{H}$, and in this talk, we will describe new techniques that let us quantify the "roughness" of such surfaces, find sharp bounds on the distortion of embeddings of $\mathbb{H}$, and estimate the accuracy of an approximate algorithm for the Sparsest Cut Problem.

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 evolution. As we will discuss in this talk, horizontal curves are very well behaved even if the underlying conformal structures degenerate in moduli space as t to T. We can describe where the metrics will have essentially settled down to the limit by time t T as opposed to regions on which the metric still has to do an infinite amount of stretching. This quantified information is essential in applications and allows us to prove a "no-loss-of-topology" result at finite time singularities of Teichmüller harmonic map flow which, combined with earlier work, yields that this geometric flow decomposes every map into a collection of branched minimal immersions and curves.This is joint work with Peter Topping

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 asymptotics of filling problems tell us about the geometry of surfaces in groups and spaces. What do minimal and nearly minimal surfaces look like in different spaces, and how is the geometry of surfaces related to the geometry of the ambient space? Our main examples will arise from geometric group theory, including nilpotent groups and symmetric spaces.

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 conformal maps when the triangulations are sufficiently fine chosen. We will also discuss the relationship between the discrete uniformization theorem and convex polyhedral surfaces in the hyperbolic 3-space. This is a joint work with J. Sun and T. Wu.

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 for manifolds with a lower bound on the Ricci curvature, and derive some important consequences : Bishop-Gromov inequality, Myers theorem, Cheeger-Gromoll splitting theorem. Then we will define the Gromov-Hausdorff distance between metric spaces which will allow us to consider limits of sequences of Riemannian manifolds, along the way we will prove Gromov’s precompactness theorem for sequences of manifolds with a Ricci lower bound. We will also see on examples what type of degeneration can occur when considering these « Ricci limit spaces », we will in particular encounter curvature blow up and volume collapsing. One of the major point in the study of these limit spaces is to understand which results on smooth manifolds with a Ricci lower bound carry on to the limit spaces, we will give an introduction to this topic by outlining the proof by Cheeger.

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 homogeneous spaces by C. Jordan, L. Bieberbach, H. J. Zassenhaus, D. Kazhdan-G. Margulis, it has been extended to the Riemannian setting by G. Margulis for manifolds of non positive curvature. The goal of these lectures is to present the recent work of V. Kapovitch and B. Wilking who gave a sharp version of the Margulis lemma under the assumption that the Ricci curvature is bounded below. Their method uses the structure of « Ricci limit spaces » explained by T. Richard during his lectures.

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 theorem on circle packing polyhedral metrics and Marden-Rodin’s proof- Thurston’s conjecture on the convergence of circle packings to the Riemann mapping and its solution by Rodin-Sullivan- Finite dimensional variational principles associated to polyhedral surfaces- A discrete conformal equivalence of polyhedral surfaces and its relationship to convex polyhedra in hyperbolic 3-space- A discrete uniformization theorem for compact polyhedral surfaces- Convergence of discrete conformality and some open problems

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 these lectures is to give some basic definitions about minimal surface theory and present some results about the construction of minimal surfaces in Riemannian 3-manifolds.

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 theorem on circle packing polyhedral metrics and Marden-Rodin’s proof- Thurston’s conjecture on the convergence of circle packings to the Riemann mapping and its solution by Rodin-Sullivan- Finite dimensional variational principles associated to polyhedral surfaces- A discrete conformal equivalence of polyhedral surfaces and its relationship to convex polyhedra in hyperbolic 3-space- A discrete uniformization theorem for compact polyhedral surfaces- Convergence of discrete conformality and some open problems

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 arguments in relation to the volume of Riemannian manifolds. Dealing with the case of surfaces first, we will focus our attention on generalisations in higher dimension and present new estimates obtained in a work in progress.

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 type of the curve, possibly changing the notion of length, or if one counts types of curves. Here, two curves are of the same type if they differ by a mapping class. I will describe some results in these directions.

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 map is unique and is called the Riemann mapping. In the 90's, Ken Stephenson, motivated by a circle packing approximation scheme suggested by Thurston (and first proved to converge by Rodin-Sullivan), predicted that the Riemann Mapping may be approximated by a different scheme, i.e., by a sequence of finite networks endowed with particular choices of conductance constants. These networks are naturally defined in terms of the contact graph of any circle packing.We will affirm Stephenson's Conjecture in a greater generality.

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 homogeneous spaces by C. Jordan, L. Bieberbach, H. J. Zassenhaus, D. Kazhdan-G. Margulis, it has been extended to the Riemannian setting by G. Margulis for manifolds of non positive curvature. The goal of these lectures is to present the recent work of V. Kapovitch and B. Wilking who gave a sharp version of the Margulis lemma under the assumption that the Ricci curvature is bounded below. Their method uses the structure of « Ricci limit spaces » explained by T. Richard during his lectures.

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 asymptotics of filling problems tell us about the geometry of surfaces in groups and spaces. What do minimal and nearly minimal surfaces look like in different spaces, and how is the geometry of surfaces related to the geometry of the ambient space? Our main examples will arise from geometric group theory, including nilpotent groups and symmetric spaces.

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 result is a construction of an infinitely differentiable convex plane domain whose Minkowski sum with any generically rotated copy of itself is not five times differentiable.

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 complex structure also admit scalar-flat Kahler ALE metrics. A local moduli space of scalar-flat Kahler ALE metrics can then be constructed, which is universal up to small diffeomorphisms. I will also discuss a formula for the dimension of the local moduli space in the case of a scalar-flat Kahler ALE surface which deforms to a minimal resolution of an isolated quotient singularity. This is joint work with Jiyuan Han.

Percolation of random nodal lines

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 homogeneous spaces by C. Jordan, L. Bieberbach, H. J. Zassenhaus, D. Kazhdan-G. Margulis, it has been extended to the Riemannian setting by G. Margulis for manifolds of non positive curvature. The goal of these lectures is to present the recent work of V. Kapovitch and B. Wilking who gave a sharp version of the Margulis lemma under the assumption that the Ricci curvature is bounded below. Their method uses the structure of « Ricci limit spaces » explained by T. Richard during his lectures.

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 homogeneous spaces by C. Jordan, L. Bieberbach, H. J. Zassenhaus, D. Kazhdan-G. Margulis, it has been extended to the Riemannian setting by G. Margulis for manifolds of non positive curvature. The goal of these lectures is to present the recent work of V. Kapovitch and B. Wilking who gave a sharp version of the Margulis lemma under the assumption that the Ricci curvature is bounded below. Their method uses the structure of « Ricci limit spaces » explained by T. Richard during his lectures.

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-Hausdorff convergence for sequences of open 3-manifolds due to Brock-Bromberg one obtains explicit upper bounds for the volume of a mapping torus in terms of the translation distance of the monodromy on Teichmueller space. We will explain Brock-Bromberg's approach to the Thurston's uniformization theorem for hyperbolic manifolds which are mapping tori. In particular the "coarse geometry" of the convex core of a quasi fuchsian manifold.

Le lieu des zéros d'un polynôme à coefficients réels de n variables est (en général) une hypersurface de l'espace affine réel de dimension n dont la topologie dépend du choix du polynôme.À quelle topologie s'attendre lorsque le polynôme est choisi au hasard ? J'expliquerai les principaux résultats que l'on a pu établir avec Damien Gayet sur cette question.