Program of courses

1st Week (June 13 to 17, 2016)

Title : An introduction to open 3-manifolds

Abstract : 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-manifolds can now beconsidered to be reasonably well-understood.By contrast, noncompact 3-manifolds remainmuch more mysterious. There is a series of examples,beginning with work of L. Antoine and J. H. C. Whitehead,which show that open 3-manifolds can exhibit wildbehavior at infinity. No comprehensive structure theoryanalogous to geometrization à la Thurston is currently availablefor these objects

In these lectures, we will focus on two aspects of the subject:

(1) constructing interesting examples, and

(2) finding sufficientconditions that rule out exotic examples, in particular inconnection with Riemannian geometry.


Title : Some aspects of minimal surface theory

Abstract : 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.


Title : Lower bounds on Ricci curvature, with a glimpse on limit spaces

Abstract : 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.

Some notes are available here

2nd Week (June 20 to 24, 2016) (tentative program, possible changes)

Title : The Margulis lemma, old and new

Abstract : 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.


    •   Feng Luo (Rutgers University)

Title : An introduction to discrete conformal geometry of polyhedral surfaces

Abstract : 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.

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


    •     Robert Young (New-York University)

Title : Quantitative geometry and filling problems

Abstract : 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.






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