LONG COURSES

Perelman's work and the geometrization conjecture by L. Bessières, G. Besson, J. Lott, and S. Maillot

• Schedule: May 2008, from monday to thursday, 10h-12h15.
• Program:
  the aim of this course is to go as far as possible in the description of Perelman's proof of the geometrization conjecture.
• Prerequisites:
  the content of the basic courses given by Z. Djadli, S. Gallot and S. Maillot.
• References:
  • G. Perelman: The entropy formula for the Ricci flow and its geometric applications. arXiv link
  • G. Perelman: Ricci flow with surgery on three-manifolds. arXiv link
  • G. Perelman: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv link
  • B. Kleiner and J. Lott: Notes on Perelman's papers. The text can be downloaded from: http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.html
  • J. Morgan and G. Tian: Ricci Flow and the Poincaré conjecture, A.M.S. and Clay Mathematics Institute, Clay Mathematics Monographs, vol. 3.
  • L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti: Suites de métriques extraites du flot de Ricci sur les variétés asphériques de dimension 3 (English version to be posted soon). arXiv link
  • Many more references may be found at: http://www.math.lsa.umich.edu/~lott/ricciflow/perelman.html

Optimal transport of measures and Ricci curvature by D. Cordero-Erausquin

• Schedule: May 19th-31st, from monday to thursday, 16h-18h
• Program:
  1. Background: Brunn-Minkowksi (BM) and Prékopa-Leindler inequalities, and optimal transport in Euclidean space (the Brenier map).
  2. Optimal transport on Riemannian manifolds and its link with Jacobi fields and Ricci curvature.
  3. First approach to (BM) type inequalities on Riemannian manifolds : a general inequality.
  4. Second approach relying on lower bounds on the Ricci curvature.
  5. Convex functionals on the Wasserstein space and Ricci curvature.
• Prerequisites: the arithmetic-geometric inequality.
• Main reference:
  • C. Villani: Topics in Optimal Transportation, AMS, Graduate Studies in Mathematics vol. 58 (2003)
• Detailed bibliography:
  • C. Villani: Optimal Transport, old and new. Can be downloaded at http://www.umpa.ens-lyon.fr/~cvillani
  • D. Cordero-Erausquin, R. McCann and M. Schmuckenschläger: two papers which can be downloaded at http://www.institut.math.jussieu.fr/~cordero/recherche.html
  • R McCann: PhD dissertation thesis which can be downloaded at http://www.math.toronto.edu/mccann

Introduction to the Ricci flow by Z. Djadli

• Schedule: May 5th-17th, from monday to thursday, 13h30-15h45.
• Program:
  the lectures will focus mainly on the analytical aspects of the Ricci flow. The goal is to provide all the tools used in the study of the Ricci flow. More specifically, the program is:
  1. ''Variational'' aspect of the Ricci flow.
  2. Maximum principles.
  3. Short time existence for the Ricci flow.
  4. The Ricci flow on 3-dimensional manifolds of positive curvature.
  5. The Ricci flow on 4-dimensional manifolds of positive curvature.
  6. The Ricci flow under pinching.
  7. If time allows, we will also adress the higher dimensional case (mainly some recent results of Böhm-Wilking and Brendle-Schoen)
• Prerequisites: the lectures given by S. Gallot
• References: B. Chow and D. Knopf, The Ricci flow: an introduction. Mathematical Surveys and Monographs vol. 110, AMS.

Introduction to Ricci curvature by S. Gallot

• Schedule: April 14th-26th, from monday to thursday, 10h-12h15.
• Program:
  1. Riemannian metrics and geodesics (a quick review).
  2. Different curvatures and their geometric and analytic interpretations (a quick review). Examples.
  3. Comparison theorems (Hadamard-Cartan, Rauch, Bishop-Gromov and Toponogov). Some links between curvature and topology.
  4. Mostow's rigidity theorem.
  5. Different definitions of Einstein manifolds.
  6. Problems of existence, uniqueness or rigidity of Einstein manifolds.
  7. Some Riemannian compactness theorems. Applications to Einstein metrics.
• Prerequisites:
  • Basic Topology and Differential Calculus.
  • Manifolds, submanifolds, tangent bundle. Differential of a map between manifolds. Differential forms.
  • Flow of a vectorfield, geodesics.
  • Connections, curvature tensor and how to compute it in classical cases (a quick review will be done).
• Main References:
  • S. Gallot, D. Hulin and J. Lafontaine: Riemannian Geometry, Universitext, Springer (Third edition 2004).
  • A. Besse: Einstein Manifolds, Ergebnisse der Math., Springer, Berlin-Heidelberg, 1987.
• Detailed bibliography for the prerequisites:
  • V.I. Arnold: Méthodes mathématiques de la mécanique classique, Moscow, Ed. Mir 1974.
  • M. Berger and B. Gostiaux: Géométrie différentielle, variétés, courbes et surfaces PUF, 1987. English translation : Differential Geometry : Manifolds, Curves and Surfaces, GTM 115, Springer.
• Detailed bibliography for the lectures:
  • T. Sakai: Riemannian Geometry, American Math. Soc., Providence, Rhode Island (1996).
  • J. Milnor: Morse theory, Princeton University Press (1963), pp. 43--123.
  • M. Do Carmo: Riemannian Geometry, Birkhaüser, Basel 1992.
  • F. Warner: Foundations of differentiable manifolds and Lie groups, Scott et Foresman, Greenville, Ill, 1971.
  • J. Cheeger and D. Ebin: Comparison theorems in Riemannian geometry, North Holland, 1975.

Topology and geometry of 3-manifolds by S. Maillot

• Schedule: April 14th-26th, from monday to thursday, 13h30-15h45.
• Program:
  1. Topology of 3-manifolds and the geometrization program
     • Prime decomposition of 3-manifolds
     • Seifert manifolds; graph manifolds
     • Torus splitting; statement(s) of the geometrization conjecture
     • The Loop and Sphere Theorems and their consequences
  2. Introduction to hyperbolic geometry
     • Hyperbolic space; classification of isometries
     • Margulis Lemma
     • Mostow rigidity
     • Finite volume hyperbolic 3-manifolds
• Prerequisites:
  • Basic algebraic topology: fundamental group, homology groups.
  • Basic Riemannian geometry: Riemannian metric, sectional curvature, geodesics
• Main references for part 1:
  • P. Scott: The geometries of 3-manifolds. Bull. London Math. Soc. 15 (1983), no. 5, 401--487.
  • M. Boileau, S. Maillot and J. Porti: Three-dimensional orbifolds and their geometric structures. Panoramas et Syntheses 15. SMF, Paris, 2003. Chapters 1--3.
• Main reference for part 2:
  • R. Benedetti and C. Petronio: Lectures on hyperbolic geometry. Universitext. Springer-Verlag, Berlin, 1992.
• Detailed bibliography:
  • J. Hempel: 3-manifolds. Reprint of the 1976 original. AMS Chelsea Publishing, Providence, RI, 2004. xii+195 pp. ISBN: 0-8218-3695-1
  • W. Jaco: Lectures on three-manifold topology. CBMS Regional Conference Series in Mathematics, 43. American Mathematical Society, Providence, R.I., 1980. xii+251 pp. ISBN: 0-8218-1693-4
  • W.P. Thurston: Three-dimensional geometry and topology. Vol. 1. Edited by Silvio Levy. Princeton Mathematical Series, 35. Princeton University Press, Princeton, NJ, 1997. x+311 pp. ISBN: 0-691-08304-5
  • M. Boileau, S. Maillot and J. Porti: Three-dimensional orbifolds and their geometric structures. Panoramas et Syntheses 15. SMF, Paris, 2003. The other chapters.
  • J.G. Ratcliffe: Foundations of hyperbolic manifolds: Second edition. Graduate Texts in Mathematics, 149. Springer, New York, 2006. xii+779 pp
  • M. Kapovich: Hyperbolic manifolds and discrete groups. Progress in Mathematics, 183. Birkhauser Boston, Inc., Boston, MA, 2001
  • A. Hatcher: Notes on Basic 3-Manifold Topology. Freely downloadable from http://www.math.cornell.edu/~hatcher/3M/3Mdownloads.html
  • S. Matveev: Algorithmic Topology and Classification of 3-manifolds Springer. Algorithms and computation in Mathematics, volume 9.

Introduction to Kähler-Einstein geometry and Kähler-Ricci flow by N. Pali

• Schedule: May 19th-31st, from monday to thursday, 13h30-15h30.
• Program:
  1. Some backgrounds: curvature notions for the tangent bundle (connection between the hermition case and the Kähler case).
  2. Proof of the Aubin-Calabi-Yau theorem.
  3. Uniquenness of the Kähler-Einstein metrics over Einstein-Fano manifolds.
  4. Functionals of the Kähler-Einstein geometry.
  5. Tian's properness of the K-energy.
  6. Basic facts about the Kähler-Ricci flow.
• Prerequisites: chapters I and V of Jean-Pierre Demailly's book, Complex analytic and algebraic geometry, available at: http://www-fourier.ujf-grenoble.fr/~demailly
• References:
  • Th. Aubin: Nonlinear Analysis on Manifolds. Monge-Ampère Equations, Springer-Verlag, Berlin, New-York, 1982.
  • Chapter V in J.-P. Demailly's book: Complex analytic and algebraic geometry (see above).
  • G. Tian: Canonical metrics in Kähler Geometry, Birkhäuser, 2000.
  • S. Bando and T. Mabuchi: Uniqueness of Einstein Kähler metrics modulo connected group actions, Adv. Stud. in Pure Math. 10, 1987, 11-40.