Master Mathématiques et applications, Parcours : Mathématiques fondamentales

This master is oriented towards geometry, with a special focus on its interaction with group theory and Lie theory. Since the work of Poincaré, Klein and others, modern geometry hes been intimately related with the notions of manifold and group. For instance, the uniformisation theorem implies that every closed topological surface carries a metric of constant curvature. The isometry group of the universal cover of such a surface is a Lie group, hence the geometry of the surface can be recovered purely from group theory. More recently Gromov and others intriduced the study of countable groups as geometric objects. The broad goal of this course is to introduce and study in some depth these objects and to show how they relate with each others. 

The first semester is mostly dedicated to understanding the construction of homogeneous spaces and the study of their geometric properties, in particular their curvature. This will be achieved through two courses, one on Riemannian geometry which studies geometry in a broader context, and one on Lie groups and algebras with a more algebraic slant. In parallel there will be a course on geometric group theory. 

The second semester will be more focused on hyperbolic geometry: we will give constructions of Riemannian manifolds of negative curvature, and conversely study the existence and unicity of such metrics on smooth manifolds.

Courses

Riemannian geometry (Enseignants : Adrien Boulanger, Nader Yeganefar)

Abstract

The goal of this course is twofold. First, it aims at introducing the first concepts of Riemannian geometry, as metrics, connections and curvature. A special focus will be made on manifold of constant curvature. Secondly, it aims at giving some prerequisite to the other lectures, especially for 'Negatively curved manifolds and lattices in Lie groups' (second semester). This course is also related to 'Mostow's rigidity theorem' (second semester).

Contents

• First definitions, examples.
• Connections and curvature
• Jacobi field, constant curvature spaces.
• Curvature -1 metrics on a topological surface.

References

• Gallot, Hulin, Lafontaine, 'Riemannian geometry', Springer-Verlag, Universitext, third edition 2004.
• do Carmo, 'Riemannian geometry', Birkhäuser

Lie theory (Enseignants: Frédéric Palesi, Jean Raimbault)

Abstract

The goal of this course is to give a thorough introduction to the basic notions and results on Lie groups and algebras, with a view towards their application in differential geometry via Riemannian symmetric spaces. After a few words on classical matrix groups, which are the main examples of Lie groups, the course is roughly divided in two halves. The first describes the general setting and gives fundamental results, in particular the Lie algebra--Lie group correspondance. The second gives the basic notions of structure theory of semisimple real Lie groups; the classification will be described but no attempt at proving it will be made. A last section introduces symmetric spaces of non-compact type and their totally geodesic subspaces. 

Contents

First part : Lie theory

• Classical matrix groups (examples)
• Abstract Lie groups, closed subgroups and homogeneous spaces
• Lie algebra and Lie correspondence.

Second part: Structure of semisimple Lie groups

• Tori, unipotents groups. Unipotent radical, reductive and semisimple groups.
• Cartan and nilpotent subalgebras, semisimple Lie algebras
• Decompositions: Iwasawa, Cartan, root subspaces. (If time permits we will give a short introduction to root systems, a description of the classification of complex Lie groups and some remarks about the real case).
• Symmetric spaces of non-compact type and totally geodesic subspaces.

References

• S. Helgason, 'Differential Geometry, Lie Groups, and Symmetric Spaces', AMS 2001 (Chapters II--VI).
• T. Gelander, 'Locally symmetric spaces', lecture notes

Hyperbolic group theory (Enseignant : Thierry Coulbois)

Abstract

This lecture aims at introducing the geometric theory of groups with a special focus on Gromov hyperbolic spaces and groups. A special focus will be made on studying classical examples, with which we will try to illustrate the different combinatorial and geometric tools. The lecture will also try to show to the students a glimps of the current research on the topic. \\

 

 

Contents

• Words and free groups
• Cayley Graphs and word distance
• Gromov hyperbolic spaces
• Quasi-isometry
• Hyperbolic groups

References

• Ghys, de la Harpe, 'Sur les Groupes Hyperboliques d’après Mikhael Gromov' Springer-Verlag, Progress in Mathematics, 1990
• Mark, Margalit, 'Office Hours with a Geometric Group Theorist', Princeton University Press, 2017

Negatively curved manifolds and lattices in Lie groups (Enseignants : Luisa Paoluzzi, Jean Raimbault)

Abstract

This course is a continuation of the first semester courses on Lie theory and Riemannian geometry. It studies in some depth locally symmetric spaces, Riemannian manifolds constructed using Lie theory. The best known example are hyperbolic manifolds, which can be characterised as manifolds having constant negative sectional curvature. The first part gives a short introduction to hyperbolic geometry, mostly independent of the general theory in the first semester courses. It culminates in the construction of compact hyperbolic manifolds using the Poincaré polyhedron theorem. The second part is a short introduction to the general theory of lattices in semisimple Lie groups and locally symmetric spaces. We concentrate on two results. First we prove the existence of Zassenhaus neighbourhoods and explain its geometric applications such as the thick/thin decomposition. Then we introduce lattices in Lie groups, finishing with the arithmetic construction in arbitrary semisimple groups.

Contents

First part: hyperbolic manifolds

• Models of hyperbolic space and its subspaces 
• Classification of isometries (loxodromic, parabolic, elliptic)
• Trigonometry, polyhedra, construction of polyhedra in low dimensions
• Poincaré polyhedron theorem and applications to the construction of hyperbolic manifolds.

Second part: lattices in semisimple Lie groups

• Correspondence between locally symmetric spaces and discrete subgroups
• Zassenhaus neighbourhoods and geometric applications
• Lattices in semisimple Lie groups, unipotent elements and cocompactness, arithmetic construction (time permitting we will prove the co-compact case of the Borel--Harish-Chandra theorem).

References

•  Ratcliffe, 'Foundations of hyperbolic manifolds', Springer 2006
• Raghunathan, 'Discrete subgroups of Lie groups', Springer 1972
• Gelander, 'Locally symmetric spaces', lecture notes

Mostow rigidity

Abstract

These lectures will focus on a very specific feature of hyperbolic geometry in high dimension named ``Mostow rigidity'' after G.D. Mostow : 'a closed manifold of constant negative curvature and dimension at least three is determined by its fundamental group up to isometries'. The proof that will be presented will rely on the dynamics of the fundamental group acting on the sphere at infinity of hyperbolic space and on quasiconformal geometry.

Contents

•  Sketch of the proof
• Convergence actions
• Boundary maps
• Quasi-Moebius maps
• Invariant line fields

References

•  Mostow, 'Quasi-conformal mappings in n-space and the rigidity of hyperbolic space forms', Inst. Hautes Etudes Sci. Publ. Math. 34 (1968).
• Margulis, 'The isometry of closed manifolds of constant negative curvature with the same fundamental group', Soviet Math. Dokl. 11 (1970).
• Dennis Sullivan, 'Discrete conformal groups and measurable dynamics', Bull. Amer. Math. Soc. (N.S.) 6 (1982).