Geometry and topology are fundamental disciplines of mathematics whose richness and vitality, evident throughout human history, reflect a deep link to our experience of the universe. They are a focal point of modern mathematics and indeed several domains of mathematics have recently shown a strong trend towards a geometrization of ideas and methods: two cases in point are mathematical physics and number theory.
Within this broad subject, the main areas of research of the group include the topological classification of 3-dimensional manifolds; quantization of Hitchin systems and the geometric Langlands program; classification of special Kähler metrics; the study of symplectic invariants, especially in dimension 4; non-linear partial differential equations in Riemannian geometry, convex geometry, and general relativity; and Hamiltonian dynamical systems.
Most of the researchers in the group are also membres of CIRGET, the Centre interuniversitaire de recherche en géométrie différentielle et topologie. The Center organizes scientific events as well as several weekly seminars.
Students interested in this program are expected to progress through three levels of courses.
Basic point-set topology, including connectedness, compactness, product spaces, separation axioms, metric spaces. The fundamental group and covering spaces. Simplicial complexes. Singular and simplicial homology. Part of the material of MATH 577 may be covered as well.
The aim of the course is to cover the basic theory of compact Riemann surfaces, or, alternately, one dimensional smooth projective curves over the complex numbers. The two terminologies reflect a very useful tension between analysis, on one hand, and geometry on the other, and the course aims to both understand these tensions and exploit them (a third, more algebraic aspect, of thinking of affine Riemann surfaces as field extensions of transcendence degree one of the complex numbers, will only be covered more tangentially). The course could at the same time serve as an introduction to some basic ideas of algebraic geometry, such as sheaves and their cohomology. The topics to be covered include the basic theory of coverings, analytic continuation, and algebraic curves; the elements of sheaves and their cohomology; differential forms on a surface, and their integration; the Dolbeault lemma, the Riemann-Roch theorem and Serre duality; Abel’s theorem, and the Jacobian of a curve. If time permits, I would hope to give some elements of the theory of theta-functions.
Lie groups, Lie algebras and their representations play an important role in many areas of pure and applied mathematics, ranging from differential geometry and geometric analysis to classical and quantum mechanics. Our goal is to give a motivated introduction to the representation theory of compact Lie groups and their Lie algebras, the essentials of which go back to the classical works of Elie Cartan and Hermann Weyl.
Groupe fondamental. Théorie des revêtements. Groupes d'homotopie de dimensions supérieures. Homologie singulière relative, homologie simpliciale, théorème d'approximation simpliciale. Relation entre le groupe fondamental et le premier groupe d'homologie. Théorème d'excision. Suite exacte de Mayer-Vietoris. Homologie des sphères, degré des applications entre sphères, applications. Théorème de Jordan-Brouwer. Complexes C.W. et discussion des théorèmes de base de la théorie de l'homotopie: théorème de Whithead, théorème de Hurewicz. Homologie cellulaire, caractéristique d'Euler. Le théorème de point fixe de Lefschetz.
Rappels de topologie et d’algèbre tensorielle. Variétés différentiables, espaces tangents, différentielle des fonctions, partitions de l’unité, tenseurs et formes différentielles, champs de vecteurs, théorème fondamental des EDO et dérivée de Lie. Intégration et théorème de Stokes, théorème de Fröbenius sur les distributions, cohomologie et théorème de DeRham. Métriques riemanniennes, connexions, dérivée covariante, géodésiques et courbure.
1. Differentiable manifolds:
Differentiable manifolds, tangent and cotangent spaces, smooth maps, submanifolds, tangent and cotangent bundles, implicit function theorem, partition of unity. Examples include real projective spaces, real Grassmannians and some classical matrix Lie groups.
2. Differential forms and de Rham cohomology:
Review of exterior algebra, the exterior differential and the definition of de Rham cohomology. The Poincaré Lemma and the homotopy invariance of de Rham cohomology. The Mayer-Vietoris sequence, computation of de Rham cohomology for spheres and real projective spaces. Finite-dimensionality results for manifolds with good covers, the Kunneth formula and the cohomology of tori. Integration of differential forms and Poincare duality on compact orientable manifolds.
3. An introduction to Riemannian geometry:
Existence of Riemannian metrics, isometric immersions, parallel transport and the Levi-Civita connection, the fundamental theorem of Riemannian geometry, Riemannian curvature. Geodesics, normal coordinates, geodesic completeness and the Hopf-Rinow Theorem.
Textbooks:
W. Boothby, An introduction to differentiable manifolds and Riemannian geometry, Academic Press.
R. Bott and L. Tu, Differential forms in algebraic topology, Springer.
A compact oriented surface is homeomorphic either to a sphere, a torus obtained from gluing opposite sides of a square, or to a surface obtained from gluing the sides of a hyperbolic polygon. The goal of the course is to indicate a similar classification in dimension 3. We will start following the notes of Hatcher, where two classical decompositions (prime and JSJ) of a 3-manifold into smaller pieces are described.
We will then follow the book of Thurston edited by Levy. Thurston conjectured that each piece carries one of the 8 model geometries. We will discuss each of the geometries, provide examples, and study their fundamental groups.
The most ubiquitous geometry is the hyperbolic geometry, on which we will focus at the end of the course. Some of the examples will be geometrically finite hyperbolic groups (obtained as surfaces from gluing finite polyhedra), and quasifuchsian groups. We will finish with a sketch of the proof of Mostow's rigidity theorem saying that for a compact hyperbolic 3-manifold, its fundamental group determines its isometry type.
Ce cours présentera une introduction générale à la théorie de groupes de Lie et la théorie d’espaces symétriques. Nous privilégierons les aspects géométriques de la théorie, ce qui rend le cours un complément naturel du cours donné par Yvan Saint-Aubin sur la théorie algébrique et combinatoire des algèbre de lie. Voici une courte description du contenu:
1. Rappels de Géométrie différentielle : Variétés différentiables; algèbre tensoriel et champs de tenseurs lisses sur une variété. Applications différentiables entre variétés; Connexions et dérivées covariantes; application exponentielle.
2. Groupes de Lie : Groupes de Lie et leurs algèbre de Lie ; exemples de base ; homomorphismes ; sous-groupes ; revêtements ; groupes de Lie simplement connexes ; l’application exponentielle ; homomorphismes continus ; sous-groupes fermés ; la correspondence de Lie. La représentation adjointe ; formes bilinéaires invariantes.
3. Algèbres de Lie : algèbres de Lie semi-simples et leur classification (résumé).
4. Espaces symétriques: Espaces localement symétriques par rapport à une connexion affine; groupes d’isométrie et espaces homogènes. Espaces globalement symétriques riemanniennes; Groupes de Lie compacts. Triplets de Lie.
Références : Les notes seront en principe suffisantes. Il y a une immense littérature sur le sujet. On pourra par exemple consulter les livres suivants :
Groupes de Lie :
1. F. Warner, Foundations of Differentiable Manifolds and Lie Groups. Springer-Verlag.
2. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, AMS.
3. S. Kobayshi and Numizu, Foundations of Differential Geometry, Vol. I, II, Interscience Pub.
4. C. Chevalley, Theory of Lie groups I, Princeton University Press.
Algèbres de Lie et la théorie des représentations :
5. J. Humphreys, Introduction to Lie Algebras and representation theory, Springer Verlag.
6. W. Fulton and J. Harris, Representation Theory, Springer-Verlag.
Anneaux de polynômes. Théorème de base et théorème des zéros de Hilbert, élimination classique. Dimension de Krull des anneaux. Localisation dans les anneaux. Variétés affines et projectives. Topologie de Zariski. Composantes irréductibles. Dimension de Krull. Schémas affines et projectifs. Paysage du local au global. Degré et multiplicité d'intersection. Théorème de Bezout pour les courbes planes et généralisations. Méthodes algorithmiques: bases de Gröbner, calcul avec les idéaux, calcul de dimension, de genre, de résolutions libres minimales à l'aide de logiciels.