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.
In this course I plan to cover some basic material of Kähler geometry, roughly in line with the first chapter of the book of Griffiths and Harris, or Claire Voisin’s book on Hodge theory and Complex algebraic geometry (book 1).
Material should include:
-Rudiments of function theory of several complex variables,
-Complex manifolds, de Rham and Dolbeault cohomology,
- Sheaf theory and cohomology theory,
-Kähler metrics, connections and curvature,
-Harmonic theory: the Hodge theorem and the Hodge decomposition,
-The Lefschetz decomposition.
Closing, if time allows, with some material on periods and Hodge structures, or maybe on Hyperkähler manifolds.
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.
This course will develop the basic elements of the theory of orderable groups and their applications to low-dimensional topology. In particular we will discuss the L-space conjecture which posits the equivalence between the left-orderability of the fundamental group of a 3-manifold and certain of its analytic and topological properties.
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.
Textbook: Allen Hatcher, Algebraic Topology.
Syllabus: CW-complexes, cellular approximation theorem. Homotopy groups, long exact sequence for a fiber bundle. Whitehead theorem. Freudenthal suspension theorem. Singular and cellular homology and cohomology. Hurewicz theorem. Mayer-Vietoris sequence. Universal coefficients theorem. Cup product, Kunneth formula, Poincare duality.
Prerequisites: MATH 576 or equivalent or permission of instructor.
If you have taken or are taking the physics GR course, the two courses should complement each other nicely. In particular, there will not be much overlap. While a considerable part of the physics course is (probably) spent on introducing differential geometry, we will assume that the students are comfortable with basic differential geometry. Exact solutions with high degree of symmetry will be studied as prototypical examples of spacetimes, but our focus will be on the properties of realistic spacetimes with no or very little symmetry.
The following topics will be treated.
• Some exact solutions, including black hole and cosmological solutions.
• Lorentzian geometry, geodesic congruences, variational characterization of geodesics.
• Singularity theorems of Penrose and Hawking. These theorems are the highlight of the course, and basically show that spacetimes cannot avoid developing singularities.
• Cauchy problem, if time permits. This result says that the state of the universe "today" completely determines what happens in the future in a certain sense.
The grading will be based on a few homework, and a course project, where the student studies a special topic and gives a presentation.
Ce cours est proposé comme une introduction à la géométrie riemannienne. Nous couvrirons les sujets classiques suivants : Variétés riemanniennes, connexions, géodésiques. Exemples de variétés riemanniennes. Courbure sectionnelle, courbure de Ricci, courbure scalaire. Lemme de Gauss, application exponentielle, théorème de Hopf-Rinow. Transport parallèle, holonomie, théorème d'irréductibilité et de De Rham. Variations première et seconde, champs de Jacobi, cut locus. Théorème de Bonnet-Myers, théorème de Synge, théorème de Cartan-Hadamard. Théorème de comparaison de Rauch, Alexandrov et Toponogov. Submersion riemannienne, espaces homogènes riemanniens, espaces symétriques, l'exemple de l'espace projectif complexe. Théorème de Hodge-De Rham. Théorème de Bochner. Volume, théorèmes de Bishop et de Heintze-Karcher. Sous-variétés, seconde forme fondamentale, équation de Gauss. Inégalités isopérimétriques. Géométrie spectrale. Théorème de finitude de Cheeger.
L'objectif du cours est de présenter les concepts principaux de la théorie des courbes et des surfaces plongées dans des espaces multidimensionnels. Dans ce cours, nous présentons les sujets suivants :
Théorie générale au sens de Frenet sur les courbes plongées dans des espaces multidimensionnels. Procédure d'orthogonalisation de Gram-Schmidt, Repaire mobile, Théorème fondamentale de la théorie des courbes dans Rn.
Théorie générale des surfaces plongées dans des espaces multidimensionnels basée sur la théorie du repaire mobile. Formules de Gauss-Weingarten et de Gauss-Codazzi, Caractérisation au moyen des formes fondamentales des surfaces.
Propriétés intrinsèques des surfaces. Courbures et lignes géodésiques, Surfaces à courbure constante, Théorème de Bauss-Bonnet.
Propriétés extrinsèques des surfaces. Courbure normale, Courbure moyenne, Points umbiliques, Direction conjuguée et lignes asymptotiques, Courbures principales et l'indicateur de Dupin.
Propriétés globales et caractérisation des surfaces. Forme différentielle extérieure, Lemme de Cartan, Théorie du repaire mobile, Représentation d'Enneper-Weierstrass des surfaces.
Location: Monday and Wednesday 14h35-15h55 BURN 1214
This course will serve as an introduction to the Hamiltonian formulation of classical mechanics, and the underlying differential geometry of symplectic and Poisson manifolds. We will cover:
- examples of mechanical systems, e.g. oscillators, pendulums and tops - Hamilton's equations of motion
- definitions and basic properties of symplectic and Poisson manifolds
- Liouville's theorem on phase space volumes
- Lagrangian submanifolds and Weinstein's neighbourhood theorem - momentum, symmetries and symplectic reduction
- integrable systems and action-angle variables
Further advanced topics may be selected based on the tastes and background of the audience. Possibilities include local normal forms and stability of equilibiria; classification of toric integrable systems via Delzant polytopes; perturbations of integrable systems and the Kolmogorov-Arnold-Moser (KAM) theorem; Arnold's conjecture on periodic orbits and rudiments of Floer theory; or links with quantum mechanics via geometric/deformation quantization.
Ce cours est une introduction au flot de Ricci. Nous étudierons les propriétés de base du flot (formules de variations, existence et unicité en temps court) et des applications du flot en géometrie et topologie (uniformisation des surfaces, les variétés de dimension 3 à courbure de Ricci positive).
After a brief overview of the basic theory of one complex variable, we introduce Riemann surfaces as one dimensional complex manifold. We discuss visually the topological and geometric classification for compact Riemann surfaces, including the Poincaré-Hopf index theorem and its connection to the Gauss-Bonnet theorem. We state the uniformisation theorem after a look at the complex automorphism group of the three simply connected Riemann surfaces, which are the unit disk, the complex number plane and its extension at infinity: the Riemann sphere. Uniformisation for the case of elliptic curves, i.e., compact quotient of the complex number plane, will be proved in detail by the classical method of Weierstrass P functions. Analogous to the P functions, we prove the existence of meromorphic functions on an arbitrary Riemman surface with the help of uniformisation. Armed with this, we identify compact Riemann surfaces with nonsingular algebraic curves either as finite ramified cover of the Riemann Sphere as viewed by Riemann or as algebraic curves in the projective space and show a classical method of de-singularizing algebraic curves. A little Hodge theory will allow us to obtain the Riemann existence theorem without resorting to the uniformisation theorem and so, via the above identification, the Riemann-Roch theorem and Serre duality by algebraic means: divisors, line bundles, sheaves and their cohomologies. There follows readily the uniformisation theorem for genus zero and one as well as other important consequences. If time allows at the end, we may enter into the theory of Stein spaces and show that non-compact Riemann surfaces are Stein or into the Abel-Jacobi theory and its vector bundle generalization by Atiyah-Bott and others (Donaldson-Uhlenbeck-Yau in higher dimensions). We may even try to look at the Higgs bundle generalization which offers a modern proof à la Carlo Simpson of the uniformisation theorem for compact Riemann surfaces.