Topologie différentielle, 2027 (provisoire)

Un cours de topologie différentielle de LebSci sur douze semaines, du 6 février au 24 avril 2027, consacré aux variétés lisses, aux valeurs régulières, à la transversalité et au degré.

Statut

Projet provisoire : séances en ligne en anglais le samedi à 18:00 (Asia/Beirut), d’une durée de 90 minutes. Les dates, les horaires, le format et le programme restent provisoires ; rien n’est confirmé. Les candidatures ne sont pas ouvertes.

Prérequis conseillés

Multivariable calculus
Linear algebra
Point-set topology (open sets, continuity, compactness, connectedness)
A first course in real analysis
Familiarity with proof-based mathematics

Topics Charts, atlases, and smooth manifolds Smooth maps and diffeomorphisms Examples: spheres, tori, and open subsets of Euclidean space

Topics Tangent vectors as velocities of curves Derivatives of smooth maps and the chain rule Tangent bundles and vector fields: definitions and examples

Topics Inverse function theorem and local immersions An embedding is an immersion that is a homeomorphism onto its image Proper injective immersions are embeddings; embeddings need not be proper

Topics Submersions and regular values Preimage theorem for manifolds without boundary: dimension and tangent space Examples: spheres and regular level sets

Topics Critical points, critical values, and measure zero in charts Sard’s theorem for smooth maps: statement and consequences Examples of regular values and critical values

Topics Examples of embeddings into Euclidean space Weak Whitney embedding theorem: every smooth n-manifold embeds in R^(2n+1), statement only Tubular neighborhoods: geometric picture for compact embedded manifolds without boundary

Topics Half-space charts and the boundary manifold Preimage theorem for maps to a manifold without boundary: regularity for both the map and its boundary restriction Compact one-manifolds are finite disjoint unions of circles and closed intervals

Topics Transversality: tangent images and submanifold tangent spaces span the target tangent space Transverse preimage theorem for manifolds without boundary Parametric transversality for smooth families transverse to a submanifold: statement and examples

Topics Counting transverse intersections modulo 2 for a compact source without boundary and a closed submanifold of complementary dimension Homotopy invariance via compact one-manifolds with boundary Mod 2 degree for maps between compact connected manifolds without boundary of equal dimension

Topics Orientations of vector spaces and manifolds Degree for maps between compact connected oriented manifolds without boundary of equal dimension Computing degree from signed preimages of a regular value

Topics Vector fields and indices of isolated zeros Poincaré-Hopf theorem for compact manifolds without boundary and vector fields with isolated zeros: statement only Euler characteristic and vector field examples on spheres and tori

Topics Review of regular values, transversality, and degree No continuous retraction from the closed n-ball onto its boundary sphere for n >= 1 Brouwer fixed point theorem for continuous self-maps of the closed n-ball: statement and connection to non-retraction

Contacts

À propos

Statut

Prérequis conseillés

Programme

février 2027

Topics

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mars 2027

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avril 2027

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Contacts

À propos

Statut

Prérequis conseillés

Programme

février 2027

sam.6Séance 1samedi 6 février 202718:00 – 19:30Variétés lisses (provisoire)

Topics

sam.13Séance 2samedi 13 février 202718:00 – 19:30Espaces tangents et différentielles (provisoire)

sam.20Séance 3samedi 20 février 202718:00 – 19:30Théorème d’inversion locale et immersions (provisoire)

sam.27Séance 4samedi 27 février 202718:00 – 19:30Submersions et valeurs régulières (provisoire)

mars 2027

sam.6Séance 5samedi 6 mars 202718:00 – 19:30Théorème de Sard (provisoire)

sam.13Séance 6samedi 13 mars 202718:00 – 19:30Plongements et théorème de Whitney (provisoire)

sam.20Séance 7samedi 20 mars 202718:00 – 19:30Variétés à bord (provisoire)

sam.27Séance 8samedi 27 mars 202718:00 – 19:30Transversalité (provisoire)

avril 2027

sam.3Séance 9samedi 3 avril 202718:00 – 19:30Théorie de l’intersection modulo 2 (provisoire)

sam.10Séance 10samedi 10 avril 202718:00 – 19:30Orientation et degré (provisoire)

sam.17Séance 11samedi 17 avril 202718:00 – 19:30Champs de vecteurs et théorème de Poincaré-Hopf (provisoire)

sam.24Séance 12samedi 24 avril 202718:00 – 19:30Révision et applications élémentaires (provisoire)