In my previous life, that is before coming to Bretagne, the land of cider and butter, I taught undergraduate classes like linear algebra (very often-I loved teaching linear algebra), calculus, point-set topology, and things like this. Since I arrived to Rennes I have been teaching every year, or maybe every second year, an M2 class, mostly about things like Riemannian geometry or differential topology:
Current teaching (Fall 2024):
Differential Topology and de Rham Cohomology
We-Fr, 9:45-11:15, starting September 11th and ending December 6th, with a break the week of Halloween
I will only assume a decent base of calculus in several variables (being scared of the implicit function theorem is ok, being so scared that one runs away or it or stops listening/reading/thinking is not), a decent basis of linear algebra (mostly multilinear algebra), and some contact with pointwise topology (what is a topology, what means Hausdorff, compact, connected and such). That being said, I will go relatively fast. Besides introducing the main players (manifolds and bundles) I will spend some time talking about things like the Frobenius theorem or about Lie groups (proving the bijection between Lie subgroups of a Lie group and Lie subalgebras of its Lie algebra, and hopefully also proving Cartan's closed subgroup theorem). That is basically the first part of the course.
The second part is about de Rham cohomology. Although there are surely people who have seen it before, I will prove Stokes' theorem. No extra prerequisites are needed for this, meaning that I will discuss all the needed algebra. The goal is to do things like the topological invariance of de Rham cohomology, invariance of domain, Poincare Duality, the Kuenneth formula, Jordan-Alexander duality, the Thom isomorphism and the Lefschetz fixed point theorem (maybe the Lefschetz trace formula). Intermixed with this, I will discuss a minimum of mod2 and integral intersection theory, that is things like the relation between the intersection of submanifolds of complementary dimension and of their Poincare duals. (When I type this here, it seems like a lot, but we will see…)
There are plenty of good books, and I will mention them in class, but I will provide note and at least for part of the class, Lee's book Introduction to Smooth manifolds is very good. A jewel that everybody should read is Milnor's book Topology from the Differentiable Point of View.
Here is what I have covered so far:
Week 1: Topological manifolds, C^0-partitions of unity, submanifolds, manifold construction lemma, projective space, manifolds as quotients, smooth manifolds and smooth maps, composition of smooth maps is smooth
Week 2: smooth partitions of unity, smooth manifold construction lemma, smooth manifolds as quotients, smooths manifolds as submanifolds, tangent space and differential, inverse mapping theorem, implicit function theorem
Week 3: SL_nR as a submanifold, submersions, immersions, embeddings, every manifold arises as a submanifold of some euclidean space, Brouwer fixed point theorem, fiber bundles, Hopf fibration, Hopf fibration is non-trivial.
First problem sheet --- due October 4th
Week 4: Canonical bundle over CP^n, tangent bundle, Whitney sum, dual bundle, pull-back, normal bundle... for every E->M there is F->M with E+F trivial. Extension of sections.
Week 5: Regular neighborhoods, Whitney approximation, nearby maps are homotopic, weak Whitney embedding theorem, general position, closed co-dimension one submanifold of R^n separate. Picard-Lindeloef for open sets in R^n.
Week 6: Picard-Lindeloef and flows for manifolds. Isotopy extension theorem. Lie bracket. Flows commute if and only if Lie bracket vanishes. Vector fields can be made coordinate vector fields if and only if Lie bracket vanishes. Distributions of k-planes, and distribution associated to a foliation.
Second problem sheet --- due October 25th
Week 7: The Frobenius theorem, Lie groups (Lie algebra, exponential, Lie group homomorphisms, bijection between Lie subgroups and Lie subalgebras, consequences of Cartan's closed subgroup theorem)
Week 8: Differential forms, exterior differential, orientability, integration of top-degree forms, Stokes' theorem, another proof of the fixed point theorem
Week 9: cochain complexes, cohomology, de Rham and compactly supported, calculation of H^*(R^1) and H^*1(S^1), cochain homotopies, homotopy invariance of de Rham cohomology, topological invariance, Poincare lemma, dimension invariance, snake lemma, statement of Mayer-Vietoris
Third problem sheet --- due November 22nd
Week 10: Proof of Mayer-Vietoris, calculation of cohomology of sphere and CP^n, Jordan-Alexander duality, finite dimensionality, beginning of proof of Euler-formula (discussion of what is a triangulation)
Week 11: Proof of Euler-formula (that was a flop), Poincare duality, corollaries of Poincare duality, proof of Poincare lemma for compactly supported cohomology, the Kuenneth formula, Poincare polynomials, mod(2)-degree is well-defined.
I also taught this class last year and, although there will be changes, here you have some key-words indicating what I covered in each class:
September 20th: Topological manifolds, partitions of unity, manifold construction lemma, complex projective space, manifolds as quotients
September 23rd: Smooth manifolds, smooth maps and diffeomorphisms, maximal atlas, smooth partition of unity, Whitehead approximation theorem for functions.
September 27thth: Tangent space via derivations.
September 29th: Inverse mapping theorem and implicite function theorem (for manifolds). Examples of submanifolds.
October 4th: SL_nR and SO_n as submanifolds. Manifolds with boundary, including implicite function theorem, and Brouwer's fixed point theorem.
October 6th: Definition of fiber bundles and vector bundles. Vectorbundle construction lemma. Proof that the Hopf fibration and the canonical bundle of CP^n are nor trivial.
October 11th: Examples of vector bundles: tangent bundle, Whitney sum, dual bundle, Alt^k-bundle, etc...
October 14th: prove that for every vector bundle E-> M there is F-> M with E+F -> M trivial, (weak) Whitney embedding theorem, general position theorem
October 18th: prove existence of regular neighborhoods and a couple of applications. Vector fields, Picard Lindeloef, creme anglaise theorem.
Octbober 20th: Thom isotopy extension theorem, normal form for non-vanishing vector fields, Lie brackets and some of its properties. Definition of Lie group.
October 25th: Lie groups, Lie algebra, exponential map, one-parameter subgroups groups, rambling about Lie groups
October 27th: Distributions, Foliations and the Frobenius theorem.
November 8th: Fast review of multilinear algebra, differential forms, uniqueness of exterior derivative
November 9th: Existence of exterior derivative, orientation, existence of volume forms, integration
November 15th: integration of forms, Stokes, Beginning of de Rham.
November 17th: cochain morphisms induce maps on cohomology, functoriality of de Rham cohomology, homotopic cochain morphisms induce the same maps, statement of "homotopic is homotopic" theorem (proof next time) and applications: topological invariance of de Rham cohomology, Poincare lemma, invariane of domain
November 22nd: prove that homotopic maps induce identical maps on de Rham cohomology. Proof of the snake lemma and of Mayer-Vietoris, both for usual de Rham and for the compactly supported one. Calculation of the cohomology of the spheres of complex projective spaces.
November 24th: I rambled a lot about things like the Whitehead manifold. Then discussed the existence of good covers and proved that the Betti numbers of manifolds of finite topology are finite.
November 29th: Proved the Euler formula (that is, Euler charateristic = Euler characteristic) and Poincare Duality.
December 1st: Proof of the Kuenneth formula and discussion of transfer.
December 6th: Proof of the Jordan-Alexander theorem, calculation of Lusternik-Schnirelmann category of CP^n and T^n.
December 8th, 13th and 15th: mod-2-degree, Borsuk-Ulam, degree, intersection form, Thom isomorphism and Thom class, Hopf-index theorem, brief sketch of the proof of the Lefschetz trace formula and Lefschetz fixed point theorem.
Past teaching:
Fondaments de la topologie différentielle and de Rham cohomology.
Géométrie Riemannienne et courbure: A basic course (24h) on riemannian geometry, starting with bundles and connections, the curvature tensor, first and second variation formulas, the theorems of Hadamard-Cartan, Cartan, Bonnet-Myers and Synge, basic Morse theory, the existence of periodic geodesics on closed manifolds, and a very little bit of comparison geometry. (taught 3 times)
Other than that I taught couple of short courses for graduate students.
Flexibility in geometry: I discussed, following Geiges's notes, the proof that open invariant differential relations satisfy the h-principle, focusing on the existence of immersions into euclidean space of the same dimension and on the sphere eversion.
Homology and cohomology of groups: Start from the beginning and prove the Eilenberg-Ganea theorem. I have note somewhere, but they are kind of sketchy.
Existence of minimal surfaces: Solution of the Plateau problem in R^n and of its generalisations if one replaces the disk by other surfaces.