An inspiring, really motivating teacher
Since moving to Rennes, I’ve mostly taught M2 classes—Riemannian geometry, differential topology, and the like—almost every year, or maybe every other year. Teaching isn’t technically part of my CNRS job description, but, in the doses that I do it, I love it. Whether I’m any good at it, is another question...
Current teaching
Nothing on the horizon...
Past teaching
Riemannian Geometry—Fall 2025.
Master 2 level class (48 hours).
The goal was to introduce manifold, tangent spaces, bundles and such and then turn to proper Riemannian geometry to prove the classics: Hadamard-Cartan, Synge, Bonnet-Myers, Bishop-Gromov. Lie groups appeared quite prominently. I then spoke about CAT(-1) spaces, Gromov-hyperbolic spaces, the boundary, stablity of quasi-geodesics and the existence of boundary maps. In a last class where probably nobody understood anything I sketched Tukia's proof of Mostow rigidity.
The goal was to introduce manifold, tangent spaces, bundles and such and then turn to proper Riemannian geometry to prove the classics: Hadamard-Cartan, Synge, Bonnet-Myers, Bishop-Gromov. Lie groups appeared quite prominently. I then spoke about CAT(-1) spaces, Gromov-hyperbolic spaces, the boundary, stablity of quasi-geodesics and the existence of boundary maps. In a last class where probably nobody understood anything I sketched Tukia's proof of Mostow rigidity.
Recommended reading
These were books I recommended at the beginning of the class. They cover the material (and much more!) until Week 10 or so.
- J. Lee, Introduction to Smooth Manifolds.
- P. Pankka and S. Souto, A course in differential topology(preliminary version avilable here).
- M. do Carmo, Riemannian Geometry.
Week-by-week syllabus
- Week 1: Topological manifolds, smooth manifolds, maximal atlas, dimension is well-defined, smooth maps between smooth manifolds, composition of smooth maps is smooth, submanifolds, submanifolds are smooth manifolds in their own right, implicit function theorem and examples of submanifolds (S^n, SL_nR, SO_n), Lie groups, getting manifolds gluing charts (RP^n, CP^n), manifolds as quotients as free proper actions (RP^n, T^n, SL^nZ\SL^nR), existence of partitions of unity.
- Week 2: Tangent space and differential. Inverse mapping theorem and implicit function theorem (preimage of a regular value is a submanifold). Every manifold can be embedded into some R^n. Fiber bundles (Hopf fibration is not trivial), vector bundles, principal bundles (Hopf fibration and frame bundle), vector bundle construction lemma. Construction of the tangent bundle, Whitney sum, dual bundle (more about this next week).
- Week 3: Sections of bundles, Whitney approximation, sections defined on closed sets can be extended to global sections, existence of Riemannian metrics, differential forms, and exterior derivative.
- Week 4: Lie bracket, Picard-Lindelöf and the flow associated with a vector field, Lie derivative of vector fields, differential forms and tensor fields, Lie derivative of vector field is given by Lie bracket, flows commute if and only if Lie bracket of corresponding vector fields vanishes, proof of Frobenius theorem, and geometric argument to prove that the distribution of complex planes in S^3 is not integrable.
- Week 5: Connections, every vector bundle admits a connection, connection induced on dual bundle, Levi-Civita connection, parallel transport, example in S^2, geodesics via ODE, local existence, geodesically complete, first variation formula (as Euler-Lagrange equation with a potential).
- Week 6: Exponential map, differential of the exponential map at 0, Lie bracket, existence of regular neighborhoods, quotient manifold theorem (for free actions of compact groups), geometry of SL_nR/SO_n via Killing form, symmetric spaces (homogeneous, geodesically complete, G/K).
- Week 7: Gauss lemma, Hopf-Rinow, curvature tensor and its symmetries, sectional curvature, Ricci curvature, scalar curvature, Jacobi equation and fields as tangent vector fields to geodesic variations.
- Week 8: Jacobi fields and differential of the exponential map, Hadamard-Cartan, calculation of the curvature of H^n, R^n and S^n, Jacobi fields in constant curvature, first variation formula (again) and second variation formula, Bonnet-Myers.
- Week 9: Existence of closed minimizing geodesics in homotopy classes, Synge's theorem, Killing fields, Noether's theorem, Claireaux's theorem, unimodular Lie groups, existence of bi-invariant metrics on compact Lie groups, calculation of Levi-Civita connection, curvature tensor and sectional curvature of bi-invariant metrics on Lie groups, proof that semisimple Lie groups with bi-invariant metrics are compact and have surjective exponential map, discussion of Killing form with proof that if G is semisimple and compact then the negative of the Killing form is an Einstein metric. Discussion (off-topic) of corollaries of Cartan's closed group theorem.
- Week 10: Index Lemma, bounds for distance to conjugate points, Rauch's theorem, volume growth of simply connected manifolds of negative curvature, quasi-isometries and Milnor-Svarc lemma, exponential growth for of pi_1 for closed manifolds of negative curvature.
- Week 11: Bishop-Gromov, manifolds of sectional curvature less or equal to K are CAT(K) spaces. Convexity of distance functions in CAT(0) spaces, finite/compact groups have fixed points, geodesics at bounded distance are parallel, maximal abelian subgroups of pi1 of negatively curved manifolds are cyclic, flat strip theorem, convex projection.
- Week 12: Boundary of a CAT(-1) space, Morse lemma, quasi-isometries induce boundary map, delta-hyperbolicity.
- Week 13: Boundary map is quasi-conformal, proof of Mostow rigidity (for hyperbolic manifolds) à la Tukia.
Homework problems
There were 4 sets of homeworks problems. Students had a week to work them out.
Differential topology and de Rham cohomology—Fall 2024.
Master 2 class (36 hours).
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…)
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…)
Recommended reading
There are plenty of good books, and I will mention them in class, and I will provide notes for at least part of the class.
- J. Lee, Introduction to Smooth Manifolds.
- J. Milnor, Topology from the differentiable point of view. A jewel.
- Notes extracted from what whas going to be A course in differential topology, my book with Pekka (preliminary version avilable here).
Week-by-week syllabus
- 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.
- 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.
- 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
- 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.
Homework problems
There were 4 sets of homeworks problems. Students had a week to work them out.
Differential topology and de Rham cohomology—Fall 2023.
Master 2 class (36 hours).
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 were oral exams at the end of the course.
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…)
Recommended reading
There are plenty of good books, and I will mention them in class, and I will provide notes for at least part of the class.
- J. Lee, Introduction to Smooth Manifolds.
- J. Milnor, Topology from the differentiable point of view. A jewel.
- Notes extracted from what whas going to be A course in differential topology, my book with Pekka (preliminary version avilable here).
Class-by-class syllabus
- 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.
Prior to 2022.
Before arriving to Rennes I actually taught things like linear algebra and such. Since being here I have only taught master level classes (36 hours) and courses de l'école doctorale, what ever that is.
Master 2 level classes, ranging from 36-48 hours of lecture time.
- Differential topology and de Rham cohomology (2022)
- Fundamentals of differential geometry and topology (2020)
- Fundamentals of differential geometry and topology (2019)
- Fundamentals of differential geometry and topology (2018)
Courses de l'école doctorale (6-8 hours for doctoral students):
- Small eigenvalues of the Laplacian. (2023)
- 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. (2022)
- Homology and cohomology of groups: Start from the beginning and prove the Eilenberg-Ganea theorem. (2016)
- Existence of minimal surfaces: Solution of the Plateau problem in R^n and of its generalisations if one replaces the disk by other surfaces. (2015)
Graduate students
Having graduate students is what I like most of my job. Here is my math genealogy.
Current
- Thibault Favier, working on his Ph.D. at the Université de Rennes since the fall of 2023. Supervised jointly with Anna Lenzhen.
- Sophie Wright, working on her Ph.D. at the University of Bristol since the fall of 2023. Supervised jointly with Viveka Erladsson.
Dealt with
- Marie Trin, graduated at the Université de Rennes in the spring of 2024 and is now a postdoc at the MPI Leipzig.
- Lucien Grillet, graduated from the Université Rennes 1 in the fall of 2022 and is now a data scientist at Chouette.
- José Andrés Rodriguez Migueles, graduated from the Université Rennes 1 in 2018. Currently a junior researcher at CIMAT.
- Cyril Lacoste, graduated from the Université Rennes 1 in 2018 and is currently professeur d'informatique en classes preparatoires scientifiques au Lycee Kerichen in Brest.
- Maxime Bergeron, graduated from the University of British Columbia in 2016 and works currently for Riskfuel.
- Nina White, graduated from the University of Michigan in 2012. She is currently a lecturer at the University of Michigan.
- Johanna Mangahas, graduated from the University of Michigan in 2010. Johannas is currently an associate professor at the University at Buffalo.
- Ian Biringer, graduated from the University of Chicago in 2009. Currently Ian is a professor at Boston College.
More students I have had something to do with
Master students (M2 at the Université Rennes)
- Khoi Nguyen Nguyen Ba, 2025. Thurston's classification of mapping classes a la Bers.
- Victor Fuentes, 2024. Classical theorems in Riemannian geometry.
- Yago Moreno, 2023. The Benjamini-Schramm theorem on the recurrence of distributional limits of planar graphs.
- Thibault Favier, 2023. Failure of Teichmueller space to be CAT(0).
- Marie Trin, 2021. Thurston's compactification of Teichmueller space via geodesic currents.
- Yulin Gong, 2020. Mostow's rigidity theorem (via BCG).
- Lucien Grillet, 2019. The Poincaré conjecture in higher dimensions.
- Cyril Lacoste, 2015. Virtual cohomological dimension of SLnZ.
- José Andrés Rodriguez Migueles, 2015. Mostow's Rigidity Theorem.
Undergraduate projects (always in Rennes)
- Mattis Chick and Christophe Martinez, 2023. Construction of the p-adics.
- Josselin Miss and Marius Breton, 2024. Galois theory.
- Margot Savy, 2025. Coverings and fundamental group.
Students (at the time) that came for some time to work with me in Rennes.
- Tommaso Cremaschi, May and June 2016.
- James Farre, May and June 2017.
- Thi Hanh Vo, February 2020.
- Rodrigo de Pool, April to June 2024.