Lecture 1

The structure of differential graded Lie algebra (dgLa) on the vector space of graphs with a wedge ordering of edges. « Zero graphs » = minus themselves. The vertex-blow-up differential, Lie bracket, its Jacobi identity. The determining properties of the complex (identities involving zero graphs).

 * Examples of nontrivial cocycles: the tetrahedron, 5-wheel + prism. The (2k+1)-wheels as markers of nontrivial cocycles originating from the Grothendieck–Teichmueller Lie algebra.

Lecture 3

The graph orientation morphism: constructing a symmetry by using a graph cocycle.

 * Endomorphisms of the space of multivector fields. The Schouten bracket, its own Jacobi identity; the master-equation. The Richardson–Nijenhuis bracket, its own Jacobi identity; the master equation.

 * The edge (the stick in the construction of graph differential). The graph orientation morphism Or takes the stick to the Schouten bracket.

 * Verifying the property of the image of a graph cocycle under Or to be a Poisson cocycle. Canonical and incidental solutions of the problem of factorization via the Jacobiator. Topological identities in the spaces of Leibniz graphs.

 * Lie algebra homomorphism: from graph cocycles to symmetries of Poisson brackets.

Lecture 4

Open problems about the graph complex and Poisson bracket symmetries: the combinatorics of cocycles and Poisson non-triviality of the Kontsevich flows.

 * Deformation quantization: the noncommutative associative star-product, its associator, the factorization problem.

 * The action of graph complex on the space of Kontsevich’s star-products.

Let W be a Coxeter group. We provide a precise description of the conjugacy classes in W, yielding an analogue of Matsumoto’s theorem for the conjugacy problem in arbitrary Coxeter groups. This extends to all Coxeter groups an important result on finite Coxeter groups by M. Geck and G. Pfeiffer from 1993. In particular, we describe the cyclically reduced elements of W, thereby proving a conjecture of A. Cohen from 1994.

To sum up: We will define complementary series for hyperbolic groups and prove their irreducibility.

More precisely: The complementary series representations are a family of unitary representations that can be realized on the Gromov boundary of the hyperbolic group. They can be viewed as a one-parameter deformation of the quasi-regular representation arising on the boundary, « sometimes » approaching the trivial representation, in a certain sense. The starting point of this work is to find a suitable scalar product in order to unitarize the complementary series. Then, a spectral estimates combined with counting estimates enable us to prove an ergodic theorem à la Bader-Muchnik to achieve the irreducibility.

Joint work with Kevin Boucher and Jean-Claude Picaud.

The work concerns a δ-hyperbolic metric space (X,d) (possibly with extra conditions) and the main assumption is that its entropy, denoted by H, is bounded above. Then, if a subgroup of its isometry group acts properly and co-compactly and if D denotes the diameter of the quotient, we will show a Bishop-Gromov type inequality on (X,d) only in terms of δ, H and D. It is a curvature-free inequality and we will explain how the bound on the entropy plays the role of a (weak version of a) lower bound on the Ricci curvature and how the δ-hyperbolicity relates to a bound on the negative part of the sectional curvature. Some consequences of this inequality are a finiteness theorem as well as a compactness result.

This is joint work with G. Courtois, S. Gallot and A. Sambusetti.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

Path geometry and CR structures on real 3-manifolds were studied by E. Cartan. There is an interesting local geometry with curvature invariants and an interesting global geometry. In particular, one can obtain flat structures studying configurations of flags. The model spaces are closed orbits of SL(3,R) and SU(2,1) in a complex flag manifold. We will review these geometries and discuss a notion of flag structure, which includes both geometries. We also review volume and Chern-Simons invariants for such geometric structures.

In this course, we will discuss the use of convex integration to construct wild weak solutions in the context of the Euler and Navier-Stokes equations. In particular, we will outline the resolution of Onsager’s conjecture as well as the recent proof of non-uniqueness of weak solutions to the Navier-Stokes equations.

Onsager’s conjecture states that weak solutions to the Euler equation belonging to Hölder spaces with Hölder exponent greater than 1/3 conserve energy, and conversely, there exit weak solutionslying in any Hölder space with exponent less than 1/3 which dissipate energy. The conjecture itself is linked to the anomalous disspoation of energy in turbulent flows, which has been called the zeroth law of turbulence.

For initial datum of finite kinetic energy, Leray has proven that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. We prove that weak solutions of the 3D Navier-Stokes equations are not unique, within a class of weak solutions with finite kinetic energy. The non-uniqueness of Leray-Hopf solutions is the subset of a famous conjecture of Ladyzenskaja in ’69, and to date, this conjecture remains open.

The mapping class group acts on the set of representations modulo conjugation of the fundamental group of n-punctured genus g-curves. For representations with values in SL(2,C), finite orbits of this action have been classified in the literature under various additional constraints. We complete this classification by the remaining case of reducible representations for g>0. This study is motivated by the following result: up to some minor technical conditions, representations modulo conjugation with values in GL(r,C) that have finite orbit under the action of the mapping class group are precisely those that appear as the monodromy of a logarithmic connection on the curve, with poles at the punctures, that admits an algebraic universal isomonodromic deformation. Both results concern a joint work with G. Cousin.