The detailed **Program** will be soon avaiable.

**Speaker**: M.-C. Arnaud

**Title**: Tonelli Hamiltonians and their integrability.

**Abstract**: Tonelli Hamiltonians model a lot of natural phenomena as mechanical systems. Those that are (smoothly) completely integrable are very well understood. But sometimes we meet Tonelli Hamiltonians that are integrable in a weak sense (their integral may be just continuous, Lipschitz or differentiable). In this course, we will introduce Tonelli Hamiltonians on the cotangent bundle of the flat n-dim torus and talk over their invariant Lagrangian submanifold and their complete integrability in the (smooth) classical sense. Then we will focus on weakly integrable Hamiltonians and explain what can be said concerning their dynamics, especially when $n=2$.

**Speaker**: G. Benedetti

**Title**: Systolic inequalities in contact and symplectic geometry.

**Abstract**: Roughly speaking, systolic geometry tries to answer the following question: How long can the shortest geodesic on a closed Riemannian manifold M be? More precisely, systolic geometry aims at providing an upper bound on the length of the shortest geodesic in terms of some geometric quantities such as the Riemannian volume of M. Following Àlvarez Paiva - Balacheff and Abbondandolo - Bramham - Hryniewicz - Salomão, we formulate an analogous question for contact manifolds N and discuss its relation with the Riemannian version when N is the unit sphere bundle of M.

If N has dimension 3, we will give such an upper bound locally around those contact forms, whose Reeb flow generates a free circle action. Then, we will present an application of this result to the study of curves with prescribed geodesic curvature on a closed oriented Riemannian surface. Finally, we will sketch a generalization (to a large extent unexplored) of local systolic geometry to odd-symplectic forms defined on circle bundles over closed manifolds of arbitrary dimension.

This is joint work with Jungsoo Kang.

**Structure of the course**:

Lecture 1

Systolic geometry: From Riemannian to contact manifold. The proof of the local contact systolic inequality in dimension 3 (Part I).

Lecture 2

The proof of the local contact systolic inequality in dimension 3 (Part II). Application to the study of curves with prescribed geodesic curvature on surfaces (Part I).

Lecture 3

Application to the study of curves with prescribed geodesic curvature on surfaces (Part II). A generalization of the systolic inequality for odd-symplectic forms.

**Speaker**: Vincent Humilière

**Title**: Action selectors from symplectic topology and applications.

**Abstract**: Actions selectors associate to any Hamiltonian function the action of one its trajectories, in a canonical and continuous way. They are very convenient tools that can be used in many different situations in symplectic topology and Hamiltonian dynamics, in particular when one wants to drop the convexity assumption on the Hamiltonian. We will sketch the construction of action selectors, give their properties and present some of their applications. In particular, we plan to discuss symplectic capacities, symplectic homogenization, variational solutions to Hamilton-Jacobi equations, and the recent developments of C⁰ symplectic geometry.

**Speaker**: Albert Fathi

T.B.A.