# Vendredi 12 janvier 2018

10h 30 Amadeu Delshams (UPC)

Shadowing of non-transversal heteroclinic chains

Abstract: We present a new result about the shadowing of non-transversal chain of heteroclinic connections based on the idea of dropping dimensions. We illustrate this new mechanism with several examples. As an application we discuss this mechanism in a simplification of a toy model system derived by Colliander et al. in the context of the cubic defocusing nonlinear Schrödinger equation. This is a joint work with Adrià Simon and Piotr Zgliczynski

Shadowing of non-transversal heteroclinic chains

Abstract: We present a new result about the shadowing of non-transversal chain of heteroclinic connections based on the idea of dropping dimensions. We illustrate this new mechanism with several examples. As an application we discuss this mechanism in a simplification of a toy model system derived by Colliander et al. in the context of the cubic defocusing nonlinear Schrödinger equation. This is a joint work with Adrià Simon and Piotr Zgliczynski

**15-17 janvier : colloque de clôture de la chaire d'Eva Miranda.**

__https://www.ceremade.dauphine.fr/~fejoz/Colloques/Geometry-Dynamics2018/index.php__# Vendredi 19 janvier 2018

10h30 Ignasi Mundet (UB)

Let X be a compact symplectic manifold and let Ham(X) denote the group of hamiltonian diffeomorphisms

of X. I will talk about the following results:

Theorem 1: there exists a constant C (depending only on the topology of X) such that any finite subgroup G

of Ham(X) has an abelian subgroup whose index in G is at most C.

Theorem 2: if b_1(X)=0 then Theorem 1 holds true replacing Ham(X) by Symp(X).

Theorem 3: for general X, Theorem 1 holds true replacing Ham(X) by Symp(X) and "abelian" by "2-step nilpotent".

In the first part of the seminar I will explain the context of these results. In particular, I will comment on similar questions for diffeomorphism groups, putting emphasis on situations where these theorems imply that the finite transformation groups in the symplectic category are much more restricted than in the smooth category. In the second part of the seminar I will explain the main ideas in the proofs of the theorems.

**Finite subgroups of Ham and Symp**

Let X be a compact symplectic manifold and let Ham(X) denote the group of hamiltonian diffeomorphisms

of X. I will talk about the following results:

Theorem 1: there exists a constant C (depending only on the topology of X) such that any finite subgroup G

of Ham(X) has an abelian subgroup whose index in G is at most C.

Theorem 2: if b_1(X)=0 then Theorem 1 holds true replacing Ham(X) by Symp(X).

Theorem 3: for general X, Theorem 1 holds true replacing Ham(X) by Symp(X) and "abelian" by "2-step nilpotent".

In the first part of the seminar I will explain the context of these results. In particular, I will comment on similar questions for diffeomorphism groups, putting emphasis on situations where these theorems imply that the finite transformation groups in the symplectic category are much more restricted than in the smooth category. In the second part of the seminar I will explain the main ideas in the proofs of the theorems.

# Vendredi 26 janvier 2018

10h30 Alfonso Sorrentino (Tor Vergata)

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where ``it the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered''.

Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures.

In this talk I shall focus on several of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture).

This talk is based on works in collaboration with V. Kaloshin and with G. Huang and V. Kaloshin.

**On the integrability of mathematical billiards**

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where ``it the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered''.

Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures.

In this talk I shall focus on several of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture).

This talk is based on works in collaboration with V. Kaloshin and with G. Huang and V. Kaloshin.