Program & abstracts

Abstract: we discuss the existence of periodic solutions, bifurcating from invariant tori, for some problems of relativistic mechanics, such as the relativistic Kepler problem in special relativity and the Schwarzschild problem of general relativity.  

Abstract: symplectic billiards - introduced by P. Albers and S. Tabachnikov in 2018 - have been recently intensively studied. After briefly introducing this dynamical system, we define the corresponding area spectrum and Mather’s beta function. The aim of the talk is to discuss possible answers to this question: for certain classes of domains, does the area spectral rigidity hold? In other words, does the area spectrum completely determine the symplectic billiard table, up to equi-affinities? We present a recent result both for a class of axially symmetric domains and a class of centrally symmetric ones. 

This result adapts and extends to symplectic billiards a paper by De Simoi, Kaloshin, and Wei (2017). Joint work with L. Baracco and O. Bernardi. 

Abstract: Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arise from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study the nonlinear Klein-Gordon equation and show that small amplitude breathers cannot exist (under certain conditions). We also construct generalized breathers, these are solutions which are periodic in time and in space are localized up to exponentially small (with respect to the amplitude) tails. This is a joint work with O. Gomide, T. Seara and C. Zeng.

Abstract: in this talk we deal, for the classical N-body problem, with the existence of action minimizing half entire expansive solutions with prescribed limit shape and initial configuration, tackling the cases of hyperbolic, parabolic and hyperbolic-parabolic arcs in a unified manner. Our approach is based on the minimization of a renormalized Lagrangian action defined on a suitable functional space. With this new strategy, we are able to confirm the already-known results of existence of both hyperbolic and parabolic solutions, and we prove for the first time the existence of hyperbolic-parabolic solutions for any prescribed asymptotic expansion in a suitable class. Associated with each element of this class, we find a viscosity solution of the Hamilton-Jacobi equation as a linear correction of the value function. Besides, we also manage to give a precise description of the growth of parabolic and hyperbolic-parabolic solutions.

This work is in collaboration with Susanna Terracini. 

Abstract: we delineate a framework for the integrability (in the sense of quasi-periodicity of the dynamics) of ODEs with symmetry which is based on reconstruction techniques from simple reduced dynamics and organizes, unifies and extends previous results from Hamiltonian and non-Hamiltonian (e.g., nonholonomic) mechanics. 

Abstract: Mathematical billiards in strictly convex domains with smooth boundaries serve as examples of twist maps on the cylinder, where the dynamics are "almost integrable" near the boundary of the domain. Building on this idea, Lazutkin demonstrated that there exists a Cantor set of positive measure containing 0, for which the billiard maps have invariant curves corresponding to these rotation numbers. Furthermore, these curves vary smoothly with respect to the rotation number in a Whitney sense. In this talk, I will present a generalization of this result for billiards with analytic boundaries, in a joint effort with Vadim Kaloshin and Frank Trujillo, inspired by recent work by Carminati, Marmi, Sauzin, and Sorrentino. This generalization reveals that the Cantor set of rotation numbers can be complexified, and its complex counterpart contains structures known as "diamonds." This result has intriguing implications for length spectral rigidity.

Abstract: we shall review some results about the relationship between Dirichlet forms and the dynamics on fractal sets. 

Abstract: in this talk we treat the restricted (n+1)-body problem with a non Newtonian homogeneous potential where the  primaries move on an arbitrary -periodic orbit. We prove that the satellite equation has infinitely many periodic solutions. These solutions are obtained as critical solutions of a family of time-dependent perturbed Lagrangian systems, bifurcating uniformly from a compact set of periodic solutions of the unperturbed Lagrangian system.