Video of seminars

Talk by Susanna Terracini, University of Torino, on 7 March 2017: In its full generality, the N-body problem of Celestial Mechanics has challenged many generation of mathematicians. It is commonly accepted, since the early works by H. Poincaré, that the periodic problem, through its associated action spectrum, carries precious information on the whole dynamics of a Hamiltonian system. Therefore, the problem of the existence and the qualitative properties of periodic and other selected trajectories for the N-body problem (from the classical celestial mechanics point of view to more recent advances in molecular and quantum models) has been extensively studied over the decades, and, more recently, new tools and approaches have given a significant boost to the field. We shall review some old an new results on the existence and classification of selected trajectories of the classical N-body problem, with an emphasis on new analytical and geometrical techniques.

Talk by Renato Calleja, Departamento de Matemáticas y Mecánica - IIMAS-UNAM (Mexico), on 11 January 2017: Many problems in Celestial Mechanics are described by conformally symplectic systems (e.g. mechanical systems with a friction proportional to the velocity). The symplectic structure provides geometric identities that are used to prove theorems of existence of quasi-periodic solutions that are in an "a-posteriori" format. The a-posteriori approach presents several advantages over the classical perturbation methods and provides results on regularity, convergence of Lindstedt series and efficient algorithms to approximate the quasi-periodic solutions numerically. I will discuss other implications of this approach including a criterion for the breakdown of analyticity and hyperbolicity [C-Figueras-12], local behaviour near invariant tori, and shape and numerical approximation of domains of analyticity of invariant tori as the friction vanishes to zero [Bustamante-C-17]. This is joint work with Alessandra Celletti, Rafael de la Llave.

Talk by Walter Craig, MacMaster University, on 12 September 2016: The evolution of vortex filaments in three dimensions is an important problem in mathematical hydrodynamics. It appears in questions on solutions of the Euler equations as well as in the fine structure of vortex filamentation in a superfluid. It is also a setting in the analysis of partial differential equations with a compelling analogy to Hamiltonian dynamical systems. I will give an analysis of a system of model equations for the dynamics of near-parallel vortex filaments in the Euler flow of a three dimensional fluid. These equations can be formulated as a Hamiltonian system of partial differential equations. My talk will describe some aspects of a phase space analysis of solutions, including the construction of periodic and quasi-periodic orbits via a version of KAM theory for PDEs, and a topological principle to count multiplicity of solutions. This is ongoing joint work with L. Corsi (Georgia Tech), C. Garcia (UNAM), C.-R. Yang (McMaster and Shantou University).

Talk by Carlangelo Liverani, University of Roma "Tor Vergata", on 27 April 2016.

Talk by Michael Walter, Stanford University, on 09 March 2016: The Kronecker coefficients are among the most fascinating objects in classical representation theory. While many of their fundamental properties remain mysterious, the past decade has brought some significant advances, in part through new and surprising connections to quantum physics and computational complexity theory. In this talk, I will give an introduction to this subject and its motivations and discuss some recent results.

Talk by Sergio Doplicher on 17 February 2016: The features of Quantum Mechanics which were new for Classical Physics, and still are definitely counter - intuitive, can all be summarized in a statement: the C* Algebra generated by all observables is non-commutative. In Quantum Field Theory, neglecting gravitational forces between elementary particles, observables measured in spacelike separate regions commute, which makes the overall C* algebra "more non-commutative". But spacetime regions are subset of Minkowski space (or of a curved Einstein manifold), a classical space. But if we do not neglect gravity among the interactions between elementary particles, spacetime cannot be described by a classical manifold in the small. It becomes a Quantum Space. We will review a simple model of Quantum Spacetime, the spectral properties of some of its geometric operators, the problem of interacting quantum fields on quantum spacetime, and close with few words on the cosmological consequences of the model.

Talk by Victor Kac on 13 January 2016: A natural framework for the theory of non-commutative Hamiltonian ODE is the double Poisson algebras, introduced in 2008 by van den Bergh. Likewise, a natural framework for non-commutative PDE is the double Poisson vertex algebras, introduced in a recent joint work with De Sole and Valeri. In my talk I will explain these notions and the related non-commutative geometry. I will explain how this is used to develop a theory of noncommutative integrable systems.

Talk by Vadim Kaloshin on 2 December 2015: In 1964, V. Arnold constructed an example of a nearly integrable deterministic system exhibiting instabilities. In the 1970s, physicist B. Chirikov coined the term for this phenomenon "Arnold diffusion", where diffusion refers to stochastic nature of instability. One of the most famous examples of stochastic instabilities for nearly integrable systems is dynamics of Asteroids in Kirkwood gaps in the Asteroid belt. They were discovered numerically by astronomer J. Wisdom. During the talk we describe a class of nearly integrable deterministic systems, where we prove stochastic diffusive behaviour. Namely, we show that distributions given by deterministic evolution of certain random initial conditions weakly converge to a diffusion process. This result is conceptually different from known mathematical results, where existence of diffusing orbits is shown. This work is based on joint papers with O. Castejon, M. Guardia, J. Zhang, and K. Zhang.

Talk by Xavier Cabre' delivered at the meeting "An afternoon of Mathematics at Tor Vergata with Louis Nirenberg", 26 June 2015

Talk by Maria J. Esteban delivered at the meeting "An afternoon of Mathematics at Tor Vergata with Louis Nirenberg", 26 June 2015

Talk by Claude Viterbo delivered at the meeting "An afternoon of Mathematics at Tor Vergata with Louis Nirenberg", 26 June 2015

Abstract: Trojan motion is a classical subject of Celestial Mechanics, from both analytical and numerical points of view. Its main problem is due to the existence of singularities corresponding to close encounters between the Trojan and primary bodies. While numerical approaches can easily overcome this issue, analytical treatments face convergence problems that pose obstructions to representing Trojan motions in terms of series expansions. The talk will focus on introducing the Trojan problem as a particular case of the Restricted 3-Body Problem. Furthermore, we will present a set of basic tools of Perturbation Theory, used to the study of problems in Celestial Mechanics via a Hamiltonian Formulation and reduction to normal forms by Lie transformations. We will also show some novel approaches for bypassing the convergence problem.