- Tingueu el micròfon apagat tota l'estona excepte durant el "Cafè i galetes".
- Podeu tenir la càmera encesa si voleu, però en cas de saturaciò de línies, caldrà que la tanqueu.
- Si voleu fer una pregunta, si us plau demaneu torn a través del xat i espereu que els moderadors us donin la paraula.
- Si és possible, entreu amb antelació. Els que no useu un compte de correu de la UPC haureu d'esperar a ser admesos per accedir-hi.

Dia: Dimecres, 4 d'octubre de 2023

Lloc: Aula S04, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

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A càrrec de: Filippo Giuliani (Politecnico de Milano)

Títol: Sobolev instability for the cubic NLS on irrational tori.

Resum: In the last two decades the study of instability in Sobolev spaces for nonlinear Hamiltonian partial differential equations on compact manifolds has drawn lots of attention in the mathematical community. A breaktrough result in this sense is due to Colliander-Keel-Staffilani-Takaoka-Tao (Invent. Math 2010), who showed the existence of solutions to the defocusing cubic NLS on the 2-dimensional square torus with arbitrarily small initial data and arbitrarily large Sobolev norms at later times. The mechanism to construct such unstable solutions is based on the study of the resonant dynamics of NLS and it has inspired several other works. However, Staffilani noticed that the same strategy would not applied for the NLS equation on 2-dimensional irrational tori, where the resonant structure is less rich.In this talk we discuss how we overcame this problem to prove Sobolev instability for the cubic NLS on irrational tori. Moroever, we present a recent result of this type where we take into account also the presence of smooth convolution potentials.

Dia: Dimecres, 25 d'octubre de 2023

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

A càrrec de: Román Moreno (UPC)

Títol: Splitting of separatrices for rapid degenerate perturbations of the classical pendulum

Resum: In this talk we will discuss the splitting distance of a rapidly perturbed pendulum

Dia: Dimecres, 29 de novembre de 2023

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

A càrrec de: Mar Giralt (IMCCE de l'Observatoire de Paris)

Títol: An Arnold diffusion mechanism for the Galileo satellites

Resum: Among the various actions that are being taken to preserve the circumterrestrial environment, end-of-life disposal solutions play a key role. In this regard, innovative strategies should be conceived not only by means of novel technologies, but also following an advanced theoretical understanding.

A growing effort is devoted to exploit natural perturbations to lead the satellites towards an atmospheric reentry. In the case of the Medium Earth Orbit region, home of the navigation satellite Galileo, the main driver is the gravitational perturbation due to the Moon, that can increase the eccentricity in the long term. In this way, the pericenter altitude gets into the atmospheric drag domain and the satellite eventually reenters.

This is a joint work with Elisa Maria Alessi (IMATI), Inma Baldomá (UPC), Marcel Guardia (UB) and Alexandre Pousse (IMATI).

Dia: Dimecres, 20 de desembre de 2023

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

A càrrec de: Marcel Guàrdia, Universitat de Barcelona

Títol: Hyperbolic dynamics and oscillatory motions in the 3 Body Problem

Resum: Consider the three body problem with positive masses. In 1922 Chazy classified the possible final motions the three bodies may possess, that is the behaviors the bodies may have when time tends to infinity. One of them is what is known as oscillatory motions, that is, solutions of the three body problem such that the liminf (as time tends to infinity) of the relative positions between bodies is finite whereas the limsup is infinite. That is, solutions for which the bodies keep oscillating between an increasingly large separation and getting closer together. The first result of existence of oscillatory motions was provided by Sitnikov for a Restricted Three Body Problem, called nowadays Sitnikov model. His result has been extended to several Celestial Mechanics models, but always with rather strong assumptions on the values of the masses. In this talk I will explain how to construct oscillatory motions for the three body problem for any value of the masses (except for the case of three equal masses). The proof relies on the construction of hyperbolic invariant sets whose dynamics is conjugated to that of the shift of infinite symbols (i.e. symbolic dynamics). That is, we construct invariant sets for the three body problem with chaotic dynamics, which moreover contain oscillatory motions.

This is a joint work with Pau Martin, Jaime Paradela and Tere M. Seara.

Beer seminar de final de quadrimestre

Dia: Dimecres, 14 de febrer de 2024

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

A càrrec de: Anna Gierzkiewicz-Pieniążek (Jagiellonian University, Cracòvia)

Títol: The Sharkovskii Theorem for multidimensional maps with attracting periodic orbits

Resum: I would like to present a computer-assisted method for proving rigorously the existence of a wide variety of periodic orbits for multidimensional maps with an attracting n-periodic orbit, based on the proof of Sharkovskii Theorem for interval maps. The method [A. G., P. Zgliczyński, J Differ Equ, 314(2022),733-751] is a simple example of using CAPD library for C++.

Dia: Dimecres, 21 de febrer de 2024

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

A càrrec de: Rohil Prasad, University of Berkeley

Títol: On the dense existence of compact invariant set

Resum: This is joint work with Dan Cristofaro-Gardiner. We show that for any monotone-area preserving diffeomorphism of a closed surface or for any Reeb flow on a closed contact 3-manifold with torsion Chern class, there exist infinitely many distinct proper compact invariant subsets whose union is dense in the manifold. No genericity assumptions are required. The former class of systems includes all Hamiltonian diffeomorphisms of closed surfaces and the latter class of systems includes the geodesic flow of any Finsler metric on a closed surface. In particular, our methods can also show that any Finsler metric on a closed surface has infinitely many non-dense geodesics, with pairwise distinct closures, whose union is dense in the surface.

A càrrec de: Mikhail Hlushchanka, University of Amsterdam

Títol: Invariant graphs for rational maps: construction, application, and open problems

Resum: Invariant graphs provide nice combinatorial models for dynamical systems under consideration. As such, they appear naturally in various aspects of complex dynamics and have multiple applications. For instance, "Hubbard trees" were used to classify all postcritically-finite polynomials in the 80s. I will present the main approaches for the construction of invariant graphs, overview some of their applications, and discuss several open combinatorial problems in this area.

Dia: Dimecres, 28 de febrer de 2024

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

A càrrec de: Jared Todd Blanchard, Standford University

Títol: Analysis of Invariant Funnels in the Circular Restricted Three-Body Problem

Resum: Scientists have identified the ocean worlds Europa and Enceladus as the most promising targets in the search for life in the solar system. Designing trajectories to such deep space targets requires balancing fuel and time constraints. Study of the Circular Restricted Three-Body Problem (CR3BP) has allowed mission designers to find very efficient trajectories that arrive in realistic time frames. Mission design work relies on using the periodic and quasi-periodic orbits that add structure to the chaotic dynamics of the CR3BP, as well as their hyperbolic invariant manifolds to intelligently sample the state space. In this presentation, we review our novel method for computing invariant funnels around non-periodic trajectories that approach the secondary body. Invariant funnels are sets of trajectories that converge in position space to a target point and can often be generated in backward time by imposing a parallel velocity condition. We review some of their applications to mission design for the ocean worlds.

Dia: Dimecres, 6 de març de 2024

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

A càrrec de: Amadeu Delshams, UPC

Títol: Shadowing of non-transversal heteroclinic chains in lattices

Resum: We study the global instability of dynamical systems on complex lattices by shadowing chains of nontransversal heteroclinic connections between several periodic orbits. The systems we consider are inspired by the so-called toy model systems (TMS) used to show the existence of energy transfer from low to high frequencies in the nonlinear cubic Schrödinger equation (NLSE) or generalizations. Using the geometric properties of the complex projective space as a base space, we generate in a natural way collections of such systems containing these chains, both in the Hamiltonian and non-Hamiltonian setting. On the other hand, we characterize the property of "block diagonal dynamics along the heteroclinic connections" that allows these chains to be shadowed, a property which in general only holds for transversal heteroclinic connections. Due to the lack of transversality, only finite chains are shadowed, as there is a dropping dimensions mechanism in the evolution of any disk close to them. The main shadowing tools used in our work are the covering relations as introduced by one of the authors.

This is a joint work with Piotr Zgliczynski, Jagiellonian University, Krakow.

Dia: Dimecres, 13 de març de 2024

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

A càrrec de: Shoya Motonaga, Ritsumeikan University

Títol: Nonintegrability and related dynamics of the hydrogen atom in a circularly polarized microwave field

Resum: We consider the hydrogen atom with a circularly polarized microwave field (the CP problem). The model is given by a periodically perturbed Hamiltonian system with two degrees of freedom whose unperturbed system is the Kepler problem. We prove that the corresponding system in a rotating coordinate is not analytically Liouville integrable such that first integrals depend on the perturbation parameter analytically. Moreover, the persistence of infinitely many periodic orbits corresponding to elliptic orbits in the Kepler problem is provided. In our approach, the subharmonic Melnikov functions for periodic orbits play key roles. We also consider the case that the period of the periodic orbit goes to infinity.

Dia: Dimecres, 20 de març de 2024

Lloc: Aula T2 (segon pis), Facultat de Matemàtiques i Informàtica, UB.

A càrrec de: Edmond Koudjinan, Institute of Science and Technology Austria (ISTA)

Títol: On the Birkhoff conjecture for nearly centrally symmetric domains

Resum: In this talk, I will discuss a recent advance on Birkhoff conjecture, namely a proof that: an integrable, nearly centrally symmetric Birkhoff billiard table is necessarily an ellipse. This is done by combining recent breakthroughs by Bialy-Mironov (who prove the conjecture for centrally symmetric Birkhoff billiard tables) and by Kaloshin-Sorrentino (who prove the conjecture for Birkhoff billiard tables close to ellipses). In particular, we shall discuss the use of a nonstandard generating function discovered by Bialy-Mironov.

Dia: Dimecres, 3 d'abril de 2024

Lloc: Aula S01, Facultat de Matemàtiques i Estadística, UPC. Pau Gargallo,14 BCN.

A càrrec de: Andrew Clarke, UPC

Títol: Chaotic properties of billiards in circular polygons

Resum: Circular polygons are closed plane curves formed by concatenating a finite number of circular arcs so that, at the points where two arcs meet, their tangents agree. These curves are strictly convex and 𝒞^{1}, but not 𝒞^{2}. We study the billiard dynamics in domains bounded by circular polygons. We prove that there is a set accumulating on the boundary of the domain in which the return dynamics is semiconjugate to a transitive shift on infinitely many symbols. Consequently the return dynamics has infinite topological entropy. In addition we give an exponential lower bound on the number of periodic orbits of large period, and we prove the existence of trajectories along which the angle of reflection tends to zero with optimal linear speed. These results are based on joint work with Rafael Ramírez-Ros.

Sessió actual.

Last updated: Sun Apr 14 09:35:43 2024