## Seminars and Colloquia by Series

Monday, October 17, 2016 - 11:00 , Location: Skiles 005 , Boris Gutkin , Georgia Tech (School of Physics) , Organizer: Rafael de la Llave
Upon quantization, hyperbolic Hamiltonian systems generically exhibit universal spectral properties effectively described by Random Matrix Theory. Semiclassically this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to single-particle systems. I will discuss an extension of this program to hyperbolic coupled map lattices with a large number of sites (i.e., particles). The crucial ingredient is a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such a symbolic dynamics can be constructed explicitly.
Monday, August 29, 2016 - 11:00 , Location: Skiles 005 , Livia Corsi , Georgia Tech - School of Math , Organizer: Livia Corsi
A metric on the 2-torus T^2 is said to be "Liouville" if in some coordinate system it has the form ds^2 = (F(q_1) + G(q_2)) (dq_1^2 + dq_2^2). Let S^*T^2 be the unit cotangent bundle.A "folklore conjecture" states that if a metric is integrable (i.e. the union of invariant 2-dimensional tori form an open and dens set in S^*T^2) then it is Liouville: l will present a counterexample to this conjecture.Precisely I will show that there exists an analytic, non-separable, mechanical Hamiltonian H(p,q) which is integrable on an open subset U of the energy surface {H=1/2}. Moreover I will show that in {H=1/2}\U it is possible to find hyperbolic behavior, which in turn means that there is no analytic first integral on the whole energy surface.This is a work in progress with V. Kaloshin.
Monday, May 2, 2016 - 11:00 , Location: Skiles 005 , J. Mireles-James , Florida Atlantic Univ. , Organizer: Rafael de la Llave
The Parameterization Method is a functional analytic framework for studying invariant manifolds such as stable/unstable manifolds of periodic orbits and invariant tori.  This talk will focus on numerical applications such as computing manifolds associated with long periodic orbits, and computing periodic invariant circles (manifolds consisting of several disjoint  circles mapping one to another, each of which has an iterate conjugate to an irrational rotation).  I will also illustrate how to combine Automatic Differentiation with the Parameterization Method to simplify problems with non-polynomial nonlinearities.
Monday, May 2, 2016 - 10:00 , Location: Skiles 005 , C.M. Groothedde , V.U. Amsterdam , Organizer: Rafael de la Llave
We shall take a look at computer-aided techniques that can be used to prove the existence of stationary solutions of radially symmetric PDEs. These techniques combine existing numerical methods with functional analytic estimates to provide a computer-assisted proof by means of the so-named 'radii-polynomial' approach.
Monday, April 25, 2016 - 11:00 , Location: Skiles 005 , Tere M. Seara , Univ Polit. Catalunya , Organizer: Rafael de la Llave
Monday, April 25, 2016 - 10:00 , Location: Skiles 005 , Marian Gidea , Yeshiva Univ. , Organizer: Rafael de la Llave
We consider a restricted four-body problem, modeling the dynamics of a light body (e.g., a moon) near a Jupiter trojan asteroid. We study two mechanisms of instability. For the first mechanism, we assume that the orbit of Jupiter is circular, and we investigate the hyperbolic invariant manifolds associated to periodic orbits around the equilibrium points. The conclusion is that the light body can undergo chaotic motions inside the Hill sphere of the trojan, or well  outside that region. For the second mechanism, we consider the perturbative effect due to the eccentricity of the orbit of Jupiter. The conclusion is that the size of the orbit of the light body around the trojan can keep increasing, or keep decreasing, or undergo oscillations. This phenomenon is related to the Arnold Diffusion problem in Hamiltonian dynamics
Monday, April 18, 2016 - 11:05 , Location: Skiles 005 , Haomin Zhou , School of Math, Georgia Tech , Organizer: Haomin Zhou
In this talk, I will present new models to describe the evolution of games. Our dynamical system models are inspired by the Fokker-Planck equations on graphs. We will present properties of the models, their connections to optimal transport on graphs, and computational examples for generalized Nash equilibria. This presentation is based on a recent joint work with Professor Shui-Nee Chow and Dr. Wuchen Li.
Tuesday, April 12, 2016 - 13:30 , Location: Skiles 005 , Richard Montgomery , Univ. California Santa Cruz , Organizer: Rafael de la Llave
Video Conference David Alcaraz confernce.  Newton's famous    three-body problem defines dynamics on the space of congruence classes of triangles in the plane.  This space is a three-dimensional non-Euclidean  rotationally symmetric metric space centered'' on  the  shape sphere. The shape sphere is  a two-dimensional sphere whose points represent   oriented similarity classes of planar triangles. We describe how the sphere arises from the three-body problem and  encodes its dynamics.    We will  see how   the classical solutions of Euler and Lagrange, and the relatively recent figure 8 solution are encoded as points or curves on  the sphere.  Time permitting, we will show how the sphere pushes us to formulate natural topological-geometric questions about three-body solutions and helps supply the answer to some of these questions.  We may take a brief foray into the planar N-body problem and  its  associated shape sphere'' :   complex projective N-2 space.
Monday, April 11, 2016 - 11:00 , Location: Skiles 005 , Angel Jorba , Univ. of Barcelona , Organizer: Rafael de la Llave
Dynamical systems have proven to be a useful tool for the design of space missions. For instance, the use of invariant manifolds is now common to design transfer strategies. Solar Sailing is a proposed form of spacecraft propulsion, where large membrane mirrors take advantage of the solar radiation pressure to push the spacecraft. Although the acceleration produced by the radiation pressure is smaller than the one achieved by a traditional spacecraft it is continuous and unlimited. This makes some long term missions more accessible, and opens a wide new range of possible applications that cannot be achieved by a traditional spacecraft. In this presentation we will focus on the dynamics of a Solar sail in a couple of situations. We will introduce this problem focusing on a Solar sail in the Earth-Sun system. In this case, the model used will be the Restricted Three Body Problem (RTBP) plus Solar radiation pressure. The effect of the solar radiation pressure on the RTBP produces a 2D family of "artificial'' equilibria, that can be parametrised by the orientation of the sail. We will describe the dynamics around some of these "artificial'' equilibrium points. We note that, due to the solar radiation pressure, the system is Hamiltonian only for two cases: when the sail is perpendicular to the Sun - Sail line; and when the sail is aligned with the Sun - sail line (i.e., no sail effect). The main tool used to understand the dynamics is the computation of centre manifolds. The second example is the dynamics of a Solar sail close to an asteroid. Note that, in this case, the effect of the sail becomes very relevant due to the low mass of the asteroid. We will use, as a model, a Hill problem plus the effect of the Solar radiation pressure, and we will describe some aspects of the natural dynamics of the sail.
Monday, March 7, 2016 - 11:00 , Location: Skiles 005 , Qingtian Zhang , Penn State University , Organizer:
Abstract: In this talk, I will present the uniqueness of conservative solutions to Camassa-Holm and two-component Camassa-Holm equations. Generic regularity and singular behavior of those solutions are also studied in detail. If time permitting, I will also mention the recent result on wellposedness of cubic Camassa-Holm equations.