## Seminars and Colloquia by Series

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.
Wednesday, March 2, 2016 - 13:00 , Location: Skiles 006 , Yiming Long , Nankai University , Organizer: Rafael de la Llave
One of the major tools in the study of periodic solutions of  Hamiltonian systems is the Maslov-type index theory for symplectic  matrix paths. In this lecture, I shall give first a brief introduction  on the Maslov-type index theory for symplectic matrix paths as well as  the iteration theory of this index. As an application of these  theories I shall give a brief survey about the existence, multiplicity  and stability problems on periodic solution orbits of Hamiltonian  systems with prescribed energy, especially those obtained in recent  years. I shall also briefly explain some ideas in these studies, and  propose some open problems.
Monday, February 15, 2016 - 11:00 , Location: Skiles 005 , Eduardo Duenez , University of Texas at San Antonio , Organizer:
The Mean Ergodic Theorem of von Neumann proves the existence of limits of (time) averages for any cyclic group K = {U^n : n \in Z} acting on some Hilbert space H via powers of a unitary transformation U.  Subsequent generalizations apply to so-called _multiple_ ergodic averages when Z is replaced by an arbitrary amenable group G, provided the image group K is nilpotent (Walsh's ergodic 2014 theorem for Z; generalization to G amenable by Zorin-Kranich).  In this talk we survey a framework for mean convergence of polynomial group actions based on continuous model theory.  We prove mean convergence of unitary polynomial Z-actions, and discuss how the full framework accomodates the most recent results mentioned above and allows generaling them.
Monday, February 1, 2016 - 11:00 , Location: Skiles 005 , Geng Chen , Georgia Tech , Organizer:
The nonlinear wave equation: u_{tt} - c(u)[c(u)u_x]_x = 0 is a natural generalization of the linear wave equation. In this talk, we will discuss a recent breakthrough addressing the Lipschitz continuous dependence of solutions on initial data for this quasi-linear wave equation. Our earlier results showed that this equation determines a unique flow of conservative solution within the natural energy space H^1(R). However, this flow is not Lipschitz continuous with respect to the H^1 distance, due to the formation of singularity. To prove the desired Lipschitz continuous property, we constructed a new Finsler type metric, where the norm of tangent vectors is defined in terms of an optimal transportation problem. For paths of piecewise smooth solutions, we carefully estimated how the distance grows in time. To complete the construction, we proved that the family of piecewise smooth solutions is dense, following by an application of Thom's transversality theorem. This is a collaboration work with Alberto Bressan.