Seminars and Colloquia by Series

Monday, April 6, 2015 - 11:00 , Location: Skiles 005 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
We present a methodology to rigorously validate a given approximation of a quasi-periodic Lagrangian torus of a symplectic map. The approach consists in verifying the hypotheses of a-posteriori KAM theory based of the parameterization method (following Rafael de la Llave and collaborators). A crucial point of our imprementation is an analytic Lemma that allows us to control the norm of periodic functions using their discrete Fourier transform. An outstanding consequence of this approach it that the computational cost of the validation is assymptotically equivalent of the cost of the numerical computation of invariant tori using the parametererization method. We pretend to describe some technical aspects of our implementation. This is a work in progress joint with Jordi-Lluis Figueras and Alejandro Luque.
Monday, March 30, 2015 - 11:00 , Location: Slikes 005 , Chungen Liu , Nankai University, China , Organizer: Chongchun Zeng
The iteration theory for Lagrangian Maslov index is a very useful tool    in studying the multiplicity of brake orbits of Hamiltonian systems.  In  this talk, we show how to use this theory to prove that there exist at    least $n$ geometrically distinct brake orbits on every $C^2$ compact convex symmetric hypersurface in $\R^{2n}$ satisfying the reversible condition. As a consequence, we show that if    the Hamiltonian function is convex and even, then Seifert conjecture of 1948 on the multiplicity of brake orbits holds for any positive integer $n$.
Monday, March 23, 2015 - 11:00 , Location: Skiles 005 , Adam Fox , Western New England Univ. , Organizer: Rafael de la Llave
The Standard Map is a discrete time area-preserving dynamical system and is one of the simplest of such systems to exhibit chaotic dynamics.  Traditional studies of the Standard Map have employed symmetric forcing functions that do not induce a net flux.  Although the dynamics of these maps is rich there are many systems which cannot be modeled with these restrictions.  In this talk we will explore the dynamics of the Standard Map when the forcing is asymmetric and induces a positive flux on the system.  We will introduce new numerical methods to study these dynamics and give an overview of how transport in the system changes under these new forces.
Monday, March 9, 2015 - 11:00 , Location: Skiles 005 , Rodrigo Trevino , Courant Inst. of Mathematical Sciences, NYU , Organizer: Rafael de la Llave
A Penrose tiling is an example of an aperiodic tiling and its vertex set is an example of an aperiodic point set (sometimes known as a quasicrystal). There are higher rank dynamical systems associated with any aperiodic tiling or point set, and in many cases they define a uniquely ergodic action on a compact metric space. I will talk about the ergodic theory of these systems. In particular, I will state the results of an ongoing work with S. Schmieding on the deviations of ergodic averages of such actions for point sets, where cohomology plays a big role. I'll relate the results to the diffraction spectrum of the associated quasicrystals.
Monday, February 23, 2015 - 11:00 , Location: Skiles 005 , Rafael Tiedra de Aldecoa , Pontificia Univ. Catolica de Chile , Organizer: Rafael de la Llave
We show that all time changes of the horocycle flow on compact surfaces of constant negative curvature have purely absolutely continuous spectrum in the orthocomplement of the constant functions. This provides an answer to a question of A. Katok and J.-P. Thouvenot on the spectral nature of time changes of horocycle flows. Our proofs rely on positive commutator methods for self-adjoint operators and the unique ergodicity of the horocycle flow. < <>>
Thursday, February 19, 2015 - 13:00 , Location: Skiles 006 , Yannan Shen , Univ. of Texas at Dallas , Organizer: Rafael de la Llave
We develop a mathematical model for  ultra-short pulse propagation in nonlinear metamaterials characterized by a weak Kerr-type nonlinearity in their dielectric response. The fundamental equation in the model is the short-pulse equation (SPE) which will be derived in frequency band gaps. We use a multi-scale ansatz to relate the SPE to the nonlinear Schroedinger equation, thereby characterizing the change of width of the pulse from the ultra short regime to the classical slow varying envelope approximation. We will discuss families of solutions of the SPE in characteristic coordinates, as well as discussing the global wellposedness of generalizations of the model that describe uni- and bi-directional nonlinear waves.
Monday, February 16, 2015 - 11:00 , Location: Skiles 005 , Lei Zhang , Georgia Institute of Technology , Organizer: Lei Zhang
We consider an atomic model of deposition of materials over a quasi-periodic medium. The atoms of the deposited material interact with the medium (a quasi-periodic interaction) and with their nearest neighbors (a harmonic interaction). This is a quasi-periodic version of the well known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show  that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type. The iterative method is not based on transformation theory as the most usual KAM theory, but it is based on a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of the equilibrium equilibrium equation.
Tuesday, February 10, 2015 - 11:00 , Location: Skiles 005 , Sara Lapan , Northwestern University , , Organizer:
Given a holomorphic map of C^m to itself that fixes a point, what happens to points near that fixed point under iteration?  Are there points attracted to (or repelled from) that fixed point and, if so, how?  We are interested in understanding how a neighborhood of a fixed point behaves under iteration.  In this talk, we will focus on maps tangent to the identity.  In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point.  This theorem from the early 1900s continues to serve as inspiration for this study in higher dimensions.  In dimension 2 our picture of a full neighborhood of a fixed point is still being constructed, but we will discuss some results on what is known, focusing on the existence of a domain of attraction whose points converge to that fixed point.
Monday, October 20, 2014 - 11:00 , Location: Skiles 005 , Fabio Difonzo , School of Mathematics, Georgia Institute of Technology , Organizer:
We consider several possibilities on how to select a Filippov sliding vector field on a co-dimension 2 singularity manifold, intersection of two co-dimension 1 manifolds, under the assumption of general attractivity. Of specific interest is the selection of a smoothly varying Filippov sliding vector field. As a result of our analysis and experiments, the best candidates of the many possibilities explored are based on the so-called barycentric coordinates: in particular, we choose what we call the moments solution. We then examine the behavior of the moments vector field at first order exit points, and show that it aligns smoothly with the exit vector field. Numerical experiments illustrate our results and contrast the present method with other choices of Filippov sliding vector field. We further present some minimum variation properties, related to orbital equivalence, of Filippov solutions for the co-dimension 2 case in \R^{3}.
Monday, October 6, 2014 - 11:00 , Location: Skiles 005 , William Gignac , School of Mathematics Georgia Inst. Technology , Organizer: Rafael de la Llave
In this talk, we will discuss a question posed by Vladimir Arnold some twenty years ago, in a subject he called "dynamics of intersections." In the simplest setting, the question is the following: given a (discrete time) holomorphic dynamical system on a complex manifold X and two holomorphic curves C and D in X which pass through a fixed point P of the system, how quickly can the local intersection multiplicies at P of C with the iterates of D grow in time? Questions like this arise naturally, for instance, when trying to count the periodic points of a dynamical system. Arnold conjectured that this sequence of intersection multiplicities can grow at most exponentially fast, and in fact we can show this conjecture is true if the curves are chosen to be suitably generic. However, as we will see, for some (even very simple) dynamical systems one can choose curves so that the intersection multiplicities grow as fast as desired. We will see how to construct such counterexamples to Arnold's conjecture, using geometric ideas going back to work of Yoshikazu Yamagishi.