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Series: CDSNS Colloquium

I will discuss a two dimensional spatial pattern formation
problem proposed by Doelman, Sandstede, Scheel, and Schneider in 2003 as
a phenomenological model of convective fluid flow . In the same work
the authors just mentioned use geometric singular perturbation theory to
show that the coexistence of certain spatial patterns is equivalent to
the existence of some heteroclinic orbits between equilibrium solutions
in a four dimensional vector field. More recently Andrea Deschenes,
Jean-Philippe Lessard, Jan Bouwe van den Berg and the speaker have
shown, via a computer assisted argument, that these heteroclinic orbits
exist. Taken together these arguments provide mathematical proof of the
existence of some non-trivial patterns in the original planar PDE. I
will present some of the ingredients of this computer assisted proof.

Series: CDSNS Colloquium

This is the 3rd Jorge Ize Memorial lecture, at IIMAS, Mexico City. We will join a videoconference of the event.

The equations governing the motion of a system consisting of a deformable body attached to a rigid body are the partial differential equations for the deformable body subject to boundary conditions that are the equations of motion for the rigid body. (For the ostensibly elementary problem of a mass point on a light spring, the dynamics of the spring itself is typically ignored: The spring is reckoned merely as a feedback device to transmit force to the mass point.) If the inertia of a deformable body is small with respect to that of a rigid body to which it is attached, then the governing equations admit an asymptotic expansion in a small inertia parameter. Even for the simple problem of the spring considered as a continuum, the asymptotics is tricky: The leading term of the regular expansion is not the usual equation for a mass on a massless spring, but is a curious evolution equation with memory. Under very special physical circumstances, an elementary but not obvious process shows that the solution of this equation has an attractor governed by a second-order ordinary differential equation. (This survey of background material is based upon joint work with Michael Wiegner, J. Patrick Wilber, and Shui Cheung Yip.) This lecture describes the rigorous asymptotics and the dimensions of attractors for the motion in space of light nonlinearly viscoelastic rods carrying heavy rigid bodies and subjected to interesting loads. (The motion of the rod is governed by an 18th-order quasilinear parabolic-hyperbolic system.) The justification of the full expansion and the determination of the dimensions of attractors, which gives meaning to these curious equations, employ some simple techniques, which are briefly described (together with some complicated techniques, which are not described). These results come from work with Suleyman Ulusoy.

Series: CDSNS Colloquium

We present a method to find KAM tori with fixed frequency in
degenerate cases, in which the Birkhoff normal form is singular.
The method provides a natural classification of
KAM tori which is based on Singularity Theory. The
method also leads to effective algorithms of computation,
and we present some numerical results up to the verge of breakdown.
This is a joint work with Alejandra Gonzalez and Rafael de la Llave.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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 $n$.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.
www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf
<http://www.mat.uc.cl/%7Ertiedra/download/Horocycles_Bordeaux_2014.pdf
<http://www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf>>

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.