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Series: CDSNS Colloquium

I will describe several
recent results with N. Nadirashvili where we construct extremal metrics
for eigenvalues on riemannian surfaces. This involves the study of a
Schrodinger operator. As an application, one gets isoperimetric
inequalities on the 2-sphere
for the third eigenvalue of the Laplace Beltrami operator.

Series: CDSNS Colloquium

We investigate the existence of quasi-periodic solutions for state-dependent delay
differential equationsusing the parameterization
method, which is different from the usual way-working on the solution
manifold. Under the assumption of finite-time differentiability of
functions and exponential dichotomy, the existence and smoothness of
quasi-periodic solutions are investigated by using contraction
arguments We also develop a KAM
theory to seek analytic quasi-periodic solutions. In contrast with the finite differentonable theory, this requires adjusting parameters. We prove that the set of parameters which guarantee the
existence of analytic quasi-periodic solutions is of positive measure.
All of these results are given in an a-posterior form. Namely, given a
approximate solution satisfying some non-degeneracy conditions, there is
a true solution nearby.

Series: CDSNS Colloquium

We
construct invariant manifolds of interior multi-spike states for the
nonlinear Cahn-Hilliard equation and then investigate the dynamics on
it. An equation for the motion of the spikes is derived. It turns out
that the dynamics of interior spikes has a global character and each
spike interacts with all the others and with the boundary. Moreover, we
show that the speed of the interior spikes is super slow, which
indicates the long time existence of dynamical multi-spike solutions in
both positive and negative time. This result is obtained through the
application of a companion abstract result concerning the existence of
truly invariant manifolds with boundary when one has only approximately
invariant manifolds.

Series: CDSNS Colloquium

In this talk we examine the typical behavior of a trajectory of a
piecewise smooth system in the neighborhood of a co-dimension 2
discontinuity manifold $\Sigma$. It is well known that (in the class
of Filippov vector fields, and under commonly occurring conditions) one may
anticipate sliding motion on $\Sigma$. However, this motion itself is not
in general uniquely defined, and recent contributions in the literature
have been trying to resolve this ambiguity either by justifying a
particular selection of a Filippov vector field or by substituting the
original discontinuous problem with a regularized one.
However, in this talk, our concern is different: we look at what we should
expect of a typical solution of the given discontinuous system in a
neighborhood of $\Sigma$. Our ultimate goal is to detect properties that
are satisfied by a sufficiently wide class of discontinuous systems and
that (we believe) should be preserved by any technique employed to
define a sliding
solution on $\Sigma$.

Series: CDSNS Colloquium

In this
talk, the existence, stability, and multiplicity of spatially
nonhomogeneous steady-state solution and periodic solutions for a
reaction–diffusion model with nonlocal delay effect and Dirichlet
boundary condition are investigated by using Lyapunov–Schmidt
reduction. Moreover, we illustrate our general results by
applications to models with a single delay and one-dimensional
spatial domain.

Series: CDSNS Colloquium

We study the existence
and branching patterns of wave trains in a two-dimensional lattice
with linear and nonlinear coupling between nearest particles and a
nonlinear substrate potential. The wave train equation of the
corresponding discrete nonlinear equation is formulated as an
advanced-delay differential equation which is reduced by a
Lyapunov-Schmidt reduction to a finite-dimensional bifurcation
equation with certain symmetries and an inherited Hamiltonian
structure. By means of invariant theory and singularity theory, we
obtain the small amplitude solutions in the Hamiltonian system near
equilibria in non-resonance and $p:q$ resonance, respectively. We
show the impact of the direction $\theta$ of propagation and obtain
the existence and branching patterns of wave trains in a
one-dimensional lattice by investigating the existence of travelling
waves of the original two-dimensional lattice in the direction
$\theta$ of propagation satisfying $\tan\theta$ is rational

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.