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Series: CDSNS Colloquium

The Allen-Cahn equation is a second order semilinear
elliptic PDE that arises in mathematical models describing
phase transitions between two constant states. The
variational structure of this equation allows us to study
energy-minimal phase transitions, which correspond to
uniformly bounded non-constant globally minimal solutions.
The set of such solutions depends heavily on the geometry
of the underlying space. In this talk we shall focus on the
case where the underlying space is a Cayley graph of a
group with the word metric. More precisely, we assume that
the group is hyperbolic and show that there exists a
minimal solution with any “nice enough” asymptotic
behaviour prescribed by the two constant states. The set in
the Cayley graph where the phase transition for such a
solution takes place corresponds to a solution of an
asymptotic Plateau problem.

Series: CDSNS Colloquium

In this talk, we will consider semiconcavity of viscosity solutions for a class of degenerate elliptic integro-differential equations in R^n. This class of equations includes Bellman equations containing operators of Levy-Ito type. Holder and Lipschitz continuity of viscosity solutions for a more general class of degenerate elliptic integro-differential equations are also provided.

Series: CDSNS Colloquium

We prove the existence of diffusion orbits drifting along
heteroclinic chains of normally hyperbolic 3-dimensional cylinders, under
suitable assumptions on the dynamics on the cylinders and on their
homoclinic/heteroclinic connections. These assumptions are satisfied in the
a priori stable case of the Arnold diffusion problem. We provide a
geometric argument that extends Birkhoff's procedure for constructing
connecting orbits inside a zone of instability for a twist map on the
annuls. This is joint work with J.-P. Marco.

Series: CDSNS Colloquium

Let f be a rational self-map
of the complex projective plane. A central problem when analyzing the
dynamics of f is to understand the sequence of degrees deg(f^n) of the
iterates of f. Knowing the growth rate and structure of this sequence in
many
cases enables one to construct invariant currents/measures for dynamical
system as well as bound its topological entropy. Unfortunately, the
structure of this sequence remains mysterious for general rational maps.
Over the last ten years, however, an approach
to the problem through studying dynamics on spaces of valuations has
proved fruitful. In this talk, I aim to discuss the link between
dynamics on valuation spaces and problems of degree/order growth in
complex dynamics, and discuss some of the positive results
that have come from its exploration.

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