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Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

Solving numerically kinetic equations requires high computing
power and storage capacity, which compels us to derive more tractable,
dimensionally reduced models. Here we investigate fluid models derived from
kinetic equations, typically the Vlasov equation. These models have a lower
numerical cost and are usually more tangible than their kinetic counterpart
as they describe the time evolution of quantities such as the density ρ,
the fluid velocity u, the pressure p, etc. The reduction procedure
naturally leads to the need for a closure of the resulting fluid equations,
which can be based on various assumptions. We present here a strategy for
building fluid models from kinetic equations while preserving their
Hamiltonian structure. Joint work with M. Perin and E. Tassi
(CNRS/Aix-Marseille University) and P.J. Morrison (University of Texas at
Austin).

Series: CDSNS Colloquium

The study of random
Hamilton-Jacobi PDE is motivated by mathematical physics, and in
particular, the study of random Burgers equations. We will show that,
almost surely, there is a unique stationary solution, which also has
better regularity than expected.
The solution to any initial value problem converges to the stationary
solution exponentially fast. These properties are closely related to the
hyperbolicity of global minimizer for the underlying Lagrangian system.
Our result generalizes the one-dimensional
result of E, Khanin, Mazel and Sinai to arbitrary dimensions. Based on
joint works with K. Khanin and R. Iturriaga.

Series: CDSNS Colloquium

The talk concerns limit behaviors of stationary measures of diffusion
processes generated from white-noise perturbed systems of ordinary
differential equations.
By relaxing the notion of Lyapunov functions associated with the
stationary Fokker-Planck equations, new existence and non-existence
results of stationary measures will be presented. As noises vanish,
concentration and limit behaviors of stationary measures will be
described with particular attentions paying to the special role played
by multiplicative noises. Connections to problems such as stochastic
stability, stochastic bifurcations, and thermodynamics limits will also
be discussed.

Series: CDSNS Colloquium

We prove results concerning the equidistribution of some
"sparse" subsets of orbits of horocycle flows on $SL(2, R)$ mod lattice.
As a consequence of our analysis, we recover the best known rate of growth
of Fourier coefficients of cusp forms for arbitrary noncompact lattices of
$SL(2, R)$, up to a logarithmic factor. This talk addresses joint work
with Livio Flaminio, Giovanni Forni and Pankaj Vishe.

Series: CDSNS Colloquium

We will consider (sub)shifts with complexity such that the difference
from n to n+1 is constant for all large n. The shifts that arise
naturally from interval exchange transformations belong to this class.
An interval exchange transformation on d intervals has at most d/2
ergodic probability measures. We look to establish the correct bound for
shifts with constant complexity growth. To this end, we give our
current bound and discuss further improvements when more assumptions are
allowed. This is ongoing work with Michael Damron.

Series: CDSNS Colloquium

I present a formalism and an computational scheme to
quantify the dynamics of grain boundary migration in polycrystalline
materials, applicable to three-dimensional microstructure data obtained
from non-destructive coarsening experiments. I
will describe a geometric technique of interface tracking using
well-established optimization algorithms and demonstrate how, when
coupled with very basic physical assumptions, one can effectively
measure grain boundary energy density and mobility of a given
misorientation type in the two-parameter subspace of boundary
inclinations. By doing away with any specific model or parameterization
for the energetics, I seek to have my analysis applicable to general
anisotropies in energy and mobility. I present results
in two proof-of-concept test cases, one first described in closed form
by J. von Neumann more than half a century ago, and the other which
assumes analytic but anisotropic energy and mobility known in advance.

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.