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Series: CDSNS Colloquium

We will consider the
Frenkel-Kontorova models and their higher dimensional generalizations
and talk about the corresponding discrete weak KAM theory. The existence
of the discrete weak KAM solutions is related to the additive
eigenvalue problem in
ergodic optimization. In particular, I will show that the discrete weak
KAM solutions converge to the weak KAM solutions of the autonomous
Tonelli Hamilton-Jacobi equations as the time step goes to zero.

Series: CDSNS Colloquium

One dimensional discrete Schrödinger operators arise naturally in modeling
the motion of quantum particles in a disordered medium. The medium is
described by potentials which may naturally be generated by certain ergodic
dynamics. We will begin with two classic models where the potentials are
periodic sequences and i.i.d. random variables (Anderson Model). Then we
will move on to quasi-periodic potentials, of which the randomness is
between periodic and i.i.d models and the phenomena may become more subtle,
e.g. a metal-insulator type of transition may occur. We will show how the
dynamical object, the Lyapunov exponent, plays a key role in the spectral
analysis of these types of operators.

Series: CDSNS Colloquium

The format of this talk is rather non-standard. It is actually a combination of two-three mini-talks. They would include only motivations, formulations, basic ideas of proof if feasible, and open problems. No technicalities. Each topicwould be enough for 2+ lectures. However I know the hard way that in diverse audience, after 1/3 of allocated time 2/3 of people fall asleep or start playing with their tablets. I hope to switch to new topics at approximately right times.The topics will probably be chosen from the list below.“A survival guide for feeble fish”. How fish can get from A to B in turbulent waters which maybe much fasted than the locomotive speed of the fish provided that there is no large-scale drift of the water flow. This is related tohomogenization of G-equation which is believed to govern many combustion processes. Based on a joint work with S. Ivanov and A. Novikov.How can one discretize elliptic PDEs without using finite elements, triangulations and such? On manifolds and even reasonably “nice” mm–spaces. A notion of rho-Laplacian and its stability. Joint with S. Ivanov and Kurylev.One of the greatest achievements in Dynamics in the XX century is the KAM Theory. It says that a small perturbation of a non-degenerate completely integrable system still has an overwhelming measure of invariant tori with quasi-periodicdynamics. What happens outside KAM tori has been remaining a great mystery. The main quantative invariants so far are entropies. It is easy, by modern standards, to show that topological entropy can be positive. It lives, however,on a zero measure set. We were able to show that metric entropy can become infinite too, under arbitrarily small C^{infty} perturbations. Furthermore, a slightly modified construction resolves another long–standing problem of theexistence of entropy non-expansive systems. These modified examples do generate positive positive metric entropy is generated in arbitrarily small tubular neighborhood of one trajectory. The technology is based on a metric theory of“dual lens maps” developed by Ivanov and myself, which grew from the “what is inside” topic.How well can we approximate an (unbounded) space by a metric graph whose parameters (degree of vertices, length of edges, density of vertices etc) are uniformly bounded? We want to control the ADDITIVE error. Some answers (the mostdifficult one is for R^2) are given using dynamics and Fourier series.“What is inside?” Imagine a body with some intrinsic structure, which, as usual, can be thought of as a metric. One knows distances between boundary points (say, by sending waves and measuring how long it takes them to reach specific points on the boundary). One may think of medical imaging or geophysics. This topic is related to the one on minimal fillings, the next one. Joint work with S. Ivanov.Ellipticity of surface area in normed space. An array of problems which go back to Busemann. They include minimality of linear subspaces in normed spaces and constructing surfaces with prescribed weighted image under the Gauss map. I will try to give a report of recentin “what is inside?” mini-talk. Joint with S. Ivanov.More stories are left in my left pocket. Possibly: On making decisions under uncertain information, both because we do not know the result of our decisions and we have only probabilistic data.

Series: CDSNS Colloquium

Given a dynamical system (in finite or infinite dimension) it is very natural to look for finite dimensional invariant subspaces on which the dynamics is very simple. Of particular interest are the invariant tori on which the dynamics is conjugated to a linear one. The problem of persistence under perturbations of such objects has been widely studied starting form the 50's, and this gives rise to the celebrated KAM theory. The aim of this talk is to give an overview of the main difficulties and strategies, having in mind the application to PDEs.

Series: CDSNS Colloquium

We classify the local dynamics near the solitons of the supercritical gKDV equations. We prove that there exists a co-dim 1 center-stable (unstable) manifold, such that if the initial data is not on the center-stable (unstable) manifold then the corresponding forward(backward) flow will get away from the solitons exponentially fast; There exists a co-dim 2 center manifold, such that if the intial data is not on the center manifold, then the flow will get away from the solitons exponentially fast either in positive time or in negative time. Moreover, we show the orbital stability of the solitons on the center manifold, which also implies the global existence of the solutions on the center manifold and the local uniqueness of the center manifold. Furthermore, applying a theorem of Martel and Merle, we have that the solitons are asymptotically stable on the center manifold in some local sense. This is a joint work with Zhiwu Lin and Chongchun Zeng.

Series: CDSNS Colloquium

The study of nonlocal transport in physically relevant systems requires
the formulation of mathematically well-posed and physically meaningful
nonlocal models in bounded spatial domains. The main problem faced by
nonlocal partial differential equations in general,
and fractional diffusion models in particular, resides in the treatment
of the boundaries. For example, the naive truncation of the
Riemann-Liouville fractional derivative in a bounded domain is in
general singular at the boundaries and, as a result, the incorporation
of generic, physically meaningful boundary conditions is not feasible.
In this presentation we discuss alternatives to address the problem of
boundaries in fractional diffusion models. Our main goal is to present
models that are both mathematically well posed
and physically meaningful. Following the formal construction of the
models we present finite-different methods to evaluate the proposed
non-local operators in bounded domains.

Series: CDSNS Colloquium

We present a general mechanism to establish the existence of diffusing
orbits in a large class of nearly integrable Hamiltonian systems. Our
approach relies on successive applications of the `outer dynamics'
along homoclinic orbits to a normally hyperbolic invariant manifold.
The information on the outer dynamics is encoded by a geometrically
defined map, referred to as the `scattering map'.
We find pseudo-orbits of the scattering map that keep moving in some
privileged direction.
Then we use the recurrence property of the `inner dynamics', restricted
to the normally hyperbolic invariant manifold, to return to those
pseudo-orbits.
Finally, we apply topological methods to show the existence of true
orbits that follow the successive applications of the two dynamics.
This method differs, in several crucial aspects, from earlier works.
Unlike the well known `two-dynamics' approach, the method relies
heavily on the outer dynamics alone.
There are virtually no assumptions on the inner dynamics, as its
invariant objects (e.g., primary and secondary tori, lower dimensional
hyperbolic tori and their
stable/unstable manifolds, Aubry-Mather sets) are not used at all.
The method applies to unperturbed Hamiltonians of arbitrary degrees of
freedom that are not necessarily convex.
In addition, this mechanism is easy to verify (analytically or
numerically) in concrete examples, as well as to establish diffusion in
generic systems.

Series: CDSNS Colloquium

We use invariant manifold
results on Banach spaces to conclude the existence of spectral
submanifolds (SSMs) in a class of nonlinear, externally forced beam
oscillations .
Reduction of the governing PDE to the SSM provides an exact
low-dimensional model which we compute explicitly. This model captures
the correct asymptotics of the full, infinite-dimensional
dynamics. Our approach is general enough to admit extensions to other
types of continuum vibrations. The model-reduction procedure we employ
also gives guidelines for a mathematically
self-consistent modeling of damping in PDEs describing structural vibrations.

Series: CDSNS Colloquium

The nonlinear Schroedinger
equation (NLS) can be derived as a formal approximating equation for the
evolution of wave packets in a wide array of nonlinear dispersive PDE’s
including the propagation of waves on the surface of an inviscid
fluid. In
this talk I will describe recent work that justifies this approximation
by exploiting analogies with the theory of normal forms for ordinary
differential equations.

Series: CDSNS Colloquium

It was shown by V.E. Zakharov that the equations for water waves can be posed as a Hamiltonian PDE, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations. This latter is joint work with C. Sulem.