Seminars and Colloquia by Series

Monday, April 24, 2017 - 11:00 , Location: Skiles 005 , Zehnghe Zhang , Rice University , Organizer: Rafael de la Llave
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.
Tuesday, April 11, 2017 - 14:05 , Location: Skiles 006 , Dmitri Burago , Penn State , Organizer: Igor Belegradek
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.
Monday, April 10, 2017 - 11:00 , Location: Skiles 006 , Prof. Michela Procesi , Dipartimento di Matematica e Fisica - Universita' di Roma Tre , Organizer: Livia Corsi
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.
Monday, March 6, 2017 - 11:00 , Location: Skiles 005 , Dr. Jiayin Jin , Georgia Tech , Organizer: Jiayin Jin
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.
Friday, March 3, 2017 - 11:00 , Location: Skiles 005 , Diego Del Castillo-Negrete , Oak Ridge National Lab. , Organizer: Rafael de la Llave
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.
Monday, January 30, 2017 - 11:00 , Location: Skiles 005 , T.M-Seara , Univ. Polit. Catalunya , Organizer: Rafael de la Llave
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.
Friday, January 13, 2017 - 11:00 , Location: Skiles 005 , Florian Kogelbauer , ETH (Zurich) , Organizer: Rafael de la Llave
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.
Wednesday, November 30, 2016 - 11:00 , Location: Skiles 005 , Prof. Eugene Wayne , Boston University , Organizer: Chongchun Zeng
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.
Wednesday, November 16, 2016 - 11:00 , Location: 006 , Prof. Walter Craig , McMaster University , Organizer: Livia Corsi
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.
Monday, October 24, 2016 - 11:06 , Location: Skiles 006 , Hongkun Zhang , U. Mass Amherst , Organizer:
We investigate deterministic superdiusion in nonuniformly hyperbolic system models in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which differs from the usual requirement that the mean square displacement grow asymptotically linearly in time. We obtain an explicit formula for the superdiffusion constant in terms of the ne structure that originates in the phase transitions as well as the geometry of the configuration domains of the systems. Models that satisfy our main assumptions include chaotic Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply to other nonuniformly hyperbolic systems with slow correlation decay rates of order O(1/n)