Georgia Institute of Technology College of Sciences

The workshop is a part of GTMAP activities of  the thematic semester on Dynamics (Fall 2017). 

This is a half day workhop on a panorama of mathematical questions in Dynamical Systems, presented by some people in the school of Mathematics and Physics.

2-2:30 Chongchun Zeng on "Wind-driven water waves and hydrodynamic instability"
2:30-3 Leonid Bunimovich on "Isospectral compression of multidimensional systems and networks"
3-3:30 Roman Grigoriev on "Streamwise Localization of Traveling Wave Solutions in Channel Flow"

3:30-4 BREAK

4-4:30 Albert Fathi on "The Pageault barrier: a tool to find the recurrent part of a dynamical system"
4:30-5 Rafael de la Llave on "Perturbations of quasi-periodic orbits: From Theory to computations."

Prof. Chongchun Zeng (GT MATH)

Talk title: Wind-driven water waves and hydrodynamic instability

Abstract:  In this talk, we consider the mathematical theory of wind-generated water waves in the framework of the interface problem between two incompressible inviscid fluids under the influence of gravity. This entails the careful study of the stability of the shear flow solutions to the interface problem of the two-phase Euler equation. Based on a rigorous derivation of the linearized equations about shear flow solutions, we obtained rigorously the linear instability criterion of Miles due to the presence of the critical layer in the steady shear flows. Our analysis is valid even in the presence of surface tension and a vortex sheet (discontinuity in the tangential velocity across the air--sea interface).  We are thus able to give a unified equation including the Kelvin--Helmholtz and quasi-laminar models of wave generation put forward by Miles.

Prof. Leonid Bunimovich (GT MATH)

Talk title: Isospectral compression of multidimensional systems and networks

Abstract: When dealing with large networks it is tempting to compress such object while keeping, as much as possible, information about it. for a great majority of real world networks all information about them is contained in their weighted adjacency. Laplace or some other matrix. I will talk about theory of isospectral transformations which  in particular allows to compress a network keep all information about spectrum and eigenvectors of these matrices. Applications of this theory allowed to advance several areas of research and extract more information from real data than various other methods. It also allows to get new types of visualization of networks via building various their skeletons. This approach/theory is applicable to undirected and to directed networks.

Prof. Roman Grigoriev (GT PHYSICS)

Talk Title: Streamwise Localization of Traveling Wave Solutions in Channel Flow

Abstract: Channel flow of an incompressible fluid at Reynolds numbers above 2400 possesses a number of different spatially localized solutions which approach the laminar flow far upstream and downstream. We use one such relative time-periodic solution that corresponds to a spatially localized version of a Tollmien-Schlichting wave to illustrate how the upstream and downstream asymptotics can be computed analytically. In particular, we show that for these spanwise uniform states the asymptotics predict exponential localization that has been observed for numerically computed solutions of several canonical shear flows but never properly understood theoretically.

Prof. Albert Fathi (GT MATH)

Talk Title: The Pageault barrier: a tool to find the recurrent part of a dynamical system

Abstract: If f is a (invertible) dynamical system on a (compact) space X. Usually one is interested in the behavior of the sequence f^n(x), for n -> infinity, but also in control problems in the Lyapunov stable states (for example a stable attracting equilibrium) and sometime in the Lyapunov saddle state (like a saddle point of a gradient system). To find these states Conley introduced an approach through chain recurrence and Lyapunov functions. The purpose of this lecture is to introduce the Pageault barrier to study these problems. This a more metric approach, meaning that we can introduce a kind of distance that allows to locate the pieces where the recurrent dynamics is occurring.

Prof. Rafael de la Llave (GT MATH)

Talk Title: Perturbations of quasi-periodic orbits: From Theory to computations.

Abstract: Since the time if Hyparco, it was known that the motion of celestial bodies is approximately given by epicycles. In modern language, expressed in Fourier series of a few frequencies, which we now call quasi-periodic. In modern language, these quasi-periodic solutions are landmarks that organize the behavior of a Hamiltonian system.

The mathematically rigorous theory of persistence of quasi-periodic solutions is rather recent. Late 50's, early 60's with the work of Kolmogorov, Arnold, Moser (KAM). The theory is rather subtle since it involves high regularity as well as number theory. We will describe some new proofs that lead to very efficient algorithms as well as provide condition numbers that guarantee that the computations are correct.

Bio]
1) Prof. Chongchun Zeng: After post-docs at NYU, and 5 years at Univ of Virginia, Chongchun Zeng joined the School of Mathematics in 2005, where he has been professor since 2009.  Zeng is a leader in use of dynamical systems tools to study PDEs.

2) Prof. Leonid Bunimovich: After training at Moscow University and the Institute of Theoretical Physics in Moscow, Bunimovich joined the School of Mathematics, where he is a Regents' professor.  World famous for the discovery of a fundamental mechanism of chaos, Bunimovch has wide ranging interests in ergodic theory, statistical mechanics, mathematical biology, and probability.

3) Prof. Roman Grigoriev obtained his MS from Moscow State University and PhD from Caltech. After two years as a postdoc at the University of Chicago, he joined the School of Physics in 2000.  His research group studies a variety of topics related to nonlinear dynamics and pattern formation in out-of-equilibrium systems. Iinterests include deterministic modeling of fluid turbulence, phase change and heat transfer in fluids, and dynamical and topological description of cardiac arrhythmias.

4) Prof. Albert Fathi was educated in Paris and Orsay. He held positions in Paris (France), Gainesville (Florida) and Lyon (France). His mathematical interests are Topology, Geometry, Dynamical Systems and PDE.

5) Prof. Rafael de la Llave joined the School of Mathematics in August 2010, after having held positions at Princeton and the Univ. of Texas at Austin. He has wide ranging interests in dynamical systems and mathematical physics, spanning from KAM theory and hyperbolic dynamics to stability of matter and optics.  He also has interest in combining analytical and numerical techniques to study dynamics.