Georgia Institute of Technology College of Sciences

 I will discuss the soliton resolution and asymptotic stability problems for the sine-Gordon equation. It is known that the obstruction to the asymptotic stability for the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds. This is joint work with Jiaqi Liu and Bingying Lu.

 We study a family of dynamical systems obtained by coupling a chaotic (Anosov) map on the two-dimensional torus -- the chaotic variable -- with the identity map on the one-dimensional torus -- the neutral variable -- through a dissipative interaction. We show that the  two systems synchronize, in the sense that the trajectories evolve toward an attracting invariant manifold, and that the full dynamics is conjugated to its linearization around the invariant manifold. When the interaction is small, the evolution of the neutral variable is very close to the identity and hence the neutral variable appears as a slow variable with respect to the fast chaotic variable: we show that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast  variable.

The seminar can also be accessed online via zoom link: Meeting ID: 961 2577 3408

Gutzwiller semi-classical quantization, Boven-Sinai-Ruelle dynamical zeta functions for chaotic dynamical systems, statistical mechanics partition functions, and path integrals of quantum field theory are often presented in ways that make them appear as disjoint, unrelated theories. However, recent advances in describing fluid turbulence by its dynamical, deterministic Navier-Stokes underpinning, without any statistical assumptions, have led to a common field-theoretic description of both (low-dimension) chaotic dynamical systems, and (infinite-dimension) spatiotemporally turbulent flows. 

I have described the remarkable experimental progress connecting turbulence to deterministic dynamics in the Sept 24, 2023 colloquium (the recoding is available on the website below). In this seminar I will use a lattice discretized field theory in 1 and 1+1 dimension to explain how temporal `chaos', `spatiotemporal chaos' and `quantum chaos' are profitably cast into the same field-theoretic framework.

https://ChaosBook.org/overheads/spatiotemporal/

The talk will also be on Zoom:   GaTech.zoom.us/j/95338851370

Quantum mechanics and diffusion on a network, in the sense of a metric graph, are locally one-dimensional, but the way the graph is connected can add multidimensional features and some strange phenomena.  Quantum graphs have been an active area of research since the 1990s.  I’ll review the subject and share some ideas about analyzing Schrödinger and heat equations on metric graphs, through the associated eigenvalue problem and the heat kernel.

This talk is based on a 2022 article with David Borthwick and Kenny Jones, and on work in progress with David Borthwick, Anna Maltsev, and Haozhe Yu.