Georgia Institute of Technology College of Sciences

The talk will survey the main definitions and properties of patchy vector fields and patchy feedbacks, with applications to asymptotic feedback stabilization and nearly optimal feedback control design. Stability properties for discontinuous ODEs and robustness of patchy feedbacks will also be discussed.

I propose a generalization of Hopf fibrations to quotient the streamwise translation symmetry of water waves and turbulent pipe flows viewed as dynamical systems. In particular, I exploit the geometric structure of the associated high dimensional state space, which is that of a principal fiber bundle. Symmetry reduction analysis of experimental data reveals that the speeds of large oceanic crests and turbulent bursts are associated with the dynamical and geometric phases of the corresponding orbits in the fiber bundle. In particular, in the symmetry-reduced frame I unveil a pattern-changing dynamics of the fluid structures, which explains the observed speed u ≈ Ud+Ug of intense extreme events in terms of the geometric phase velocity Ug and the dynamical phase velocity Ud associated with the orbits in the bundle. In particular, for oceanic waves Ug/Ud~-0.2 and for turbulent bursts Ug/Ud~0.43 at Reynolds number Re=3200.

This is a continuation of the talk from last week.

An essential feature of the theory of 3-manifolds fibering over the circle is that they often admit infinitely many distinct structures as a surface bundle. In four dimensions, the story is much more rigid: a given 4-manifold admits only finitely many fiberings as a surface bundle over a surface. But how many is “finitely many”? Can a 4-manifold possess three or more distinct surface bundle structures? In this talk, we will survey some of the beautiful classical examples of surface bundles over surfaces with multiple fiberings, and discuss some of our own work. This includes a rigidity result showing that a class of surface bundles have no second fiberings whatsoever, as well as the first example of a 4-manifold admitting three distinct surface bundle structures, and our progress on a quantitative version of the “how many?” question.