Georgia Institute of Technology College of Sciences

Suppose n runners are running on a circular track of circumference 1, with all runners starting at the same time and place. Each runner proceeds at their own constant speed. We say that a runner is lonely at some point in time if the distance around the track to the nearest other runner is at least 1/n. For example, if there two runners then there will come a moment when they are at anitpodal points on the track, and at this moment both runners are lonely. The lonely runner conjecture asserts that for every runner there is a point in time when that runner is lonely. This conjecture is over 50 years old and remains widely open.

A coprime matching of two sets of integers is a matching that pairs every element of one set with a coprime element of the other set. We present a recent partial result on the lonely runner conjecture. Coprime matchings of intervals of integers play an central role in the proof of this result.

Joint work with Fei Peng

In recent years tremendous progress was made in understanding the ``inviscid damping" phenomenon near shear flows and vortices, which are steady states for the 2d incompressible Euler equation, especially at the linearized level. However, in real fluids viscosity plays an important role. It is natural to ask if incorporating the small but crucial viscosity term (and thus considering the Navier Stokes equation in a high Reynolds number regime instead of Euler equations) could change the dynamics in any dramatic way. It turns out that for the perturbative regime near a spectrally stable monotonic shear flows in an infinite periodic channel (to avoid boundary layers and long wave instabilities), we can prove uniform-in-viscosity inviscid damping. The proof introduces techniques that provide a unified treatment of the classical Orr-Sommerfeld equation in a way analogous to Rayleigh equations. 

Morse theory establishes a celebrated link between classical gradient dynamics and the topology of the
underlying phase space. It provided the motivation for two independent developments. On the one hand, Conley's
theory of isolated invariant sets and Morse decompositions, which is a generalization of Morse theory, is able
to encode the global dynamics of general dynamical systems using topological information. On the other hand,
Forman's discrete Morse theory on simplicial complexes, which is a combinatorial version of the classical
theory, and has found numerous applications in mathematics, computer science, and applied sciences.
In this talk, we introduce recent work on combinatorial topological dynamics, which combines both of the
above theories and leads as a special case to a dynamical Conley theory for Forman vector fields, and more
general, for multivectors. This theory has been developed using the general framework of finite topological
spaces, which contain simplicial complexes as a special case.

We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is also an indication that Ivrii's Conjecture about the measure set of periodic orbits may not be true.