# Applied/ACMS/absS22

# ACMS Abstracts: Spring 2022

### Alex Townsend (Cornell)

Title: What networks of oscillators spontaneously synchronize?

Abstract: Consider a network of identical phase oscillators with sinusoidal coupling. How likely are the oscillators to spontaneously synchronize, starting from random initial phases? One expects that dense networks of oscillators have a strong tendency to pulse in unison. But, how dense is dense enough? In this talk, we use techniques from numerical linear algebra, computational algebraic geometry, and dynamical systems to derive the densest known networks that do not synchronize and the sparsest ones that do. We will find that there is a critical network density above which spontaneous synchrony is guaranteed regardless of the network's topology, and prove that synchrony is omnipresent for random networks above a lucid threshold. This is joint work with Martin Kassabov, Steven Strogatz, and Mike Stillman.

### Geoffrey Vasil (Sydney)

Title: The mechanics of a large pendulum chain

Abstract: I’ll discuss a particular high-dimensional system that displays subtle behaviour found in the continuum limit. The only catch is that it formally shouldn’t, which raises a few questions. When is a discrete system large enough to be called continuous? When are approximate (broken) symmetries good enough to be treated like the real thing? When and why does a fluid approximation work as well as we like to assume? What does all this say about observables and the approach to equilibria? The particular system I have in mind is a large ideal pendulum chain, and it’s cousin the continuous flexible string. I propose that the pendulum chain is a perfect model system to study notoriously difficult phenomena such as vortical turbulence, waves, cascades and thermalisation, but with many fewer degrees of freedom than a three-dimensional fluid.