Against most odds, Martin Wechselberger and Georg Gottwald were able to hold the annual Sydney Dynamics Group (SDG) conference for 2020 in person. Not only was the pandemic threatening to throw all plans to the wind, but the Jemby Rinjah Eco Lodge in the Blue Mountains, where the conference is usually held in NSW, was tragically burnt down during the 2019-2020 bushfire season. So by December, having spent most of the year researching and teaching from our makeshift offices at home, we were especially grateful to have the opportunity to come together and collaborate over four days in Jervis Bay. 

The first day consisted of postgraduate talks - one-slide presentations followed by discussion sessions. Lachlan Burton started the conference off with his research on the 3-body problem, specifically looking at classifying structure in the interactions between two heavy bodies and one satellite. He showed how from any initial condition, three different satellite trajectories are possible: escape from, collision with, or quasiperiodic orbit around a heavy body. He coloured these in a high-resolution (1GB!) `orbit type diagram' where the chaotic dynamics were evident in the dense mixing of these initial conditions. The following presentations covered a broad range of topics, including instability criteria for standing wave solutions to the nonlinear Schrödinger equation by Mitchell Curran, social media sentiment derived from vocabulary usage by Cathy Liu, and how the generalised spheroidal equation appears in the dynamics of Lagrange tops (spinning about their axis with fixed base points), Kerr black holes (rotating with no electrical charge), and many systems that display quantum monodromy by Sean Dawson.

Eric Hester showed us the remarkable effectiveness of signed-distance coordinates in resolving the complexity of multi-scaled boundary layer dynamics. These coordinates are a generalisation of polar coordinates to arbitrary boundaries, where an orthogonal system between an arbitrary point and the boundary is constructed. James Chok presented his bold pursuit in constructing a basis of the vector space of weight matrices output in machine learning simulations using Hahn polynomials. And Michael Denes' work on identifying coherent structures in oceanographic data led to discussion about the broader unresolved debate on how to define such structures in the first place. 

The extended talks by postdocs and senior staff on the next three mornings were followed by Q&A afternoon sessions. Sasha Fish gave an animated talk on a property of group actions that arises as a stronger form of topological mixing in dynamical systems: the Glasner property. An example is the action by multiplication of the natural numbers on the one-dimensional torus: \(nA\) is epsilon-dense for \(n\) a natural number, \(A\) an infinite subset of the torus, and any positive epsilon. Sasha presented numerous open problems concerning different systems. Robby Marangell updated us on his research on the Maslov index, specifically his current attempts to extend the theory to non-Hamiltonian systems, and Georg Gottwald presented a machine learning approach to spectrally decomposing the Koopman operator using extended dynamic mode decomposition. Lauren Smith showed some nice results concerning synchronisation conditions of oscillator networks with sparse internal coupling. Specifically, by subdividing the network and applying collective coordinate reduction, Lauren needs only statistical properties of the full sparse network to determine features such as the critical coupling strength required for full synchronisation, without which lowest degree nodes remain unsynchronised with the main cluster.

Sean Gasoriek gave an interactive presentation on magnetic billiards. He presented numerous open problems relating to the algebraic integrability of systems of different geometries, comparing with results for regular analogues. He also presented the inverse problem of constructing billiards from their caustics. Myself and my supervisor Geoffrey Vasil gave a joint presentation on our search for an analogue of the Laplace-Beltrami operator on high-density metric graphs. We showed that in some cases, like for a high-density spider web, the operator we have derived reduces to a recognisable continuous counterpart. This enables us to, for example, determine why and how a random, high-density, metric-graph triangulation of a drum behaves differently to the drum itself. Martin Weschelberger closed out the conference with a talk on navigating invariant manifolds of multi-timescale problems in order to determine fixed points of the corresponding dynamical system. An open question relates to how to find these manifolds in the first place.

All in all, the workshop was a great success we can only hope that conditions improve in 2021, and that domestic and international colleagues may join us for the next SDG conference!