The eNuggets Schedule

It is well known that the Laplace-Beltrami equation on the sphere admits separation of variables in a number of coordinate systems. I will discuss an approach that relates this to (super-) integrable systems. When considering the corresponding commuting operators and their joint spectrum this leads to the study of Fuchsian equations and their degenerations. Many questions about the relationship of these three objects are still open.

We will present some questions regarding billiards within quadrics.

About a decade ago, there was a spark of interest in studying PDE solutions by analytically continuing the solutions into the complex plane and studying the formation and dynamics of singularities. Interesting results were obtained, but the technical challenge of these methods slowed progress. In recent years, new asymptotic methods have been devised to study these singularities, and use them to describe PDE solutions. I will show some recent results of mine in this area, and we can discuss other contexts in which this may be useful.

I will discuss some progress & challenges using the parametrisation method in Fenichel Theory as well as Pontryagin and Rodygin Theory on slow-fast averaging. 

We will show that a fractional Laplacian can arise in a random walk model when the walker takes a Sibuya distributed number of steps each time step. The result holds for walks on graphs, giving fractional graph Laplacians, on unbounded one-dimensional domains where the fractional Laplacian is equivalent to a Riesz derivative. We conjecture the result to also be true on bounded domains where the spectral fractional Laplacian arises. This enables the formulation of random walks in a potential field with heavy tailed jump distributions. We show this for time-dependent potentials and present an open problem for space-dependent potentials.

I introduce a persistent random walk model with finite velocity and self-reinforcing directionality that models the organisation of exponential runs into truncated power-law flights. A hyperbolic PDE can be formulated to describe the time evolution of the random walk probability density function. I will show some interesting results and outstanding problems as well as discuss many possible extensions of this model.

A stellarator is a promising device for the procurement of fusion energy. The idea is to precisely design a magnetic field so that the charged particles in the fusion plasma are confined. The shape and strength of the magnetic field is critical to ensure good confinement. Many properties imposing restrictions on both the shape and strength of the magnetic field have been proposed that would guarantee good confinement. However, for the vast majority of properties, the question remains: do magnetic fields with these ideal property exist?

In my talk I will describe and motivate martingale optimal transport problem and discuss its applications and possible further developments.

The question of sampling probability measures on manifolds (which correspond to the level set of a function) has received considerable interest over the last decade due to application areas of molecular dynamics and Bayesian learning. Typical approaches to addressing this question involve dealing with constrained stochastic dynamics. Constraints for ‘purely’ spatial stochastic differential equations are well understood, both from an analysis and numerical perspective. In this talk I will focus on adding constraints in the ‘phase-space’ to the so-called Langevin dynamics which is a popular dynamics that arises in applications. In addition to explaining some unexpected results, I will also discuss several open problems associated with constraints on the phase space.

I will explain the dynamical approach to the task of finding patterns in sets of positive density in Z^d. That approach involves recurrence. We will sketch the proof of recurrence along the polynomial times using the spectral measure and bounds on the exponential sums. For finding infinite patterns, we upgrade the notion of recurrence to become "twisted" by an action of a group (of symmetries of a particular polynomial multi-variable map). Our open problems will concern either the case of rather small group of symmetries (the best possible sum-product phenomenon) or multiple correlation case (volumes of all d-simplexes formed by set of positive density in Z^d) for which the spectral measure does not exist.

I will pose some questions about analysing dynamics with varying scales in space and time

...and (do) I feel fine?
Let's reflect on SDG and the dynamical systems community in Australia/New Zealand as a whole. What has to happen next? What needs to be done? What has to go?