ePosters
ePosters are very short, single-slide presentations!
Session 1 #1
Lachlan Burton
Restricted Three-body Problem with Dissipation
The restricted Three-body problem exhibits a rich non-hyperbolic dynamics, which can manifest as, e.g, a fat-tailed distribution of the survival time of scattering orbits, an intricate basin boundary structure, and a mixed phase space. The addition of a dissipative mechanism qualitatively changes the above, and I will present some results in this area.
Session 1 #2
Mitch Curran
The Maslov Index for Linearised NLS
The Maslov index is a tool that has been used to generalise the classical Sturm oscillation theorem to systems of differential equations, that is, to the Morse Index Theorem. For a family of selfadjoint differential operators, one can construct a path of Lagrangian planes in a symplectic Hilbert space by using Green’s second identity to relate the operators to their traces on a family of shrinking (star-shaped) domains. A notion of winding exists for paths in the Lagrangian Grassmannian; this is the Maslov index. One computes it by tracking nontrivial intersections of the path with a fixed reference plane. Therefore, choosing the reference plane to encode the boundary conditions allows one to count eigenvalues, provided the winding is monotone. Using these ideas, I am looking to prove an instability criterion for standing waves of the Nonlinear Schrödinger equation on a compact interval, where monotonicity of the winding no longer holds.
Session 1 #3
Sean Dawson
What do the Lagrange Top and Kerr Black Holes have in Common?
In short: both systems involve the Teukolsky master equation (TME), also known as the generalised spheroidal equation (GSE). This is the most general form of the confluent Heun equation; a second order ODE with two finite regular singularities and one irregular singularity at infinity. By symplectic reduction, the Lagrange top with added Harmonic potential (originally on phase space T*SO(3)) descends down to an integrable system on T*S2. Separating the Schrödinger equation on the reduced manifold in spherical coordinates gives the GSE. The GSE is fascinating because it appears in many systems that display quantum monodromy.
Session 1 #4
Eric Hester
Signed-distance coordinates: Geometric machinery for arbitrary boundary layers
Boundary layers have fascinated mathematicians, scientists, and engineers since 1904, when Prandtl first invoked them to resolve the longstanding paradox of D'Alembert. While the generic mechanism behind boundary layers is well understood--multiple scales emerge from singular perturbations--the details are technical, and compound in complexity with additional effects. I will outline how signed-distance coordinates help simplify boundary layers with arbitrary physics, in arbitrary geometries, to arbitrary asymptotic orders.
Session 1 #5
Wenqi Yue
Analyzing finite-size effects on coupled oscillator systems through homogenization with collective coordinate
Certain coupled oscillator systems of finite size exhibit behaviours that are distinctively different to their thermodynamic limit. In this project we attempt to generalize the collective coordinate framework for deriving reduced dynamics that captures the dynamics of a finite system, including the effects of finite system size. For the Kuramoto-Sakaguchi model we begin by examining whether the contributions from the non-synchronized, rogue oscillators can be approximated by stochastic noises through homogenization from time-scale separation. The approximation by stochastic noises and their dependence on system parameters are investigated numerically. We then attempt to generalize the collective coordinate framework incorporating the contributions as stochastic noises and derive the corresponding reduced dynamics, which are compared to dynamics of the full system.
Session 2 #1
James Chok
Machine Learning using Orthogonal Polynomials
I will discuss my work with Geoff Vasil on machine learning and orthogonal polynomials. When training a neural network, vector data is input into a sequence of adjustable affine functions and nonlinear transformations. This process works well for visual data, and many other applications are currently being developed. However, little attention is paid to the abstract vector basis associated with a given data format. I will discuss how we might consider machine learning in a basis-independent way. In particular, I have found neural networks can be trained using a basis of discrete orthogonal Hanh polynomials. These special functions many interesting mathematical properties that can aid the analysis of the linear operations inherent in the machine learning process.
Session 2 #2
Michael Denes
Identifying Transport Gateways in Ocean Fronts
Ocean flows are dominated by coherent structures, like eddies, gyres, and fronts, with lifetimes longer than typical dynamical timescales. Ocean fronts in particular are boundaries between distinct water masses, and current literature focusses on searching for steep gradients in water mass properties, like temperature and salinity, to identify their boundaries. In this talk I will discuss a number of preliminary results in identifying a front by its dynamics using the numerical adaptive transfer operator method which approximates the eigenfunctions of the dynamic Laplace operator. These eigenfunctions partition phase space into minimally mixing or dynamically distinct regions. I will also discuss ideas we have to identify gateways of transport across these fronts.
Session 2 #3
Cathy Liu
Temporal trend of happiness in social media data
In recent years, social media have become a valuable resource in evaluating public mood. The project aims to undertake a stepwise methodology to analyze tweets containing the word “climate” between November 2019 and June 2020. It involves selecting tweets from Twitter, data preprocessing and application of a simple algorithm to get the sentiment analysis. We investigate how the sentiment of the climate debate varied during the Bushfire season of 2019/2020 by analysing Twitter messages from Australia. We find that happiness has an increasing trend in early and after Bushfire, and has a decreasing trend in height of bushfire.
Session 2 #4
Christopher Rock
Links between higher-order Cheeger problems and Laplace-Beltrami eigenvalues
We discuss some inequalities relating higher-order Cheeger constants to higher eigenvalues of the Laplace-Betrami operator. We also obtain some inequalities relating the higher-order dynamic Cheeger constants, a numerical description of the coherent sets in the phase space of a dynamical system, to the eigenvalues of the dynamic Laplacian.
Session 2 #5
Sam Jelbart
Discontinuity in Geometric Singular Perturbation Theory
Many singularly perturbed ODEs in e.g. mechanics, electronics and mathematical biology become discontinuous as a perturbation parameter tends to zero. This is problematic for the application of well-developed analytical frameworks like geometric singular perturbation theory (GSPT), which rely on a smooth connection to the singular limit dynamics. This ePoster focuses on the resolution of this issue via a geometric method of desingularisation known as blow-up. We are able to conjecture a rather general result which has analogues in the theory of regularised piecewise-smooth systems, namely, that every singular perturbation problem of the kind outlined above has an invariant slow-manifold which can be described via GSPT applied to a desingularised system after blow-up.