Applied Mathematics Seminars in 2021

Abstract: Spatial growth plays an important role in controlling pattern formation in many different types of physical systems. Specific examples include directional quenching in alloy melts, growing interfaces in biological systems, moving masks in ion milling, eutectic lamellar crystal growth, and traveling reaction fronts, and such growth processes have been shown to select and control spatially periodic patterns, while mediating defects. One of the simplest mechanisms of growth is a directional "quench" which travels across a domain, suppressing patterns in one part of the domain, and exciting them in the other. In this talk we will discuss how a linear phase diffusion equation with a nonlinear boundary condition posed on the half-plane can be used to characterize the formation of stripes oblique to the quenching interface. We use rigorous analysis, formal asymptotics, and numerical continuation to characterize stripe selection for various quenching speeds and stripe angles. Of particular interest, we find that the slow-growth, small-angle regime is governed by the glide-motion of a dislocation defect at the quench interface. Finally, we compare our results numerically to stripe formation in a quenched anisotropic Swift-Hohenberg equation, a prototypical model of pattern formation.

Abstract: This talk will be in two parts. The first part will be introductory, and will address the question: Given an ordinary differential equation (ODE) with certain physical/geometric properties (for example a preserved measure, first and/or second integrals), how can one preserve these properties under discretization? The second part of the talk will cover some more recent work, and address the question: How can one deduce hard to find properties of an ODE from its discretization?

Abstract: Population process in general setting, where each individual reproduces and dies depending on the state (such as age and type) of the individual as well as the entire population, offers a more realistic framework to population modelling. Formulating the population dynamics as a measure-valued stochastic process allows us to incorporate such dependence. The asymptotics, namely the law of large numbers and the central limit theorem, can be obtained. Some examples, including sexual reproduction and the spread of viral infection will be given.

Schedule: Online at 9am. Zoom link to be sent to mailing list.

Abstract: The well-known game of Rock-Paper-Scissors can be used as a simple model of competition between three species. When modelled in continuous time using differential equations, the resulting system contains a heteroclinic cycle between three equilibrium solutions. The game can be extended in a symmetric fashion by the addition of two further strategies (‘Lizard’ and ‘Spock’): now each strategy is dominant over two of the four other strategies, and is dominated by the remaining two. The differential equation now contains a set of coupled heteroclinic cycles forming a heteroclinic network. In this talk I will discuss how we study the dynamics near this heteroclinic network. In particular, I will show how we are able to identify regions of parameter space in which arbitrarily long periodic sequences of visits are made to the neighbourhoods of the equilibria, and how these regions form a complicated pattern in parameter space.

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Abstract: The asymptotic approach is suggested for the description of interacting surface and internal oceanic solitary waves. This approach allows one to describe a stationary moving wave patterns consisting of two plane solitary waves moving at an angle to each other. The results obtained within the approximate asymptotic theory is validated by comparison with the exact two-soliton solution of the Kadomtsev–Petviashvili equation. The suggested approach is equally applicable to a wide class of non-integrable equations too. As an example, the asymptotic theory is applied to the description of wave patterns in the 2D Benjamin–Ono equation describing internal waves in the infinitely deep ocean containing a relatively thin one of the layers.

Abstract: Matrices and tensors that appear in computational mathematics are so often well-approximated by low-rank objects. Since random ("average") matrices are almost surely of full rank, mathematics needs to explain the abundance of low-rank structures. We will give various methodologies that allow one to begin to understand the prevalence of compressible matrices and tensors and we hope to reveal underlying links between disparate applications. We will also show how the appearance of low-rank structures can be used in function approximation, fast transforms, and partial differential equation (PDE) learning.

Abstract: Traditional Indigenous marriage rules have been studied extensively since the mid-1800s. Despite this, they have historically been cast aside as having very little utility. Here, I will walk through some of the interesting mathematics of the Gamilaraay system and show that, instead, they are in fact a very clever construction. Indeed, the Gamilaraay system dynamically trades off kin avoidance -- to minimise incidence of recessive diseases -- against pairwise cooperation, as understood formally through Hamilton's rule.