Pure & Applied Mathematics Seminar

Mathematical Model for Simulating Ice Shelf Breakup

A mathematical model for simulating ice-shelf breakup under wave forcing is presented in three linked studies. The first derives a frequency-domain solution and applies a transfer-matrix method to handle connected ice–water segments efficiently. The second introduces a time-domain model for a bounded ice shelf and solves its eigenvalue problems numerically. The third develops the main result: a time-domain breakup model using a numerical multi-segment formulation and a breakup-threshold procedure that updates geometry as fractures occur. Together, these studies provide a coherent framework from frequency-domain analysis to numerical simulation of ice-shelf breakup.

Intra-disciplinary bridges for multi-dimensional patterns

The perspective of pattern formation has been successful in drawing from and helping advance multiple areas of mathematics, including dynamical systems, partial differential equations and numerical computing. Formal asymptotic and rigorous approaches such as spatial dynamics have been highly successful over the past years to study/prove the existence and stability of patterns in one spatial dimension. They have also been extended to higher dimensions under certain geometries: such as cylinderical, channel-like domains, etc. They are also useful in understanding invasion fronts, localised patterns, spiral waves and defects in 1D. However, the extension of the wealth of the above mentioned approaches to the analysis of patterns in 2D/3D is not straightforward. A non-exhaustive list of examples of situations that are resistant to analysis, and yet very relevant in diverse applications are: patterns formed with more than one preferred lengthscale, aperiodic patterns, multi-dimensional defects, spatial localisation without radial symmetry, patterns in heterogeneous domains, patterns in the presence of a dynamic bifurcation parameter, patterns in lattice systems and non-local systems.

However in all of these examples, we are able to obtain numerical approximations to equilibria of the associated governing PDE, either through an initial-boundary value problem approach (time-stepping) or via a root-finding approach (numerical continuation). Since it is a non-objective function if numerical computability equals proof of existence :) I will talk about using the approach of computer assisted proofs, to prove the existence of a solution to the full PDE, in a small neighbourhood of a numerical computed solution, as a certification problem. If time permits, I want to also introduce a second bridge from pattern formation to the area of computational/numerical algebraic geometry, which allows us to obtain a complete set of equilibria (closed over the set of complex numbers) of all finite-equilibria for polynomial reductions of the PDEs that govern a specific pattern forming scenario.

Speakers

Noah Cresp, Joshua Knight, Samuel Walsh

Noah Cresp: The Drug Diffusion Problem: Comparison of Analytic Methods

This project will investigate the modelling of drug diffusion in layered biological tissue where separate media have distinct physical properties. This closely reflects a real-world drug delivery problem, where tissues such as muscle and fat will have different absorption characteristics. Classical techniques often struggle with interface conditions, so I will be comparing two methods of obtaining solutions. Namely: the Laplace transform, and the Unified Transform Method. The aim of the project is to assess their accuracy, computational efficiency, and practical applicability. Numerical tools will be used to visualise and benchmark solutions, enabling a comparison of each model’s strengths and limitations. This work will highlight both the potential and the challenges of applying analytic techniques to multilayer diffusion, and will identify open questions that can be pursued in future honours or PhD research.

Joshua Knight: Finite Difference Time Domain Simulations For Maxwell's Equations

The Finite Difference Time Domain (FDTD) Yee Algorithm is used to approximate solutions to Maxwell's equations in one and two dimensions. Problems in free space or with a region of higher electric permitivity and problems involving reflecting and absorbing boundary condition were also investigated. The one dimensional solution was compared to the D'Alembert solution in order to analyse the errors and their dependence on spatial grid size.

Samuel Walsh: Solution of Interface Problems for The Linearised Benjamin-Bona-Mahony Equation on Star-Shaped Networks

This talk will present the solution to the Linearised Benjamin-Bona-Mahony Equation (LBBM) applied to an arbitrary star shaped network of semi-infinite leads with appropriate interface conditions. The solution technique involving the Unified Transform Method (UTM) will be discussed and results for specific initial conditions visualised. Finally, we discuss an obstacle to both the numerical evaluation and asymptotic analysis of formulae obtained via UTM in earlier work on LBBM and present a novel resolution.

Teaching an old dog some new tricks: from discrete solitons and vortices, to rogue waves, flat bands and PINNs in variants of the discrete nonlinear Schrodinger model

In this talk we will revisit nonlinear dynamical lattices, starting with an overview of their physical applications in atomic, optical, mechanical and metamaterial systems. Then we will focus most notably on the prototypical discrete nonlinear Schrodinger (DNLS) model, one of the countless areas where Mark Ablowitz's insights and contributions have shaped our understanding and produced numerous (with many still ongoing) research directions. We will explore at first some of its main features, including discrete solitons, discrete vortices and related structures in 1d, 2d and 3d, not only in square, but also in other lattice patterns (hexagonal, honeycomb, etc.). If time permits, motivated by recent experimental and mathematical developments, we will consider some interesting recent variants on the theme, such as, e.g., what happens in Kagome' lattices. This will motivate a discussion of the notion of so-called flat bands and compactly supported nonlinear states therein. We will discuss simple methods of producing lattices with flat bands, and some experimental implementations thereof in electrical circuits. Compactly supported nonlinear states will also be seen to arise from nonlinearly dispersive variants of the model recently proposed in the context of the mathematical analysis of turbulent cascades. We will also touch upon extreme events and so-called rogue waves in integrable and non-integrable variants of the model. We will end with some machine-learning inspired touches of how to "discover" such lattices from data, using Physics-Informed Neural Networks (PINNs) and how to improve PINNs using the symmetries of the lattice.

Symmetry: From Graphs to Groups and Back

Groups arise naturally as symmetries of objects: for example symmetries of chemical molecules, polygons or the Rubik’s Cube. These objects can often be represented using graphs, which may also be viewed as networks encoding pairwise relationships.

From a graph, it is straightforward to extract a group—its automorphism group, consisting of all symmetries of the graph. The reverse question is more subtle: given a finite group, can we associate a graph to it in a natural way? The answer is yes, and exploring this question reveals deep connections between algebra and combinatorics.

Along the way, I will discuss graph-theoretic problems that lead to natural algebraic generalizations, highlighting the interplay between graph theory and group theory.