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.

Learning Theory for Neural Operators

In this talk, we present results on the approximability and data requirements necessary to learn surrogates of nonlinear mappings between infinite-dimensional spaces. Such surrogate models have a wide range of applications and can be used in uncertainty quantification and parameter estimation problems in fields such as classical mechanics, fluid mechanics, electrodynamics, and earth sciences. Our analysis shows that, for certain neural network architectures, empirical risk minimization based on noisy input-output pairs can overcome the curse of dimensionality. Additionally, we provide a numerical comparison to other approaches including classical constructive methods.

The Gromov-Hausdorff distance between spheres

The Gromov-Hausdorff distance is a fundamental tool in Riemannian geometry (through the topology it generates) and is also utilized in Applied Geometry and Topological Data Analysis as a metric for expressing the stability of methods which process geometric data (e.g. hierarchical clustering and persistent homology barcodes via the Vietoris-Rips filtration). In fact, distances such as the Gromov-Hausdorff distance or its Optimal Transport variants (i.e. the so-called Gromov-Wasserstein distances) are nowadays often invoked in Machine Learning Applications.

Whereas it is often easy to estimate the value of the Gromov-Hausdorff distance between two given metric spaces, its precise value is rarely easy to determine. Some of the best estimates follow from considerations related to both the stability of persistent homology features and to Gromov's filling radius. However, these turn out to be non-sharp.

In this talk, I will describe these estimates and also results which permit calculating the precise value of the Gromov-Hausdorff between pairs of spheres (endowed with their usual geodesic distance). These results involve lower bounds which arise from a certain version of the Borsuk-Ulam theorem that is applicable to discontinuous maps, and also matching upper bounds which are induced from specialized constructions of (a posteriori optimal) "correspondences" between spheres.

Hidden Multiple Time-Scales Dynamics

Many biological and physical systems evolve on several disparate timescales, i.e. the observed dynamics have distinct temporal features that can be attributed to processes evolving on different timescales. In mathematical terms, such multi-time-scale models can be considered as singular perturbation problems.

The powerful tools of Geometric Singular Perturbation Theory (GSPT) have been widely used to analyse complicated dynamical phenomena in many areas ranging from mathematical physiology and neuroscience and chemical reaction systems to mechanics, fluid dynamics and nonlinear wave problems.

During the last years, the toolbox of GSPT has been adapted and extended to situations where the slow-fast structures and the resulting applicability of GSPT are somewhat hidden. Problems of this type include singularly perturbed systems in non-standard form, with different multi-time-scale structure in distinct regions of phase space, depending singularly on more than one parameter, and limiting on non-smooth systems as a parameter tends to zero.

In situations where several distinct scalings must be used, the blow-up method has been established as a powerful tool for matching of the different scaling regimes. Roughly speaking, blow-up replaces a singular point by a higher-dimensional "sphere of directions." This resolves the local geometry and helps track how trajectories transition between different scaling regimes. In this talk I will survey some of these developments mostly in the context of specific models.

Ranks of elliptic curves—a computational and statistical perspective

Elliptic curves reside at the intersection of many areas of mathematics. The rank of an elliptic curve over the rational numbers measures the size of its group of rational points—intuitively, it counts the number of independent points needed to generate all rational solutions. A fundamental question remains: do curves of arbitrarily large rank exist? In this talk, we approach this question from a broad point of view. We present large-scale computations and compare the data to a heuristic linear algebra model inspired by what we know about the arithmetic of elliptic curves.

Recurrence in Combinatorics: A Dynamical Perspective

We will present a dynamical approach, originating in work of Furstenberg, to classical recurrence problems in combinatorics. After introducing the correspondence principle, we show how certain combinatorial questions can be reformulated as problems in harmonic analysis and reduced to the study of exponential sums with polynomial phases. As an illustration, we outline a proof of the Furstenberg–Sárközy theorem and discuss more recent joint results with Björklund, Buliński and Skinner.

Weighted Alexandrov-Fenchel and Minkowski inequalities in hyperbolic space

Alexandrov-Fenchel and Minkowski inequalities are fundamental in differential geometry and convex geometry. In recent years, their weighted versions have attracted substantial attention, especially in the space forms setting. In this talk, I will first briefly survey weighted geometric inequalities in space forms. I will then present my recent joint work with Yong Wei on sharp weighted Minkowski and Alexandrov-Fenchel type inequalities for closed, static-convex hypersurfaces in hyperbolic space. These inequalities compare weighted curvature integrals with quermassintegrals and include weights such as $\Phi^k$ and $\cosh^k r$ for any $k\geq 1$, where $\Phi=\cosh r -1$. Our results unify and extend a number of previously known inequalities in the literature. This is joint work with Yong Wei.

Partial 3D shape comparisons

Scientists have access to a wide range of digital sensors that allow them to report at multiple time and length scales on the subjects of their studies. Finding efficient algorithms to describe and compare the shapes included in those reports has become a central problem in data science. Those algorithms have gained from developments in computational geometry and in machine learning. In this talk I will present another source of support to further improve those algorithms. Using techniques from statistical physics, I show that we can define a possibly partial correspondence between 3D shapes, with a cost associated with this correspondence that serves as a measure of the similarity of the shapes. I will illustrate the effectiveness of this approach on synthetic data as well as on real anatomical data. This is joint work with Dr Henri Orland, IPHT, CEA, Saclay, France.

Generative digital coke microstructure methods and applications for sustainable steel making

Metallurgical coke is a porous carbon fuel used in blast furnaces for iron production. To operate efficiently, high-quality coke must balance strength and reactivity, which are influenced by its microstructure and composition. Although much research has focused on the relationship between coke properties and blast furnace performance, there is limited understanding of how optimal coke microstructures may differ from those produced by current methods, underscoring the need for further research.

A considerable challenge in improving coke manufacture is that experimental methods are both time- and energy-intensive; thus, the development of computational algorithms to generate digital samples for simulating material properties is a crucial step to reduce iteration time (for industry testing) and enable rapid exploration of potential microstructure improvements. This talk outlines the microstructure classification and simulation techniques required to generate representative digital coke microstructures, and how both algorithmic and hardware acceleration strategies are utilised to enable these digital samples to be generated at sizes matching those of real coke samples. The current and potential future applications of this quantitative modelling methodology as an alternative to energy- and time-intensive experimental frameworks for improving coke quality, and the possible positive impacts for sustainable steelmaking, are explored.