Applied Mathematics Seminars in 2025

Abstract: Mathematical models are created for a variety of reasons. One of the most common is to give insight into the operation of systems. This can be enabled through the optimisation of model structures and the inference of model parameters - to see whether the model captures the behaviour of the system. Some models can address the question of what may happen or change in a system under different circumstances. Others are more tuned to look at how the system may operate to enact these changes.

Some deterministic and stochastic modelling and parameter inference paradigms are explored through the lens of protein transport in cells, particularly the insulin-stimulated translocation of the glucose transporter GLUT4. Such systems have relatively sparse data available.

A bottom-up approach is outlined showing model construction and hypothesis testing using from different sources to simultaneously constrain the model parameters. Different methods of fit assessment and model comparisons are discussed, allowing insights to be made into the operation of the cellular system.

Abstract: Decision-making with computational models often involves two interrelated tasks—Bayesian inference (calibrating models from data) and stochastic optimisation (optimising decisions under uncertainty). These tasks typically require repeated evaluations of the underlying computational model, which can be prohibitively expensive when the model is based on PDEs with high-dimensional inputs. Operator learning offers a promising alternative by constructing surrogate models that learn the mapping from input fields to PDE solutions, enabling rapid and repeated evaluations. However, for downstream decision-making tasks, not only the accuracy of the operator approximation but also the fidelity of its derivatives are crucial, as gradients drive outer-loop optimisation and inference procedures.

In this talk, we present derivative-informed operator learning—an approach that incorporates derivative information during training to improve both the predictive and differential accuracy of learned surrogates. We discuss the theoretical underpinnings of this framework and practical training strategies that enable the efficient solution of computationally intensive outer-loop tasks.

Abstract: In this talk, I explore models of information propagation and opinion diffusion over networks, such as the Independent Cascade and Friedkin–Johnsen models, where nodes iteratively update their states through interactions with their neighbours. I first discuss the fundamental behaviour of these dynamics, for example, how properties of the underlying graph influence the patterns, speed, and eventual equilibrium of spreading. Next, I discuss optimisation problems built on these dynamics, particularly influence maximisation, which asks: given a budget for “seeding” nodes, how can we choose them to maximise eventual spread? I highlight key algorithmic approaches and hardness results. Finally, I present an emerging and nuanced connection between opinion dynamics and graph neural networks (GNNs), showing how insights from the former can inform the design, interpretability, and robustness of the latter.

Abstract: By performing a systematic study of the Hénon map, we find low-period sinks for parameter values extremely close to the classical ones. This raises the question whether or not the well-known Hénon attractor is a strange attractor, or simply a stable periodic orbit. Using results from our study, we conclude that even if the latter were true, it would be practically impossible to establish this by computing trajectories of the map. This is joint work with Zbigniew Galias.

Abstract: With recent advances in experimental tools and wearable devices, it has become increasingly easy to collect large volumes of time-series data. In this talk, I will demonstrate how such data can be used to uncover hidden structure in complex biological systems. First, I will introduce GOBI (General Model-based Inference), a simple yet scalable method for inferring regulatory networks from time-series data. GOBI can infer both gene-regulatory and ecological networks, surpassing traditional causality methods such as Granger and CCM. Next, I will present Density-PINN (a physics-informed neural network) that infers the shape of the time-delay distribution governing interactions within a network. This inferred distribution helps identify the number of pathways responsible for signaling responses to antibiotics, addressing a long-standing question about the primary sources of cell-to-cell heterogeneity under stress. Finally, I will show how combining mathematical modeling with machine learning enables analysis of large-scale sleep–wake time-series measured by smartwatches. This approach allowed us to develop personalized sleep-wake schedules that mitigate daytime sleepiness and reduce depression risk, and I will briefly discuss translation into a practical app for broader use. In particular, this algorithm has been serviced globally via the Galaxy Watch since last August, demonstrating how mathematical models can be translated into practical applications.

Abstract: Most mathematical models of contagion—whether for diseases, computer viruses, or ideas—focus on a single agent spreading in isolation. Reality is far more complex: pathogens interact within hosts, behaviors shape disease transmission, and multiple contagions often overlap in space and time. These interacting contagions can amplify, suppress, or fundamentally alter each other’s dynamics.

In this talk, I will introduce the idea of contagions as a unifying concept across biology and society, with examples from disease ecology and the One Health perspective, where human, animal, and environmental health are deeply interconnected. I will outline the first mathematical steps in modeling these interactions, beginning with extensions of classical epidemic models and network-based frameworks. Even simple models reveal surprising phenomena: shifts in epidemic thresholds, changes in persistence or/and the order of phase transitions, and unexpected outcomes when contagions couple together.

Abstract: In this talk, I will discuss resonant orbits and the phenomenon of mean-motion resonance overlapping, which generate natural heteroclinic, propellant-free pathways that spacecraft can follow to change their semimajor axis. This is useful for low-fuel and/or low-energy space mission trajectory design. I will start by describing some recent related results from the Earth-Moon system, with applications to missions throughout cislunar space. Then, moving to the more complicated systems of the outer planets, which have multiple moons, I will discuss secondary resonant phenomena which occur inside the normally hyperbolic invariant manifolds formed by unstable mean motion resonant orbit families. The secondary resonances inside these manifolds are found to overlap in some cases, resulting in a total change of structure for the more unstable members of such families—precisely the orbits most useful for outer planet tour design.

Abstract: Mathematical models describing the collective behavior of large populations of coupled phase oscillators can be found in various fields of physics, chemistry, and biology. In the thermodynamic limit, when the number of oscillators tends to infinity and the distribution of oscillator parameters converges to some probability density, it is often observed that after a sufficiently long transient, the state of the population approaches some statistical equilibrium. In this talk, we describe how the properties of this equilibrium can be used to reconstruct system parameters of the underlying network. The effectiveness of the approach is demonstrated by its application to so-called chimera states in networks of phase oscillators with nonlocal coupling.

Abstract: The coordinated behaviour of populations of cells plays a central role in tissue growth and renewal. Cells react to their microenvironment by modulating processes such as movement, growth and proliferation, and signalling. Alongside experimental studies, computational models offer a useful means by which to investigate these processes. To this end a variety of cell-based modelling approaches have been developed, ranging from lattice-based cellular automata to lattice-free models that treat cells as point-like particles or extended shapes. However, it remains unclear how these approaches compare when applied to the same biological problem, and what differences in behaviour are due to different model assumptions and abstractions. In this talk, we present a series of studies which demonstrate how modelling frameworks and parameter choice influence simulation output and study outcome.

Abstract: Atherosclerotic plaques are fatty, cellular lesions that form in artery walls and can lead to heart attack or stroke. Plaques are initiated by blood-borne lipid particles (“bad cholesterol”) that become trapped in the artery wall and trigger an immune reaction. Subsequent plaque progression involves a complex interplay between these lipids and the cells that are recruited to the lesion site.

The two main cell types in plaques are macrophages and smooth muscle cells. Macrophages are specialised immune cells that are recruited to the early plaque to ingest and remove retained lipids. Smooth muscle cells (SMCs) are artery wall-resident cells that are recruited to the plaque when ineffective lipid removal by macrophages leads to the localised accumulation of lipid and dead cells. SMCs form a protective cap over this hazardous material but can also undergo a harmful transition towards a defective macrophage-like phenotype.

In this talk, I will discuss two different models for the cell and lipid dynamics in plaques. The first is a reaction-diffusion type model of macrophage migration and lipid ingestion in the early plaque. We consider how the capacity of macrophages to remove lipid by emigration influences the trajectory of the early plaque and shapes its early spatial structure. The second model proposes a system of ODEs to study SMC phenotype transition in the mid-stage plaque. We consider how the exposure of cap SMCs to uncleared lipid can drive this transition and accelerate plaque progression. These two models represent a small part of a larger body of recent modelling research that aims to provide novel mechanistic insight into the formation of dangerous atherosclerotic plaques.

Abstract: We develop a set of sufficient conditions for guaranteeing that an integrable system with symmetry group \(K\) on a Poisson manifold \(M\) descends to an integrable system on a dense open subspace of the quotient Poisson space \(M/K\) and on its symplectic leaves. We shall focus on the simplest examples given by reductions of 'master systems' on cotangent bundles of compact Lie groups. Time permitting, we shall also deal with applications associated with Heisenberg doubles and with moduli spaces of flat connections. In almost all examples, the term `integrability’ refers to degenerate integrability, alias superintegrability.

The talk is aimed to be self-contained in the sense that all necessary notions will be defined. Details and references can be found in the recent preprint arXiv:2507.12051 based on collaboration with Maxime Fairon (Dijon).

Abstract: Atherosclerosis is a chronic inflammatory disease in which lipids and cellular material accumulate in the artery wall to form plaques. In this talk, we present a series of partial differential equation models for early atherosclerotic plaque development. These models use a multiphase framework with a time-varying spatial domain to represent the growing plaque. Using these models along with associated approximate ODE submodels, we explore a number of phenomena underlying early plaque development. We focus in this talk on 1) the balance between cell death and the recycling of dead cellular material, 2) the competing effects of pro-atherogenic low density lipoproteins (LDL) and atheroprotective high density lipiproteins (HDL), and 3) the onset of cell necrosis, and how these processes affect plaque composition and growth.

Abstract: Dissipative relativistic hydrodynamics is expected to describe the thermalised behaviour of strongly coupled fluids such as the quark-gluon plasma forming inside particle colliders. This late-time behaviour is accurately described by a hydrodynamic series expansion in small gradients. Surprisingly, this hydrodynamic expansion is accurate much earlier than expected, even when the systems are still quite anisotropic.

This early success is intimately related to the asymptotic nature of the hydrodynamic expansion. The theory of exponential asymptotics explicitly shows how transient non hydrodynamic modes are encoded in the late-time expansion, and how to include these modes to obtain information about the early non-equilibrium behaviour of the system.

In this talk I will review the theory of resurgence and its role in simple models of relativistic hydrodynamics and beyond.

Abstract: Under appropriate conditions, a magnetic field can be viewed as a non-autonomous Hamiltonian system. This fact is a foundational aspect of how we think about the topology of magnetic field lines. In particular, the Hamiltonian structure of magnetic fields has guided the conceptual design of devices for the magnetic confinement of plasma. In this talk, we give a classification of magnetic fields that admit a Hamiltonian structure using the so called monodromy representation. We explore some examples of magnetic fields with no Hamiltonian structure. Of particular interest is the consequences for the field line topology. Ultimately, these non-Hamiltonian magnetic fields open new possibilities for the design of magnetic fields for plasma confinement. This is joint work with David Perrella (ICMAT, Madrid) and David Pfefferle (UWA, Perth).

Abstract: We propose a notion of conditioned stochastic stability of invariant measures on repellers: we consider whether quasi-ergodic measures of absorbing Markov processes, generated by random perturbations of the deterministic dynamics and conditioned upon survival in a neighborhood of a repeller, converge to an invariant measure in the zero-noise limit. Under suitable choices of the random perturbation, we find that equilibrium states on uniformly expanding/hyperbolic repellers are conditioned stochastically stable. This is joint work with Bernat Bassols Cornudella, Jeroen S.W. Lamb.

Abstract: This work investigates the impact of surfactant on the interfacial stability of a two-layer shear flow in a channel. The surfactant molecules can get adsorbed at the interface or form micellar aggregates when their concentration is beyond a critical value. A mathematical model is formulated that describes the hydrodynamics and transport of surfactant. The effect of surfactants on the stability of the flow is investigated via a linear stability analysis, and the identified instabilities are followed into the nonlinear regime by carrying out numerical computations of a lubrication-type system. The range of validity of the asymptotic model is estimated by carrying out comparisons with direct numerical simulations (DNS) of the full system of governing equations.

Abstract: Anderson acceleration with window size m is a nonlinear convergence acceleration mechanism for fixed-point iterative methods that is widely used in scientific computing, optimization and machine learning. For many applications AA(m) dramatically improves the convergence speed, both when iteration counts are small, and asymptotically for large iteration counts. Nevertheless, very little is known about how to bound or quantify the improvement in asymptotic convergence speed provided by windowed AA(m). In this talk I will give an overview of what is known about the asymptotic convergence speed of windowed AA(m) and highlight some recent results and open problems.

Abstract: As generative models become increasingly powerful and pervasive, the ability to unlearn specific data, whether due to privacy concerns, legal requirements, or the correction of harmful content—has become critically important. Unlike conventional training, where data is accumulated and knowledge reinforced, unlearning aims to selectively remove the influence of particular data points without retraining from scratch. As model sizes increase, efficient unlearning algorithms become essential, since the cost of training scales accordingly. In this talk, we will begin with some intuitive baseline algorithms, examine their shortcomings, and then present two fast, new algorithms that utilize loss gradient orthogonalization to unlearn. Along the way, we will also explore how the generated samples transform after unlearning.

Abstract: We present a Bayesian based approach for some classical wave-propagation inverse problems involving reconstructing characteristics of one or more objects from scattered-field data obtained by illuminating the objects with plane waves. Our method is based on incorporating the data into a Bayesian framework and sampling the resulting posterior probability distribution using a Markov Chain Monte Carlo (MCMC) method. The mathematical model associated with the forward problem is the Helmholtz equation, posed in the unbounded region outside the objects, whose morphology is parametrised by a vector in a high-dimensional space associated with the Bayesian prior distribution. The key to our approach is to accelerate the MCMC sampling using an efficient surrogate for the forward problem.

Abstract: Cone Beam CT is a classic example of a linear(ized) inverse problem that has become widely used, particularly in guiding radiation therapy. A particular challenge in lung cancer radiation therapy is dealing with patient respiratory motion, which has motivated the development of 4D CBCT. By incorporating hardware control and reasonable patient models adaptive 4DCBCT was developed and tested in the ADAPT clinical trial to achieve 4DCBCT with 85% less imaging dose and 65% less imaging time. These results motivated additional hardware developments that now form the basis of a novel Quantum CT modality that can be used for 3D imaging during treatment delivery and is also well suited to rural lung cancer screening, albeit with additional constraints and approaches required to best solve the associated inverse problem.

Abstract: We consider a product of random Matrices M_n = X_0 ... X_{n-1}. The (X_i) are i.i.d. and generate a Zariski dense semi-group. We are interested in the limit behaviour of M_n. I will first give a quick overview of historical results, whose proofs rely on moment assumptions for the logarithm of the norm. Then I will state some of my results obtained using the pivoting technique, putting emphasis on the optimality of moment assumptions. Most notably, the law of large numbers for the spectral radius and the coefficients hold with a first moment assumption only and a contraction result equivalent to the simplicity of the Lyapunov spectrum holds without moment assumptions. If time permits, I will give an idea of the proof of these results.

Abstract: We describe the convergence to equilibrium for systems with holes by establishing limit theorems in the asymptotic regime. Joint with Atnip, Froyland, Gonzalez-Tokman and Nakano.

Abstract: In this talk, I will discuss a Dolgopyat-type estimate for a class of partially hyperbolic systems on the 2-Torus, and its consequences: the CLT, its higher order corrections, and exponential decay of correlations for a class of three-dimensional skew products. This is a joint work with Roberto Castorrini.

Abstract: Signomials generalize polynomials by allowing arbitrary real exponents, at the expense of restricting the resulting function to the positive orthant. In this talk, I present a novel convex relaxation hierarchy of lower bounds for signomial optimization and discuss how it can be used to study nonlinear dynamics. The hierarchy is derived through the newly-defined concept of signomial rings and a signomial Positivstellensatz based on conditional "sums of arithmetic-geometric exponentials" (SAGE). Numerical examples are provided to illustrate the performance of the hierarchy on problems in chemical engineering and optimal control.

Abstract: The multi-scale nature of the natural world is mirrored by the biological systems that seek to predict and control it. To explore how the brain has approached this suite of control problems we employ theoretical and empirical techniques – leveraging biophysical spiking and mean field models, and statistical modelling – showing how cells coordinate to form larger neural assemblies through the lense of dynamical systems theory. We extended these insights to key features of neural anatomy that inform our understanding of neural dynamics across phylogeny and behaviour, arousal, consciousness, its deficits in disease, and its potential axes for manipulation in clinical treatments.

Abstract: Eigenfunctions in closed quantum systems with a chaotic classical limit are equidistributed following the quantum ergodicity theorem. In contrast, in open scattering systems one finds that resonance states have a multifractal structure. How is this structure generated from the underlying classical dynamics? We answer this by generalizing Ulam’s matrix approximation of the Perron-Frobenius operator. It gives rise to a conditionally invariant measure describing resonance states in the semiclassical limit. Numerical examples are a dielectric cavity and the three-disk scattering system.

Abstract: Selfconsistent transfer operators (STOs) are nonlinear operators describing the thermodynamic limit of mean-field coupled maps, and are the discrete time analogues of McKean-Vlasov equations. They allow to model and study the emergent behaviour of interacting chaotic units in discrete time and, more generally, of systems whose evolution is not Hamiltonian or for which there are no evident conserved quantities. The study of the dynamics of STOs gives information on the emergent behaviour in the thermodynamic limit and, in particular, on the existence and stability of equilibrium measures. Studies of STOs have mostly been limited to the case of small coupling where the STO is close to a linear operator. In this talk, I will describe recent results providing sufficient conditions for the existence and stability of fixed points in the strong coupling regime where the STO presents “genuinely” nonlinear dynamics with bounded basins of attraction and multiple fixed points.

Abstract: We study the small amplitude linearization of the Korteweg de Vries equation on the line, but with a defect at x=0 represented by a network of finite intervals adjoined at that point, scattering waves. For a representative collection of examples, we obtain explicit contour integral representations of the solution and obtain existence and unicity results for piecewise smooth data. We also discuss extensions to more complex metric graph domains and introduce a serial version of the unified transform method which may be more efficient for such problems.

Abstract: The Swift–Hohenberg equation (SHE) is a partial differential equation that explains how patterns emerge from a spatially homogeneous state. It has been widely used in the theory of pattern formation. Following a recent study by Bramburger and Holzer, we consider the discrete SHE on deterministic and random graphs. The two families of the discrete models share the same continuum limit in the form of a nonlocal SHE on a circle. The analysis of the continuous system, parallel to the analysis of the classical SHE, shows bifurcations of spatially periodic solutions at critical values of the control parameters. However, the proximity of the discrete models to the continuum limit does not guarantee that the same bifurcations take place in the discrete setting in general, because some of the symmetries of the continuous model do not survive discretization. We use the center manifold reduction and normal forms to obtain precise information about the number and stability of solutions bifurcating from the homogeneous state in the discrete models on deterministic and sparse random graphs. Moreover, we present detailed numerical results for the discrete SHE on the nearest-neighbor and small-world graphs.

Abstract: The Lévy Flight Foraging Hypothesis is a widely accepted dogma which asserts that animals using search strategies allowing for long jumps, also known as Lévy flights, have an evolutionary advantage over those animals using a foraging strategy based on continuous random walks modelled by Brownian motion. However, recent discoveries suggest that this popular belief may not be true in some geometric settings. In this talk we will explore some of the recent progress in this direction which combines Riemannian geometry with stochastic analysis to create a new set of properties for diffusion processes.

Abstract: This research investigates the synergistic effects of combining Vaccinia (VV) and Vesicular stomatitis (VSV) oncolytic viruses for cancer treatment. The mathematical model is based on an experiment by Le Boeuf et al., where the combination of VV and VSV demonstrated synergistic oncolytic activity in vitro and in vivo. The model’s solution long-term behavior has been analyzed, allowing us to predict the outcome of the combination, a global sensitivity analysis has been conducted to determine the parameters that most significantly influence the treatment outcome and several numerical simulations have been carried out to investigate the effect of different factors. Optimal time delays (\tau1 and \tau2 representing optimal virus initial delivery of VV and VSV), to minimize tumor cell count during virus administration, have been determined. Joint work with Raluca Eftimie, Anotida Madzvamuse, Rachid Ouifki, Amina Eladdadi and Helen Byrne

Abstract: ln this talk I will explore the interplay between classical—as well as modern—analytical techniques in applied mathematics and high performance computing. When not completely disjointed (or indeed one completely absent), these two approaches are often deployed with a verification-oriented mindset. By contrast, I will strive to convince you of the power of analytically-informed computational approaches (or indeed computationally-informed analytical methods) that draw elements from mathematical modelling, differential equations, asymptotic and complex analysis, and control theory. Interfaces will be the main theme of the talk not only as the intersection space between different methodologies, but also in view of the problems motivating this work, which include interfacial flow problems such as elucidating the dynamics of droplets, bubbles and liquid films in non-trivial (and highly non-linear!) regimes as the setup for hybrid technique development.