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Maria Reboredo Prado
Stratosphere-Troposphere Interactions: Understanding how the Upper Atmosphere Shapes the Weather and Climate We See.
It is well known that the stratosphere – the second layer of the atmosphere – can exert a powerful influence on weather and climate at the surface. This influence is particularly important in winter, when the strength of the stratospheric polar vortex — a belt of fast-moving winds encircling the pole — can vary dramatically. If the impacts of these variations on the lower atmosphere were better understood and more accurately represented in forecast models, they could provide early signals of winter weather patterns, improving predictions on subseasonal to seasonal timescales. Yet, the mechanisms by which changes high in the atmosphere produce a response near the surface remain poorly understood. 
In this talk, I will use an idealised atmospheric model to test how much of this downward effect can be explained by large-scale balanced dynamics. The model allows individual processes – such as atmospheric cooling, surface friction, and boundary effects – to be switched on or off, making it possible to assess their relative roles in shaping the surface response. A key improvement on previous theoretical models is that the displacement of the tropopause — the boundary between the troposphere and stratosphere – emerges naturally as part of the flow’s response to forcing. Overall, this framework provides a controlled setting for identifying the dynamical mechanisms that transmit stratospheric disturbances to the surface.

Orson Hart
Inertialess instability of viscosity-stratified Couette flow
In this talk, we discuss the stability of viscosity-stratified Couette flow. By neglecting inertia, we isolate the destabilising effect of viscosity stratification on a fluid configuration consisting of three layers of immiscible, incompressible fluid. We show how the linear stability analysis can be simplified by considering symmetric flow configurations and, by considering disturbances of arbitrary wavelength, we identify a new instability mechanism that is only induced by finite-wavelength disturbances.

Vedanta Thapar

Embedding complex networks with the random walk first return time distribution

We consider the problem of network embedding, specifically we propose the first return time distribution (FRTD) of a random walk as an interpretable and mathematically grounded node embedding. The FRTD assigns a probability mass function to each node, allowing us to define a distance between any pair of nodes using standard metrics for discrete distributions. We present several arguments to motivate the FRTD embedding. First, we show that FRTDs are strictly more informative than eigenvalue spectra, yet insufficient for complete graph identification, thus placing FRTD equivalence between cospectrality and isomorphism. Second, we argue that FRTD equivalence between nodes captures structural similarity. Third, we empirically demonstrate that the FRTD embedding outperforms manually designed graph metrics in network alignment tasks. Finally, we show that random networks that approximately match the FRTD of a desired target also preserve other salient features. Together these results demonstrate the FRTD as a simple and mathematically principled embedding for complex networks.

Chun Lam Li

Neural network augmentation of the Maxey-Riley equation for Particle-Laden Flows
Fluid-particle interactions in low-Reynolds-number confined flows govern many biological and engineering processes, including microthrombi transport in cerebral microvessels and particle separation in microfluidic devices. Fully resolved direct numerical simulations (DNS) are accurate but costly for small particle sizes and large numbers of particles, limiting the simulation and analysis of particle-laden flows. We develop a machine learning (ML) model based upon the Maxey-Riley (M-R) equation to provide fast predictions of particle motion. A key feature is that the ML model is physics-based; neural networks learn corrections to an M-R based model to improve its predictions beyond the conditions for which it was derived. We consider a single spherical particle moving between two parallel plates, with plane Poiseuille flow as the undisturbed flow. The problem is parametrized by the flow Reynolds number, particle-to-fluid density ratio, confinement ratio (particle diameter to channel height), and initial wall-normal position. The model is trained on particle velocities from DNS sampled at discrete time points, and evaluated on trajectories that were not used for training. The ML model improves prediction accuracy of particle trajectories over the uncorrected M-R baseline across the tested physical parameters, at a far lower computational cost than DNS.

work with Justin Sirignano¹, and Sarah L. Waters¹
1. Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK

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The acquisition and maintenance of the correct cellular pattern and tissue architecture is essential for organ function in multicellular organisms. Beyond generating the required cellular diversity, developing tissues need to attain the appropriate morphology. Tissue architecture is built through symmetry breaking instabilities such as folding, branching, buckling or budding. Studying single-cell responses alone is not enough to reveal the mesoscale physical and biochemical processes that regulate tissue organization and morphology over time and scale. In our lab we investigate how the interaction between the extracellular matrix (ECM) and tissues regulates processes during development and cancer, with a focus on ECM mechanical properties. While most research in this domain has concentrated on the ECM's elasticity as a primary determinant of cell and tissue behaviour, it is important to note that the ECM possesses both viscous and elastic properties. I will present our findings demonstrating that the passive viscoelastic properties of the ECM regulate tissue architecture and patterning both during development and cancer. Specifically, we show that ECM viscoelasticity influences the spatial and temporal organization of multicellular tissues in breast and intestinal organoids. Overall, our work highlights the critical role of viscoelasticity in driving morphological symmetry breaking instabilities, a fundamental process in morphogenesis and oncogenesis, and suggests ways of controlling tissue through ECM mechanics.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards.

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The rapid growth of the battery industry has created an urgent need for faster process development, reduced experimental burden, and improved manufacturing efficiency. Mathematical modelling provides a powerful framework for understanding and optimising processes across the battery manufacturing chain.

This talk will explore how modelling can support battery manufacturing across a range of levels of complexity, from high-fidelity multiphysics simulations to reduced-order models, numerical solutions of governing transport equations, and simpler analytical tools used directly by engineers. Using examples from electrode manufacture, it will demonstrate how different approaches can be applied according to the question being addressed, balancing accuracy, computational effort and practical utility. The talk will show how these complementary techniques can be used to gain process insight, guide experimentation, explore large design spaces and support decisions from early-stage development through to industrial scale-up.

Drawing on examples from industrial practice, the talk will illustrate how modelling can shorten development timelines, reduce reliance on costly experimental programmes, and enable more effective scale-up of battery manufacturing processes. It concludes with a discussion of emerging opportunities for applied mathematics in next-generation battery production and digital manufacturing.

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Evolution underlies the transformation of a normal cell to a cancer, yet learning the parameters defining this dynamic process from single-timepoint bulk samples is an open challenge. To understand how cancer cells evolve in vivo, we must rely on naturally occurring, heritable lineage tracing markers that encode the evolutionary history of a population of cells. Here, I shall introduce our work on identifying selectively neutral “epigenetic barcodes” and employing them as a molecular clock. By coupling this process with mathematical modelling and Bayesian inference, we characterised the evolutionary history of almost 2000 lymphoid cancers (Gabbutt et al., 2025). Across a broad range of cancer types, we demonstrated that tumour growth rates and malignancy ages differed by orders of magnitude. In 2 independent cohorts of patients with chronic lymphocytic leukaemia (CLL), a typically indolent and slow growing cancer, the inferred growth rates were highly prognostic. I shall further discuss recent work applying this approach to resolve the clonal relationship between acute myeloid leukaemia (AML) blasts and differentiated neutrophils in patients without bone marrow failure. 

In this session, we will discuss how best to use the Christmas vacation to consolidate your learning, to prepare for collections, and of course to have a rest!

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

Professor Pereyra will talk about; 'Physics-informed deep generative models: Applications to computational sensing'

This talk introduces a novel mathematical and computational framework for constructing high-dimensional Bayesian inversion methods that leverage state-of-the-art generative denoising diffusion models as highly informative priors. A central innovation is the construction of physics-informed generative models using Langevin diffusion processes and Markov chain Monte Carlo (MCMC) sampling techniques to develop stochastic neural network architectures capable of near-exact sampling. The obtained networks are modular and composed of interpretable layers that are directly related to statistical image priors and data likelihoods derived from forward observation models. The layers encoding the data likelihood function are designed for flexibility, enabling scene and instrument model parameters to be specified at inference time and seamlessly integrated with pre-trained foundational generative priors. To achieve high computational efficiency, we employ adversarial model distillation, which yields excellent sampling performance with as few as four Markov chain Monte Carlo steps, even in problems exceeding one million dimensions. Our approach is validated through non-asymptotic convergence analysis and extensive numerical experiments in computational image and video restoration. We conclude by discussing unsupervised training strategies that allow the models to be fine-tuned directly from measurement data, thereby bypassing the need for clean reference data.

The talk is based on recent work in physics-informed generative AI for Bayesian imaging: https://arxiv.org/abs/2503.12615 (ICCV 2025), which uses a distilled latent Stable Diffusion XL model trained on five billion clean images as a zero-shot prior, and  https://arxiv.org/pdf/2507.02686, which integrates pixel-based diffusion models with deep unfolding and diffusion distillation (TMLR 2025). The extension to video restoration is presented in https://arxiv.org/abs/2510.01339 (ICLR 2025). Our approach to unsupervised training of diffusion models is introduced in https://arxiv.org/abs/2510.11964.

This talk is divided in three parts. First, I review structural properties and examples of geometries with parallel skew torsion. Second, I take a closer look at those geometries where the torsion is in addition closed; this equips every tangent space with the structure of a metric Lie algebra. These manifolds always locally split as a product into well understood factors, allowing a complete local classification. Third, I investigate various G-structures defining a geometry of this type, with a focus on almost Hermitian structures, extending recent results of Barbaro-Pediconi-Tardini. This is joint work with Andrei Moroianu.

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The apicomedial actomyosin network is crucial for generating mechanical forces in epithelial cells. Pulsatile, oscillatory behaviour of this contractile network occurs in multiple organisms and contexts, for example during mouse embryo compaction or Xenopus neurulation. However, the emergence and control of such pulsed contractions are not fully understood. Here, we study pulsed contractions of larval epithelial cells (LECs) during development of the abdomen in fruit flies. Specifically, we combine in vivo 4D microscopy and simulations of an active elastomer to investigate subcellular spatial patterns of actomyosin dynamics. Our model quantitatively matches in vivo observations and identifies cell geometry and polarity as key contributors to in vivo actomyosin dynamics. Moreover, our findings support the notion that spatiotemporal oscillatory behaviour of the actomyosin network is an emergent property, rather than driven by upstream signalling.

I will additionally give a brief overview of other dynamic phenomena in cell biology that my collaborators and I have studied, such as ultradian (i.e. faster-than-circadian) oscillations in gene expression.

Throughout, I will highlight open challenges for data-driven approaches in cell biology.

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