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Underneath large glaciers and ice sheets, water flows through a permeable network of cavities and channels, held open by melting the ice above balancing the downwards flow of ice. Dissipation within the water flow is a significant enough source of heat that instabilities can develop if the flow rate is high enough, eroding large channels that rapidly drain water from the glacier bed. I will present a model for the system, and discuss the linear problem, observational evidence for the stability criterion, and the non-linear interactions that rapidly become the dominant control on channel spacing. Recently, there has been some discussion of slowing glaciers down by pumping water out from under them - I will consider the viability of this strategy in view of the results in this talk.

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Dr Peter Braam is going to talk about; 'State time geometry: causal performance profiles and optimal data transport in parallel execution'

The increasing complexity of parallel architectures and heterogeneous microarchitectures makes predicting and optimising program performance notoriously difficult. For Optimal Data Transport, we present a discrete variant of the Wasserstein–Fisher–Rao metric that quantifies the true cost of data layout transformations and movement across memory hierarchies. For Causal Performance Profiles, we introduce the Lyons–Gregg Signature, which combines the ideas of Terry Lyons' rough path signatures with hardware performance counters (eBPF) to capture cross-correlated, causal bottlenecks in execution streams. Both arose from State Time Geometry (STG), a model for stateful program execution on computing infrastructure, first modelled as a dynamical system governing state-values over the space of memory addresses. The address space generalises to geometric objects defining infrastructure and leads to the metric. The state transitions of parallel executions become a Grothendieck quantum field theory over the infrastructure and carry the statistical model for the Lyons-Gregg Signature. The central theme is that an intuitive faithful model is not doomed by complexity but forms a geometric domain in which both theoretical and engineering perspectives are simplified.

(In a companion lecture in the Computing Laboratory at 11:00 on Nov 13, we will discuss STG's underlying categorical and geometric structure and its relationship to programming languages and formal methods)

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This session will look at how you can get the most out of your lectures and tutorials. We’ll talk about how to prepare effectively, make lectures more productive, and understand what tutors expect from you during tutorials. You’ll leave with practical tips to help you study more confidently and make your learning time count.

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

Every population consists of individuals who vary in their traits, and each trait may, or may not, be associated with frailty or fitness. Variation in frailty and fitness traits makes population studies prone to selective depletion bias (SDB). The issue is widespread across scientific domains. When an ageing cohort exhibits declining mortality, is it individuals becoming healthier or selective depletion of the frail? In an epidemic, when growth in cumulative infections decelerates, is it individuals cautiously changing behaviour or selective depletion of the most susceptible? In microbial populations, when mutations increase population vulnerability to stress, it is individuals becoming more vulnerable or mutant populations having higher variance in fitness? In each case, the first explanation invokes individuals changing, while the second recognises that populations change due to selection on pre-existing variation. While the former are intuitive and widely adopted, explanations that rely on selective depletion are more neutral and less commonly considered due to cognitive biases and challenges in estimating all variation that matters. 

Remodelling selection (ReMS) is proposed as a general strategy of study design and analysis to address the SDB problem. I will show how the approach has been employed in specific case studies and how it has been formalised in general.

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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.

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