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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

Christian Alber will talk about 'Multiscale Spectral Generalised Finite Elements: Preconditioning and Model Reduction Beyond SPD'

Multiscale PDEs with heterogeneous, highly oscillatory coefficients can be challenging for standard numerical methods. This talk presents a framework for the design and analysis of the Multiscale Spectral Generalised Finite Element Method (MS-GFEM), a partition of unity method using optimal local approximation spaces constructed from eigenproblems. These spaces are closely related to GenEO coarse spaces in domain decomposition.  Under suitable assumptions, we establish nearly exponential local approximation rates and describe their dependence on problem parameters. We also discuss how the resulting coarse spaces can be used in two-level domain decomposition preconditioners. Applications to Helmholtz and Stokes problems illustrate the approach beyond the symmetric positive definite setting, with numerical experiments complementing the analysis.

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Professor Rob Scheichl will talk about: 'Efficient localised model reduction for multiscale PDEs via Grassmannian Interpolation'

Multiscale Spectral Generalized Finite Element Methods (MS-GFEM) are a powerful new discretisation method for general variational problems that satisfy a Gårding-type inequality, including strongly non-Hermitian problems. The construction of optimal approximation spaces is localised and requires no a priori regularity assumptions. The global approximation error is controlled by the local errors, which are rigorously shown to decay nearly exponentially. The optimality hinges on an SVD of the local restriction operator in a suitable, coefficient-dependent inner product on an oversampled patch. Compactness of this operator in the space of a-harmonic functions guarantees spectral accuracy akin to Weyl asymptotics for the Laplacian. As such, MS-GFEM can be seen as an “hp-version” of Localized Orthogonal Decomposition (LOD). Given the coefficient function the local approximation spaces can be constructed efficiently. In this talk, I will show how we can use Grassmannian interpolation of the resulting local approximation spaces on sparse grids in high dimensions to derive localized model order reduction methods that inherit the nearly exponential spatial and parametric convergence of MS-GFEM and of sparse grids, respectively. I will show some numerical experiments confirming the theoretical results in the context of elliptic problems.

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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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We extend the construction of BPHZ renormalised models in regularity structures via the spectral gap inequality to the fractional heat operator. Its Green’s function fails to be smooth across the entire time-zero hyperplane, violating the classical assumption that kernels are smooth away from the origin. The proof requires a delicate balance between the gain in regularity provided by the spectral gap approach and the limited spatial decay of the kernel. 

Joint work with L. Broux and R. Steele.

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Your chance to ask Mathematrix DPhil students about the process of applying to PhD programs, including written stages and interviews!

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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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