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We introduce a systematic framework for constructing emergent local commuting Hamiltonians in (d+1) dimensions from symmetries defined on a d-dimensional boundary. Starting from a generic d-dimensional symmetry, we gauge it by introducing layers of gauge fields carrying dual symmetries, which can in turn be gauged. The fixed points of this iterative gauging procedure yield (d+1)-dimensional stabilizer Hamiltonians built from local Gauss-law constraints. These Hamiltonians encode the emergent bulk order while realizing the original symmetry on the boundary.

I will show how, starting in one dimension, this construction reproduces familiar two-dimensional topological codes, such as the toric code, and naturally realizes all gapped boundary conditions as different one-dimensional symmetric phases at the boundary. The framework extends to higher-form and subsystem symmetries in arbitrary dimensions, yielding, for example, three-dimensional fracton models.

If time permits, I will also discuss how these symmetries can be classified within the framework of tensor networks.

Based on arXiv: 2403.07575, 2410.09044, and 2203.12563.
 

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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!

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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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.

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. 

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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