In my talk, I will discuss a new result providing a positive answer to a natural problem about continuous families of projections in II_1-factors. The problem is naturally viewed through the lens of “W*-bundles”, and our proof is via a novel technique which utilises free probability theory in a uniform manner across these bundles. This leads to the notion of selflessness for W*-bundles, which also provides a number of other regularity properties for these objects, such as strict comparison, real rank zero, and stable rank one. This is joint work with David Jekel and Stuart White.

Jian-Qing Zheng will talk about: 'Generative Models on the Space of Diffeomorphisms: A Deformation-Centric Framework for Multi-Organ Anatomy'

Generative models for images are typically formulated in pixel space, where the geometric structure of the underlying objects is not directly represented. For anatomical data, a more natural representation is provided by the deformation that maps one anatomical configuration to another, rather than by the intensities themselves. The set of such deformations forms a structured, non-Euclidean space, and working in this space changes how registration, generation, and representation learning can be approached. In this talk, a framework will be presented in which deformations, rather than images, are treated as the primary modeling object. Image registration is recast as the problem of recovering a deformation between two anatomies, and is extended to the multi-organ setting by modeling deformations of several organs jointly with their geometric couplings. A diffusion-based generative model is then introduced that operates directly on deformations, so that each generated sample is, by construction, an interpretable transformation of a real anatomy. The framework is extended into a foundation model trained across multiple modalities and anatomical regions, and is evaluated on medical imaging tasks including few-shot segmentation, registration, and phenotype-conditioned anatomical prediction.

There has been substantial progress in the construction of eigenvarieties and $p$-adic families of automorphic forms, and their relationship with Selmer groups and ($p$-adic) $L$-functions. In this talk I will introduce some of these constructions, starting with modular forms, and the concept of complete $p$-adic rigidity: the non-existence of nontrivial $p$-adic deformations. I will explain some of the techniques used to study the geometry of eigenvarieties, and how these specialise to show that certain noncuspidal 'Saito—Kurokawa' points are completely $p$-adically rigid. If time permits, I will also briefly outline how similar strategies may be used to construct $p$-adic families through cuspidal, nonholomorphic Saito—Kurokawa points and to produce nontrivial Selmer classes predicted by the Bloch—Kato conjecture. 

Professor Jinchao Xu will talk about; 'Neural Networks and Classical Numerical Methods: A Theoretical Perspective'

This talk compares neural network-based methods with classical numerical methods from a theoretical perspective. Through several representative examples, we examine both the potential and the limitations of deep neural networks in scientific computing and, more broadly, in machine learning. We begin by comparing ReLU deep neural networks with polynomials and piecewise polynomial spaces, focusing on their structures and expressive power. We then revisit the curse of dimensionality and discuss whether deep neural networks truly offer advantages over traditional numerical methods for high-dimensional problems. Next, we consider the use of deep neural networks for solving partial differential equations, with particular emphasis on the challenge of achieving high accuracy. Finally, we examine multigrid methods and explore whether their underlying principles can help us better understand, design, and train deep neural network models with possible implications for broader AI applications.

This is a Joint OxPDE & Numerical Analysis Seminar 

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This is a joint OxPDE and Numerical Analysis seminar. 

We consider a cross diffusion system of two populations, often called the Busenberg-Travis system. The two species are transported by the same pressure gradient with Darcy’s law, modeling overcrowding effect (populations tend to move away from regions of high pressure). However, their mobility is different: the first species moves with mobility 1, whereas the second moves with mobility \nu. The difficulty to prove existence is to prove strong compactness of each densities, which we achieve with a variant of the div-curl lemma applied to evolution PDEs.

Many biological reaction-diffusion systems are multiscale: in some regions molecules are abundant, while in others only a few are present. Where numbers are low, intrinsic noise is significant, and a stochastic model such as Gillespie's algorithm is needed to capture the fluctuations and rare events that shape the behaviour. Where numbers are high, this approach is too expensive, and a continuum PDE is sufficient.

Hybrid methods aim to apply each description where it is appropriate, but most require an explicit spatial interface separating the stochastic and deterministic regions. The Spatial Regime Conversion Method (SRCM) avoids this. Each region of space carries both discrete particles and continuous PDE mass, and moves mass between them through conversion events as local concentrations change. The method therefore adapts automatically as the system evolves, resolving stochastic detail wherever intrinsic noise matters and using the cheaper PDE everywhere else, with no fixed interface to track.

In this talk I introduce the method and show how it works, then illustrate it on examples including epidemic spread and a Turing instability driven by noise, where it reproduces the stochastic behaviour that a continuum model alone cannot capture.

Networks are fascinating because of their ability to describe complex structures found in a broad variety of systems, from arts and humanities, via the life sciences to the physical science and mathematics. Perhaps even more startling is the variety of approaches that different disciplines have contributed to the study of networks. All of these approaches have a common goal: finding simplicity in complexity. Yet complexity science has no single overarching theory of what simplicity means and how and why it can be found. In this talk I will present some well known methods and results to highlight different approaches to finding simplicity that computer science, physics and mathematics have developed. I will then highlight some less-known connections and argue that an overarching theory of simplicity may be within reach. 

The question is this:  can one effectively decide whether two given groups have isomorphic profinite completions? Thanks to Bridson and Wilton, it is known that the answer is `no' in general, even for finitely presented residually finite groups. However, if the groups are (and are given to be) virtually polycyclic, then the answer is 'yes'. This is not really surprising, as a lot is known both about the profinite completions of such groups and about how they are determined up to isomorphism; but it may be instructive to see how it is done.

The random geometric graph G(n,S^d,p) is obtained by placing n random points independently and uniformly on the unit sphere S^d, and connecting two points whenever they are sufficiently close, with the threshold chosen so that each edge appears with probability p. The underlying geometry of the model creates correlations between edges, making its behavior richer than that of the corresponding binomial random graph G(n,p).

A striking recent application of these correlations is due to Ma, Shen, and Xie, who used high-dimensional random geometric graphs to obtain an exponential improvement over Erdős’s celebrated lower bound for R(k,Ck), where C>1 is fixed. I will discuss a simplification of their approach using Gaussian random geometric graphs, leading to a much shorter analysis and sharper quantitative bounds.

I will then turn to a complementary question: when does the geometry disappear? More precisely, for which dimensions d is G(n,S^d,p) statistically indistinguishable from G(n,p)? This problem, introduced by Bubeck, Ding, Eldan, and Rácz, has attracted considerable interest across probability, theoretical computer science, and high-dimensional statistics. They conjectured that the threshold is governed by the signed triangle count, namely d≍n^3p^3 up to logarithmic factors. I will outline a proof of this conjecture for a wide range of p.

This talk is based on joint work with Zach Hunter and Aleksa Milojevic.

In the 2010s, Cornulier and Tessera presented an algorithm deciding whether a Lie group has exponential or polynomially bounded Dehn function. I will discuss the highlights of their work, and then focus on the following question: in case the Dehn function is polynomially bounded, what is the degree of the bounding polynomials? The heart of the matter in this context is the geometric relation between a (completely) solvable group and its largest nilpotent quotient. I will outline the basics of this geometry, and present a new method that exploits it to give (in some cases) better bounds on the degree of the bounding polynomials.

Joint with Gabriel Pallier.

This talk by Peter Huston gives an overview of the motivation for and classification of fusion 1-categories and 2-categories. In particular, we will review how fusion 1-categories naturally arise in operator algebras from the subfactor classification programme, which furnishes exotic examples of fusion category, such as the Haagerup subfactor, which are inaccessible by other approaches. Fusion 2-categories are a categorification of fusion 1-category, arising naturally from the study of TQFT in 4D, or as quantum symmetries of fusion 1-categories. We will outline the classification of fusion 2-categories. In particular, we will see that, while fusion 1-categories are wild in the sense that they cannot be constructed from lower dimensional data like finite groups, fusion 2-categories are comparatively tame, expressible in terms of braided fusion 1-categories and extension theory.

It is well known that twice the square of the maximum of a reflected Brownian bridge, starting and ending at zero, has the same distribution as the random variable $S=\sum_{n=1}^\infty \frac{e_n}{n^2}$, where $e_1, e_2, \ldots$ is a sequence of independent standard exponential random variables, and that twice the square of the maximum of a standard Brownian excursion (i.e. a Brownian bridge, starting and ending at zero, conditioned to stay positive) has the same distribution as $S+S'$, where $S'$ is an independent copy of $S$. (The random variables $S$ and $S+S'$ are in fact closely related to the Riemann zeta function.) In this talk, I will present a conjectural generalisation of these identities in law, which relates maximal heights of non-intersecting reflected Brownian bridges and non-intersecting Brownian excursions to absorption times for discrete Whittaker processes. The latter are a family of Markov chains on reverse plane partitions which are closely related to the Toda lattice.  This work is motivated by an attempt to understand the large scale behaviour of discrete Whittaker processes, in particular the question of whether they belong to the KPZ universality class, which we now conjecture to be the case based on this apparent connection with non-intersecting Brownian bridges.

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This is a joint OxPDE and Numerical Analysis seminar.

Take a break at the end of term with some Mathematrix crafts and sweet treats! Supplies for watercolor and origami will be provided, and you are welcome to bring your own crafts. 

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Professor Boffi will talk about: 'Fictitious domain approach to FSI: theoretical results and implementation details'

In this talk I will review the main aspects of our fictitious domain - distributed Lagrange multiplier - approach to the approximation of fluid-structure interaction problems. Theoretical results include the analysis of the continuous problem in a linearized setting and the stability of the discrete scheme in space and time.

I will give details on some implementation aspects related to the treatment/integration of the coupling terms and I will propose a multigrid strategy for the solution of the discrete system.

The uncertainty in future market dynamics is an important consideration when developing strategies for hedging derivatives, particularly data driven strategies such as deep hedging. Deep market generators can produce higher fidelity training data than classical models, but, like those, typically require frequent recalibration to new market data. The resulting strategies are thus susceptible to underperformance if there is a mismatch (distributional shift) between training data and live data. We present a framework to train a modified deep hedger which displays a form of ambiguity aversion, henceforth termed an Ambiguity-Averse Deep Hedger (AADH). The modeller has full control over exactly which aspects of distributional shifts the AADH is to be robust to, through selection of features relevant to the trading strategy which are used to cluster the training data, allowing for the evaluation of a loss function motivated by the theory of smooth ambiguity aversion.

Stochastic search is ubiquitous in biology and ecology, from synaptic transmission and intracellular signaling to predators seeking prey and the spread of disease. In dynamic systems like these, the number of 'searchers' is rarely constant: new agents may be recruited while others can abandon the search. Despite the ubiquity of these dynamics, their combined influence on search times remains largely unexplored. In this talk we will introduce a general framework for stochastic search in which agents progressively join and leave the process, a mechanism we term 'dynamic redundancy and mortality'. Under minimal assumptions on the underlying search dynamics, our framework yields the exact distribution of the first-passage time to a target region and further reveals surprising connections to stochastic search with stochastic resetting, wherein a single searcher is randomly 'reset' to its initial state. We will then treat the target region as a queue, which we show has interarrival times governed by a thinned nonhomogeneous Poisson process. Altogether this work provides a rigorous foundation for studying stochastic search processes with a fluctuating number of searchers. This work is in collaboration with Dr. Aanjaneya Kumar (Santa Fe Institute) and José Giral-Barajas (Imperial College London).

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We study reinforcement learning in the physical world, where the underlying dynamics evolve according to an unknown stochastic differential equation, while only discrete-time data are available. Existing RL algorithms typically ignore this SDE structure, which can limit their effectiveness in physical-world settings. We develop a systematic approach for adapting existing RL algorithms to this setting with minimal modifications, by leveraging the smoothness of the underlying continuous-time dynamics. In particular, for the LQR setting, we show that our framework can recover the exact continuous-time optimal control with only discrete-time information. We further identify a fundamental trade-off between discretization error and statistical error that is intrinsic to RL in the physical world. Finally, we extend the framework to mean-field optimal control.

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Tumor growth and angiogenesis drive complex spatiotemporal variation in micro-environmental oxygen levels. Previous experimental studies have observed that cancer cells exposed to chronic hypoxia retained a phenotype characterized by enhanced migration and reduced proliferation, even after being shifted to normoxic conditions, a phenomenon which we refer to as hypoxic memory. However, because dynamic hypoxia and related hypoxic memory effects are challenging to measure experimentally, our understanding of their implications in tumor invasion is quite limited. Here, we propose a novel phenotype-structured partial differential equation modeling framework to elucidate the effects of hypoxic memory on tumor invasion along one spatial dimension in a cyclically varying hypoxic environment. We incorporated hypoxic memory by including time-dependent changes in hypoxic-to-normoxic phenotype transition rate upon continued exposure to hypoxic conditions. Our model simulations demonstrate that hypoxic memory significantly enhances tumor invasion without necessarily reducing tumor volume. This enhanced invasion was sensitive to the induction rate of hypoxic memory, but not the dilution rate. Further, shorter periods of cyclic hypoxia contributed to a more heterogeneous profile of hypoxic memory in the population, with the tumor front dominated by hypoxic cells that exhibited stronger memory. Overall, our model highlighted the complex interplay between hypoxic memory and cyclic hypoxia in shaping heterogeneous tumor invasion patterns.

Keywords: Tumor invasion, cyclic hypoxia, hypoxic memory, phenotype-structured model

Applications involving optimization over graphs include molecular design, graph neural network verification, neural architecture search, etc. This talk discusses formulating graph spaces using mixed-integer optimization and incorporating application-specific constraints. We discuss computational challenges with these mixed-integer optimization formulations and zoom in on the practical implications for these applications. We mention what has been done (by both ourselves and others) and what other research still needs to be done.

Co-authors: Shiqiang Zhang, Yilin Xie, Christopher Hojny, Juan Campos, Jixiang Qing, Christian Feldmann, David Walz, Frederik Sandfort, Miriam Mathea, Calvin Tsay

This talk is hosted by Rutherford Appleton Laboratory, Harwell Campus

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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.