In 2008, Toms constructed a counterexample to the Elliott conjecture: a pair of simple, separable, nuclear and unital C*-algebras which are indistinguishable by the Elliott invariant, but are not isomorphic. The key to distinguishing this pair of carefully crafted C*-algebras lies with a rather refined invariant called the Cuntz semigroup. Consequently, Toms’s counterexample highlighted the importance of the Cuntz semigroup to the classification of C*-algebras.

In this talk, we will discuss the Cuntz semigroup in the context of graph C*-algebras, a highly diverse class of mostly non-simple C*-algebras. In particular, we will accentuate how the highly organised structure of a unital graph C*-algebra is reflected in its Cuntz semigroup and if enough time permits, mention properties of unital graph C*-algebras that are revealed by these Cuntz semigroups.

Mr Oluwadamilola (Dami) Fasina will talk about; 'Learning with tensor paraproducts'

We discuss computational (Neural FIM) and analytical (tensor paraproducts) tools for learning structure of sets. In the first situation we focus on learning the metric amongst elements of a statistical manifold. To do so, we design a neural network which enables one to compute the Fisher information metric (FIM), so long the Jensen-Shannon divergences amongst probability distributions on the statistical manifold are preserved during training. In the second situation we focus on analyzing the structure of function compositions through separation of its low and high frequency components. This is accomplished by elaborating on J.M. Bony’s celebrated work on paraproducts by discretizing and allocating distinct scaling parameters along each dimension of the support of a function composition (with a prescribed regularity), permitting finer analytical control. A consequence of this extension is highlighted with a discussion of the regularity gains of kernels of integral operators. 

We consider a Kolmogorov-type PDE corresponding to a particle under white noise force. We are interested in stopping the process at a fixed position i.e. imposing Dirichlet conditions at a side boundary. We construct a simple Gaussian heat kernel inside the domain and investigate a boundary-layer kernel connected to some work by McKean. We show that this boundary layer heat kernel has a novel jump condition. We outline a polynomial expansion of for the heat kernels and then construct a Volterra equation for solving the original problem. The novel jump leads to a periodic structure of the Volterra equation.

Dehn Surgery is an operation on 3-manifolds that is ubiquitous in the field of low-dimensional topology. Concretely, given a link L in a 3-manifold N, Dehn surgery produces many new 3-manifolds. A classical result of Lickorish-Wallace states that from one fixed 3-manifold N one can obtain all other 3-manifolds by Dehn surgery using some link in N. However it remains unclear, in general, which manifolds can be obtained by Dehn surgery using a fixed manifold N and a fixed link L. I will discuss how one can algorithmically decide this question and then discuss applications of this algorithm. 

TBA 

Denote by p(k) the limit, as n tends to infinity, of the probability that a random permutation on n letters has some invariant set of size k. For example, p(1) = 1 - 1/e. I will discuss the asymptotic behaviour of p(k). Joint work with Mehtaab Sawhney.

Donaldson--Thomas invariants are virtual counts of coherent sheaves on complex Calabi–Yau 3-folds, where Brav–Bussi–Joyce's shifted Darboux theorem plays an important role. In this talk, I will present a Darboux-type theorem in characteristic $p>2$ for $(-1)$-shifted symplectic forms equipped with an "infinitesimal" structure suggested by Toën and Robalo. This result may be viewed as a first step towards exploring Donaldson–Thomas theory in positive characteristic.

The boundedness of Fourier multipliers on non-commutative $L_p$-spaces ($1 < p < \infty$) is a fundamental problem in non-commutative analysis. Building on the non-commutative Cotlar identity introduced by Mei and Ricard (2017), which yields $L_p$-boundedness ($1 < p < \infty$) of Hilbert transforms on amalgamated free products of finite von Neumann algebras, their approach relies heavily on freeness in the underlying free product structure.

In this talk, Xiaoqi Lu introduces a new strategy that overcomes this limitation. Our approach combines a generalized Cotlar identity, which holds on suitable subspaces and captures non-freeness information, with an additional condition related to the property of Rapid Decay to control the remaining components. From this framework, we establish the $L_p$-boundedness ($1 < p < \infty$) of Rademacher-type Hilbert transforms on graph products of finite von Neumann algebras. This unified framework extends earlier results for free products of finite von Neumann algebras and for graph products of groups acting on right-angled buildings. This is a joint work with Runlian Xia.

Recent advances at the interface of stochastic analysis, rough path theory, stochastic filtering, stochastic control, and mean-field systems have led to a rapidly developing framework for analyzing stochastic dynamics conditioned on common/observation noise. This mini course  will survey how rough stochastic differential equations, introduced in 2021 by A. Hocquet, K. Lê and the speaker, lead to a unifying perspective across several areas of applied probability. (Additional coauthors include F. Bugini, J. Dause, W. Stannat, H. Zhang and P.Zorin-Kranich).

A central challenge in applied mathematics is to extract predictive structure from data generated by complex dynamical systems. Koopman operator methods provide a principled framework for this task by embedding nonlinear dynamics into a linear operator acting on observables, reducing analysis and forecasting to questions about spectral approximation.

In this talk, I will present recent results on the analysis of data-driven Koopman methods, with an emphasis on when spectral quantities can be reliably approximated from finite data. I will describe a general framework that connects operator-theoretic properties of the Koopman operator with the behaviour of practical algorithms, clarifying phenomena such as spectral pollution and the role of continuous spectra. I will also discuss fundamental limitations: there exist classes of dynamical systems for which finite data cannot recover meaningful spectral information, placing intrinsic constraints on what Koopman-based approaches can achieve. Building on this, I will show how spectral approximation errors translate into quantitative bounds for forecasting, capturing how approximation and statistical errors propagate over time and ultimately limit long-term prediction. These results have implications for applications including fluid dynamics, molecular systems, and geophysical flows. I will conclude by highlighting open problems at the intersection of operator theory, numerical analysis, and scientific machine learning.

TBA

Negative-form symmetries arise when one extends the usual p-form dictionary below ordinary zero-form symmetries. Conceptually, however, they are different: the action of (-n)-form symmetries on a QFT modifies the parameters or background data that defines the QFT, as opposed to acting on the extended operators of the theory. For example, (-1)-form symmetries are implemented by spacetime-filling topological operators that act on a theory by shifting its theta-angle. I will review recent work arxiv:2606.05543 that has begun to develop the machinery of (-2)-form symmetries, which act of a QFT by modifying the anomaly inflow data – equivalently the SymTFT action – thereby relating QFTs whose ordinary global symmetries differ by anomaly data.

Dr Lindon Roberts will talk about: 'Optimization Algorithms for Bilevel Learning with Applications to Imaging'

Many imaging problems, such as denoising or inpainting, can be expressed as variational regularization problems. These are optimization problems for which many suitable algorithms exist. We consider the problem of learning suitable regularizers for imaging problems from example (training) data, which can be formulated as a large-scale bilevel optimization problem. 

In this talk, I will introduce new deterministic and stochastic algorithms for bilevel optimization, which require no or minimal hyperparameter tuning while retaining convergence guarantees. 

This is joint work with Mohammad Sadegh Salehi and Matthias Ehrhardt (University of Bath), and Subhadip Mukherjee (IIT Kharagpur).

The global Langlands-Kottwitz method seeks to express Frobenius-Hecke traces on the cohomology of Shimura varieties in terms of (twisted) orbital integrals; the latter are central objects in local harmonic analysis which enter the Arthur-Selberg trace formula. While this method is well studied, we present a new local analogue: a formula relating the cohomology of local Shimura varieties to twisted orbital integrals. This local formula bridges the point-counting formula for global Shimura varieties with the point-counting formula for Igusa varieties. As an application of our local formula, we propose a new approach, based on categorical Langlands, towards Rapoport's vanishing conjecture on certain twisted orbital integrals. This conjecture is itself a key ingredient in the global Langlands-Kottwitz method for a non-quasi-split prime. This is joint work with Rong Zhou.

TBA

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.

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This is a joint OxPDE and Numerical Analysis seminar. 

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.

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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.