Past

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. 

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.

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.

TBA

This is a joint OxPDE and Numerical Analysis seminar. 

This is a Joint OxPDE & Numerical Analysis Seminar 

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 

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. 

In this talk, I present an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost Kähler manifolds. The formula expresses the mass in terms of the total Hermitian scalar curvature and topological data associated with the underlying almost complex structure, extending a result of Hein and LeBrun in the Kähler ALE case. The proof is based on a spin-c adaptation of Witten's proof of the positive mass conjecture in the spin case and is therefore distinct from previous complex-geometric methods. In dimension 4, I show that one can prove a positive mass theorem and a Penrose-type inequality for asymptotically Euclidean (AE) almost Kähler manifolds using this formula.

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.

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.

Topological Tverberg theory seeks r-fold analogues of classical nonembeddability results. Given a simplicial complex K, the central question is whether every continuous map from K into R^d necessarily identifies r points lying on r pairwise disjoint faces of K. The corresponding collection of faces is called a Tverberg r-partition. Perhaps surprisingly, the existence of such partitions depends on the arithmetic properties of r.

The admissible-prescribable conjecture proposes a refinement of this theory by predicting exactly which face dimensions must occur in Tverberg r-partitions. The conjecture has been verified in a number of cases, using tools such as shelling constructions and discrete Morse theory to determine the homotopy type of the relevant configuration spaces.

In this talk, we present counterexamples that settle the remaining open cases and disprove the conjecture in full generality. Our approach combines a diagrammatic description of configuration spaces with techniques from the theory of homotopy colimits of covers, allowing us to equivariantly reduce these spaces. We then show how methods from differential and PL topology, including the r-fold Whitney trick and surgery of intersections techniques, can be employed to construct the desired counterexamples.

This talk is based on forthcoming joint work with Pavle Blagojević and Florian Frick.

Modern ML methods for derivatives sit at a delicate interface between market prices, implied-volatility (IV) surfaces, and the simulated environments produced by market generators. To date, these models have largely operated in one of two coordinate systems: price space, where markets quote and no-arbitrage constraints are most naturally enforced, and IV space, where surfaces are smoothed, regularized, and evaluated. This talk presents a technique that unifies learning across both coordinates — using gradients from each via a differentiable Jäckel operator and a low-vega gating mechanism — enabling end-to-end batch training without the error-prone, expensive, hand-engineered filtering usually needed to discard incompatible IV values. I will present PIVOT (Price-Implied Volatility Operator Transform), a differentiable Jäckel IV operator that preserves the accuracy of the standard "Let's Be Rational" (LBR) solver in the forward pass while supplying implicit gradients through the Black–Scholes/Black-76 price map. This gives neural volatility-surface models a principled bridge between price-space and IV-space objectives, with explicit handling of the low-vega singular regime. Second, I will  present Fast-Vollib ( https://pypi.org/project/fast-vollib/), a CUDA-accelerated option-pricing library with NumPy, PyTorch, and JAX interfaces, built for high-throughput pricing-label generation in AI/ML batch training.

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.

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).

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.

Katherine Pearce is going to talk about: 'Randomized Algorithms for Tensor CUR Approximations in Attention Mechanisms'

Attention mechanisms are a central component of transformer models that capture contextual relationships between tokens in large language models. Although many of the underlying computations (e.g., query, key, and value embeddings in multi-head attention) are inherently multi-way, classical transformer models are built on matrix-based formulations. In this talk, we discuss several ways that tensorial structure can be imposed on and exploited in attention mechanisms of transformer models. We describe how tensor-based attention can capture higher-order contextual relationships among tokens. We then explore how randomized algorithms to compute tensor CUR decompositions may be used to accelerate computations in tensor-based attention and reduce storage requirements.

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.

I am going to talk on a work aimed at formalising approximation procedures in physics.  The main new model-theoretic tool in this work is the notion of the ultraproduct in the classes of emerging metric structures which generalises the ultraproduct  of general structures developed by J.Kiesler. In particular, the structure of Minkowski spacetime with the action of the Lorentz group is an emerging metric ultraproduct of certain finite structures invariant under the action of appropriate finite groups. Also, it is shown that any compact simple Lie group is representable as emerging metric ultraproduct of finite groups.
 

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).

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.

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.

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.

Professor Ruoyu Sun will talk about: 'Understanding and Improving LLM Training via Hessian and Spectral Analysis' 

In the first part, we investigate the approximate block-diagonal Hessian structure of neural networks. We identify the conditions under which this structure emerges and give the first rigorous proofs based on random matrix theory. From this structural perspective, we explain why Adam works far better than SGD on Transformers. Following this structural guideline, we design the memory-efficient optimizer Adam-mini; Normuon is another optimizer developed under the same principle.

 In the second part, we adopt a spectral perspective to study and refine normalization layers for neural network training. We propose a preconditioning (PC) layer, an advanced weight-centric module built with low-degree polynomial preconditioning for scalable spectral control. Theoretically, for deep linear networks, we prove that bounding each layer's singular values ensures geometric convergence of gradient descent to global minima. Empirically, PC delivers consistent efficiency gains over a standard Transformer baseline in Llama2-1B pretraining.

Nilsequences are sequences coming from Lie groups which play the role of additive characters in higher order Fourier analysis. In this talk, I will define these and give some basic examples without assuming any prior knowledge. I'll use this to state an equidistribution result due to Green and Tao, and compare what happens in this setting to the familiar case of sequences in the torus.
 

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. 

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.

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. 

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.

The celebrated additive kinematic formula expresses the mean volume of the Minkowski sum of two compact convex subsets of the Euclidean space placed at random. What about non convex subsets? What about other Lie groups than the Euclidean space? In a joint work with Andreas Bernig, we prove additive kinematic formulas for compact subanalytic sets of the Euclidean space and of the 3-sphere. The key is to generalize the Minkowski sum of convex bodies by a notion of convolution of subanalytic sets introduced by Schapira in the late 80s using Euler characteristic computations. The above will of course be an excuse to discuss integral geometric formulas and constructible functions.

In this talk, I will focus on a new construction of boundary correlators (or wavefunction coefficients in dS) that highlights simplicity at all spins and automatically imposes the conservation of boundary currents. This new construction is formulated in twistor space, a complex projective space that encodes solutions to equations of motion as holomorphic data. This is done via an isomorphism called the Penrose transform. First, I will discuss the case of AdS_3 and AdS_5, where bulk-to-boundary correlators naturally arise in minitwistor space. Then, I will show how in (A)dS₄ one can construct bulk correlation functions using only twistors, dual twistors, and the infinity twistor as building blocks. The relation to coordinate space arises now via nested Penrose transforms. The boundary limit of these correlators yields CFT correlators/wavefunction coefficients that satisfy the expected Ward identities. Finally, I will briefly discuss how this can be generalized to AdS_5 boundary correlators using ambitwistors.

Mechanistic ordinary differential equation (ODE) models are a powerful tool to study dynamic biological systems. However, their predictive power is constrained by gaps, biases, and inconsistencies in the literature. They typically also require quantitative time-lapse data for training, which is time-consuming to collect. At the same time, machine-learning approaches can capture complex patterns from data, but they are often harder to interpret and typically require large training datasets. Hybrid scientific machine learning (SciML) models offer a promising way to combine the strengths of both approaches by integrating mechanistic models with flexible data-driven modules. 
Despite this promise, the use of SciML in biology remains limited by insufficient infrastructure. Dedicated software is needed because coding end-to-end differentiable workflows for gradient-based training of hybrid models is technically challenging. In addition, model exchange is hindered by the lack of a standardized, reproducible format for specifying SciML training problems, analogous to the PEtab standard for ODE models. To address these challenges, we developed PEtab-SciML, an extension of the PEtab format, and implemented support for it in the state-of-the-art modeling toolboxes PEtab.jl and AMICI. In this seminar, I will introduce the PEtab-SciML format. Using real-data examples, I will show how PEtab-SciML enables the integration of diverse data modalities into dynamic model training; such as learning the kinetic parameters of an ODE model from omics and protein sequence data. I will also show how it supports machine-learning-based black-boxing of complex model components, such as quarantine strength in an SIR model. Finally, I will show how PEtab-SciML enables the use of efficient training strategies, such as curriculum learning, that make SciML models easier to train and apply in practice. 

Stability is the prototypical model theoretic dividing line. One interpretation is that a binary relation is stable if it is "close to unary": if the question $(x,y)\in E$ can be answered, at least most of the time, by knowing enough information about $x$, and separately enough information about $y$.

One natural question is asking how this can generalize to ternary (and higher-arity) relations. The connection to hypergraph regularity suggests an approach to identifying ternary stable-like properties, and also that there should be several versions, since a ternary relation could be almost unary, or almost binary, or a combination of these properties.

In this talk, I'll survey some of what we know about several of these "stable-like" ternary notions.

In many prediction and decision problems, the relevant inputs are path-valued covariates rather than static feature vectors. This paper studies asymptotic theory and empirical applications for path regression using signatures. We first establish \(L^2\) approximation rates for truncated signature representations. We prove a minimax-optimal approximation rate over a class of smooth coefficient functionals of observable It\^{o} diffusions. Building on this approximation theory, we then develop asymptotic results for three signature-based learning procedures: Signature-OLS, Signature-LASSO, and Signature-Logistic. These results establish asymptotic normality for least-squares path regression, sparse recovery for high-dimensional signature regression, and latent-score consistency for binary-response classification. Extensive empirical studies cover three real-data applications: foreign-exchange realized-volatility forecasting from intraday price paths, battery end-of-life prediction from early HPPC pulse paths, and epileptic seizure detection from short EEG windows. The empirical results show that signatures provide informative representations of path-valued covariates relative to handcrafted features.

Understanding the behaviour of L-functions of modular forms is a very classical and yet open problem. The Bloch-Kato conjecture predicts that the order of vanishing of the L-function of a modular form should be given by the rank of certain Bloch-Kato Selmer groups. In order to give a lower bound to these ranks in certain cases where the L-function vanishes, Bellaiche and Chenevier developed a clever strategy where they construct classes in the Selmer group via the geometry of points corresponding to certain lifts of modular forms on higher dimensional eigenvarieties. This strategy was successfully adapted for ordinary modular forms by Berger and Betina to give a lower bound in terms of the smoothness of Saito-Kurokawa points on a genus 2 Siegel eigenvariety. We generalise this work to finite slope and crucially infinite slope forms which are not seen on the Coleman-Mazur eigencurve - here we must develop the machinery of small parabolic eigenvarieties for the problem to be well defined. As a result we get new results towards the Bloch—Kato conjecture for infinite slope forms.

The ε-energy is a regularisation of the Dirichlet energy introduced by Tobias Lamm. Like the famous Sacks-Uhlenbeck regularisation this greatly improves the existence and regularity theory. When we take the limit of a sequence of ε-harmonic maps with the parameter ε decreasing to 0 these converge, in the standard bubbling sense, to harmonic maps, which we hope to extract information about. I will talk about some recent results for these sequences, being when we might hope to have no loss of energy and no neck forming and what sort of harmonic maps we can obtain in the limit.