Past

Professor Andrew Saxe will talk about; 'Demystifying depth: principles of learning in deep neural networks' 

Deep neural networks have revolutionized artificial intelligence, yet their inner workings remain poorly understood. This talk presents mathematical analyses of the nonlinear dynamics of learning in several solvable deep network models, offering theoretical insights into the role of depth. These models reveal how learning algorithms, data structure, initialization schemes, and architectural choices interact to produce hidden representations that afford complex generalization behaviours. A recurring theme across these analyses is a neural race: competing pathways within a deep network vie to explain the data, with an implicit bias toward shared representations. These shared representations in turn shape the network’s capacity for systematic generalization, multitasking, and transfer learning. I will show how such principles manifest across diverse architectures—including feedforward and linear attention networks. Together, these results provide analytic foundations for understanding how environmental statistics, network architecture, and learning dynamics jointly structure the emergence of neural representations and behaviour.

I will describe a circle-valued Morse theory for simplicial complexes. The central objects of study are partial matchings which admit certain zigzag cycles; these cyclic matchings lift canonically to acyclic matchings on the infinite cyclic cover of the underlying simplicial complex. From the lifted acyclic matchings, we obtain a finitely generated Morse chain complex defined over the Novikov ring, which consists of power series in one variable with finite negative support. We then establish a quasi-isomorphism between this Morse-Novikov complex and the simplicial chain complex of the cyclic cover, duly completed over the Novikov ring. As a pleasant consequence, we can define new computable invariants to detect (obstructions to) the fiberedness of tame knots.

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).

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.

We discuss key results and milestones achieved while studying certain families of Diophantine equations as well as touching on open problems. We note that this is an overview of a large body of work involving multiple collaborators, including: A. Argáez-García (UADY), M. Bennett (UBC), N. Coppola (Padova), M. Curcó-Iranzo (Utrecht), S. Siksek (Warwick), M. Khawaja (Warwick) and Ö. Ülkem (Academia Sinica).

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

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

Quantum field theory is hard for several reasons, for example one can rarely compute perturbation series (=Feynman integrals) at large loop order, and even if you can, the series diverges. Conversely, intrinsically non-perturbative approaches like the functional renormalization group require approximations that are often not easy to control, or have unclear relations to perturbative computations.
Tropical field theory is a new approach for solving these issues for a generic theory without restricting to unphysical boundary cases. It keeps almost all qualitative and combinatorial features of perturbative QFT (in particular all non-planar diagrams, renormalization, relative numerical importance of Feynman integrals, and divergence of perturbation series), while at the same time reducing the analytic complexity, and establishing a rigorous connection to non-perturbative functional/path integral methods of QFT. Based on 2512.21091 with Erik.

Choosing where to stop an iteration or how far to increase a model complexity parameter is a recurring problem in numerical computation and data analysis. Typical symptoms are diminishing returns, a noise-dominated floor, and overfitting---accordingly, many heuristics seek an elbow or plateau beyond which further effort is not worthwhile.  AAA rational approximation provides a sharp instance of this difficulty when constructing rational approximations from noisy data, where the error often decreases rapidly at first and then fluctuates in a noisy band.  Standard AAA has no mechanism to recognize this regime and may continue iterating until a preset degree cap is reached. We thus propose noiseChop, a noise-aware stopping rule designed to run online alongside AAA. The method is inspired by Chebfun's standardChop but is tailored to AAA by using quantities already available during the iteration---a monotone envelope of the $\infty$-norm nonlinear error and the linearized error from the Loewner least squares step.  
The method first detects evidence of stagnation and then selects an early cutoff degree that achieves good accuracy without chasing noise. Numerical tests illustrate robust behavior across several functions, sample sets, and noise levels. The method is soon to be available as an optional feature in Chebfun's AAA code.

Hydrogels are soft, highly absorbent porous materials which are commonly used in pharmaceutical applications such as in soft contact lenses, drug delivery and wound healing. They are commonly modelled as hyperelastic materials with an additional chemical force driving the influx of water into the gel. In this talk, I will show how taking the “osmosis-dominated limit” (i.e. the regime where chemical forces dominate over elastic, which is the relevant limit for most commonly used hydrogels) can simplify the PDEs governing hydrogel dynamics. In the linear case, I will show the swelling problem can be entirely decoupled from the solid mechanics problem. In the nonlinear case, I will show the coupling is sufficiently weak as to enable a simplified solution procedure by finite element methods.

I will give an update on a proposed model theory for directed limits and colimits of first-order structures, originally motivated by applications to commutative algebra and the model theory of valued fields. To illustrate the usefulness of the formalism, I will prove a new general AKE theorem in mixed characteristic in a language with a cross-section of the value group and a lift of the residue field.

I will also discuss connections with other approaches to this topic, including pro- and ind-definable sets, infinitary logic, Feferman's local functors, accessible functors, and ultraproducts, some of which I have not discussed previously.

The Kervaire conjecture was formulated around 1963 after a conversation between Kervaire and Baumslag. It states that adding a generator and then a relator to a non-trivial group always yields a non-trivial group. To this day, the conjecture remains unproven in its most general form; however, it has been shown under certain additional hypotheses, either on the new relator or on the original group. For instance, the result holds for locally indicable groups and for locally residually finite groups. In this talk, I will explain Klyachko’s proof of the conjecture for torsion-free groups, which uses a funny property of the sphere known as the Car Crash Theorem, and van Kampen pictures. I will also discuss how these techniques were generalised by Fenn and Rourke to study equations over torsion-free groups defined by a large class of words (amenable words).

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. 

TBA

This is a joint OxPDE and Numerical Analysis seminar.

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.

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.

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.

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.

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.

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).