Cryptocurrencies such as Bitcoin have only recently become a significant part of the financial landscape. Many billions of dollars are now traded daily on limit order-book markets such as Binance, and these are probably among the most open, liquid and transparent markets there are. They therefore make an interesting platform from which to investigate myriad questions to do with market microstructure. I shall talk about a few of these, including live-trading experiments to investigate the difference between on-paper strategy analysis (typical in the academic literature) and actual trading outcomes. I shall also mention very recent work on the new Hyperliquid exchange which runs on a blockchain basis, showing how to use this architecture to obtain datasets of an unprecendented level of granularity. This is joint work with Jakob Albers, Mihai Cucuringu and Alex Shestopaloff.

We discuss ongoing work with Joseph Leung in which we obtain estimates for sums of Fourier coefficients of GL(2) and certain GL(3) automorphic forms along the values of irreducible binary cubics.

Feferman proved in 1962 that any arithmetical theorem is a consequence of a suitable transfinite iteration of uniform reflections. This result is commonly known as Feferman's completeness theorem. The talk aims to give one or two new proofs of Feferman's completeness theorem that, we hope, shed new light on this mysterious and often overlooked result.

Moreover, one of the proofs furnishes sharp bounds on the order types of well-orders necessary to attain completeness.

(This is joint work with Fedor Pakhomov and Dino Rossegger.)

Circadian clocks generate autonomous daily rhythms of gene expression in anticipation of daily sunlight and temperature cycles in a variety of organisms. The simples and best characterised of all circadian clocks in nature is the cyanobacterial clock, the core of which consists of just 3 proteins - KaiA, KaiB and KaiC - locked in a 24-h phosphorylation-dephosphorylation loop. Substantial progress has been made in understanding how cells generate and sustain this rhythm, but important questions remain: how does the clock maintain resilience in the face of internal and external fluctuations, how is the clock coupled to other cellular processes and what dynamics arise from this coupling? We address these questions using an interdisciplinary approach combining time-lapse microscopy and modelling. In this talk, I will first characterise the clock's free-running robustness and explore how the clock buffers environmental noise and genetic mutations. Our stochastic model predicts how the clock filters out such noise, including fast light fluctuations, to keep time while remaining responsive to environmental shifts, revealing also that the wild-type operates at a noise optimum. Next, I will focus on how the clock interacts with the other major cellular cycle, the cell division cycle. Our single-cell data shows that the clock couples to the division rate and expression of cell cycle-dependent factors using both frequency modulation and amplitude modulation strategies, with implications for cell growth and cell size control. Our findings illustrate how simple systems can exhibit complex dynamics, advancing our understanding of the interdependency between gene circuits and cellular physiology.  
 

Complex representations of p-adic groups are in many ways well-understood. The category has Bernstein's decomposition into blocks, and for many groups each block is known to be equivalent to modules over a Hecke algebra. In particular, the unipotent block of GLn (the block containing the trivial representation) is equivalent to the modules over an extended affine hecke algebra of type A. Over \bar{Fl} the situation is more complicated in the general case: the Bernstein block decomposition can fail (eg for SP8), and there is no longer in general an equivalence with the Hecke algebra. However, some groups, such as GLn and its inner forms, still have a Bernstein decomposition. Furthermore, Vigernas showed that the unipotent block of GLn contains a subcategory that is equivalent to modules over the Schur algebra, a mild extension of the Hecke algebra with much of the same theory, and this subcategory generates the unipotent block under extensions. Building on this work, we show that the derived category of the unipotent block of GLn is triangulated-equivalent to the perfect complexes over a dg-enriched Schur algebra. We prove this by combining general finiteness results about Schur algebras with the well-known structure of the l-modular unipotent blocks of GLn over finite fields.

This session is aimed at first-year undergraduates preparing for Prelims exams. A panel of lecturers will share key advice on exam technique and revision strategies, and a current student will offer practical tips from their own experience. This event complements the Friday@2 session in Week 1 on Dealing with Exam Anxiety.

The accurate semantic segmentation of individual tree crowns within remotely sensed data is crucial for scientific endeavours such as forest management, biodiversity studies, and carbon sequestration quantification. However, precise segmentation remains challenging due to complexities in the forest canopy, including shadows, intricate backgrounds, scale variations, and subtle spectral differences among tree species. While deep learning models improve accuracy by learning hierarchical features, they often fail to effectively capture fine-grained details and long-range dependencies within complex forest canopies.

This seminar introduces PerceptiveNet, a novel model that incorporates a Logarithmic Gabor-implemented convolutional layer alongside a backbone designed to extract salient features while capturing extensive context and spatial information through a wider receptive field. The presentation will explore the impact of Log-Gabor, Gabor, and standard convolutional layers on semantic segmentation performance, providing a comprehensive analysis of experimental findings. An ablation study will assess the contributions of individual layers and their interactions to overall model effectiveness. Furthermore, PerceptiveNet will be evaluated as a backbone within a hybrid CNN-Transformer model, demonstrating how improved feature representation and long-range dependency modelling enhance segmentation accuracy.

In this seminar, we explore the quantitative convergence of wide deep neural networks with Gaussian weights to Gaussian processes, establishing novel rates for their Gaussian approximation. We show that the Wasserstein distance between the network output and its Gaussian counterpart scales inversely with network width, with bounds apply for any finite input set under specific non-degeneracy conditions of the covariances. Additionally, we extend our analysis to the Bayesian framework, by studying exact posteriors for neural networks, when endowed with Gaussian priors and regular Likelihood functions, but we also provide recent advancements in quantitative approximation of trained networks via gradient descent in the NTK regime. Based on joint works with A. Basteri, and A. Agazzi and E. Mosig.

Given a mod $p$ Galois representation, one often wonders whether it arises by reducing a $p$-adic one, and whether these lifts are suitably 'well-behaved'. In this talk, we discuss how ideas from homotopy theory aid the study of Galois deformations, reviewing work of Galatius-Venkatesh.

When dealing with PDEs arising in fluid mechanics, bounded-energy weaksolutions are, in many cases, the only type of solutions for which one can guarantee global existence without imposing any restrictions on the size of the initial data or forcing terms. Understanding how to construct such solutions is also crucial for designing stable numerical schemes.

In this talk, we will explain the strategy for contructing weak solutions for the Navier-Stokes system for viscous compressible flows, emphasizing the difficulties encountered in the case of non-isotropic viscous stress tensors. In particular, I will present some results obtained in collaboration with Didier Bresch and Maja Szlenk.

We consider quantum electrodynamics in 2+1 dimensions (QED3) with N matter fields and Chern-Simons level k. For small values of k and N, this theory describes various experimentally relevant systems in condensed matter, and is also conjectured to be part of a web of non-supersymmetric dualities. We compute the scaling dimensions of monopole operators in a large N and k expansion, which appears to be extremely accurate even down to the smallest values of N and k, and allows us to find dynamical evidence for these dualities and make predictions about the phase transitions. For instance, we combine these estimates with the conformal bootstrap to predict that the notorious Neel-VBS transition (QED3 with 2 scalars) is tricritical, which was recently confirmed by independent lattice simulations. Lastly, we propose a novel phase diagram for QED3 with 2 fermions, including duality with the O(4) Wilson-Fisher fixed point.

Given an FP-infinity subgroup G of SL(2,C), we are interested in the asymptotic behavior of the cohomology of G with coefficients in an irreducible complex representation V of SL(2,C). We prove that, as the dimension of V grows, the dimensions of these cohomology groups approximate the L2-Betti numbers of G. We make no further assumptions on G, extending a previous result of W. Fu. This yields a new method to compute those Betti numbers for finitely generated hyperbolic 3-manifold groups. We will give a brief idea of the proof, whose main tool is a completion of the universal enveloping algebra of the p-adic Lie algebra sl(2, Zp).

Given an FP-infinity subgroup G of SL(2,C), we are interested in the asymptotic behavior of the cohomology of G with coefficients in an irreducible complex representation V of SL(2,C). We prove that, as the dimension of V grows, the dimensions of these cohomology groups approximate the L2-Betti numbers of G. We make no further assumptions on G, extending a previous result of W. Fu. This yields a new method to compute those Betti numbers for finitely generated hyperbolic 3-manifold groups. We will give a brief idea of the proof, whose main tool is a completion of the universal enveloping algebra of the p-adic Lie algebra sl(2, Zp).

A consequence of the works of Costello and Lurie is that the Hochschild chain complex of a Calabi-Yau category admits the structure of a framed E_2 algebra (the genus zero operations). I will describe a new algebraic point of view on these operations which admits generalizations to the setting of relative
Calabi-Yau structures, which do not seem to fit into the framework of TQFTs. In particular, we obtain a generalization of string topology to manifolds with boundary, as well as interesting operations on Hochschild homology of Fano varieties. This is joint work with Chris Brav.

The Complexity Cluster Event Organisers Professor Gui-Qiang G. Chen, Professor Helen Byrne, and Professor Mohit Dalwadi, cordially invite you to attend a Complexity Cluster Research Workshop on Tuesday, 20th May 2025, in the Glen Callater Room, H B Allen Centre, Keble College.

Complexity Cluster Research Workshop
Venue: Glen Callater Room, H B Allen Centre, Keble College
Date: Tuesday, 20th May 2025
Organisers: 
Professor Helen Byrne
Professor Gui-Qiang G. Chen
Professor Mohit Dalwadi
 

Programme:
4.00pm ̶ 4.15pm: Coffee, Drinks & Refreshments
4:15pm ̶ 4:40pm: Professor Didier Bresch (CNRS and Universite Savoie Mont-Blanc, France): Mathematical Topics around Granular Media
4:45pm – 5:10pm: Dr. Keith Chambers (Mathematical Institute, University of Oxford): Structured Population Models to Explore Lipid-Driven Macrophage Heterogeneity in Early Atherosclerotic Plaques
5:15pm ̶ 5:40pm: Coffee, Drinks & Refreshments
5:40pm ̶ 6:05pm: Dr. Tara Trauthwein (Department of Statistics, University of Oxford): Approximation Results for Large Networks
6:05pm-6:30pm: Isaac Newell (OxPDE, Mathematical Institute, University of Oxford): The Gauss Equation for Isometric Embeddings of Regularity in W^1+2/3,3∩ C¹
6:35pm-7:00pm: Discussion

For abstracts please click the file here: Complexity-Cluster_Workshop_20250212_Final_0.pdf

Quantized universal enveloping algebras admit an intriguing class of (unbounded) Hilbert space representations obtained via their cluster structure. In these so-called positive representations the standard generators act by (essentially self-adjoint) positive operators. 

The aim of this talk is to discuss some analytical questions arising in this context, and in particular to what extent these representations can be understood using the theory of locally compact quantum groups in the sense of Kustermans and Vaes. I will focus on the simplest case in rank 1, where many of the key features (and difficulties) are already visible. (Based on work in progress with Kenny De Commer, Gus Schrader and Alexander Shapiro). 

The two-point Chowla conjecture predicts that $\sum_{x<n<2x} \lambda(n)\lambda(n+1) = o(x)$ as $x\to \infty$, where $\lambda$ is the Liouville function (a $\{\pm 1\}$-valued multiplicative function encoding the parity of the number of prime factors). While this remains an open problem, weaker versions of this conjecture are known. In this talk, we outline an approach initiated by Helfgott and Radziwill, which reformulates the problem in terms of bounding the eigenvalues of a certain matrix.

The Complexity Cluster Event Organisers Professor Gui-Qiang G. Chen, Professor Helen Byrne, and Professor Mohit Dalwadi, cordially invite you to attend a Complexity Cluster Research Workshop on Tuesday, 20th May 2025, in the Glen Callater Room, H B Allen Centre, Keble College.

Complexity Cluster Research Workshop
Venue: Glen Callater Room, H B Allen Centre, Keble College
Date: Tuesday, 20th May 2025
Organisers: 
Professor Helen Byrne
Professor Gui-Qiang G. Chen
Professor Mohit Dalwadi
 

Programme:
4.00pm ̶ 4.15pm: Coffee, Drinks & Refreshments
4:15pm ̶ 4:40pm: Professor Didier Bresch (CNRS and Universite Savoie Mont-Blanc, France): Mathematical Topics around Granular Media
4:45pm – 5:10pm: Dr. Keith Chambers (Mathematical Institute, University of Oxford): Structured Population Models to Explore Lipid-Driven Macrophage Heterogeneity in Early Atherosclerotic Plaques
5:15pm ̶ 5:40pm: Coffee, Drinks & Refreshments
5:40pm ̶ 6:05pm: Dr. Tara Trauthwein (Department of Statistics, University of Oxford): Approximation Results for Large Networks
6:05pm-6:30pm: Isaac Newell (OxPDE, Mathematical Institute, University of Oxford): The Gauss Equation for Isometric Embeddings of Regularity in W^1+2/3,3∩ C¹
6:35pm-7:00pm: Discussion

For abstracts please click the file here: Complexity-Cluster_Workshop_20250212_Final_0.pdf

The Complexity Cluster Event Organisers Professor Gui-Qiang G. Chen, Professor Helen Byrne, and Professor Mohit Dalwadi, cordially invite you to attend a Complexity Cluster Research Workshop on Tuesday, 20th May 2025, in the Glen Callater Room, H B Allen Centre, Keble College.

Complexity Cluster Research Workshop
Venue: Glen Callater Room, H B Allen Centre, Keble College
Date: Tuesday, 20th May 2025
Organisers: 
Professor Helen Byrne
Professor Gui-Qiang G. Chen
Professor Mohit Dalwadi
 

Programme:
4.00pm ̶ 4.15pm: Coffee, Drinks & Refreshments
4:15pm ̶ 4:40pm: Professor Didier Bresch (CNRS and Universite Savoie Mont-Blanc, France): Mathematical Topics around Granular Media
4:45pm – 5:10pm: Dr. Keith Chambers (Mathematical Institute, University of Oxford): Structured Population Models to Explore Lipid-Driven Macrophage Heterogeneity in Early Atherosclerotic Plaques
5:15pm ̶ 5:40pm: Coffee, Drinks & Refreshments
5:40pm ̶ 6:05pm: Dr. Tara Trauthwein (Department of Statistics, University of Oxford): Approximation Results for Large Networks
6:05pm-6:30pm: Isaac Newell (OxPDE, Mathematical Institute, University of Oxford): The Gauss Equation for Isometric Embeddings of Regularity in W^1+2/3,3∩ C¹
6:35pm-7:00pm: Discussion

For abstracts please click the file here: Complexity-Cluster_Workshop_20250212_Final_0.pdf

In this talk I will present two new results concerning differentiability of Lipschitz maps between Carnot groups. The former is a suitable adaptation of Pansu-Rademacher differentiability theorem to general Radon measures. More precisely we construct a suitable bundle associated to the measure along which Lipschitz maps are differentiable, very much in the spirit of the results of Alberti-Marchese. The latter is the converse of Pansu’s theorem. Namely, let G be a Carnot group and μ a Radon measure on G. Suppose further that every Lipschitz map between G and H, some other Carnot group, is Pansu differentiable μ-almost everywhere. We show that μ must be absolutely continuous with respect to the Haar measure of G. This is a joint work with Guido De Philippis, Andrea Marchese, Andrea Pinamonti and Filip Rindler.

This new sub-Riemannian result will be an excuse to present and discuss the techniques employed in Euclidean spaces to prove the converse of Rademacher's theorem.

Conformal welding, the process of glueing together Riemann surfaces along their boundaries, has recently played a prominent role in probability theory. In this talk, I will discuss two examples, namely the welding associated with random Jordan curves (SLE(k) loops) and particularly their limit as k tends to zero, and the welding of random trees (such as the CRT).

TBA

Predictions of complex flows remain a significant challenge for engineering systems. Computationally affordable predictions of turbulent flows generally require Reynolds-Averaged Navier–Stokes (RANS) simulations and Large-Eddy Simulation (LES), the predictive accuracy of which can be insufficient due to non-Boussinesq turbulence and/or unresolved multiphysics that preclude qualitative fidelity in certain regimes. For example, in turbulent combustion, flame–turbulence interactions can lead to inverse-cascade energy transfer, which violates the assumptions of many RANS and LES closures. We present an adjoint-based, solver-embedded data assimilation method to augment the RANS and LES equations using trusted data. This is accomplished using Python-native flow solvers that leverage differentiable programming techniques to construct the adjoint equations needed for optimization. We present applications to shock-tube ignition delay predictions, turbulent premixed jet flames, and shock-dominated nonequilibrium flows and discuss the potential of adjoint-based approaches for future machine learning applications.

Several physical models are available to understand the dynamics of fluid mixtures, including the so-called Baer-Nunziato models. The partial differential equations associated with these models look like those of Navier-Stokes, with the addition of new relaxation terms. One strategy to obtain these models is homogenisation: starting from a mesoscopic mixture, where two pure fluids satisfying the compressible Navier-Stokes equations share the space between them, a change of scale is performed to obtain a macroscopic mixture, where the two fluids can coexist at any point in space.

This problem concerns the study of the Navier-Stokes equations with strongly oscillating initial data. We'll start by explaining some results in this framework, in one dimension of space and on the torus, for barotropic fluids. We will then detail the various steps involved in demonstrating homogenisation. Finally, we'll explain how to adapt this reasoning to homogenisation for perfect gases, with and without heat conduction.

TBA

Numerically solving PDEs typically requires compressing infinite information into a finite system of algebraic equations. Pragmatically, we usually follow a recipe: “Assume solutions of form X; substitute into PDE Y; discard terms by rule Z.” In contrast, Lanczos’s pioneering “tau method” prescribes modifying the PDE to form an exact finite system. Crucially, any recipe-based method can be viewed as adding a small equation correction, enabling us to compare multiple schemes independently of the solver. 

This talk also addresses a paradox: PDEs often admit infinitely many solutions, but finite systems produce only a finite set. When we include a “small” correction, the missing solutions are effectively hidden. I will discuss how tau methods frame this perspective and outline proposals for systematically studying and optimising various residuals.

We study a multi-agent setting in which brokers transact with an informed trader. Through a sequential Stackelberg-type game, brokers manage trading costs and adverse selection with an informed trader. In particular, supplying liquidity to the informed traders allows the brokers to speculate based on the flow information. They simultaneously attempt to minimize inventory risk and trading costs with the lit market based on the informed order flow, also known as the internalization-externalization strategy. We solve in closed form for the trading strategy that the informed trader uses with each broker and propose a system of equations which classify the equilibrium strategies of the brokers. By solving these equations numerically we may study the resulting strategies in equilibrium. Finally, we formulate a competitive game between brokers in order to determine the liquidity prices subject to precommitment supplied to the informed trader and provide a numerical example in which the resulting equilibrium is not Pareto efficient.

If $E/\mathbb{Q}$ is an elliptic curve, and $F/\mathbb{Q}$ is a finite Galois extension, then $E(F)$ is not merely a finitely generated abelian group, but also a Galois module. If we fix a finite group $G$, and let $F$ vary over all $G$-extensions, then what can we say about the statistical behaviour of $E(F)$ as a $\mathbb{Z}[G]$-module? In this talk I will report on joint work with Adam Morgan, in which we investigate the simplest non-trivial special case of this very general question. Our work has surprising connections to questions about frequency of failure of the Hasse principle for genus 1 hyperelliptic curves, and to work of Heath-Brown on 2-Selmer group distributions in quadratic twist families.