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).

Given a smooth geometrically connected curve C over a perfect field k and a smooth commutative group scheme G defined over the function field K of C, one can consider isomorphism classes of G-torsors locally trivial at completions of K coming from closed points of C. They form a generalized Tate-Shafarevich group which specializes to the classical one in the case when k is finite. Recently, these groups have been studied over other base fields k as well, for instance p-adic or number fields. Surprisingly, finiteness can be proven in some cases but there are also quite a few open questions which I plan to discuss  in my talk.

Recently, much attention has been paid to the intersection between coarse geometry and graph theory, giving rise to the fresh, exciting new field aptly known as ‘coarse graph theory’. One aspect of this area is the study of so-called ‘fat minors’, a large-scale analogue of the usual idea of a graph minor.

In this talk, I will introduce this area and motivate some interesting questions and conjectures. I will then sketch a proof that a finitely presented group is either virtually planar or contains arbitrarily ‘fat’ copies of every finite graph.

No prior knowledge or passion for graph theory will be assumed in this talk.

I introduce modal group theory, where one investigates the class of all groups using embeddability as a modal operator. By employing HNN extensions, I demonstrate that the modal language of groups is more expressive than the first-order language of groups. Furthermore, I establish that the theory of true arithmetic, viewed as sets of Gödel numbers, is computably isomorphic to the modal theory of finitely presented groups. Finally, I resolve an open question posed by Sören Berger, Alexander Block, and Benedikt Löwe by proving that the propositional modal validities of groups constitute precisely the modal logic S4.2.

The notion of a superconformal structure on a supermanifold goes back some forty years. I will discuss some recent work that shows how these structures and their deformations govern supersymmetric and superconformal field theories in geometric fashion. A superconformal structure equips a supermanifold with a sheaf of dg commutative algebras; the tangent sheaf of this dg ringed space reproduces the Weyl multiplet of conformal supergravity (equivalently, the superconformal stress tensor multiplet), in any dimension and with any amount of supersymmetry. This construction is uniform under twists, and thus provides a classification of relations between superconformal theories, chiral algebras, higher Virasoro algebras, and more exotic examples.
 

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.

Shape optimization is a rich field at the intersection of analysis, optimization, and engineering. It seeks to determine the optimal geometry of structures to minimize performance objectives, subject to physical constraints—often modeled by Partial Differential Equations (PDEs). Traditional approaches commonly assume that these constraints admit a unique solution for each candidate shape, implying a single-valued shape-to-solution map. However, many real-world structures exhibit multistability, where multiple stable configurations exist under identical physical conditions.

This research departs from the single-solution paradigm by investigating shape optimization in the presence of multiple solutions to the same PDE constraints. Focusing on a neo-Hookean hyperelastic model, we formulate an optimization problem aimed at controlling the energy jump between distinct solutions. Drawing on bifurcation theory, we develop a theoretical framework that interprets these solutions as continuous branches parameterized by shape variations. Building on this foundation, we implement a numerical optimization strategy and present numerical results that demonstrate the effectiveness of our approach.

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.

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.

Junior Strings is a seminar series where DPhil students present topics of common interest that do not necessarily overlap with their own research area. This is primarily aimed at PhD students and post-docs but everyone is welcome.

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.

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.

Let G be an infinite discrete group; e.g. free group, surface groups, or hyperbolic 3-manifold group.

Finite dimensional unitary representations of G of fixed dimension are usually very hard to understand. However, there are interesting notions of convergence of such representations as the dimension tends to infinity. One notion — strong convergence — is of interest both from the point of view of G alone but also through recently realized applications to spectral gaps of locally symmetric spaces. For example, this notion bypasses (unconditionally) the use of Selberg's Eigenvalue Conjecture in obtaining existence of large area hyperbolic surfaces with near-optimal spectral gaps. 

The talk is a broadly accessible discussion on these themes, based on joint works with W. Hide, L. Louder, D. Puder, J. Thomas, R. van Handel.

I am going to start by reviewing axioms of quantum mechanics, which in fact give a description of a Hilbert space. I will argue that the language that Dirac and his followers developed is that of continuous logic and the form of axiomatisation is that of "algebraic logic" in the sense of A. Tarski's cylindric algebras. In fact, Hilbert spaces can be seen as a continuous model theory version of cylindric algebras.

During the talk I will describe my research on host-pathogen interactions during lung infections. Various modelling approaches have been used, including a hybrid multiscale individual-based model that we have developed, which simulates pulmonary infection spread, immune response and treatment within in a section of human lung. The model contains discrete agents which model the spatio-temporal interactions (migration, binding, killing etc.) of the pathogen and immune cells. Cytokine and oxygen dynamics are also included, as well as Pharmacokinetic/Pharmacodynamic models, which are incorporated via PDEs. I will also describe ongoing work to develop a continuum model, comparing the spatial dynamics resulting from these different modelling approaches.  I will focus in the most part on two infectious diseases: Tuberculosis and COVID-19.

This talk will be a case study on the recently discovered boundary Carrollian conformal algebra (BCCA) in theoretical physics. It is an infinite-dimensional subalgebra of an abelian extension of the Witt algebra. A striking feature of this is that it is not integer graded; this already puts us in a rather novel setting, since infinite-dimensional Lie algebras almost exclusively appear with integer grading in physics. But this means that there is new ground to be broken in this direction of research. In this talk, I will present some very early results from our attempt at studying the representations of the BCCA. Any thoughts and comments are very welcome as they could be immensely helpful for us to navigate these unfamiliar waters!

The self-energies in Quantum Electrodynamics (QED) are not only fundamental physical quantities but also well-suited for investigating the mathematical structure of perturbative Quantum Field Theory. In this talk, I will discuss the QED self-energies up to the fourth order in the loop expansion. Going beyond one loop, where the integrals can be expressed in terms of multiple polylogarithms, we encounter functions associated with an elliptic curve, a K3 surface and a Calabi-Yau threefold. I will review the method of differential equations and apply it to the scalar Feynman integrals appearing in the self-energies. Special emphasis will be placed on the concept of canonical bases and on how to generalize them beyond the polylogarithmic case, where they are well understood. Furthermore, I will demonstrate how canonical integrals may be identified through a suitable integrand analysis.

Geometry and topology call tell us about the shape of data. In this talk, I will give an introduction to my work on learning stratified spaces from samples, look at the use of persistent homology in cell biophysics, and apply persistence in understanding material structures.

We will review physics-informed neural networks (NNs) and summarize available extensions for applications in computational science and engineering. We will also introduce new NNs that learn functionals and nonlinear operators from functions and corresponding responses for system identification. 

These two key developments have formed the backbone of scientific machine learning that has disrupted the path of computational science and engineering and has created new opportunities for all scientific domains. We will discuss some of these opportunities in digital twins, autonomy, materials discovery, etc.

Moreover, we will discuss bio-inspired solutions, e.g., spiking neural networks and neuromorphic computing.

It’s a busy and stressful term for a lot of us so come and take a break and do some colouring and origami with us. Venting is very much encouraged.

In the last three decades, a fruitful way to approximate the area functional in low codimension is to interpret submanifolds as the nodal sets of maps (or sections of vector bundles), critical for suitable physical energies or well-known lagrangians from gauge theory. Inspired by the situation in codimension two, where the abelian Higgs model has provided a successful framework, we look at the non-abelian SU(2) model as a natural candidate in codimension three. In this talk we will survey the new key difficulties and some recent partial results, including a joint work with D. Parise and D. Stern and another result by Y. Li.

In this talk, we consider the Cauchy problem for a number of semilinear PDEs, subject to initial data distributed according to a family of Gaussian measures.  

We first discuss how the flow of Hamiltonian equations transports these Gaussian measures. When the transported measure is absolutely continuous with respect to the initial measure, we say that the initial measure is quasi-invariant. 

In the high-dispersion regime, we exploit quasi-invariance to build a (unique) global flow for initial data with negative regularity, in a regime that cannot be replicated by the deterministic (pathwise) theory.  

In the 0-dispersion regime, we discuss the limits of this approach, and exhibit a sharp transition from quasi-invariance to singularity, depending on the regularity of the initial measure. 

We will also discuss how the same techniques can be used in the context of stochastic PDEs, and how they provide information on the invariant measures for their flow. 

This is based on joint works with  J. Coe (University of Edinburgh), J. Forlano (Monash University), and M. Hairer (EPFL).

TBC

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