Hopf-Galois theory provides an analogue to Galois theory for non-Galois extensions, of particular interest are separable but not necessarily normal extensions of fields. A Hopf-Galois structure consists of a cocommutative Hopf algebra over the base field and an action of this on the top field satisfying various conditions, this functions much like a Galois group though an extension may admit multiple distinct Hopf-Galois structures or none at all. As in classical Galois theory, we have a notion of the Hopf-Galois correspondence; the natural fix map from the Hopf sub-algebras of such a structure to the intermediate fields of the extension is always injective and inclusion reversing but, unlike for the Galois correspondence, it is not in general surjective. Much work has been done to determine when the Hopf-Galois correspondence is surjective and the skew brace has proved to be a fruitful tool in the Galois case. We introduce the skew bracoid, a generalisation of the skew brace that corresponds to Hopf-Galois extensions on separable extensions of fields and comprises two groups connected by a compatible transitive action. We then outline how the skew bracoid may be of use in the study of the Hopf-Galois correspondence in the separable case, providing some preliminary results.
SubRiemannian geometry represents a vast generalisation of Riemannian geometry and it is meant to model motions that are permitted only along a limited set of directions at any point. The aim of this talk is to give an intuition of how subRiemannian geometry naturally arises from modelling different mathematical and physical problems (e.g. optimal control, image processing, thermodynamics). Surprisingly, for such a setting many fundamental mathematical objects are not yet defined or understood. If time allows, I will present what obstacles appear when trying to extend the most basic tools of geometric analysis and differential geometry to subRiemannian manifolds.
Computational models are revolutionizing our understanding of infectious disease spread. This presentation explores how integrating mobile phone data, global mobility patterns, socioeconomic strata and epidemiological records can enhance our ability to characterize and predict epidemic dynamics across spatio-temporal scales. I will show how these data can be used to: (i) uncover the initial phases (i.e., cryptic spreading) of the COVID-19 pandemic globally; (ii) quantify social inequalities in the adoption of non-pharmaceutical interventions in a large metropolitan area; and (iii) improve the realism of traditional epidemic models by accounting for multiple dimensions beyond age in the stratification of contact patterns.
In recent years, many models in mathematical physics have been encoded into graphical models, which are more accessible through the lens of probability theory. These graphical models often exhibit a natural percolation structure. One such model is the Random Loop Model introduced by Daniel Ueltschi. Peter Mühlbacher showed that the loop threshold for the Random Loop Model with θ=1 is larger than the percolation threshold. This is due to so-called blocking events in graphs with uniformly bounded degree. The proof primarily relies on a coupling method.
In my talk, I will introduce the model and the basic proof techniques. Furthermore, I will discuss a recent result where we generalize the method to obtain new results for general trees.
I will explain why the tree case differs from the case of a general graph. If time permits, I will use the Galton-Watson case to illustrate how the coupling in the proof works.
This talk is based on joint work with V. Betz, M. Kraft, B. Lees and C. Mönch
Constant mean curvature (CMC) surfaces are special geometric variational objects, closely related to minimal surfaces. The key properties of a CMC surface are its area, mean curvature, genus, and index. The index of a CMC surface measures its stability: the index counts how many ways one can perturb the surface to decrease the area while keeping the enclosed volume constant. In this talk we discuss relationships between these key properties. In particular we present recent joint work with Ben Sharp, where we bound the index of CMC surfaces linearly from above by genus and the correct scale-invariant quantity involving mean curvature and area.
The quest to understand out-of-equilibrium behaviour of complex quantum systems represents one of the frontiers of contemporary quantum science. For a long time, the prevailing belief has been that complex quantum systems, comprising many interacting degrees of freedom, all suffer the same inevitable fate: that of thermalisation, whereby the system relaxes towards a featureless thermal state, completely "forgetting" its initial condition. However, a flurry of recent works has unearthed a new paradigm of behaviour in many well-known physical systems, including Rydberg atoms, lattice gauge theories, and certain kinds of frustrated magnets. Such systems have been understood to possess a subtle breakdown of ergodicity, now commonly known as "quantum many-body scars". Quantum many-body scars exhibit fascinating properties, such as extreme sensitivity to initial conditions: while a system initialised randomly undergoes chaotic dynamics and thermalisation, specific initial conditions can result in persistent dynamical revivals, surpassing native thermalisation timescales. The discovery of quantum many-body scars has not only deepened our understanding of many-body quantum mechanics, but it also has direct practical relevance for improving the control over the delicate physical phenomena underpinning quantum technologies. In this talk, I will present a pedagogical overview of this fascinating new field of physics, highlighting a few of the remaining mysteries for theory and future experiments.
When and how should we intervene to manage an emerging infectious disease most effectively? Deciding when to enforce or relax non-pharmaceutical interventions (NPIs) based on real-time outbreak surveillance data is a central challenge in infectious disease epidemiology. Practical surveillance data, often characterised by reporting delays and infection under-ascertainment, can misinform decision-making. This may lead to mistimed NPIs that fail to control disease spread or allow harmful epidemic peaks that overwhelm healthcare capacities.
In this talk, I will introduce EpiControl, a novel model-predictive control algorithm designed to optimise NPI decisions by jointly minimising cumulative future risks and costs across stochastic epidemic projections. I will demonstrate how this algorithm outperforms data-insensitive strategies while also discussing the intrinsic limitations of surveillance quality, disease growth rates, and decision frequency in flattening epidemic peaks or reducing endemic oscillations. Additionally, I will present my ongoing research on integrating population behaviour into the policy-making framework.
Spin of a particle is an integral notion in Quantum theory. Understanding this idea has directly influenced the development and enrichment of many topics in pure mathematics for the last century, including Algebra, Lie theory, Representation theory, Operator theory, Differential geometry, to name a few. In this talk I shall give an overview of the mathematical aspects of Spin, and discuss various pure mathematical ideas associated to it.
We study various formulation of zero-sum games between a singular-controller and a stopper with a finite-time horizon, where the underlying process is a multi-dimensional controlled stochastic differential equation evolving in an unbounded domain. We prove that such games admit a value and present an optimal strategy for the stopper. In some cases, we show the game's value is the maximal solution, in a suitable Sobolev class, of a variational inequality of 'min-max' type with both obstacle and gradient constraint. Under stricter assumptions, we provide an optimal strategy for the controller and establish a connection between the space derivative of the value function and the solution of an optimal stopping problem with absorption.
Hil Meijer (University of Twente, NL),
Title: Synchrony across the brain; a harmonic balance approach to delay-coupled oscillators
Abstract: Delays are a natural component of computational models of large-scale brain dynamics. Delays combined with local synaptic activity typically lead to oscillations, but the question is whether synchrony or some out-of-phase solution is stable. Here we present a machinery using harmonic balance and accounting for symmetries to look for instabilities of the synchronous solution. We first analyse a simple model on a ring where ``travelling waves'' with activity jumping to the nearest or next-nearest neighbour appear. Employing numerical continuation, we also track which pattern exists as we change the delay. For stability of the asynchronous solutions, we rely on simulations. We then move on to the Wilson-Cowan model with similar results, and highlight some of the additional numerical challenges.
Davide Liessi (University of Udine, IT).
Title: Stability of periodic orbits of delay equations
Abstract: The local stability properties of periodic orbits of a delay differential equation or of a renewal equation can be studied by computing the Floquet multipliers, i.e the eigenvalues of the monodromy operators of the linearized equation. These operators can be approximated via pseudospectral collocation, resulting in a matrix whose eigenvalues can be computed with standard methods. In this seminar I will recall the key ideas of the Floquet theory for delay equations, based on the sun-star perturbation theory, and I will present the pseudospectral approximation method, along with some examples showing its effectiveness.
