Below are the future seminars organised by the School of Mathematics.
Please note that only some seminar series are advertised here.
Dependent Choice (DC) is one the most useful choice principles with many equivalents (including the Downward Löwenheim–Skolem and the Baire Category Theorem). When we violate the Axiom of Choice via symmetric extensions we often want to preserve at least that much. In this talk we will discuss a few older results about the preservation of DC in generic and symmetric extensions, and we will present a recent breakthrough from a work-in-progress with Jonathan Schilhan.
Title:
Oscillatory neural dynamics in a phase-amplitude framework
Abstract
Model reduction techniques can provide useful insight into the dynamics behaviour of high dimensional oscillatory systems such as networks of neurons or neural field models. In this talk we will discuss the recently introduced phase-isostable framework which extends the classical technique of phase reduction to include a notion of a distance from limit cycle. This allows for representation of off cycle trajectories, the description of a greater variety of dynamics and greater accuracy in capturing the behaviour of the full model. We will highlight how this framework can be utilised to reveal bifurcations of phase-locked states in discrete networks and how this can be extended to networks with conduction delays and networks where the node oscillations are induced by delays. If time allows, we will also see how the framework can be applied to continuum neural field models to investigate instabilities of oscillatory phase waves to more exotic patterned states.
References:
[1] R Nicks, R Allen and S Coombes 2024 Insights into oscillator network dynamics using a phase-isostable framework, Chaos, Vol 34, 013141
[2] R Nicks, R Allen and S Coombes 2024 Phase and amplitude responses for delay equations using harmonic balance, Physical Review E, Vol 110, L012202
Abstract
In this study, the real-world application is concerned with the determination of a space-dependent index of refraction of an optical anti-reflection coating. The model is based on solving an inverse coefficient identification problem for the 1D Helmholtz equation. The additional data necessary for the inversion can be the full complex reflection coefficient or its absolute value only, measured for many wavenumbers. The numerical method is based on a FDM
direct solver combined with a nonlinear Tikhonov regularization. It is shown that, in general, the knowledge of the full complex reflection coefficient is necessary to determine uniquely a spacewise continuous index of refraction. When only the absolute value of the reflection coefficient is used as input data, constraints need to be imposed, e.g., the knowledge of the full integrated refraction index, additional smoothness assumptions on the index of refraction, or more reflectance data measured for many wavelengths. Apart from this insight into the uniqueness of solution of the inverse problem, the principal conclusion is that a better fit of the reflectance measured data is obtained by using a continuously varying index of refraction than when this coefficient is sought as a piecewise constant function.
An opportunity to meet and discuss research ideas. More detail to follow.
This is discussion forum joint between Leeds School of Maths and Imperial College London's Department of Maths on the subject of supporting international students. We'll be joined from Imperial by Dr. Charlotte Kestner, their deputy head of Maths, Dr. Chris Hallsworth, their Director of Undergraduate Studies, and Inkeri Hibbins, their head of the student hub (which is roughly equivalent to our Student Information Service).
Our big increase in the proportion of international students this year has changed some of the dynamics of many aspects of our degree programmes, such as small group teaching, and pastoral care. Imperial's undergraduate Maths cohort has routinely been over 60% international for many years now, and hopefully this forum will give all of us a chance to learn from each other and reflect on our own experiences.
