Surfactants are chemicals that preferentially reside at interfaces. Once surfactant molecules have adsorbed to an interface, they reduce the surface tension between the two neighbouring fluids and may induce fluid flow. Surfactants have many household applications, such as in cleaning products and cosmetics, as well as industrial applications, like mineral processing and agriculture. Thus, understanding the dynamics of surfactant solutions is particularly important with regards to improving the efficacy of their applications as well as highlighting how they work. In this seminar, we will explore the spreading of a droplet over a substrate, in which there is constant injection of liquid and soluble surfactant through a slot in the substrate. Firstly, we will see how the inclusion of surfactant alters the spreading of the droplet. We will then investigate the early- and late-time behaviour of our model and compare this with numerical simulations. We shall conclude by briefly examining the effect of changing the geometry of the inflow slot.

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The centre of the universal enveloping algebra of a complex semisimple Lie algebra has been understood for a long time since the pioneering work of Harish-Chandra. In contrast, the centres of the equivalent notions in characteristic p are still yet to be computed explicitly. In this talk, Zhenyu Yang and Rick Chen will present an explicit basis for the centre of the restricted enveloping algebra of sl_2, constructed from explicit calculations combined with techniques from non-commutative rings and Morita equivalences. They will then explain how to generalise the argument to compute the centre of the distribution algebra of the second Frobenius kernel of the algebraic group SL_2. This work was part of their summer project under the supervision of Konstantin Ardakov.

How many vectors are needed to simultaneously generate $m$ complete flags in $\mathbb{R}^d$, in the worst-case scenario?  A classical linear algebra fact, essentially equivalent to the Bruhat cell decomposition for $\text{GL}_d$, says that the answer is $d$ when $m=2$.  We obtain a precise answer for all values of $m$ and $d$.  Joint work with Federico Glaudo and Chayim Lowen.

In the 1990s, physicists introduced an ideal way to count curves inside a Calabi-Yau 3-fold, called the Gopakumar-Vafa (GV) theory. Building on several previous attempts, Maulik-Toda recently gave a mathematical rigorous definition of the GV invariants. We expect that the GV invariants and the Gromov-Witten (GW) invariants are related by an explicit formula, but this stands as a challenging open problem. In this talk, I will explain recent mathematical developments on the GV theory, especially for local curves, including the cohomological chi-independence theorem and the GV/GW correspondence in a special case.

In this talk, we will explore the emerging role of generative AI in mathematical research. Building on insights from the “Malliavin–Stein experiment”, carried out in collaboration with Charles-Philippe Diez and Luis Da Maia, we will discuss our experience and reflect on how AI might influence the way mathematics is conceived, proven, and created.

Your chance to ask Mathematrix DPhil students about the process of applying to PhD programs, including written stages and interviews! 

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Wave energy has the theoretical potential to meet global electricity demand, but it remains less mature and less cost-competitive than wind or solar power. A key barrier is the absence of engineering convergence on an optimal wave energy converter (WEC) design. In this work, I demonstrate how geometry optimisation can deliver step-change improvements in WEC performance. I present methodology and results from optimisations of two types of WECs: an axisymmetric point-absorber WEC and a top-hinged WEC. I show how the two types need different optimisation frameworks due to the differing physics of how they make waves. For axisymmetric WECs, optimisation achieves a 69% reduction in surface area (a cost proxy) while preserving power capture and motion constraints. For top-hinged WECs, optimisation reduces the reaction moment (another cost proxy) by 35% with only a 12% decrease in power. These result show that geometry optimisation can substantially improve performance and reduce costs of WECs.

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Blood vessels are among the most vital structures in the human body, forming intricate networks that connect and support various organ systems. Remarkably, during early embryonic development—before any blood vessels are visible—their precursor cells are arranged in stereotypical patterns throughout the embryo. We hypothesize that these patterns guide the directional growth and fusion of precursor cells into hollow tubes formed from initially solid clusters. Further analysis of cells within these clusters reveals unique organization that may influence their differentiation into endothelial and blood cells. In this work, I revisit the problem of pattern formation through the lens of active matter physics, using both developing embryonic systems and in vitro cell culture models where similar patterns are observed during tissue budding. These different systems exhibit similar patterning behavior, driven by changes in cellular activity, adhesion and motility.

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WCMB, University of Oxford and University of Bristol

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The last decade has seen tremendous progress in our ability to generate realistic-looking data, be it images, text, audio, or video. In this presentation, we will look at the closely related problem of quantifying realism, that is, designing functions that can reliably tell realistic data from unrealistic data. This problem turns out to be significantly harder to solve and remains poorly understood, despite its prevalence in machine learning and recent breakthroughs in generative AI. Drawing on insights from algorithmic information theory, we discuss why this problem is challenging, why a good generative model alone is insufficient to solve it, and what a good solution would look like. In particular, we introduce the notion of a universal critic, which unlike adversarial critics does not require adversarial training. While universal critics are not immediately practical, they can serve both as a North Star for guiding practical implementations and as a tool for analyzing existing attempts to capture realism.

We study a very general class of first-order linear hyperbolic
systems that both become weakly hyperbolic and contain singular
lower-order coefficients at a single time t = 0. In "critical" weakly
hyperbolic settings, it is well-known that solutions lose a finite
amount of regularity at t = 0. Here, we both improve upon the analysis
in the weakly hyperbolic setting, and we extend this analysis to systems
containing critically singular coefficients, which may also exhibit
modified asymptotics and regularity loss at t = 0.

In particular, we give precise quantifications for (1) the asymptotics
of solutions as t approaches 0, (2) the scattering problem of solving
the system with asymptotic data at t = 0, and (3) the loss of regularity
due to the degeneracies at t = 0. Finally, we discuss a wide range of
applications for these results, including weakly hyperbolic wave
equations (and equations of higher order), as well as equations arising
from relativity and cosmology (e.g. at big bang singularities).

This is joint work with Bolys Sabitbek (Ghent).

to follow

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University of Cambridge