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What do the roots of random polynomials look like? Classical works of Erdős-Turán and others show that most roots are near the unit circle and they are approximately rotationally equidistributed. We will begin with an understanding of why this happens and see how ideas from extremal combinatorics can mix with analytic and probabilistic arguments to show this. Another main feature of random polynomials is that their roots tend to "repel" each other. We will see various quantitative statements that make this rigorous. In particular, we will study the smallest separation $m_n$ between pairs of roots and show that typically $m_n$ is on the order of $n^{-5/4}$. We will see why this reflects repulsion between roots and discuss where this repulsion comes from. This is based on joint work with Oren Yakir.

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A discussion on how race and ethnicity interact with the concept of merit in academia, based on sections from the book 'Misconceiving Merit' by Blair-Loy and Cech. 

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In this talk, we will explore three flow configurations that illustrate the behaviour of slow-moving viscous fluids in confined geometries: viscous gravity currents, fracturing of shear-thinning fluids in a Hele-Shaw cell, and rectangular channel flows of non-Newtonian fluids. We will first develop simple mathematical models to describe each setup, and then we will compare the theoretical predictions from these models with laboratory experiments. As is often the case, we will see that even models that are grounded in solid physical principles often fail to accurately predict the real-world flow behaviour. Our aim is to identify the primary physical mechanisms absent from the model using laboratory experiments. We will then refine the mathematical models and see whether better agreement between theory and experiment can be achieved.

Several interesting and emerging problems in statistics, machine learning and optimal transport can be cast as minimisation of (entropy-regularised) objective functions defined on an appropriate space of probability distributions.  Numerical methods have historically focused on linear objective functions, a setting in which one has access to an unnormalised density for the distributional target.  For nonlinear objectives, numerical methods are relatively under-developed; for example, mean-field Langevin dynamics is considered state-of-the-art.  In the nonlinear setting even basic questions, such as how to tell whether or not a sequence of numerical approximations has practically converged, remain unanswered.  Our main contribution is to present the first computable measure of sub-optimality for optimisation in this context.  

Joint work with Clémentine Chazal, Heishiro Kanagawa, Zheyang Shen and Anna Korba.

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The development of a complex functional multicellular organism from a single cell involves tightly regulated and coordinated cell behaviours coupled through short- and long-range biochemical and mechanical signals. To truly comprehend this complexity, alongside experimental approaches we need mathematical and computational models, which can link observations to mechanisms in a quantitative, predictive, and experimentally verifiable way. In this talk I will describe our efforts to model aspects of embryonic development, focusing in particular on the planar polarised behaviours of cells in epithelial tissues, and discuss the mathematical and computational challenges associated with this work. I will also highlight some of our work to improve the reproducibility and re-use of such models through the ongoing development of Chaste (https://github.com/chaste), an open-source C++ library for multiscale modelling of biological tissues and cell populations.

This week's Fridays@2 will be a panel discussion focusing on what it is like to pursue a research degree. The panel will share their thoughts and experiences in a question-and-answer session, discussing some of the practicalities of being a postgraduate student, and where a research degree might lead afterwards.

We consider particles following a diffusion process in a multi-well potential and attracted by their barycenter (corresponding to the particle approximation of the Wasserstein flow of a suitable free energy). It is well-known that this process exhibits phase transitions: at high temperature, the mean-field limit has a single stationary solution, the N-particle system converges to equilibrium at a rate independent from N and propagation of chaos is uniform in time. At low temperature, there are several stationary solutions for the non-linear PDE, and the limit of the particle system as N and t go to infinity do not commute. We show that, in the presence of multiple stationary solutions, it is still possible to establish local convergence rates for initial conditions starting in some Wasserstein balls (this is a joint work with Julien Reygner). In terms of metastability for the particle system, we also show that for these initial conditions, the exit time of the empirical distribution from some neighborhood of a stationary solution is exponentially large with N and approximately follows an exponential distribution, and that propagation of chaos holds uniformly over times up to this expected exit time (hence, up to times which are exponentially large with N). Exactly at the critical temperature below which multiple equilibria appear, the situation is somewhat degenerate and we can get uniform in N convergence estimates, but polynomial instead of exponential.

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Pfaffian functions, and by extension Pfaffian and semi-Pfaffian sets, play a crucial role in various areas of mathematics, including o-minimal theory. Incidence combinatorics has recently experienced a surge of activity, fuelled by the introduction of the polynomial partitioning method of Guth and Katz. While traditionally restricted to simple geometric objects such as points and lines, focus has shifted towards incidence questions involving higher dimensional algebraic or semi-algebraic sets. We present a generalization of the polynomial partitioning method to semi-Pfaffian sets and illustrate how this leads to Pfaffian generalizations of classic results in incidence geometry, such as the Szemerédi-Trotter Theorem. Finally, we outline an application of semi-Pfaffian geometry and Khovanskii's bound to the robustness of neural networks.

This session will explore practical ways to manage your time effectively as a student. We’ll discuss how to find the right balance between revising and working on problem sheets, tools and strategies to help you plan your workload, and how to set realistic priorities. We’ll also talk about what kind of study balance makes sense over the Christmas break. Come along to pick up useful tips for staying organised, focused, and on top of your studies.

This session is likely to be most relevant for first-year undergraduates, but all are welcome.

Microbial communities contain many evolving and interacting bacteria, which makes them complex systems that are difficult to understand and predict. We use theory – including game theory, agent-based modelling, ecological network theory and metabolic modelling - and combine this with experimental work to understand what it takes for bacteria to succeed in diverse communities. One way is to actively kill and inhibit competitors and we study the strategies that bacteria use in toxin-mediated warfare. We are now also using our approaches to understand the human gut microbiome and its key properties including ecological stability and the ability to resist invasion by pathogens (colonization resistance). Our ultimate goal is to both stabilise microbiome communities and remove problem species without the use of antibiotics.

As discovered by Perelman, the study of ancient Ricci flows which are $\kappa$-noncollapsed is a crucial prerequisite to understanding the singularity behaviour of more general Ricci flows. In dimension three, these so-called "$\kappa$-solutions" have been fully classified through the groundbreaking work of Brendle, Daskalopoulos, and Šešum. Their classification result can be extended to higher dimensions, but only for those Ricci flows that have uniformly positive isotropic curvature (PIC), as well as weakly-positive isotropic curvature of the second type (PIC2); it appears the classification result fails with only minor modifications to the curvature assumption. Indeed, with the alternative assumption of non-negative curvature operator, a rich variety of new examples emerge, as recently constructed by Buttsworth, Lai, and Haslhofer; Haslhofer himself has conjectured that this list of non-negatively curved $\kappa$-solutions is now exhaustive in dimension four. In this talk, we will discuss some recent progress towards resolving Haslhofer's conjecture, including a compactness result for non-negatively curved $\kappa$-solutions in dimension four, and a symmetry improvement result for bubble-sheet regions. This is joint work with Anusha Krishnan and Timothy Buttsworth.