Past

Successful navigation of the Covid-19 pandemic is predicated on public cooperation with safety measures and appropriate perception of risk, in which emotion and attention play important roles. Signatures of public emotion and attention are present in social media data, thus natural language analysis of this text enables near-to-real-time monitoring of indicators of public risk perception. We compare key epidemiological indicators of the progression of the pandemic with indicators of the public perception of the pandemic constructed from ∼20 million unique Covid-19-related tweets from 12 countries posted between 10th March and 14th June 2020. We find evidence of psychophysical numbing: Twitter users increasingly fixate on mortality, but in a decreasingly emotional and increasingly analytic tone. Semantic network analysis based on word co-occurrences reveals changes in the emotional framing of Covid-19 casualties that are consistent with this hypothesis. We also find that the average attention afforded to national Covid-19 mortality rates is modelled accurately with the Weber–Fechner and power law functions of sensory perception. Our parameter estimates for these models are consistent with estimates from psychological experiments, and indicate that users in this dataset exhibit differential sensitivity by country to the national Covid-19 death rates. Our work illustrates the potential utility of social media for monitoring public risk perception and guiding public communication during crisis scenarios.

We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $P=\{P_{u,v}| u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y,z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w:E\to(0,\infty)$ is said to induce $P$ if for every $u,v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many 2-connected metrizable graphs.
 

We consider a chain of oscillators with one particle in contact with a thermostat at temperature T. The thermostat is modeled by a Langevin dynamics or a renewal of the velocity with a gaussian random variable with variance T. The dynamics of the oscillators is perturbed by a random exchange on velocities between nearest neighbor particles.
The (thermal) energy has a macroscopic superdiffusive behavior governed by a fractional heat equation (i.e. with a fractional Laplacian). The microscopic thermostat impose a particular boundary condition to the fractional Laplacian, corresponding to certain probabilities of transmission/reflection/absorption/creation for the corresponding superdiffusive Levy process.
This is from a series of works in collaboration with Tomazs Komorowski, Lenya Ryzhik, Herbert Spohn.

We will survey an analogy between random integers and random permutations, which goes back to works of Erdős and Kac and of Billingsley.
This analogy inspired results and proofs about permutations, originating in the setting of integers, and vice versa.
Extensions of this analogy will be described, involving the generalized Ewens measure on permutations, based on joint work with D. Elboim.
If time permits, an analogous analogy, this time between random polynomials over a finite field and random permutations, will be discussed and formalized, with some applications.
 

For an ordinary commutative Noetherian ring R we would define the singularity category to be the quotient of the (derived category of) finitely generated modules modulo the (derived category of) fg projective modules [``the bounded derived category modulo compact objects’’]. For a ring spectrum like C^*(BG) (coefficients in a field of characteristic p) it is easy to define the module category and the compact objects, but finitely generated objects need a new definition. The talk will describe the definition and show that the singularity category is trivial exactly when G is p-nilpotent. We will go on to describe the singularity category for groups with cyclic Sylow p-subgroup.

The idea of assigning weights to local coordinate functions is used in many areas of mathematics, such as singularity theory, microlocal analysis, sub-Riemannian geometry, or the theory of hypo-elliptic operators, under various terminologies. In this talk, I will describe some differential-geometric aspects of weightings along submanifolds. This includes a coordinate-free definition, and the construction of weighted normal bundles and weighted blow-ups. As an application, I will describe a canonical local model for isotropic embeddings in symplectic manifolds. (Based on joint work with Yiannis Loizides.)

I will discuss a class of generalized global symmetries, which we call “Chern-Weil global symmetries,” that arise ubiquitously in gauge theories. The Noether currents of these Chern-Weil global symmetries are given by wedge products of gauge field strengths and their conservation follows from Bianchi identities, so they are not easy to break. However, exact global symmetries should not be allowed in a consistent theory of quantum gravity. I will explain how these symmetries are typically gauged or broken in string theory. Interestingly, many familiar phenomena in string theory, such as axions, Chern-Simons terms, worldvolume degrees of freedom, and branes ending on or dissolving in other branes, can be interpreted as consequences of the absence of Chern-Weil symmetries in quantum gravity, suggesting that they might be general features of quantum gravity.

In this session, Ben Fehrman and Markus Upmeier will give their thoughts on how to deliver a good talk for a conference or a seminar and tips for what to do and what to avoid. There will be a particular emphasis on how to give a good talk online. 

Crystal Structure Prediction aims to reveal the properties that stable crystalline arrangements of a molecule have without stepping foot in a laboratory, consequently speeding up the discovery of new functional materials. Since it involves producing large datasets that themselves have little structure, an appropriate classification of crystals could add structure to these datasets and further streamline the process. We focus on geometric invariants, in particular introducing the density fingerprint of a crystal. After exploring its computations via Brillouin zones, we go on to show how it is invariant under isometries, stable under perturbations and complete at least for an open and dense space of crystal structures.

The well-known Schur-Weyl duality provides a link between the representation theories of the general linear group $GL_n$ and the symmetric group $S_r$ by studying tensor space $(\mathbb{C}^n)^{\otimes r}$ as a ${(GL_n,S_r)}$-bimodule. We will discuss a few variations of this idea which replace $GL_n$ with some other interesting algebraic object (e.g. O$_n$ or $S_n$) and $S_r$ with a so-called diagram algebra. If time permits, we will also briefly look at how this can be used to define Deligne's category which 'interpolates' Rep($S_t$) for any complex number $t \in \mathbb{C}$.

Human life expectancy has been increasing steadily over the last century but this has resulted in an increasing incidence of age-related chronic diseases. Over 60% of people over the age of 65 will suffer from more than one disease at the same time (multimorbidity) and 25-50% of those over 80 years old develop frailty, defined as an accumulation of deficits and loss of reserve. Multimorbidity and frailty have complex medical needs and are strongly associated with disability and hospitalization. However, current treatments are suboptimal with problems of polypharmacy due to the fact that each disease is treated individually. Geroprotectors target fundamental mechanisms of ageing common to multiple age-related diseases and shows promise in delaying the onset of multimorbidity and frailty in animal models. However, their clinical testing in patients has been challenging due to the high level of complexity in the mode of action of geroprotectors and in the way multimorbidity and frailty develop.

 The talk will give an overview of these problems and make the case for the use of AI approaches to solve some of those complex issues with a view of designing appropriate clinical trials with geroprotectors to prevent age-related multimorbidity and frailty and extend healthspan.

The interplay between fluid flows and fractures is ubiquitous in Nature and technology, from hydraulic fracturing in the shale formation to supraglacial lake drainage in Greenland and hydrofracture on Antarctic ice shelves.

In this talk I will discuss the above three examples, focusing on the scaling laws and their agreement with lab experiments and field observations. As climate warms, the meltwater on Antarctic ice shelves could threaten their structural integrity through propagation of water-driven fractures. We used a combination of machine learning and fracture mechanics to understand the stability of fractures on ice shelves. Our result also indicates that as meltwater inundates the surface of ice shelves in a warm climate, their collapse driven by hydrofracture could significantly influence the flow of the Antarctic Ice Sheets. 

Recent advances in computing power and the potential to make more realistic assumptions due to increased flexibility have led to the increased prevalence of simulation models in economics. While models of this class, and particularly agent-based models, are able to replicate a number of empirically-observed stylised facts not easily recovered by more traditional alternatives, such models remain notoriously difficult to estimate due to their lack of tractable likelihood functions. While the estimation literature continues to grow, existing attempts have approached the problem primarily from a frequentist perspective, with the Bayesian estimation literature remaining comparatively less developed. For this reason, we introduce a widely-applicable Bayesian estimation protocol that makes use of deep neural networks to construct an approximation to the likelihood, which we then benchmark against a prominent alternative from the existing literature.
 

Computer-based simulation of partial differential equations (PDEs) involves approximating the unknowns and relies on suitable description of geometrical entities such as the computational domain and its properties. The Finite Element Method (FEM) is by large the most popular technique for the computer-based simulation of PDEs and hinges on the assumption that discretized domain and unknown fields are both represented by piecewise polynomials, on tetrahedral or hexahedral partitions. In reality, the simulation of PDEs is a brick within a workflow where, at the beginning, the geometrical entities are created, described and manipulated with a geometry processor, often through Computer-Aided Design systems (CAD), and then used for the simulation of the mechanical behaviour of the designed object. This workflow is often repeated many times as part of a shape optimisation loop. Within this loop, the use of FEM on CAD geometries (which are mainly represented through their boundaries) calls then for (re-) meshing and re-interpolation techniques that often require human intervention and result in inaccurate solutions and lack of robustness of the whole process. In my talk, I will present the mathematical counterpart of this problem, I will discuss the mismatch in the mathematical representations of geometries and PDEs unknowns and introduce a promising framework where geometric objects and PDEs unknowns are represented in a compatible way. Within this framework, the challenges to be addressed in order to construct robust PDE solvers are many and I will discuss some of them. Mathematical results will besupported by numerical validation.

This talk will be three short stories on the general theme of elastic
instabilities in soft solids. First I will discuss the inflation of a
cylindrical cavity through a bulk soft solid, and show that such a
channel ultimately becomes unstable to a finite wavelength peristaltic
undulation. Secondly, I will introduce the elastic Rayleigh Plateau
instability, and explain that it is simply 1-D phase separation, much
like the inflationary instability of a cylindrical party balloon. I will
then construct a universal near-critical analytic solution for such 1-D
elastic instabilities, that is strongly reminiscent of the
Ginzberg-Landau theory of magnetism. Thirdly, and finally, I will
discuss pattern formation in layer-substrate buckling under equi-biaxial
compression, and argue, on symmetry grounds, that such buckling will
inevitably produce patterns of hexagonal dents near threshold.

Totally geodesic submanifolds are perhaps one of the easiest types of submanifolds of Riemannian manifolds one can study, since a maximal totally geodesic submanifold is completely determined by any one of its points and the tangent space at that point. It comes as a bit of a surprise then that classification of such submanifolds — up to an ambient isometry — is a nightmarish and widely open question, even on such a manageable and well-understood class of Riemannian manifolds as symmetric spaces.

We will discuss the theory of totally geodesic submanifolds of symmetric spaces and see that any maximal such submanifold is homogeneous and thus can be completely encoded by some Lie algebraic data called a 'Lie triple'. We will then talk about the duality between symmetric spaces of compact and noncompact type and discover that there is a one-to-one correspondence between totally geodesic submanifolds of a symmetric space and its dual. Finally, we will touch on the known classification in rank one symmetric spaces, namely in spheres and projective/hyperbolic spaces over real normed division algebras. Time permitting, I will demonstrate how all this business comes in handy in other geometric problems on symmetric spaces, e. g. in classification of isometric cohomogeneity one actions.

Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…

The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.

We will discuss a generalisation of hyperbolic groups, from the group actions point of view. By studying torsion, we will see how this can help to answer questions about ordinary hyperbolic groups.

I will introduce product structure theory of graphs and show how families of graphs that have such a structure admit short adjacency labeling scheme and small induced universal graphs. Time permitting, I will talk about another recent application of product structure theory, namely vertex ranking (coloring).

A quantum circuit defines a discrete-time evolution for a set of quantum spins/qubits, via a sequence of unitary 'gates’ coupling nearby spins. I will describe how random quantum circuits, where each gate is a random unitary matrix, serve as minimal models for various universal features of many-body dynamics. These include the dynamical generation of entanglement between distant spatial regions, and the quantum "butterfly effect". I will give a very schematic overview of mappings that relate averages in random circuits to the classical statistical mechanics of random paths. Time permitting, I will describe a new phase transition in the dynamics of a many-body wavefunction, due to repeated measurements by an external observer.

A wide variety of fixed-point iterative methods for the solution of nonlinear operator equations in Hilbert spaces exists. In many cases, such schemes can be interpreted as iterative local linearisation methods, which can be obtained by applying a suitable preconditioning operator to the original (nonlinear) equation. Based on this observation, we will derive a unified abstract framework which recovers some prominent iterative methods. It will be shown that for strongly monotone operators this unified iteration scheme satisfies an energy contraction property. Consequently, the generated sequence converges to a solution of the original problem.

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We consider the random directed graph $D(n,p)$ with vertex set $\{1,2,…,n\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p = 1/n + \lambda n-4/3$ for $\lambda \in \mathbb{R}$. We show that, within this critical window, the strongly connected components of $D(n,p)$, ranked in decreasing order of size and rescaled by $n-1/3$, converge in distribution to a sequence $(C_1,C_2,\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs in the sense of an $L^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(C_1,C_2,\ldots)$ contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.

Point cloud registration is the task of finding the transformation that aligns two data sets. We make the assumption that the data lies on a low-dimensional algebraic variety.  The task is phrased as an optimization problem over the special orthogonal group of rotations. We solve this problem using Riemannian optimization algorithms and show numerical examples that illustrate the efficiency of this approach for point cloud registration. 

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The science of cities seeks to understand and explain regularities observed in the world's major urban systems. Modelling the population evolution of cities is at the core of this science and of all urban studies. Quantitatively, the most fundamental problem is to understand the hierarchical organization of cities and the statistical occurrence of megacities, first thought to be described by a universal law due to Zipf, but whose validity has been challenged by recent empirical studies. A theoretical model must also be able to explain the relatively frequent rises and falls of cities and civilizations, and despite many attempts these fundamental questions have not been satisfactorily answered yet. Here we fill this gap by introducing a new kind of stochastic equation for modelling population growth in cities, which we construct from an empirical analysis of recent datasets (for Canada, France, UK and USA) that reveals how rare but large interurban migratory shocks dominate city growth. This equation predicts a complex shape for the city distribution and shows that Zipf's law does not hold in general due to finite-time effects, implying a more complex organization of cities. It also predicts the existence of multiple temporal variations in the city hierarchy, in agreement with observations. Our result underlines the importance of rare events in the evolution of complex systems and at a more practical level in urban planning.

arXiv link: https://arxiv.org/abs/2011.09403

Topological data analysis is a growing area of research where topology and geometry meets data analysis. Many data science problems have a geometric flavor, and thus computational tools like persistent homology and Mapper were often found to be useful. Domains of applications include cosmology, material science, diabetes and cancer research. We will discuss some main tools of the field and some prominent applications.

Spacetimes with compact directions play an important role in supergravity and string theory. The simplest such example is the Kaluza-Klein spacetime, where the compact space is a flat torus. An interesting question to ask is whether this spacetime, when viewed as an initial value problem, is stable to small perturbations of initial data. In this talk I will discuss the global, non-linear stability of the Kaluza-Klein spacetime to toroidal-independent perturbations and the particular nonlinear structure appearing in the associated PDE system.

The goal of sequential learning is to draw inference from data that is gathered gradually through time. This is a typical situation in many applications, including finance. A sequential inference procedure is `anytime-valid’ if the decision to stop or continue an experiment can depend on anything that has been observed so far, without compromising statistical error guarantees. A recent approach to anytime-valid inference views a test statistic as a bet against the null hypothesis. These bets are constrained to be supermartingales - hence unprofitable - under the null, but designed to be profitable under the relevant alternative hypotheses. This perspective opens the door to tools from financial mathematics. In this talk I will discuss how notions such as supermartingale measures, log-optimality, and the optional decomposition theorem shed new light on anytime-valid sequential learning. (This talk is based on joint work with Wouter Koolen (CWI), Aaditya Ramdas (CMU) and Johannes Ruf (LSE).)
 

For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.

I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).

I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).

Chowla's conjecture from the 1960s is the assertion that the Möbius function does not correlate with its own shifts. I'll discuss some recent works where with collaborators we have made progress on this conjecture.

Veering triangulations are a special class of ideal triangulations with a rather mysterious combinatorial definition. Their importance follows from a deep connection with pseudo-Anosov flows on 3-manifolds. Recently Landry, Minsky and Taylor introduced a polynomial invariant of veering triangulations called the taut polynomial. During the talk I will discuss how and why it is connected to the Alexander polynomial of the underlying manifold.  

We will discuss confinement in 4d N=1 theories obtained after soft supersymmetry breaking deformations of 4d N=2 Class S theories. Confinement is characterised by a subgroup of the 1-form symmetry group of the theory that is left unbroken in a massive vacuum of the theory. The 1-form symmetry group is encoded in the Gaiotto curve associated to the Class S theory, and its spontaneous breaking in a vacuum is encoded in the N=1 curve (which plays the role of Seiberg-Witten curve for N=1) associated to that vacuum. Using this proposal, we will recover the expected properties of confinement in N=1 SYM theories, and the theories studied by Cachazo, Douglas, Seiberg and Witten. We will also recover the dependence of confinement on the choice of gauge group and discrete theta parameters in these theories.

We investigate whether Swampland constraints on the low-energy dynamics of weakly coupled string vacua in AdS can be related to inconsistencies of their putative holographic duals or, more generally, recast in terms of CFT data. In the main part of the talk, we shall illustrate how various swampland consistency constraints are equivalent to a negativity condition on the sign of certain mixed anomalous dimensions. This condition is similar to established CFT positivity bounds arising from causality and unitarity, but not known to hold in general. Our analysis will include LVS, KKLT, perturbative and racetrack stabilisation, and we shall also point out an intriguing connection to the Distance Conjecture. In the final part we will take a complementary approach, and show how a recent, more rigorous CFT inequality maps to non-trivial constraints on AdS, mentioning possible applications along the way.

Speaker: Katherine Staden
Introduced by: Frances Kirwan
Title: Inducibility in graphs
Abstract: What is the maximum number of induced copies of a fixed graph H inside any graph on n vertices? Here, induced means that both edges and non-edges have to be correct. This basic question turns out to be surprisingly difficult, and it is not even known for all 4-vertex graphs H. I will survey the area and discuss some key results, ideas and techniques -- combinatorial, analytical and computer-assisted.

Speaker: Pierre Haas
Introduced by: Alain Goriely
Title: Shape-Shifting Droplets
Abstract: Experiments show that small oil droplets in aqueous surfactant solution flatten, upon slow cooling, into a host of polygonal shapes with straight edges and sharp corners. I will begin by showing how plane (and rather plain) geometry explains the sequence of these polygonal shapes. I will go on to show that geometric considerations of that ilk cannot however explain the three-dimensional polyhedral shapes that the initially spherical droplets evolve through while flattening. I will conclude by showing that the experimental data agree with the predictions of a model based on a partial phase transition of the oil near the droplet edges.

Buildings are geometric structures useful in understanding certain classes of groups. In a series of papers written during the 1980s, Ronan and Smith developed the theory of “presheaves on buildings”. By constructing a coefficient system consisting of kP-modules (where P is the stabiliser of a given simplex), and computing the sheaf homology, they proved several results relating the homology spaces with the irreducible G-modules. In this talk we discuss their methods as well as our implementation of the algorithms, which has allowed us to efficiently compute the irreducible representations of some groups of Lie type.

Our current approach to cancer treatment has been largely driven by finding molecular targets, those patients fortunate enough to have a targetable mutation will receive a fixed treatment schedule designed to deliver the maximum tolerated dose (MTD). These therapies generally achieve impressive short-term responses, that unfortunately give way to treatment resistance and tumor relapse. The importance of evolution during both tumor progression, metastasis and treatment response is becoming more widely accepted. However, MTD treatment strategies continue to dominate the precision oncology landscape and ignore the fact that treatments drive the evolution of resistance. Here we present an integrated theoretical, experimental and clinical approach to develop treatment strategies that specifically embrace cancer evolution. We will consider the importance of using treatment response as a critical driver of subsequent treatment decisions, rather than fixed strategies that ignore it. Through the integrated application of drug treatments and drug holidays we will illustrate that, evolutionary therapy can drive either tumor control or extinction. Our results strongly indicate that the future of precision medicine shouldn’t be in the development of new drugs but rather in the smarter evolutionary application of preexisting ones.

Abstract: Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e. removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimise prices' changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.
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Composite materials make up over 50% of recent aircraft constructions. They are manufactured from very thin fibrous layers  (~10^-4 m) and even  thinner resin interfaces (~10^-5 m). To achieve the required strength, a particular layup sequence of orientations of the anisotropic fibrous layers is used. During manufacturing, small localised defects in the form of misaligned fibrous layers can occur in composite materials, adding an additional level of complexity. After FE discretisation the model exhibits multiple scales and large spatial variations in model parameters. Thus the resultant linear system of equations can be very ill-conditioned and extremely large. The limitations of commercially available modelling tools for solving these problems has led us to the implementation of a robust and scalable preconditioner called GenEO for parallel Krylov solvers. I will discuss using the GenEO coarse space as an effective multiscale model for the fine-scale displacement and stress fields. For the coarse space construction, GenEO computes generalised eigenvectors of the local stiffness matrices on the overlapping subdomains and builds an approximate coarse space by combining the smallest energy eigenvectors on each subdomain via a partition of unity.

A link for this talk will be sent to our mailing list a day or two in advance.  If you are not on the list and wish to be sent a link, please contact @email.

In March 2020, as COVID-19 cases started to surge for the first time in the UK, a team spanning UCL engineers, University College London Hospital (UCLH) intensivists and Mercedes Formula 1 came together to design, manufacture and deploy non-invasive breathing aids for COVID-19 patients. We reverse engineered and an off-patent CPAP (continuous positive airways pressure) device, the Philips WhisperFlow, and changed its design to minimise its oxygen utilisation (given that hospital oxygen supplies are under extreme demand). The UCL-Ventura received regulatory approvals from the MHRA within 10 days, and Mercedes HPP manufactured 10,000 devices by mid-April. UCL-Ventura CPAPs are now in use in over 120 NHS hospitals.

In response to international need, the team released all blueprints open source to enable local manufacture in other countries, alongside a support package spanning technical, manufacturing, clinical and regulatory components. The designs have been downloaded 1900 times across 105 countries, and around 20 teams are now manufacturing at scale and deploying in local hospitals. We have also worked closely with NGOs, on a non-profit basis, to deliver devices directly to countries with urgent need, including Palestine, Uganda and South Africa.

Phase transitions are present in a wide array of systems ranging from traffic to machine learning algorithms. In this talk, we will relate the concept of phase transitions to the convexity properties of the associated thermodynamic energy. Motivated by noisy stochastic gradient descent in supervised learning, we will consider the problem of understanding the thermodynamic limit of exchangeable weakly interacting diffusions (AKA propagation of chaos) from an energetic perspective. The strategy will be to exploit the 2-Wasserstein gradient flow structure associated with the thermodynamic energy in the infinite particle setting. Using this perspective, we will show how the convexity properties of the thermodynamic energy affects the homogenization limit or the stability of the log-Sobolev inequality.

The purpose of this talk is to provide a quick introduction to the buzzwords in the title. Then I'll discuss some (mostly unexplored) conjectures and thoughts.

We survey some recent progress in understanding stationary reflection at successors of singular cardinals and its influence on cardinal arithmetic:

1) In joint work with Yair Hayut, we reduced the consistency strength of stationary reflection at $\aleph_{\omega+1}$ to an assumption weaker than $\kappa$ is $\kappa^+$ supercompact.

2) In joint work with Yair Hayut and Omer Ben-Neria, we prove that from large cardinals it is consistent that there is a singular cardinal $\nu$ of uncountable cofinality where the singular cardinal hypothesis fails at nu and every collection of fewer than $\mathrm{cf}(\nu)$ stationary subsets of $\nu^+$ reflects at a common point.

The statement in the second theorem was not previously known to be consistent. These results make use of analysis of Prikry generic objects over iterated ultrapowers.

The asymptotic cone of a metric space X is what you see when you "look at X from infinitely far away". The asymptotic cone therefore captures much of the large scale geometry of the metric space. Furthermore, the construction often produces a smooth space from a discrete one, allowing us to apply the techniques of calculus. Notably, Gromov used asymptotic cones in his proof that finitely generated groups of polynomial growth are virtually nilpotent.

In the talk I will define asymptotic cones using the language of ultrafilters and ultralimits. We will then look at the particular cases of asymptotic cones of virtually nilpotent groups and hyperbolic metric spaces. At the end, we will prove a result of Gromov which relates the fundamental group of the asymptotic cone to the filling order of the underlying metric space.

The dimer model, a classical model of statistical mechanics, is the uniform distribution on perfect matchings of a graph. In two dimensions, one can define an associated height function which turns the model into a random surface (with specified boundary conditions). In the 1960s, Kasteleyn and Temperley/Fisher found an exact "solution" to the model, computing the correlations in terms of a matrix called the Kasteleyn matrix. This exact solvability was the starting point for the breakthrough work of Kenyon (2000) who proved that the centred height function converges to the Dirichlet (or zero boundary conditions) Gaussian free field. This was the first proof of conformal invariance in statistical mechanics.

In this talk, I will focus on a natural modification of the model where one allows the vertices on the boundary of the graph to remain unmatched: this is the so-called monomer-dimer model, or dimer model with free boundary conditions. The main result that we obtain is that the scaling limit of the height function of the monomer-dimer model in the upper half-plane is the Neumann (or free boundary conditions) Gaussian free field. Key to this result is a somewhat miraculous random walk representation for the inverse Kasteleyn matrix, which I hope to discuss.

Joint work with Marcin Lis (Vienna) and Wei Qian (Paris).

We analyze spectral properties of generic quantum operations, which describe open systems under assumption of a strong decoherence and a strong coupling with an environment. In the case of discrete maps the spectrum of a quantum stochastic map displays a universal behaviour: it contains the leading eigenvalue \lambda_1 = 1, while all other eigenvalues are restricted to the disk of radius R<1. Similar properties are exhibited by spectra of their classical counterparts - random stochastic matrices. In the case of a generic dynamics in continuous time, we introduce an ensemble of random Lindblad operators, which generate Markov evolution in the space of density matrices of a fixed size. Universal spectral features of such operators, including the lemon-like shape of the spectrum in the complex plane, are explained with a non-hermitian random matrix model. The structure of the spectrum determines the transient behaviour of the quantum system and the convergence of the dynamics towards the generically unique invariant state. The quantum-to-classical transition for this model is also studied and the spectra of random Kolmogorov operators are investigated.

It is sometimes possible to represent a complicated polytope as a projection of a much simpler polytope. To quantify this phenomenon, the extension complexity of a polytope $P$ is defined to be the minimum number of facets in a (possibly higher-dimensional) polytope from which $P$ can be obtained as a (linear) projection. In this talk, we discuss some results on the extension complexity of random $d$-dimensional polytopes (obtained as convex hulls of random points on either on the unit sphere or in the unit ball), and on the extension complexity of polygons with all vertices on a common circle. Joint work with Matthew Kwan and Yufei Zhao

Many complex networks depend upon biological entities for their preservation. Such entities, from human cognition to evolution, must first encode and then replicate those networks under marked resource constraints. Networks that survive are those that are amenable to constrained encoding, or, in other words, are compressible. But how compressible is a network? And what features make one network more compressible than another? Here we answer these questions by modeling networks as information sources before compressing them using rate-distortion theory. Each network yields a unique rate-distortion curve, which specifies the minimal amount of information that remains at a given scale of description. A natural definition then emerges for the compressibility of a network: the amount of information that can be removed via compression, averaged across all scales. Analyzing an array of real and model networks, we demonstrate that compressibility increases with two common network properties: transitivity (or clustering) and degree heterogeneity. These results indicate that hierarchical organization -- which is characterized by modular structure and heavy-tailed degrees -- facilitates compression in complex networks. Generally, our framework sheds light on the interplay between a network's structure and its capacity to be compressed, enabling investigations into the role of compression in shaping real-world networks.

arXiv link: https://arxiv.org/abs/2011.08994