Past

We review old and new properties of systems of linear partial differential equations with constant coefficients. We discuss solvability in different function classes, to observe very different solution spaces. We examine the existence of vector potentials in the different spaces, by which we mean systems Av=0 with the property v=Bu, where A and B are linear PDE operators with constant coefficients. Properties of the systems and their solutions are examined both from linear algebra and algebraic geometry angles. A special class of operators that are examined is that of constant rank operators, which are prevalent in the nonlinear analysis of compensated compactness theory. We will discuss some of the challenges of extending this theory to non-constant rank operators.

This talk is about the mathematics behind an artistic project focusing on the vibrations of soap films.

The Onsager framework for linear irreversible thermodynamics provides a thermodynamically consistent model of mass transport in a phase consisting of multiple species, via the Stefan–Maxwell equations, but a complete description of the overall transport problem necessitates also solving the momentum equations for the flow velocity of the medium. We derive a novel nonlinear variational formulation of this coupling, called the (Navier–)Stokes–Onsager–Stefan–Maxwell system, which governs molecular diffusion and convection within a non-ideal, single-phase fluid composed of multiple species, in the regime of low Reynolds number in the steady state. We propose an appropriate Picard linearisation posed in a novel Sobolev space relating to the diffusional driving forces, and prove convergence of a structure-preserving finite element discretisation. The broad applicability of our theory is illustrated with simulations of the centrifugal separation of noble gases and the microfluidic mixing of hydrocarbons.

Black hole horizons are normally at finite spatial distance from the exterior region, but when they are degenerate (or extreme as they are usually referred to in this case) the spatial distance becomes infinite. One can still fall into the black hole in finite proper time but the crossing sphere is replaced by an "internal infinity". Near to the horizon of an extreme Kerr black hole, the scattering properties of test fields bear some similarities to what happens at an asymptotically flat infinity. This observation triggered a natural question concerning the peeling behaviour of test fields near such horizons. A geometrical tool known as the Couch-Torrence inversion is particularly well suited to studying this question. In this talk, I shall recall some essential notions on the peeling of fields at an asymptotically flat infinity and describe the Couch-Torrence inversion in the particular case of extreme Reissner-Nordström black holes, where it acts as a global conformal isometry of the spacetime. I will then show how to extend this inversion to more general spherically symmetric extreme horizons and describe what results can be obtained in terms of peeling. This is a joint ongoing project with Jack Borthwick (University of Besançon) and Eric Gourgoulhon (Paris Observatory).

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

Recently, Terence Tao used a new quantitative approach to infer that certain ‘slightly supercritical’ quantities for the Navier–Stokes equations must become unbounded near a potential blow-up time. In this talk I’ll discuss a new strategy for proving quantitative bounds for the Navier–Stokes equations, as well as applications to behaviours of potentially singular solutions. This talk is based upon joint work with Christophe Prange (CNRS, Cergy Paris Université).

I will introduce a class of mean-field games under forward performance and for general risk preferences. Players interact through competition in fund management, driven by relative performance concerns in an asset diversification setting. This results in a common-noise mean field game. I will present the value and the optimal policies of such games, as well as some concrete examples. I will also discuss the partial information case, i.e.. when the risk premium is not directly observed. 

Via Poincaré duality, fundamental groups of aspherical manifolds have (appropriately shifted) isomorphisms between their homology and cohomology. In a 1973 Inventiones paper, Bieri and Eckmann defined a broader notion of a Duality Group, where the isomorphism between homology and cohomology can be twisted by what they called a Dualizing Module. Examples of these groups in geometric group theory (after passing to a finite-index subgroup) include $GL(n,\mathbb{Z})$, mapping class groups, and automorphism groups of free groups.

In work-in-progress with Thomas Wasserman we are looking into the following puzzle: the examples of duality groups that we know of that do not come from manifolds all have classifying spaces that satisfy a weaker local condition called the Cohen-Macaulay property. These spaces also satisfy weaker (twisted) versions of Poincaé duality via their local homology sheaves (or local cohomology cosheaves), and we are attempting to understand more about the links between these geometric versions of duality and the algebraic notion of a duality group. The goal of the talk is to explain more about the words used in the above paragraphs and say where we have got to so far.


 

A loop of Hamiltonian diffeomorphisms of a symplectic manifold $X$ defines, by clutching, a symplectic fibration over the two-sphere with fibre $X$.  We prove that the integral cohomology of the total space splits additively, answering a question of McDuff, and extending the rational cohomology analogue proved by Lalonde-McDuff-Polterovich in the late 1990’s. The proof uses a virtual fundamental class of moduli spaces of sections of the fibration in Morava K-theory. This talk reports on joint work with Mohammed Abouzaid and Mark McLean.

Non-smooth optimization is a core ingredient of many imaging or machine learning pipelines. Non-smoothness encodes structural constraints on the solutions, such as sparsity, group sparsity, low-rank and sharp edges. It is also the basis for the definition of robust loss functions such as the square-root lasso.  Standard approaches to deal with non-smoothness leverage either proximal splitting or coordinate descent. The effectiveness of their usage typically depend on proper parameter tuning, preconditioning or some sort of support pruning. In this work, we advocate and study a different route. By over-parameterization and marginalising on certain variables (Variable Projection), we show how many popular non-smooth structured problems can be written as smooth optimization problems. The result is that one can then take advantage of quasi-Newton solvers such as L-BFGS and this, in practice, can lead to substantial performance gains. Another interesting aspect of our proposed solver is its efficiency when handling imaging problems that arise from fine discretizations (unlike proximal methods such as ISTA whose convergence is known to have exponential dependency on dimension). On a theoretical level, one can connect gradient descent on our over-parameterized formulation with mirror descent with a varying Hessian metric. This observation can then be used to derive dimension free convergence bounds and explains the efficiency of our method in the fine-grids regime.

The line operators in a 2+1D TQFT form an algebraic structure called a modular tensor category (MTC). There is a natural action of a Galois group on MTCs which maps a given TQFT to other 'Galois conjugate' TQFTs. I will describe this Galois action and give several examples of Galois conjugate TQFTs. Galois action on a unitary TQFT can result in a non-unitary TQFT. I will derive a sufficient condition under which unitarity is preserved. Finally, I will describe the invariance of 0-form and 1-form symmetries of TQFTs under Galois action.    

*Note the different room location (L2) to usual Fridays@4 sessions*

This week is Mental Health Awareness Week. To mark this, Rebecca Reed from Siendo will deliver a session on mental health and wellbeing. The session will cover the following things: 

- The importance of finding a balance with achievement and managing stress and pressure.
- Coping mechanisms work with stresses at work in a positive way (not seeing all stress as bad).
- The difficulties faced in the HE environment, such as the uncertainty felt within jobs and research, combined with the high expectations and workload. 

I'll review 2104.13932, where we analyze the supersymmetric index of N=4 SU(N) Super Yang-Mills using the Bethe Ansatz approach, expressing it as a sum and concentrating on some family of contributions to the sum. We show that in the large N limit each term in this family corresponds to the contribution of a different euclidean black hole to the partition function of the dual gravitational theory. By taking into account non-perturbative contributions (wrapped D3-branes), we further show a one to one match between the contributions of the gravitational saddles and this family of contributions to the index, both at the perturbative and non-perturbative levels. I'll end with some new results regarding the Bethe Ansatz expansion and the information one could extract from it.

The statistical analysis of data which lies in a non-Euclidean space has become increasingly common over the last decade, starting from the point of view of shape analysis, but also being driven by a number of novel application areas. However, while there are a number of interesting avenues this analysis has taken, particularly around positive definite matrix data and data which lies in function spaces, it has increasingly raised more questions than answers. In this talk, I'll introduce some non-Euclidean data from applications in brain imaging and in linguistics, but spend considerable time asking questions, where I hope the interaction of statistics and topological data analysis (understood broadly) could potentially start to bring understanding into the applications themselves.

We can learn a lot about an integral domain by studying the Galois group of its fraction field. These groups are generally quite complicated and hard to understand, but their representations, so-called Galois representations, contain more easily accessible information. These also play the lead in many important theorems and conjectures of modern maths, such as the Modularity theorem and the Langlands programme. In this talk we give a quick introduction to Galois representations, motivated by lots of examples aimed at a general algebraist audience, and talk about some open problems.

Understanding the molecular mechanisms that control the biology of health and disease requires development of models that traverse multiple scales of organisation in order to encapsulate the relationships between genes and linking to observable phenotypes. Measuring, parameterising and simulating the entire system that determines these phenotypes in exhaustive detail is typically impossible due to the underlying biological complexity, our limited knowledge and the paucity of available data. For example, approximately one third of human genes are poorly characterised and most genes perform multiple functions, which manifest according to the surrounding biochemical context. Indeed, new functions continue to emerge even for deeply studied genes. Therefore, simplifying abstractions in concert with empirical analysis of matched genome-scale and descriptive data are valuable strategies to fill knowledge gaps relevant to a focused biomedical question or hypothesis.

Epithelial plasticity is a key driver of cancer progression and is associated with the most life-threatening phenotypes; specifically, metastasis and drug resistance. Computational methods developed in my group enable modelling the molecular control of important cancer phenotypes. We applied a machine learning approach for genome-wide context-specific biochemical interaction network inference (CoSNI) to map gene function for the Epithelial to Mesenchymal Transition cell programme (EMT_MAP), predicting new mechanisms in control of cancer invasion. Analysis of patient data with EMT_MAP and our NetNC algorithm [Cancers 2020;12:2823; https://github.com/overton-group/NetNC] enabled discovery of candidate renal cancer prognostic markers with clear advantages over standard statistical approaches. NetNC recovers the network-defined signal in noisy data, for example distinguishing functional EMT Transcription Factor targets from ‘neutral’ binding sites and defining biologically coherent modules in renal cancer drug response time course data. These and other approaches, including SynLeGG (Nucleic Acids Research 2021;49:W613-8, www.overton-lab.uk/synlegg) and an information-theoretic approach to causality (GABI) offer mechanistic insights and opportunity to predict candidate cancer Achilles’ heels for drug discovery. Computational results were validated in follow-up experiments, towards new clinical tools for precision oncology.

The discrete Fourier transform is fundamental in modern communication systems.  It is used to generate and process (i.e. modulate and demodulate) the signals transmitted in 4G, 5G, and wifi systems, and is always implemented by one of the fast Fourier transforms (FFT) algorithms.  It is possible to generalize the FFT to work correctly on input vectors with periodic missing values.   I will consider whether this has applications, such as more general transmitted signal waveforms, or further applications such as spectral density estimation for time series with missing data.  More speculatively, can we generalize to "recursive" missing values, where the non-missing blocks have gaps?   If so, how do we optimally recognize such a pattern in a given time series?

Last summer, there was a lot of activity regarding an old conjecture of van der Waerden, culminating in its solution by Bhargava, and including joint work by Sam Chow and myself on which I want to report in this talk: We showed that the number of irreducible monic integer polynomials of degree n, with coefficients in absolute value bounded by H, which have Galois group different from S_n and A_n, is of order of magnitude O(H^{n-1.017}), providing that n is at least 3 and is different from 7,8,10. Apart from the alternating group and excluding degrees 7,8,10, this establishes the aforementioned conjecture to the effect that irreducible non-S_n polynomials are significantly less frequent than reducible polynomials.

The Langlands program is a far-reaching collection of conjectures that relate different areas of mathematics including number theory and representation theory. A fundamental problem on the representation theory side of the Langlands program is the construction of all (irreducible, smooth, complex or mod-$\ell$) representations of p-adic groups. I will provide an overview of our understanding of the representations of p-adic groups, with an emphasis on recent progress including joint work with Kaletha and Spice that introduces a twist to the story, and outline some applications.

That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will argue that such direct solvers have several compelling qualities, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments.

After a general introduction to the field, particular attention will be paid to a set of recently developed randomized algorithms that construct data sparse representations of large dense matrices that arise in scientific computations. These algorithms are entirely black box, and interact with the linear operator to be compressed only via the matrix-vector multiplication.

Macroscopic Transport in Heterogeneous Porous Materials

Lucy Auton

Solute transport in porous materials is a key physical process in a wide variety of situations, including contaminant transport, filtration, lithium-ion batteries, hydrogeological systems, biofilms, bones and soils. Despite the prevalence of solute transport in porous materials, the effect of microstructure on flow and transport remains poorly understood and improving our understanding of this remains a major challenge.  In this presentation, I consider a two-dimensional microstructure comprising an array of obstacles of smooth but arbitrary shape, the size and spacing of which can vary along the length of the porous medium, allowing for anisotropy.  I use a nontrivial extension to classical homogenisation theory via the method of multiple scales to rigorously upscale the novel problem involving cells of varying area. This results in simple effective continuum equations for macroscale flow and transport where the effect of the microscale geometry on the macroscopic transport and removal is encoded within these simple macroscale equations via effective parameters such as an effective local anisotropic diffusivity and an effective local adsorption rate.  For a simple example geometry I exploit the two degrees of microstructural freedom in this problem, obstacle size and obstacle spacing, to investigate scenarios of uniform porosity but heterogenous microstructure, noting the impact this heterogeneity has on filter efficiency. 

This model constitutes the development of the core framework required to consider other crucial problems such as solute transport within soft porous materials for which there does not currently exist a simple macroscale model where the effective diffusivity and removal depend on the microstructure. Further, via this methodology I will  derive a  bespoke model for fluoride and arsenic removal filters. With this model I will be able to optimise the design of fluoride-removal filters which are being deployed across rural India. The design optimisation will both increase filter lifespan and reduce filter cost, enabling more people to access safe drinking water

Averaged interface conditions: evaporation fronts in porous media

Ellen Luckins

Homogenisation methods are powerful tools for deriving effective PDE models for processes incorporating multiple length-scales. For physical systems in which interface processes are crucial to the overall system, we might ask how the microstructure impacts the effective interface conditions, in addition to the PDEs in the bulk. In this talk we derive an effective model for the motion of an evaporation front through porous media, combining homogenisation and boundary layer analysis to derive averaged interface conditions at the evaporation front. Our analysis results in a new effective parameter in the boundary conditions, which encodes how the shape and speed of the porescale evaporating interfaces impact the overall drying process.

We derive quantitatively the weak and strong Harnack inequality for kinetic Fokker--Planck type equations with a non-local diffusion operator for the full range of the non-locality exponents in (0,1).  This implies Hölder continuity.  We give novel proofs on the boundedness of the bilinear form associated to the non-local operator and on the construction of a geometric covering accounting for the non-locality to obtain the Harnack inequalities.  Our results apply to the inhomogeneous Boltzmann equation in the non-cutoff case.

We will give a fairly self contained introduction to acylindrically hyperbolic groups, using mapping class groups as a motivating example. This will be a mainly expository talk, the aim is to make my topology seminar talk in week 5 more accessible to people who are less familiar with these topics.

The first edition of Thompson’s famous book On Growth and Form appeared in 1917. It has subsequently been regarded as a foundational work in mathematical biology and a revolutionary contribution to the field of morphology. Most existing literature credits Thompson as a lone genius who produced the 793 pages of the 1917 edition and 1116 pages of the 1942 edition. Thompson’s correspondence presents a very different picture of this tome as one arising from extensive and ongoing – perhaps sometimes unwitting? – collaboration.

G2 structure manifolds are a key ingredient in supersymmetric compactifications on seven-manifolds. We will discuss the fact that G2 structure manifolds admit refinements in the form of almost contact (3-) structures.  In fact, there are infinite dimensional spaces of these structures. We will discuss topological and differential geometric aspects of (the space of) these refinements. We will then explore applications in physics, including supersymmetry enhancement. This is based on 2101.12605.

The string 2-group is a fundamental object in string geometry, which is a refinement of spin geometry required to describe the spinning string. While many models for the string 2-group exist, the construction of a representation for it is new. In this talk, I will recall the notion of strict 2-group, and then give two examples: the automorphism 2-group of a von Neumann algebra, and the string 2-group. I will then describe the representation of the string 2-group on the hyperfinite III_1 factor, which is a functor from the string 2-group to the automorphism 2-group of the hyperfinite III_1 factor.

The sinoatrial node (SAN) is the pacemaker region of the heart.
Recently calcium signals, believed to be crucially important in heart
rhythm generation, have been imaged in intact SAN and shown to be
heterogeneous in various regions of the SAN. However, calcium imaging
is noisy, and the calcium signal heterogeneity has not been
mathematically analyzed to distinguish meaningful signals from
randomness or to identify signalling regions in an objective way. In
this work we apply methods of random matrix theory (RMT) developed for
financial data and used for analysis of various biological data sets
including β-cell collectives and EEG data. We find eigenvalues of the
correlation matrix that deviate from RMT predictions, and thus are not
explained by randomness but carry additional meaning. We use
localization properties of the eigenvectors corresponding to high
eigenvalues to locate particular signalling modules. We find that the
top eigenvector captures a common response of the SAN to action
potential. In some cases, the eigenvector corresponding to the second
highest eigenvalue appears to yield a possible pacemaker region as its
calcium signals predate the action potential. Next we study the
relationship between covariance coefficients and distance and find
that there are long range correlations, indicating intercellular
interactions in most cases. Lastly, we perform an analysis of nearest
neighbor eigenvalue distances and find that it coincides with the
universal Wigner surmise. On the other hand, the number variance,
which captures eigenvalue correlations, is a parameter that is
sensitive to experimental conditions. Thus RMT application to SAN
allows to remove noise and the global effects of the action potential
and thereby isolate the correlations in calcium signalling which are
local. This talk is based on joint work with Chloe Norris with a
preprint found here:
https://www.biorxiv.org/content/10.1101/2022.02.25.482007v1.

In this talk, I will introduce the BPS cohomology of the moduli space of Higgs bundles on a smooth projective curve of rank r and degree d using cohomological Donaldson-Thomas theory. The BPS cohomology and the intersection cohomology coincide when r and d are coprime, but they are different in general. We will see that the BPS cohomology does not depend on d. This is a generalization of the Hausel-Thaddeus conjecture to non-coprime case. I will also explain that Toda's χ-independence conjecture (and hence the strong rationality conjecture) for local curves can be proved in the same manner. This talk is based on a joint work with Naoki Koseki and another joint work with Naruki Masuda.

For graphs $G$ and $H$, we say $G \stackrel{r}{\to} H$ if every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. The blowup Ramsey number $B(G \stackrel{r}{\to} H;t)$ is the minimum $n$ such that $G[n] \stackrel{r}{\to} H[t]$. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given $G$, $H$ and $r$ such that $G \stackrel{r}{\to} H$, there exist constants $a=a(G,H,r)$ and $b=b(H,r)$ such that for all $t \in \mathbb{N}$, $B(G \stackrel{r}{\to} H;t) \leq ab^t$. They conjectured that there exist some graphs $H$ for which the constant $a$ depending on $G$ is necessary. We prove this conjecture by showing that the statement is true when $H$ is a $3$-chromatically connected, which includes all cliques on $3$ or more vertices. We are also able to show perhaps surprisingly that for any forest $F$ there is $f(F,t)$ such that  for any $G \stackrel{r}{\to} H$, $B(G \stackrel{r}{\to} H;t)\leq f(F,t)$ i.e. the function does not depend on the ground graph $G$. This is joint work with Robert Hancock.

Co-occurrence networks formed by bipartite projection are widely studied in many contexts, including politics (bill co-sponsorship), bibliometrics (paper co-authorship), ecology (species co-habitation), and genetics (protein co-expression). It is often useful to focus on the backbone, a binary representation that includes only the most important edges, however many different backbone extraction models exist. In this talk, I will demonstrate the "backbone" package for R, which implements many such models. I will also use it to compare two promising null models: the fixed degree sequence model (FDSM) that imposes hard constraints, and the stochastic degree sequence model (SDSM) that imposes soft constraints, on the bipartite degree sequences. While FDSM is more statistically powerful, SDSM is more efficient and offers a close approximation.

One core aim of (supervised) machine learning is to approximate an unknown function given a dataset containing examples of input-output pairs. Real-world examples of such functions include the mapping from an image to its label or the mapping from a molecule to its energy. For a variety of such functions, while the precise mapping is unknown, we often have knowledge of its properties. For example, the label of an image may be invariant to rotations of the input image. Generally, such properties formally correspond to the function being equivariant to certain actions on its input and output spaces. This has led to much research on building equivariant function classes (aka neural networks). In this talk, we survey this growing field of equivariance in deep learning for a mathematical audience, motivating the need for equivariance, covering concrete examples of equivariant neural networks, and offering a learning theoretic perspective on the benefits of equivariance. 

In the search for a complete description of quantum mechanical and
gravitational phenomena, we are inevitably led to consider observables on
boundaries at infinity. This is the common mantra that there are no local
observables in quantum gravity and gives rise to the tantalising possibility
of a purely boundary--or holographic--description of physics in the
interior. The AdS/CFT correspondence provides an important working example
of these ideas, where the boundary description of quantum gravity in anti-de
Sitter (AdS) space is an ordinary quantum mechanical system-- in particular,
a Lorentzian Conformal Field Theory (CFT)--where the rules of the game are
well understood. It would be desirable to have a similar level of
understanding for the universe we actually live in. In this talk I will
explain some recent efforts that aim to understand the rules of the game for
observables on the future boundary of de Sitter (dS) space. Unlike in AdS,
the boundaries of dS space are purely spatial with no standard notion of
locality and time. This obscures how the boundary observables capture a
consistent picture of unitary time evolution in the interior of dS space. I

will explain how, despite this difference, the structural similarities
between dS and AdS spaces allow to forge relations between boundary
correlators in these two space-times. These can be used to import
techniques, results and understanding from AdS to dS.

The aim of this course is to give an introduction to the regularity theory of non-smooth spaces with lower bounds on the Ricci Curvature. This is a quickly developing field with motivations coming from classical questions in Riemannian and differential geometry and with connections to Probability, Geometric Measure Theory and Partial Differential Equations.

In the lectures we will focus on the non collapsed case, where much sharper results are available, mainly adopting the synthetic approach of the RCD theory, rather than following the original proofs.

The techniques used in this setting have been applied successfully in the study of Minimal surfaces, Elliptic PDEs, Mean curvature flow and Ricci flow and the course might be of interest also for people working in these subjects.

In this talk, I would like to discuss Deligne’s version of Geometric Class Field theory, with special emphasis on the correspondence between rigidified 1-dimensional l-adic local systems on a curve and 1-dimensional l-adic local systems on Pic with certain compatibilities. We should like to give a sense of how this relates to the OG class field theory, and how Deligne demonstrates this correspondence via the geometry of the Abel-Jacobi Map. If time permits, we would also like to discuss the correspondence between continuous 1-dimensional l-adic representations of the etale fundamental group of a curve and local systems.

We will review some open questions about automorphisms of free groups, give some partial answers, and explain the deformation spaces of trees that they act on, as well as the geometry of these spaces arising from the Lipschitz metric. This will be a gentle introduction to the topic, focused on introducing the concepts.

 We develop a probabilistic framework for analysing model-based reinforcement learning in the episodic setting. We then apply it to study finite-time horizon stochastic control problems with linear dynamics but unknown coefficients and convex, but possibly irregular, objective function. Using probabilistic representations, we study regularity of the associated cost functions and establish precise estimates for the performance gap between applying optimal feedback control derived from estimated and true model parameters. We identify conditions under which this performance gap is quadratic, improving the linear performance gap in recent work [X. Guo, A. Hu, and Y. Zhang, arXiv preprint, arXiv:2104.09311, (2021)], which matches the results obtained for stochastic linear-quadratic problems. Next, we propose a phase-based learning algorithm for which we show how to optimise exploration-exploitation trade-off and achieve sublinear regrets in high probability and expectation. When assumptions needed for the quadratic performance gap hold, the algorithm achieves an order (N‾‾√lnN) high probability regret, in the general case, and an order ((lnN)2) expected regret, in self-exploration case, over N episodes, matching the best possible results from the literature. The analysis requires novel concentration inequalities for correlated continuous-time observations, which we derive.

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Dr Lukasz Szpruch

The integral of mean curvature squared is a conformal invariant of surfaces reintroduced by Willmore in 1965 whose study exercised a tremendous influence on geometric analysis and most notably on minimal surfaces in the last years.

On the other hand, the Loewner energy is a conformal invariant of planar curves introduced by Yilin Wang in 2015 which is notably linked to SLE processes and the Weil-Petersson class of (universal) Teichmüller theory.

In this presentation, after a brief historical introduction, we will discuss some recent developments linking the Willmore energy to the Loewner energy and mention several open problems.

Joint work with Yilin Wang (MIT/MSRI)

Topological defects in quantum field theory can be understood as a generalised notion of symmetry, where the operation is not required to be invertible. Duality transformations are an important example of this. By considering defects of various dimensions, one is naturally led to more complicated algebraic structures than just groups. So-called 2-groups are a first instance, which arise from invertible defects of codimension 1 and 2. Without invertibility one arrives at so-called "fusion categories”. I would like to explain how one can "gauge" such non-invertible symmetries in the case of topological field theories, and I will focus on results in two and three dimensions. This talk is based on joint work with Nils Carqueville, Vincentas Mulevicius, Gregor Schaumann, and Daniel Scherl.

We define a perturbatively calculable quantity—the on-shell correlator—which furnishes a unified description of particle dynamics in curved spacetime. Specializing to the case of flat and anti-de Sitter space, on-shell correlators coincide precisely with on-shell scattering amplitudes and boundary correlators, respectively. Remarkably, we find that symmetric manifolds admit a generalization of on-shell kinematics in which the corresponding momenta are literally the isometry generators of the spacetime acting on the external kinematic data. These isometric momenta are intrinsically non-commutative but exhibit on-shell conditions that are identical to those of flat space, thus providing a common language for computing and representing on-shell correlators which is agnostic about the underlying geometry. 

As applications of these tools, we compute n-point scalar correlators in AdS in terms of isometric momenta. In many cases, the results are direct lifts of flat-space expressions. We provide field-theoretic proofs of color-kinematics duality and BCJ relations in AdS at n-points in biadjoint scalar theory and the nonlinear sigma model. We discuss possible extensions to generic curved spacetimes without symmetry.

I will explain in what sense the theory of finite tensor categories is a categorification of the theory of finite dimensional algebras. In particular, I will introduce finite module categories, review a key result of Ostrik, and present Morita theory for finite categories. I will give many examples to illustrate these ideas. Then, I will explain the elementary properties of finite braided tensor categories. If time permits, I will also mention my own work, which consists in categorifying these ideas once more!

Dynamical systems models are a powerful tool for analyzing interactions in ecosystems and their intrinsic properties such as stability and resilience. The human intestinal microbiome is a complex ecosystem of hundreds of microbial species, critical to our health, and when in a disrupted state termed dysbiosis, is involved in a variety of diseases.  Although dysbiosis remains incompletely understood, it is not caused by single pathogens, but instead involves broader disruptions to the microbial ecosystem.  Dynamical systems models would thus seem a natural approach for analyzing dysbiosis, but have been hampered by the scale of the human gut microbiome, which constitutes hundreds of thousands of potential ecological interactions, and is profiled using sparse and noisy measurements. Here we introduce a combined experimental and statistical machine learning approach that overcomes these challenges to provide the first comprehensive and predictive model of microbial dynamics at ecosystem-scale. We show that dysbiosis is characterized by competitive cycles of interactions among microbial species, in contrast to the healthy microbiome, which is stabilized by chains of positive interactions initiated by resistant starch-degrading bacteria. To achieve these results, we created cohorts of “humanized” gnotobiotic mice via fecal transplantation from healthy and dysbiotic human donors, and subjected mice to dietary and antibiotic perturbations, in the densest temporal interventional study to date. We demonstrate that our probabilistic machine learning method achieves scalability while maintaining interpretability on these data, by inferring a small number of modules of bacterial taxa that share common interactions and responses to perturbations. Our findings provide new insights into the mechanisms of microbial dysbiosis, have potential implications for therapies to restore the microbiome to treat disease, and moreover offer a powerful framework for analyzing other complex ecosystems.

Lahars and huaicos are potent natural hazards that threaten lives and livelihoods. They comprise debris-laden fluid that flows rapidly down slopes, bulking up considerably as they progress. Owing to their rapid onset and the significant threat that they pose to communities and infrastructures, it is important to be able to predict their motion in order to assess quantitatively some of the impacts that they may cause. In this seminar I will present mathematical models of these flows and apply them to various natural settings, drawing on examples from Peru and the Philippines.  Along the way I will show some informative, idealised solutions, the susceptibility of these flows to roll wave instabilities, ways to prevent ill-posedness and how to include measured topography in the computation.

We would like to share two challenges that, if solved, could improve our domestic lives.  

 Firstly, having appliances that are as unobtrusive as possible is a strong desire, unwanted noise can cause a negative impact on relaxation.  A key target for refrigerators is low sound level, a key noise source is the capillary tube.  The capillary tube effects the phase change that is required for the refrigerant to be in the gaseous state in the evaporator for cooling.  Noise is generated during this process due to two phases being present within the flow through the tube.  The challenge is to create a numerical model and analysis of refrigerant flow properties in order to estimate the acoustic behaviour.

 Secondly, we would like to maximise the information that can be gathered from our new range of connected devices.  By analysing the data generated during usage we would like to be able to predict faults and understand user behaviour in more detail.  The challenge regarding fault prediction is the scarcity of the failure data and the impact of false positives.  Due to the number of units in the field, a relatively small fraction of false positives can remove the ROI from such an initiative.  We would like to understand if advanced machine learning methods can be used to reduce this risk.

As a model for the primes, in this talk I will address such statistical questions for the sequence of squarefree numbers, i.e., numbers not divisible by the square of any prime, among other related ``sifted'' sequences called B-free numbers. I hope to further motivate and explain our main result that shows, unconditionally, that short interval counts of squarefree numbers do satisfy Gaussian statistics, answering several questions of R.R. Hall.

The Skolem Problem asks to decide whether a linearly recurrent sequence (LRS) over the rationals has a zero term.  It is sometimes considered as the halting problem for linear loops.   In this talk we will give an overview of two current approaches to establishing decidability of this problem.  First, we observe that the Skolem Problem for LRS with simple characteristic roots is decidable subject to the $p$-adic Schanuel conjecture and the exponential-local-global principle.  Next, we define a set $S$ of positive integers such that (i) $S$ has positive lower density and (ii) The Skolem Problem is decidable relative to $S$, i.e., one can effectively determine the set of all zeros of a given LRS that lie in $S$.

The talk is based on joint work with Y. Bilu, F. Luca, J. Ouaknine, D. Pursar, and J. Nieuwveld.  

In space dimension 2 we present a finite element complex for the deformation operator acting on vectorfields and the linearized curvature operator acting on symmetric 2 by 2 matrices. We also present the tools that were used in the construction, namely the BGG diagram chase and the framework of finite element systems. For this general framework we can prove a de Rham theorem on cohomology groups in the flat case and a Bianchi identity in the case with curvature.

Latent Dirichlet Allocation (LDA) is capable of analyzing thousands of documents in a short time and highlighting important elements, recurrences and anomalies. It is generally used in linguistics to study natural language: its word analysis reveals the theme(s) of a document, each theme being identified by a specific vocabulary or, more precisely, by a particular statistical distribution of word frequency.
In the climatologists' use of LDA, the document is a daily weather map and the word is a pixel of the map. The theme with its corpus of words can become a cyclone or an anticyclone and, more generally, a 'pattern'  that the scientists term motif. Artificial intelligence – a sort of incredibly fast robot meteorologist – looks for correlations both between different places on the same map, and between successive maps over time. In a sense, it 'notices' that a particular location is often correlated with another location, recurrently throughout the database, and this set of correlated locations constitutes a specific pattern.
The algorithm performs statistical analyses at two distinct levels: at the word or pixel level of the map, LDA defines a motif, by assigning a certain weight to each pixel, and thus defines the shape and position of the motif;  LDA breaks down a daily weather map into all these motifs, each of which is assigned a certain weight.
In concrete terms, the basic data are the daily weather maps between 1948 and nowadays over the North Atlantic basin and Europe. LDA identifies a dozen or so spatially defined motifs, many of which are familiar meteorological patterns such as the Azores High, the Genoa Low or even the Scandinavian Blocking. A small combination of those motifs can then be used to describe all the maps. These motifs and the statistical analyses associated with them allow researchers to study weather phenomena such as extreme events, as well as longer-term climate trends, and possibly to understand their mechanisms in order to better predict them in the future.

The preprint of the study is available as:
 Lucas Fery, Berengere Dubrulle, Berengere Podvin, Flavio Pons, Davide Faranda. Learning a weather dictionary of atmospheric patterns using Latent Dirichlet Allocation. 2021. ⟨hal-03258523⟩ https://hal-enpc.archives-ouvertes.fr/X-DEP-MECA/hal-03258523v1