Past

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 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

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.

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!

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.

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.

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
 

We study the definability of valuation rings in ordered fields (in the language of ordered rings). We show that any henselian valuation ring that is definable in the language of ordered rings is already definable in the language of rings. However, this does not hold when we drop the assumption of henselianity.

This is joint work with Philip Dittmann, Sebastian Krapp and Salma Kuhlmann.

Groupoid C*-algebras and twisted groupoid C*-algebras are introduced by Renault in the late ’70. Twisted groupoid C*-algebras have since proved extremely important in the study of structural properties for large classes of C*-algebras. On the other hand, Steinberg algebras are introduced independently by Steinberg and Clark, Farthing, Sims and Tomforde around 2010 which are a purely algebraic analogue of groupoid C*-algebras. Steinberg algebras provide useful insight into the analytic theory of groupoid C*-algebras and give rise to interesting examples of *-algebras. In this talk, I will first recall some relevant background on topological groupoids and twisted groupoid C*-algebras, then I will introduce twisted Steinberg algebras which generalise the Steinberg algebras and provide a purely algebraic analogue of twisted groupoid C*-algebras. If I have enough time, I will further introduce pair of algebras which consist of a Steinberg algebra and an algebra of locally constant functions on the unit space, it is an algebraic analogue of Cartan pairs

The characteristic polynomial of a random Hermitian matrix induces naturally a field on the real line. In the case of the Gaussian Unitary ensemble (GUE), this fields is expected to have a very special correlation structure: the logarithm of this field is log-correlated and its maximum is at the heart of a conjecture from Fyodorov and Simm predicting its asymptotic behavior.   As a first step in this direction, we obtained in collaboration with R. Butez and O. Zeitouni, a central limit theorem for the logarithm of the characteristic polynomial of the Gaussian beta Ensembles and for a certain class of random Jacobi matrices. In this talk, I will explain how the tridiagonal representation of the GUE and orthogonal polynomials techniques allow us to analyse the fluctuations of the characteristic polynomial.

Given a variety defined over a field of characteristic zero and an algebraically integrable foliation of corank less than or equal to two, we show the existence of a categorical quotient, defined on the non-empty open subset of algebraically smooth points, through which every invariant morphism factors uniquely. Some applications to quotients by connected groups will be discussed.
 

We investigate the maximum distance of a rank-two tensor to rank-one tensors. An equivalent problem is given by the minimal ratio of spectral and Frobenius norm of a tensor. For matrices the distance of a rank k matrix to a rank r matrices is determined by its singular values, but since there is a lack of a fitting analog of the singular value decomposition for tensors, this question is more difficult in the regime of tensors.
 

iii) identify a special class of correlated compressors based on the idea of random permutations, for which we coin the term PermK. The use of this technique results in the strict improvement on the previous MARINA rate. In the low Hessian variance regime, the improvement can be as large as √n, when d > n, and 1 + √d/n, when n<=d, where n is the number of workers and d is the number of parameters describing the model we are learning.