Past

We study the problem of super-resolution using TV norm minimisation, where we recover the locations and weights of non-negative point sources from a few samples of their convolution with a Gaussian kernel. A practical approach is to solve the dual problem. In this talk, we study the stability of solutions with respect to the solutions to the dual problem. In particular, we establish a relationship between perturbations in the dual variable and the primal variables around the optimiser. This is achieved by applying a quantitative version of the implicit function theorem in a non-trivial way.

Deformation quantization modules or DQ-modules where introduced by M. Kontsevich to study the deformation quantization of complex Poisson varieties. It has been advocated that categories of DQ-modules should provide invariants of complex symplectic varieties and in particular a sort of complex analog of the Fukaya category. Hence, it is natural to aim at describing the functors between such categories and relate them with categories appearing naturally in algebraic geometry. Relying, on methods of homotopical algebra, we obtain an analog of Orlov representation theorem for functors between categories of DQ-modules and relate these categories to deformations of the category of quasi-coherent sheaves.
 

We consider a sparse multi-task regression framework for fitting a collection of related sparse models. Representing models as nodes in a graph with edges between related models, a framework that fuses lasso regressions with the total variation penalty is investigated. Under a form of generalised restricted eigenvalue assumption, bounds on prediction and squared error are given that depend upon the sparsity of each model and the differences between related models. This assumption relates to the smallest eigenvalue restricted to the intersection of two cone sets of the covariance matrix constructed from each of the agents' covariances. In the case of a grid topology high-probability bounds are given that match, up to log factors, the no-communication setting of fitting a lasso on each model, divided by the number of agents.  A decentralised dual method that exploits a convex-concave formulation of the penalised problem is proposed to fit the models and its effectiveness demonstrated on simulations. (Joint work with Sahand Negahban and Patrick Rebeschini)

Multiple scales analysis is a powerful asymptotic technique for problems where the solution depends on two scales of widely different sizes. Standard multiple scales involves the introduction of a macroscale and microscale which are assumed to be independent. A common (and usually acceptable) assumption is that when considering behaviour on the microscale, the macroscale variable can be taken as constant, however there are instances where this assumption is not valid. In this talk, I will explain one such situation, that is, when considering conductive-radiative thermal transfer within a solid matrix with spherical perforations and discuss the appropriate measures when converting the radiative boundary condition into multiple-scales form.
 

Quantum fields can sometimes have negative energy density.  In gravitational contexts, this threatens to permit both causality violations (such as traversable wormholes, warp drives, and time machines) and violations of the Second Law for black holes.  I will discuss the thermodynamic principles that rule out such pathological situations.  These principles have led us to an interesting lower bound on the energy flux, even for field theories in flat spacetime! This Quantum Null Energy Condition has now been proven for all relativistic field theories.  I will give an intuitive argument explaining why such ``quantum energy conditions'' ought to hold. 
 

Complexity Science aims to understand what is that makes some systems to be "more than the sum of their parts". A natural first step to address this issue is to study networks of pairwise interactions, which have been done with great success in many disciplines -- to the extend that many people today identify Complexity Science with network analysis. In contrast, multivariate complexity provides a vast and mostly unexplored territory. As a matter of fact, the "modes of interdependency" that can exist between three or more variables are often nontrivial, poorly understood and, yet, are paramount for our understanding of complex systems in general, and emergence in particular. 
In this talk we present an information-theoretic framework to analyse high-order correlations, i.e. statistical dependencies that exist between groups of variables that cannot be reduced to pairwise interactions. Following the spirit of information theory, our approach is data-driven and model-agnostic, being applicable to discrete, continuous, and categorical data. We review the evolution of related ideas in the context of theoretical neuroscience, and discuss the most prominent extensions of information-theoretic metrics to multivariate settings. Then, we introduce the O-information, a novel metric that quantify various structural (i.e. synchronous) high-order effects. Finally, we provide a critical discussion on the framework of Integrated Information Theory (IIT), which suggests an approach to extend the analysis to dynamical settings. To illustrate the presented methods, we show how the analysis of high-order correlations can reveal critical structures in various scenarios, including cellular automata, Baroque music scores, and various EEG datasets.

References:
[1] F. Rosas, P.A. Mediano, M. Gastpar and H.J. Jensen, ``Quantifying High-order Interdependencies via Multivariate Extensions of the Mutual Information'', submitted to PRE, under review.
https://arxiv.org/abs/1902.11239
[2] F. Rosas, P.A. Mediano, M. Ugarte and H.J. Jensen, ``An information-theoretic approach to self-organisation: Emergence of complex interdependencies in coupled dynamical systems'', in Entropy, vol. 20 no. 10: 793, pp.1-25, Sept. 2018.
https://www.mdpi.com/1099-4300/20/10/793

 

The principal ideal theorem (1930) ascertained that any number field K embeds into a finite extension, called the Hilbert class field of K, in which every ideal of the original field became principal -- however the Hilbert class field itself will not necessarily have class number 1. The class field tower problem asked whether iteratively taking Hilbert class fields must stabilize after finitely many steps. In 1964, it was finally answered in the negative by Golod and Shafarevich who produced infinitely many examples and pioneered the framework that is still the most common setting for deciding when a number field will have an infinite class field tower.

In this talk, I will sketch the proof of their cohomological result and explain how it settled the class field tower problem.

In this talk, I will present our recent progress collaborated with Prof. Gui-Qiang G. Chen and Prof. Paolo Secchi on two kinds of characteristic discontinuities: relativistic vortex sheets in three-dimensional Minkowski spacetime and multi-dimensional thermoelastic contact discontinuities.
 

A group is said to satisfy the Tits Alternative if its finitely generated subgroups exhibit a striking dichotomy: they are either "big" (they contain a non-abelian free subgroup) or "small" (they are virtually soluble). Many groups of geometric interest have been shown to satisfy the Tits Alternative: linear groups, mapping class groups of hyperbolic surfaces, etc. In this talk, I will explain how one can use ideas from group actions in negative curvature to prove such a dichotomy. In particular, I will show how one can prove a strengthening of the Tits Alternative for a large class of Artin groups. This is joint work with Piotr Przytycki.

Neural networks are undoubtedly successful in practical applications. However complete mathematical theory of why and when machine learning algorithms based on neural networks work has been elusive. Although various representation theorems ensures the existence of the ``perfect’’ parameters of the network, it has not been proved that these perfect parameters can be (efficiently) approximated by conventional algorithms, such as the stochastic gradient descent. This problem is well known, since the arising optimisation problem is non-convex. In this talk we show how the optimization problem becomes convex in the mean field limit for one-hidden layer networks and certain deep neural networks. Moreover we present optimality criteria for the distribution of the network parameters and show that the nonlinear Langevin dynamics converges to this optimal distribution. This is joint work with Kaitong Hu, Zhenjie Ren and Lukasz Szpruch. 

Partition Lie algebras are generalisations of rational differential graded Lie algebras which, by a recent result of Mathew and myself, govern the formal deformation theory of algebro-geometric objects in finite and mixed characteristic. In this talk, we will take a closer look at these new gadgets and discuss some of their applications in algebra and topology

Supergravity backgrounds are an essential ingredient in string theory or field theories via AdS/CFT. The simplest example of a 4d, N=1 background is the product of four-dimensional Minkowski space with a seven-dimensional manifold with G_2 holonomy in M-theory. For more complicated backgrounds where we allow non-zero fluxes, the supersymmetry conditions can be rephrased in terms of G-structure data. The geometry of these backgrounds is often complicated and their general features are not well understood.

In this talk, I will define the analogue of G_2 geometry for generic 4d, N=1 backgrounds with flux in both type II and eleven-dimensional supergravity. The geometry is characterised by a G-structure in 'exceptional generalised geometry' that includes G_2 structures and Hitchin's generalised geometry as subcases. Supersymmetry is then equivalent to integrability of the structures, which appears as an involutivity condition and a moment map for diffeomorphisms and gauge transformations. I will show how this works in a few simple examples and discuss how this can be used to understand general properties of supersymmetric backgrounds.

 

A panel discussion on non-academic careers for mathematicians with PhDs, featuring Katia Babbar (AI Wealth Technologies & QuantBright), Jara Imbers (Risk Management Solutions) and Tom Hawes (Smith Institute).
 

Cardiac fibrosis plays a significant role in the disruption of healthy electrical signalling in the heart, creating structural heterogeneities that induce and stabilise arrhythmia.  However, a proper understanding of the consequences of cardiac fibrosis must take into account the complex and highly variable patterns of its spatial localisation in the heart, which significantly affects the extent and manner of its impacts on cardiac wave propagation. In this work we present a methodology for the algorithmic generation of fibrotic patterns via Perlin noise, a technique for computationally efficient generation of textures in computer graphics.

Our approach works directly from image data to create populations of pattern realisations that all resemble the target image under a set of metrics. Our technique thus serves as a type of data enrichment, enabling analysis of how variability in the precise placement of fibrotic structures modulates their electrophysiological impact. We demonstrate our method, and the types of analysis it can enable, using a widely referenced histological image of four different types of microfibrotic structure. Our generator and Bayesian tuning method prove flexible enough to successfully capture each of these very distinct patterns.

We demonstrate the importance of this tool, by presenting 2D simulations overlayed on the generated images that highlight the effects of microscopic variability on the electrophysiological impact of fibrosis. Finally, we discuss the application of our methodology to the increasingly available imaging data of fibrotic patterning on a more macroscopic scale, and indeed to other areas of science underpinned by image based modelling and simulation.    

Dimension reduction is an overarching theme in data science: we enjoy finding informative patterns, features or substructures in large, complex data sets. Within the field of network science, an important problem of this nature is to identify core-periphery structure. Given a network, our task is to assign each node to either the core or periphery. Core nodes should be strongly connected across the whole network whereas peripheral nodes should be strongly connected only to core nodes. More generally, we may wish to assign a non-negative value to each node, with a larger value indicating greater "coreness." This type of problem is related to, but distinct from, commumnity detection (finding clusters) and centrality assignment (finding key players), and it arises naturally in the study of networks in social science and finance. We derive and analyse a new iterative algorithm for detecting network core-periphery structure.

Using techniques in nonlinear Perron-Frobenius theory we prove global convergence to the unique solution of a relaxed version of a natural discrete optimization problem. On sparse networks, the cost of each iteration scales linearly with the number of nodes, making the algorithm feasible for large-scale problems. We give an alternative interpretation of the algorithm from the perspective of maximum likelihood reordering of a new logistic core--periphery random graph model. This viewpoint also gives a new basis for quantitatively judging a core--periphery detection algorithm. We illustrate the algorithm on a range of synthetic and real networks, and show that it offers advantages over the current state-of-the-art.

This is joint work with Francesco Tudisco (Strathclyde)

In Formula 1 engineers strive to produce the fastest car possible for their drivers. A lap simulation provides an objective evaluation of the performance of the car and the subsequent lap time achieved. Using this information, engineers aim to test new car concepts, determine performance limitations or compromises, and identify the sensitivity of performance to car setup parameters.

The latest state of the art lap simulation techniques use optimal control approaches. Optimisation methods are employed to derive the optimal control inputs of the car that achieve the fastest lap time within the constraints of the system. The resulting state trajectories define the complete behaviour of the car. Such approaches aim to create more robust, realistic and powerful simulation output compared to traditional methods.

In this talk we discuss our latest work in this area. A dynamic vehicle model is used within a free-trajectory solver based on direct optimal control methods. We discuss the reasons behind our design choices, our progress to date, and the issues we have faced during development. Further, we look at the short and long term aims of our project and how we wish to develop our mathematical methods in the future.

I will discuss the arithmetic significance of Fourier expansions of modular forms at cusps. I will talk about joint work with F. Brunault, where we determine the number field generated by Fourier coefficients of newforms at a cusp. Then I will discuss joint work with A. Saha and K. Česnavičius where we find denominator bounds for Fourier expansions at cusps and apply these bounds to a conjecture on the Manin constants of elliptic curves.

The problem of model uncertainty in financial mathematics has received considerable attention in the last years. In this talk I will follow a non-parametric point of view, and argue that an insightful approach to model uncertainty should not be based on the familiar Wasserstein distances. I will then provide evidence supporting the better suitability of the recent notion of adapted Wasserstein distances (also known as Nested Distances in the literature). Unlike their more familiar counterparts, these transport metrics take the role of information/filtrations explicitly into account. Based on joint work with M. Beiglböck, D. Bartl and M. Eder.

Likely instabilities in stochastic hyperelastic solids

L. Angela Mihai

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

E-mail: @email.uk

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

One of the major steps in the adaptive finite element methods (AFEM) is the adaptive selection of the next partition. The process is usually governed by a strategy based on carefully chosen local error indicators and aims at convergence results with optimal rates. One can formally relate the refinement of the partitions with growing an oriented graph or a tree. Then each node of the tree/graph corresponds to a cell of a partition and the approximation of a function on adaptive partitions can be expressed trough the local errors related to the cell, i.e., the node. The total approximation error is then calculated as the sum of the errors on the leaves (the terminal nodes) of the tree/graph and the problem of finding an optimal error for a given budget of nodes is known as tree approximation. Establishing a near-best tree approximation result is a key ingredient in proving optimal convergence rates for AFEM.

The classical tree approximation problems are usually related to the so-called h-adaptive approximation in which the improvements a due to reducing the size of the cells in the partition. This talk will consider also an extension of this framework to hp-adaptive approximation allowing different polynomial spaces to be used for the local approximations at different cells while maintaining the near-optimality in terms of the combined number of degrees of freedom used in the approximation.

The problem of conformity of the resulting partition will be discussed as well. Typically in AFEM, certain elements of the current partition are marked and subdivided together with some additional ones to maintain desired properties of the partition like conformity. This strategy is often described as “mark → subdivide → complete”. The process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This hints at the possibility to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step would be necessary. This talk will present such a strategy together with theoretical results about its near-optimal performance.

It is nowadays well understood that the multidimensional isentropic Euler system is desperately ill–posed. Even certain smooth initial data give rise to infinitely many solutions and all available selection criteria fail to ensure both global existence and uniqueness. We propose a different approach to well–posedness of this system based on ideas from the theory of Markov semigroups: we show the existence of a Borel measurable solution semiflow. To this end, we introduce a notion of dissipative solution which is understood as time dependent trajectories of the basic state variables - the mass density, the linear momentum, and the energy - in a suitable phase space. The underlying system of PDEs is satisfied in a generalized sense. The solution semiflow enjoys the standard semigroup property and the solutions coincide with the strong solutions as long as the latter exist. Moreover, they minimize the energy (maximize the energy dissipation) among all dissipative solutions.

Leighton's Theorem states that if two finite graphs have a common universal cover then they have a common finite cover. I will present a new proof of this using groupoids, and then talk about two generalisations of the theorem that can also be tackled with this groupoid approach: one gives us control over the local structure of the common finite cover, and the other deals with graphs of spaces.