# Research seminars

This seminar series is hosted in the Centre for Mathematical and Statistical Modelling and is run by the four research groups in the Department of Mathematics. Joining instructions and additional information are in the events’ descriptions below. You are also welcome to join our Statistics, Data Science, and Learning mailing list, please contact Dr Ben Parker.

## Seminar programme 2023

## 01 Feb: Alexander Lvovsky (Oxford), "Quantum Physics: from paradox to technology"

Start at 16:00 in-person on-campus in TOWA-203.

ABSTRACT: TBA.

## 18 Jan: Hao Ni (UCL), "Path development network with finite-dimensional Lie group"

Start at 16:00 in-person on-campus in TOWA-203.

ABSTRACT: Path signature, a mathematically principled and universal feature of sequential data, boosts the performance of deep learning-based models in various sequential data tasks as a complimentary feature. However, it suffers from the curse of dimensionality when the path dimension is high. To tackle this problem, we propose a novel, trainable path development layer, which exploits representations of sequential data with the help of finite-dimensional matrix Lie groups. Our proposed layer, analogous to recurrent neural networks (RNN) but possessing an explicit, simple recurrent unit, can alleviate the gradient issues of RNNs with suitably chosen Lie groups. As a continuous time series model, our layer proves to be advantageous for irregular time series modelling. Empirical results on a range of datasets (e.g., characters, audio and images) show that the development layer consistently and significantly outperforms signature features in terms of accuracy and dimensionality. Moreover, the compact hybrid model (obtained by stacking one-layer LSTM with the development layer) achieves state-of-the-art against various RNN and continuous time series models. In addition, our layer enhances the performance of modelling dynamics constrained to Lie groups. The code is available at https://github.com/PDevNet/DevNet.git.

## Seminar programme 2022

## 9 Dec: Ivan Nourdin (Luxembourg), "Spectral central limit theorem for additive functionals of Gaussian fields"

Start at 15:00 in-person on-campus in TOWA-203.

ABSTRACT: We consider a centered, continuous, stationary, Gaussian field on the Euclidean space and a sequence of non-linear additive functionals of the field. Since the pioneering works from the 80s by Breuer, Dobrushin, Major, Rosenblatt, Taqqu and others, central and non-central limit theorems for this kind of functionals have never ceased to be refined. The common intuition is that the limit is Gaussian when we have short-memory and non-Gaussian when we have long-memory and the Hermite rank R is different from 1. Our goal in this talk is to explain why this intuition is not always true. For that, we introduce a spectral central limit theorem, which highlights a variety of situations where the limit is Gaussian in a long-memory context with R different from 1. Our main mathematical tools are the Malliavin-Stein method and Fourier analysis. The talk is based on a joint work with Leonardo Maini (University of Luxembourg).

## 30 Nov: Teemu Pennanen (King's), "Efficient allocations in double auction markets"

Start at 15:00 in-person on-campus in TOWA-203.

ABSTRACT: This talk proposes a simple descriptive model of discrete-time double auction markets for divisible assets. As in the classical models of exchange economies, we consider a ﬁnite set of agents described by their initial endowments and preferences. Instead of the classical Walrasian-type market models, however, we assume that all trades take place in a centralized double auction where the agents communicate through sealed limit orders for buying and selling. We ﬁnd that, under nonstrategic bidding, double auction clears with zero trades precisely when the agents’ current holdings are on the Pareto frontier. More interestingly, the double auctions implement Adam Smith’s “invisible hand” in the sense that, when starting from disequilibrium, repeated double auctions lead to a sequence of allocations that converges to individually rational Pareto allocations.

## 16 Nov: Louis Yudowitz (Queen Mary), "Ricci Flow, the Poincare Conjecture, and Bubbles"

Start at 16:00 in-person on-campus in TOWA-203.

ABSTRACT: Introduced by Richard Hamilton in 1982, Ricci flow has been used to solve a variety of problems in geometry and topology. Arguably the most notable of these is Perelman's proof of the Poincare conjecture in the early 2000s. I will give a general introduction to Ricci flow and overview of Perelman's proof. I will also discuss why Perelman's arguments don't work in higher dimensions and some recent joint work with Reto Buzano which aims to address some of these issues.

## 2 Nov: Mikolaj Kasprazak (Luxembourg and MIT), "How good is your Gaussian approximation of the Bayesian posterior?"

Start at 15:00 in-person on-campus in TOWA-203.

ABSTRACT: Practitioners who perform statistical inference need good point estimates and uncertainties. Bayesian inference provides those via the mean and variance of the posterior distribution. However, for most models, expectations with respect to the posterior and computationally intractable. Users are therefore forced to use approximations. The Bayesian Central Limit Theorem (BCLT) provides a way of approximating the posterior with a Gaussian distribution. This approximation is, however, only valid asymptotically, for infinitely many data points. In order to confidently use it in practice, we need finite-sample bounds on its quality. And to understand the quality of posterior mean and variance estimates, we need bounds on divergences that control those quantities. Our work provides the first closed-form, finite-sample bounds for the BCLT that do not require strong log concavity of the posterior. We bound the total variation distance, the 1-Wasserstein distance, which controlls the difference in means, and another distance that controls the difference in variances. This is joint work with Ryan Giordano (MIT) and Tamara Broderick (MIT). A prepliminary version of the paper is available here: arXiv:2209.14992.

## 19 Oct: Yani Pehova (LSE), "Extremal questions about graphs"

Start at 15:00 in-person on-campus in TOWA-203.

ABSTRACT: Extremal graph theory, in very few words, is the study of extrema of graph parameters subject to constraints. For example, a very early result in the area is Mantel's theorem which gives the maximum number of edges (a graph parameter) in a graph with no triangles (a constraint). In this talk I will outline some main questions from extremal graph theory, present a few popular open problems and theorems, and give sketches of some central methods in the field.

## 5 Oct: Elena Boguslavskaya (Brunel), "On optimal stopping problems for strong Markov processes"

Start at 15:00 in-person on-campus in TOWA-203.

ABSTRACT: We present a method to solve inverse optimal stopping problems in infinite horizon for strong Markov processes. Solving an optimal problem, especially in multiple dimensions, is a very difficult task. We attempt to solve the inverse problem. In particular, we assume the existence of a particular function which defines the optimal stopping region. More precisely the optimal stopping region is formed by all the points at which the above function is nonnegative. To solve the inverse optimal stopping problem, we use a new integral transform which can be built on any suitable random variable. It turns out that this integral transform if applied to the above function (which defines the optimal stopping region), leads us to recover the original optimal stopping problem. This method is especially effective if applied to polynomial functions. We illustrate our results with a few examples. The method we propose allows us to explore the correspondence between optimal stopping regions and the original optimal stopping problems. This method avoids complicated differential or integro-differential equations which arise if the standard methodology is used.

## 21 Sep: Ercan Sönmez (Klagenfurt), "On characterizations of discrete and continuous long-range percolation"

Start at 15:00 in-person on-campus in TOWA-203.

ABSTRACT: Random graphs are used as models for networks, in particular capturing their complexity by describing local and probabilistic rules according to which vertices are connected to each other. A quite famous model is long-range percolation in which the vertex set is either assumed to be Z^d or R^d and the probability of an edge between two arbitrary vertices asymptotically has a polynomial decay in their distances. The focus is in studying graph-theoretical properties of the aforementioned models, such as the behavior of the random walk on the graph and the shortest path between an arbitrary pair of two vertices. In particular, recent results on continuous long-range percolation are presented.

## 25 May: The MSM Centre Lauch Event (from 2 - 5:30pm)

Start at 14:00 on-campus in LECT-061.

The launch event for the Centre for Mathematical and Statistical Modelling will run from 2-5:30pm and feature talks by three inspiring speakers:

- **Man Lai Tang** (Brunel): *Stats and surveys with sensitive questions*

- **Sébastien Guenneau** (Imperial): I*nvisibility cloaks across the scales*

- **Martins Bruveris** (Onfido): *How to train a face recognition system?*

There will be time for informal discussion between the talks and a reception (from 5:30pm). This event will be a part of the Brunel Research Festival, and also welcome audiences from other departments and centres. Further details on the event arrangements will be available shortly at the registration page (TBA).

## 16 May: Georges Habib (Beirut), "Biharmonic Steklov operators on differential forms"

Start at 15:30 in-person on-campus in LECT-061 (room TBC)

with "livestreaming" via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: In this talk, we extend the Biharmonic Steklov operator to differential forms. The definition is motivated by the extension of the Serrin problem to differential forms. We study the spectral properties of this operator and show that it has a discrete spectrum consisting of eigenvalues with finites multiplicities. We then estimate its lowest eigenvalue in terms of geometric quantities and relate it to other boundary problems, Dirichlet, Neumann and Robin.

## 6 Apr: Mohammad Reza Hesamzadeh (KTH), "Generalized FTRs for hedging inter-nodal pricing risk"

Start at 15:30 via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: In liberalised wholesale power markets, risk averse market participants seek to enter into hedge contracts referenced to their local pricing node to offset the price and volume risks they face. In typical power markets, a range of inter-temporal hedge products emerge to meet these needs, such as swaps, caps, or load-following hedges. But some market participants also face risks from trading across pricing locations. To facilitate the hedging of these risks, many liberalised wholesale power markets make available an inter-nodal hedging instrument known as fixed-volume Financial Transmission Rights (FTRs). But, with the increasing penetration of intermittent and peaking generation, very few market participants trade in a fixed volume of electricity; most market participants cannot effectively hedge their inter-nodal pricing risks using fixed-volume instruments. To solve this problem, we propose a generalization of the concept of FTRs. Specifically, we propose that for each conventional hedge contract traded in the market, such as a Cap or Floor, a matching generalized Financial Transmission Right is made available to the market. We show that traders can form a portfolio of these generalized FTRs (G-FTRs) which matches the supply curve of a generator or the demand curve of a load. We show that, given access to a full range of G-FTRs, traders can offer generators, loads, and the system operator the hedge contracts they need to fully insulate themselves from all risk while taking the market risk on themselves. This approach offers the potential for a substantial improvement in the effectiveness of hedging of inter-nodal pricing risk in liberalised wholesale electricity markets.

## 23 Mar: Goran Peskir (Manchester), "Sticky Feller diffusions"

Start at 15:30 via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: We consider a Feller branching diffusion process $X$ with drift $c$ having $0$ as a slowly reflecting (sticky) boundary point with a stickiness parameter $1/\mu \in (0,\infty)$. We show that (i)

the process $X$ can be characterised as a unique weak solution to the SDE system

\begin{align} \h{8pc} \notag

&dX_t = (b X_t\! +\! c)\;\! I(X_t\! >\! 0)\, dt! +\! \sqrt{2 a

X_t}\, dB_t \\[2pt] \notag &I(X_t\! =\! 0)\, dt = \tfrac{1}{\mu}

\, d\ell_t^0(X)

\end{align}

where $b \in I\!\!R$ and $0<c<a$ are given and fixed, $B$ is a standard Brownian motion, and $\ell^0(X)$ is a diffusion local time process of $X$ at $0$, and (ii) the transition density function of $X$ can be expressed in the closed form by means of a convolution integral involving a new special function and a modified Bessel function of the second kind. The new special function embodies the stickiness of $X$ entirely and reduces to the Mittag-Leffler function when $b=0$. We determine a (sticky) boundary condition at zero that characterises the transition density function of $X$ as a unique solution to the Kolmogorov forward/backward equation of $X$.

Letting $\mu \downarrow 0$ (absorption) and $\mu \uparrow \infty$ (instantaneous reflection) the closed-form expression for the transition density function of $X$ reduces to the ones found by Feller (1951) and Molchanov (1967) respectively. The results derived for sticky Feller diffusions translate over to yield closed-form expressions for the transition density functions of (a) sticky Cox-Ingersoll-Ross processes and (b) sticky reflecting Vasicek processes that can be used to model slowly reflecting interest rates.

## 2 Mar: Julien Randon Furling (Paris 1, Panthéon Sorbonne), "Measuring heterogeneity in complex systems: the example of urban segregation"

Start at 15:30 in-person on-campus in LECT-061 (room TBC)

with "livestreaming" via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: This talk will present an approach developed recently to measure heterogeneity in complex systems. It was introduced to reveal multiscalar spatial dissimilarities in cities, from the most local scale to the metropolitan one, and urban segregation will indeed be taken here as an exemplar to explain and illustrate the new method. Think, for instance, of a statistical variable that may be measured at different scales in a city, e.g. ethnic group proportions, social housing rate, income distribution, or public transportation network density. Then, to any point in the city there corresponds a sequence of values for the variable, as one zooms out around the starting point, all the way up to the whole city – as if with a varifocal camera lens. The sequences thus produced encode spatial dissimilarities in a precise manner: how much they differ from perfectly random sequences is indeed a signature of the underlying spatial structure, and the analysis of this signature allows us to measure certain properties of the spatial structure.

References:

Segregation through the multiscalar lens, Olteanu, M., Randon-Furling, J., & Clark, W. A. V. (2019) PNAS, 116(25), 12250-12254.

From urban segregation to spatial structure detection, Randon-Furling, J., Olteanu, M., & Lucquiaud, A. (2020). EPB: Urban Analytics and City Science, 47(4), 645-661.

Measuring and Visualizing Patterns of Ethnic Concentration: The Role of Distortion Coefficients, de Bézenac, C., Clark, W. A., Olteanu, M., & Randon‐Furling, J. (2021). Geographical Analysis, 12271

## 23 Feb: Thomas Guhr (Duisburg-Essen), "Non-stationarity and generic features in complex systems"

Start at 15:30 via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: Non-stationarity, i.e. the seemingly erratic change of important properties, is a characteristic features of most complex systems. Equilibrium methods as in standard statistical mechanics do not apply, and new approaches are called for. We present two complementary approaches: first, an analysis that identifies different operational states of complex systems and, second, a random matrix model explaining the heavy tails of multivariate amplitude distributions, i.e. the emergence of certain generic features. We illustrate our findings with examples from finance.

## 9 Feb: Maggie Chen (Cardiff), "Behaviour fingerprint of the Bitcoin market"

Start at 15:30 in-person on-campus in LECT-061 (room TBC)

with "livestreaming" via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT:Although Bitcoin has grown to be the largest and best-known cryptocurrency, we still have fairly limited deep understanding about it, in particular, the behavioral fingerprint of Bitcoin traders. In this paper, we examine a complete set of Bitcoin transactions to study the features and activities of Bitcoin traders. Using K-means clustering analysis, among other approaches, we establish a typology of traders by learning their trading patterns and drawing upon trader classifications in mainstream finance such as informed vs. uninformed, retail vs. institutional, small vs large traders and similar categories. The distinguished trader types are often highly indicative of the traders’ specific strategy, behavior and impact on risk and market structure. Mining through millions of transaction records, we find that less than 1 percent of Bitcoin traders contribute to more than 95 percent of the market`s trading volume. These Bitcoin whales often demonstrate irregular trading volumes, size and timing patterns. In addition, we are able to propose five distinctive trader types and three of them present unique trading behavior patterns that could be highly indicative to the market.

## 26 Jan: Nelly Villamizar (Swansea), "The dimension of spline functions on tetrahedral vertex stars"

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: In the talk we will discuss new results and open question in connection to the study of the space of piecewise polynomial (or spline) functions defined on polyhedral cells. These cells are the union of 3-dimensional polytopes sharing a common vertex, so that the intersection of any two of the polytopes is a face of both. We will consider the problem of finding new bounds on the dimension of this spline space using commutative and homological algebra. The analysis of the smoothness properties of the spline functions on a given polyhedral complex involves the study of certain ideals generated by powers of linear forms, which in turn are related to ideals of fat points via inverse systems. The fat point scheme that comes from dualizing polyhedral cells is particularly well-suited and leads to the exact dimension in many cases of interest that will also be presented in the talk.

## Seminar programme 2021

## 15 December: Elena Boguslavskaya (Brunel), "Trading multiple mean reversion"

Start at 15:00 via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: How should one construct a portfolio from multiple mean-reverting assets? Should one add an asset to portfolio even if the asset has zero mean reversion? We consider a position management problem for an agent trading multiple mean-reverting assets. We solve an optimal control problem for an agent with power utility, and present an explicit solution for several important special cases and a semi-explicit solution for the general case. The nearly explicit nature of the solution allows us to study the effects of parameter misspecification, and derive a number of properties of the optimal solution.

## 1 December: Ayanendranath Basu (Kolkata), "Robust statistical data analysis: The minimum distance view"

Start at 15:00 via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: In many ways, estimation by an appropriate minimum distance method is one of the most natural ideas in statistics. Such procedures comprise a large class of estimation techniques which includes the method of maximum likelihood as a special case, and therefore can be viewed as an extension of the latter. These techniques can also be extended to the hypothesis testing paradigm, where they may be viewed as extensions of the likelihood ratio and other likelihood based tests. Unlike the likelihood based methods, most other minimum distance methods have come up much later in the statistical literature, but once they have caught on, research in this area has progressed at a very fast pace. aided by such considerations (e.g., robustness), where the method of maximum likelihood is deficient. In this lecture we will provide an overview of the development of the different distance based methods in statistics, and their applications.

## 24 November: Xiaochuan Yang (Brunel), "Topological thresholds, extreme values, and statistics of arithmetic functions through the lens of Stein's method"

Start at 15:30 in-person on-campus in LECT-061

with "livestreaming" via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: Chen-Stein method is a set of probabilistic tools for proving Poisson approximation with explicit error bounds. The method is flexible and well suited in situation where the underlying randomness exhibits complicated dependence structure. I survey this method through a wide range of problems I have worked on recently.

Given a large point cloud in Euclidean d-space sampled from some unknown probability distribution, we are concerned with the topological and geometric properties of the support of the unknown distribution. Typically one draws balls of same radius around each point and consider their union set parametrised by the radius. A sharp threshold is the parameter at which the likelihood of some topological property such as connectivity changes from extremely unlikely to extremely likely. Understanding asymptotic distributions of these thresholds is fundamental for statistical inference of the topology of data. I will explain how to relate two thresholds, namely the connectivity and the coverage thresholds, to extreme values of spatial networks generated by the point cloud. The distribution of extreme values is intimately related to the occurrence of certain rare events, which we approximate by Poisson distributions with the Chen-Stein method.

Another use of the method arises in probabilistic number theory. Here we choose at random a positive integer at most n. How many prime divisors are there for this random integer? It turns out that one can approximate accurately the distribution of the number of prime divisors by a Poisson distribution, then apply the classical Berry-Esseen bound between Poisson and a normal distribution with the same mean and variance, thereby quantifying a central limit theorem of Erdös and Kac. We also study the distribution of the largest prime divisor of this random integer through Poisson approximation.

This talk is based on papers with Louis Chen (NUS), Arturo Jaramillo (CIMAT) and Mathew Penrose (Bath).

## 17 November: Nick Baskerville (Bristol) and Diego Granziol (Oxford), "Appearance of random matrix theory in deep learning"

Start at 15:00 via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: We show using stochastic Lanczos quadrature methods that the global spectral statistics of deep neural network Hessians differ significantly from prior theoretical analysis which invariably results in (spiked perturbations of) the Wigner semi-circle and Marchenko-Pastur laws. However, investigating the local spectral statistics of the loss surface Hessians of artificial neural networks, we discover agreement with universal Gaussian Orthogonal Ensemble statistics across several network architectures and datasets. These results shed new light on the applicability of Random Matrix Theory to modelling neural networks and suggest a role for it in the study of loss surfaces in deep learning.

## 3 November: Lashi Bandara (Brunel), "When functional calculus, harmonic analysis, and geometry party together..."

Start at 15:00 in-person on-campus in LECT-061

with "livestreaming" via Zoom: https://bruneluniversity.zoom.us/j/95070805299

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: Functional calculus emerged in the latter half of last century as a convenient tool particularly in the analysis of partial differential equations. In the last thirty years, harmonic analysis has entered the picture to interact with functional calculus in an extraordinarily fruitful way. More recently, geometry has crashed the scene, with an abundance of interesting and important problems, which can be effectively dealt with using the tools coming from functional calculus and harmonic analysis. Moreover, there are fascinating geometric interpretations associated with the latter tools, although these investigations are still in their infancy.

The goal of this talk will be to flesh out a brief narrative of the journey of functional calculus, how it came to interact with harmonic analysis, and the party they've been recently having together with geometry. It will culminate with state-of-the-art results, but the beginnings will be humble, starting with the Fourier series! For the majority of the talk, no background will be assumed beyond Hilbert spaces, self-adjoint operators, and the spectrum of an operator.

## 20 October: Haiying Wang (Connecticut), "Nonuniform negative sampling and log odds correction with rare events data"

- Meeting ID: 950 7080 5299 (use Passcode as communicated per email)

ABSTRACT: We investigate the issue of parameter estimation with nonuniform negative sampling for imbalanced data. We first prove that, with imbalanced data, the available information about unknown parameters is only tied to the relatively small number of positive instances, which justifies the usage of negative sampling. However, if the negative instances are subsampled to the same level of the positive cases, there is information loss. To maintain more information, we derive the asymptotic distribution of a general inverse probability weighted (IPW) estimator and obtain the optimal sampling probability that minimizes its variance. To further improve the estimation efficiency over the IPW method, we propose a likelihood-based estimator by correcting log odds for the sampled data and prove that the improved estimator has the smallest asymptotic variance among a large class of estimators. It is also more robust to pilot misspecification. We validate our approach on simulated data as well as a real click-through rate dataset with more than 0.3 trillion instances, collected over a period of a month. Both theoretical and empirical results demonstrate the effectiveness of our method.

## 30 June: Naomi Arnold (QMUL) and Richard Clegg (QMUL), "Temporal network models"

Start at 13:30 via Zoom: https://bruneluniversity.zoom.us/j/94345153758

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: Two short talks on inference for time-varying networks.

1) Distinguishing mixtures of network growth models that vary in time

2) Studying an alt-right social network through the lens of temporal graphs

## 2 June: David Woods (Southampton), "Statistical learning through designed experiments"

Start at 16:00 via Zoom: https://bruneluniversity.zoom.us/j/94345153758

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: Statistically designed experiments are the “gold standard” for learning about products, processes and systems through the collection of data. By deliberately introducing controlled variability, whilst working to minimise uncontrolled variation, we can establish causal relationships, screen for important variables and build predictive models. I will describe how design of experiments and statistical modelling can go beyond the usual factorial design and response surface methodology. In particular, I will present methodology for (i) experiments with dynamic input variables; (ii) design to learn unknown parameters in empirical and first-principle nonlinear models; and (iii) Bayesian nonparametric learning through sequential experimentation. Where possible, methods will be illustrated on relevant examples.

## 19 May: Alex Bespalov (Birmingham), "Adaptive stochastic Galerkin methods for parametric PDEs"

Start at 16:00 via Zoom: https://bruneluniversity.zoom.us/j/94345153758

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: Parametric partial differential equations (PDEs) occur in optimisation problems and in mathematical models with inherent uncertainties (e.g., groundwater flow models). Adaptive algorithms are indispensable when solving a particularly challenging class of parametric problems represented by PDEs whose inputs depend on infinitely many uncertain parameters. For this class of problems, adaptive algorithms have been shown to yield approximations that are immune to the curse of dimensionality -- an exponential growth of the computational cost as the dimension of the parameter space increases. In particular, adaptivity is the key to efficient stochastic Galerkin finite element methods (SGFEMs), where approximations are represented as finite (sparse) generalised polynomial chaos expansions with spatial coefficients residing in finite element spaces. While in the simplest (so-called single-level) SGFEM all spatial coefficients reside in the same finite element space, a more flexible multilevel construction allows spatial coefficients to reside in different finite element spaces.

In this talk, we first give an overview of existing adaptive SGFEM algorithms and the associated theoretical results. Then, we focus on a recently proposed adaptive algorithm for computing multilevel stochastic Galerkin finite element approximations. We will discuss the convergence and rate optimality properties of the proposed algorithm and demonstrate its performance in numerical experiments.

## 12 May: Florian Pausinger (Belfast), "Persistent homology and its use in digital image analysis"

Start at 16:00 via Zoom: https://bruneluniversity.zoom.us/j/94345153758

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: Persistent homology is an algebraic tool for quantifying topological features of shapes and functions, which has recently found wide applications in data and shape analysis. In the first part of this talk, I aim to present the underlying algebraic ideas and basic concepts of this very active field of research. In the second part I sketch an application in digital image analysis.

## 21 April: Keming Yu (Brunel), "How the first/second year of UG Maths "rapid" help me to solve some problems in Statistics and Data Science"

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: The professional networking website LinkedIn and McKinsey Global Institute have listed skills in Machine Learning, AI, Statistics and Data Science as the top job demand. And FORTUNE (https://fortune.com/) identified that Statistics degrees lead careers in its Best and Worst Graduate Degrees for Jobs. I regularly receive queries about our MSc and PhD in SDS, from students/people who want to apply for courses here. I also recently talked to TNE students. They include "as the courses are in a Maths dept., how much Maths do I need?"; "My background is not in Maths but did few courses on it, can we do a good degree here?". My answer to them is that they need a university Maths year 1 and 2 "rapid" (regression + algebra + probability + integration + differentiation). Basic 'rapid' can apply for our SDA courses, but good 'rapid' may achieve a lot. This talk aims to demonstrate some of examples that the first/second year of UG Maths "rapid" do help me to develop new methods for modern statistical modelling and data analysis with good research impact.

## 7 April: Corina Constantinescu (Liverpool), "**Recent applications of insurance risk theory**"

Start at 16:00 via Zoom: https://bruneluniversity.zoom.us/j/98613510007

- Meeting ID: 986 1351 0007 (use Passcode as communicated per email)

ABSTRACT:In this talk we present three topics that we tackle using methods developed in the collective insurance risk literature. Specifically, we discuss insurance mechanisms for 1) mortgage lending, 2) car driving behaviour, and 3) poverty reduction.

## 24 March: Jeanne Houwing-Duistermaat (Leeds), "Probabilistic partial least squares methods for data integration"

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: Many studies collect multiple omics datasets to gather novel insights about different stages of biological processes. For joint modelling of these datasets, several data integration methods have been developed. These methods address high dimensionality of the datasets, within and across datasets correlation, and the presence of heterogeneity among datasets due to measuring different biological levels and using different technologies to measure them. Most methods, however, neither provide statistical evidence for a relationship between the datasets nor identify relevant variables that contribute to this relationship. We propose a probabilistic latent variable modelling framework for inferring the relationship between two omics datasets. Latent variable methods reduce dimensionality and capture correlations by forming components that are linear combinations of the variables. The correlation structure is modelled by joint and data specific components. We propose maximum likelihood estimation of the parameters and formulate a test statistic for the null hypothesis of no relationship between the datasets. We evaluate our methods via a simulation. Under the null hypothesis, the test statistic appears to approximately follow the normal distribution. Our method outperforms existing methods for small and heterogeneous datasets in terms of selecting relevant variables and prediction accuracy. We illustrate the methods by analysing omics datasets from a population cohort and a case control study. The obtained results are reproducible and biologically interpretable.

## 17 March: Rui Paulo (Lisbon), "Variable Selection in the presence of factors: a model selection perspective"

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: In the context of a Gaussian multiple regression model, we address the problem of variable selection when in the list of potential predictors there are factors, i.e., categorical variables. We adopt a model selection perspective, that is, we approach the problem by constructing a class of models, each corresponding to a particular selection of active variables. The methodology is Bayesian and proceeds by computing the posterior probability of each of these models. We highlight the fact that the set of competing models depends on the dummy variable representation of the factors, an issue already documented by Fernandez et al. (2002) in a particular example but that has not received any attention since then. We construct methodology that circumvents this problem and that presents very competitive frequentist behavior when compared with recently proposed techniques. Additionally, it is fully automatic, in that it does not require the specification of any tunning parameters. [Joint work with Gonzalo Garcia-Donato.]

## 24 February: Arianna Salili-James (Brunel), "Taking shape: Analysis and classification of object outlines using shape analysis "

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: There is no doubt that humans and other animals can not only recognise shapes but also differentiate shape from form. The latter is what aids us in classifying objects by eye, which is usually done in conjunction with more detailed descriptions of the object, such as its texture or patterns within. Subsequently, this leads to two questions which carve the basis of our work: (i) can an algorithm classify shapes in the same way a human does, and (ii) is the shape of an object alone sufficient for performing analysis and classification? To answer these questions, we will investigate and compare various non-linear and linear methods that quantify differences between shapes on a series of real-world datasets before moving to machine learning to perform classification. All the while, we will be working hand in hand with specialists in these fields, ranging from botanists to pottery historians. We are interested in analysing the shape of an object; crucially this means that we will not be studying landmarks as is often done with real-world data. Instead, we define the shapes as closed curves in R2 and work with the shape space of these curves using methods that are seldom applied to real-world data. This work is done in collaboration with Stephen Marsland (Victoria University of Wellington) and Armand Leroi (Imperial College London).

## 10 February: Ricardo Silva (UCL), "A new class of algorithms for general instrumental variable models"

- Meeting ID: 943 4515 3758 (use Passcode as communicated per email)

ABSTRACT: In causal inference for observational studies, we lack a randomised control for a treatment of interest. However, we sometimes have access to an instrumental variable, which is a type of (pseudo) randomisation that doesn’t fully control the treatment. In general, the causal effect still remains unidentifiable, but it may be possible to bound it. Until recently, no methods existed for the case where the treatment is a continuous variable. We will present an algorithm that facilitates this type of inference, while allowing for a continuum of causal assumptions that trade-off constraints on the unmeasured confounding structure against the informativeness of the bounds. Joint work with Niki Kilbertus (Helmholtz AI) and Matt Kusner (UCL).

## Seminar programme 2020

## 29 July: Kangrui Kwang (Turing Institute), "Nonstationary Nonseparable Random Fields"

Start at 15:30 via Zoom (joining instructions follow below)

ABSTRACT: Kernel based methods are widely used in both machine learning and Bayesian statistics. Standard kernels, like Gaussian and Matern kernels, are both stationary and separable. We introduce a flexible method to create nonstationary and non-separable kernels based on an infinite mixture of convolved stochastic processes. When the mixing process is stationary but the convolution function is nonstationary we arrive at non-separable kernels with constant non-separability that are available in closed form. When the mixing is nonstationary and the convolution function is stationary we arrive at non-separable random fields that have varying non-separability and better preserve local structure. We show how a single Gaussian process (GP) with these random fields can computationally and statistically outperform both separable and existing nonstationary non-separable approaches such as treed GPs and deep GP constructions.

Zoom link: https://bruneluniversity.zoom.us/j/99894030917- Meeting ID: 998 9403 0917

- Passcode: 2108619563

## 8 July: Arseni Goussev (Portsmouth), "Diffraction-in-time problem and quantum backflow"

Start at 15:30 via Zoom (joining instructions follow below).

ABSTRACT: In quantum mechanics, the direction of the probability density flow is not necessarily the same as that of velocity. One manifestation of this counter-intuitive fact is the so-called quantum backflow effect, in which the probability of finding a particle "on the left" increases with time despite the particle's velocity pointing "to the right." In my talk, I will provide a brief review of the quantum backflow problem, and will show how it can be generalized to quantum states with position-momentum correlations.

References: A. Goussev, arXiv:2002.03364 (2020); A. Goussev, PRA 99, 043626 (2019)

Zoom link: https://bruneluniversity.zoom.us/j/94925713575

- Meeting ID: 949 2571 3575

- Password: 8988773309

## 26 June: Shash Virmani (Brunel), "Simulating quantum systems classically"

Start at 15:30 via Zoom (joining instructions follow below).

ABSTRACT: In quantum mechanics, the direction of the probability density flow is not necessarily the same as that of velocity. One manifestation of this counter-intuitive fact is the so-called quantum backflow effect, in which the probability of finding a particle "on the left" increases with time despite the particle's velocity pointing "to the right." In my talk, I will provide a brief review of the quantum backflow problem, and will show how it can be generalized to quantum states with position-momentum correlations.

References: A. Goussev, arXiv:2002.03364 (2020); A. Goussev, PRA 99, 043626 (2019)

Zoom link: https://bruneluniversity.zoom.us/j/94925713575

- Meeting ID: 949 2571 3575

- Password: 8988773309

## 12 June: Jia Wei Lim (Brunel), “A Two-Phase Dynamic Contagion Model for COVID-19"

Start at 15:30 via Zoom (joining instructions follow below).

ABSCTRACT: In this talk, I will propose a continuous-time stochastic intensity model, namely the two-phase dynamic contagion process, for modelling the epidemic contagion of COVID-19. The dynamic contagion process is a branching process which is an extension of the classical Hawkes process commonly used for modelling earthquakes and more recently financial contagion. In contrast to most existing models, this model allows randomness to the infectivity of individuals rather than a constant reproductive rate as assumed by standard models. Key epidemiological quantities such as the distribution of final epidemic size and expected epidemic duration have been derived and estimated based on real data for various regions and countries. The time that governmental intervention became effective is estimated and the results are consistent with the actual time of intervention and incubation time of the disease. The purpose of this talk will be to introduce the model and to show that this model could potentially be a valuable tool in the modelling of COVID-19 infection. Similar variations of this model could also be used in more general epidemiological modelling.

Join Zoom Meeting: https://bruneluniversity.zoom.us/j/98209870169

- Meeting ID: 982 0987 0169

- Password: 6659222909

## 27 May: Stephen Langdon (Brunel), "Hybrid numerical-asymptotic methods for high frequency wave scattering"

Start at 15:30 via Zoom (joining instructions follow below).

ABSTRACT: Linear wave scattering problems (e.g. for acoustic, electromagnetic and elastic waves) are ubiquitous in science and engineering applications. However, conventional numerical methods for such problems (e.g. FEM or BEM with piecewise polynomial basis functions) are prohibitively expensive when the wavelength of the scattered wave is small compared to typical lengthscales of the scatterer (the so-called "high frequency" regime). This is because the solution possesses rapid oscillations which are expensive to capture using conventional approximation spaces. In this talk we outline recent progress in the development of "hybrid numerical-asymptotic" methods. These methods use approximation spaces containing oscillatory basis functions, carefully chosen to capture the high frequency asymptotic behaviour, leading to a significant reduction in computational cost.

Join Zoom Meeting: https://bruneluniversity.zoom.us/j/92367721405

- Meeting ID: 923 6772 1405

- Password: 7129906480

## 13 May: Paresh Date (Brunel), "Forecasting crude oil futures prices using the Kalman filter and macroeconomic news sentiment"

Start at 15:30 via Zoom (joining instructions follow below).

ABSTRACT: This presentation is split into three parts. The first part is a brief tutorial on how a commodity futures market operates, and the role played by stochastics in pricing of financial derivatives, including commodity futures. In the second part, I will introduce a recursive conditional mean estimator for linear Gaussian dynamic systems, popularly called the Kalman filter (KF), and its application in modelling the movement of crude oil futures prices. Finally, I will describe how a KF-based model can be modified using quantitative measures of macroeconomic news sentiment to enhance the futures price prediction. I will also present a summary of our findings from numerical experiments on modelling and predicting crude oil futures prices using the Kalman filter and news sentiment scores.

Join Zoom Meeting at https://bruneluniversity.zoom.us/j/98340190032

- Meeting ID: 983 4019 0032

- Password: 0154229543

## 29 April: Dalia Chakrabarty (Brunel), “Learning of High-dimensional Functions, given Discontinuous Data - Shallowness Suffices"

Start at 16:00 via Teams.

ABSCTRACT: I will be discussing a new methodology that permits the Bayesian supervised learning of generic tensor-valued functional relationships between two random variables, at least one of which is high-dimensional and distributed discontinuously. The ultimate aim is the prediction of either variable, at test data on the other. We will model the sought function as a random realisation from a tensor-variate, non-stationary stochastic process, each parameter of the covariance structure of which, is made to adapt to the discontinuity of the sought function. This is proven to be permitted by compounding multiple scalar-valued stationary processes with the mother tensor-valued non-stationary process. That 2 layers suffice in this learning strategy, is proven in the method. The talk will be motivation-heavy.

## 21 February, Martin Richter (Nottingham), "A microwave realization of the chiral orthogonal, unitary, and symplectic ensembles"

Start at 13:00 in TOWA-203

ABSTRACT: Random matrix theory has proven very successful in the understanding of the spectra of chaotic systems. Depending on symmetry with respect to time reversal and the presence or absence of a spin 1/2 there are three ensembles, the Gaussian orthogonal (GOE), Gaussian unitary (GUE), and Gaussian symplectic (GSE) one. With a further particle-antiparticle symmetry the chiral variants of these ensembles, the chiral orthogonal, unitary, and symplectic ensembles (the BDI, AIII, and CII in Cartan's notation) appear. A microwave study of the chiral ensembles is presented using a linear chain of evanescently coupled dielectric cylindrical resonators [1]. In all cases the predicted repulsion behavior between positive and negative eigenvalues for energies close to zero could be verified.

[1] https://arxiv.org/abs/1909.12886

## 7 February, Ben Parker (Brunel), "Optimal Design of Experiments on Networks"

Start at 11:00 in TOWA-203

ABSTRACT: Experimental design is the area of statistics which attempts to gain the most information out of an experiment or study, with minimal experimental effort. This talk provides a general introduction to a particular problem in this field for a general mathematical audience. We address the problem of how to optimally design experiments for subjects who are connected by a network structure. We also argue that there is a wide class of experiments that can be reformulated into a problem of design on a network. By regarding experimental design as a problem in network science, we can improve experimental design algorithms for large networks, and also find designs even when there is no obvious network relationship.

### Selected past seminars

- 22 Nov 2019: Yuri ANTIPOV (Louisiana State University, USA), “Diffraction by wedges: higher order boundary conditions, integral transforms, vector Riemann-Hilbert problems, and Riemann surfaces
- 22 Feb 2019: Ritabrata DUTTA (Warwick), "Bayesian inference of spreading processes on networks"
- 23 Nov 2018: Alfred KUME (Kent), "Matrix distributions and statistical shape analysis"
- 9 Nov 2018: Jens BOLTE (Royal Holloway), "Weyl quantisation of Hamiltonian flows on toroidal phase spaces"
- 12 Oct 2018: Tiago PEIXOTO (Bath), "Reconstructing networks with heterogeneous and unknown errors"
- 9 Feb 2018: Karthik BHARATH (Nottingham), "Distribution and sampling of warp maps for curve alignment"
- 8 Dec 2017: Stephen MARSLAND (Massey University, New Zealand), "Automated birdsong recognition"
- 9 May 2017: Sonja HOHLOCH (Antwerp, Belgium), "New aspects of homoclinic Floer homology"
- 14 March 2017: Yuliya MISHURA (Kyiv, Ukraine), "Option pricing with fractional stochastic volatility and discontinuous payoff function of polynomial growth"
- 18 Oct 2016: Cécile MAILLER (Bath), "Non-extensive condensation in reinforced branching processes"
- 5 May 2016: Alexander SHEN (Montepellier, France), "Randomness and complexity"
- 9 Feb 2016: Martin BAUER (Vienna, Austria), "The Fisher-Rao metric on the space of smooth densities"