**Department Colloquia**

18th October: Dr Colin Torney (University of Glasgow)Title: ** Cues and decision-making in collective systems**

Date: 3.30pm Wednesday 18th October

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Animal groups in nature are a classic example of a complex system in which individual behavior and social interaction scale to produce a collective response to external stimuli. In these systems there is an interplay between leadership, imitation, and environmental cues that determines the accuracy of group decisions. In this talk I will present some stylized models of information flow in interacting systems and show how evolution may drive these systems to unresponsive states. I will also discuss the methods we're using to investigate these questions in the field and lab, including tools to collect video footage, computational methods to locate animals within images, and statistical techniques to infer behavioral rules from movement data.

Title: Multiscale modelling of Lithium batteries

Date: 3.30pm Wednesday 1st November

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: The development of theoretical methods to correlate the chemical and structural properties of materials in energy storage devices is of crucial importance for a coherent interpretation of the experimental data and for their optimization. I will present how multiscale mathematical models, which combine microstructures, reaction kinetics and mass transport, can predict battery performances.

Title: To approximate or not to approximate, that is the question

Date: 3.30pm Wednesday 15th November

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: I will consider practical approximation in high dimensions and ask when we should approximate. I will give a quick overview of ideas in neural networks related to concentration of measure which are being developed by Gorban and Tyukin in Leicester. I will then talk about sparse grid approximation using smooth kernels, with some theoretical results related to interpolation and quasi-interpolation with Gaussians. As a byproduct of this work a new set of polynomials related to Hermite polynomials have been invented. This work is in collaboration with Xingping Sun, Alex Kushpel and more recently Simon Hubbert. I will make reference to applications of the sparse grid technology to solution of PDEs in 4 dimensions, with the question - Is this high?

Title: Understanding the Complex Dynamics of Faraday Pilot Waves

Date: 3.30pm Wednesday 22nd November

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Faraday pilot waves are a newly discovered hydrodynamic structure that consists a bouncing droplet which creates, and is propelled by, a Faraday wave. These pilot waves can behave in extremely complex ways and result in dynamics mimicking quantum mechanics. I will show some of this fascinating behaviour and will present a surface wave-droplet fluid model that captures many of the features observed observed in experiments, focussing on the statistical emergence of complex states.

Title: Diffraction of hydroelastic waves by a vertical cylinder

Date: 3.30pm Wednesday 29th November

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Linear problem of wave diffraction is studied for a circular vertical cylinder mounted at the sea bed and piercing the fluid surface covered by ice plate of infinite extent. The ice plate is modeled by a thin elastic plate of constant thickness clamped to the surface of the cylinder. One-dimensional incident hydroelastic wave of small amplitude propagates towards the cylinder and is diffracted on the cylinder. Deflection of the ice plate and the bending stresses in it are determined by two methods: (a) using the integral Weber transform in radial direction, (b) using the vertical modes for the fluid of constant depth with the rigid bottom and elastic upper boundary. The solution by the second method is straightforward but we cannot prove that the solution is complete because the properties of the vertical modes are not known yet. The solution by the Weber transform is more complicated but this solution is unique. In this talk we will show that these two solutions are identical. This result justifies the method of the vertical modes in the hydroelastic wave diffraction problems.

Title: Electro-Magneto Statics by a Functional Analysis Toolbox

Date: 3.30pm Wednesday 17th January

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: We will give a simple introduction to Maxwell equations. Concentrating on the static case, we will present a proper L^2-based solution theory for bounded weak Lipschitz domains in three dimensions. The main ingredients are a functional analysis toolbox and a sound investigation of the underlying operators gradient, rotation, and divergence. This FA-toolbox is useful for all kinds of partial differential equations as well..

Title: TBA

Date: 3.30pm Wednesday 28th February

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: TBA

**Applied Analysis**

17th October: Prof Ernesto Estrada (Department of Mathematics and Statistics)Title: Communicability geometry and transport in networks

Date: 3pm Tuesday 17th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: I will show how a geometry emerges from the communicability function of a network (graph). Then, I will study some examples in which "information" is claimed to flow through the shortest path but for which we show that it seems to flow through the shortest communicability path. Such communicability paths are considered as the shortest paths in a communicability distance-weighted graph. The examples we will discuss include flow of water in brain networks and the flow of cars in rush hour in different world cities. In both cases I will present theoretical and empirical results based on real-world situations.

Title: Characterising Submonolayer Deposition via the Visibility Graph

Date: 3pm Tuesday 24th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Submonolayer deposition (SD) is a term used to describe the initial stages of processes, such as molecular beam epitaxy, in which particles are deposited onto a surface, diffuse and form large-scale structures. We discuss a mean-field model of the process under the assumption of fixed rate deposition by investigating the effects of variations in the critical island size on a (SD) model using the visibility graph. Using methods from network theory and spectral graph theory, we derive results that combine the information contained in the island size distributions and spacial distributions.

Title: The Discrete Coagulation-Fragmentation System

Date: 3pm Tuesday 21st November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: In many situations in nature and industrial processes clusters of particles can combine into larger clusters or fragment into smaller clusters. The evolution of these particles can be described by differential equations known as coagulation-fragmentation equations. In the discrete size case it is assumed that the mass of each cluster is a natural number and a cluster of mass n consists of n identical units. The main part of the talk will concentrate on the case of pure discrete fragmentation. Here, the theory of substochastic C_0-semigroups can be used to obtain results relating to the existence of a unique, positive, mass conserving solution. The full coagulation-fragmentation system, where the coagulation coefficients may be time-dependent, will also be briefly examined.

Title: A distributional approach to point interactions in one dimensional quantum mechanics

Date: 3pm Tuesday 28th November

Venue: Livingstone Tower, 4th floor, room LT412

Abstract: Physicists often use regularization and renormalization procedures to deal with singular potentials. Though this approach is intuitive, it generally lacks mathematical consistency, leading sometimes to ambiguous results. Although these procedures are common in quantum field theory, they also arise in quantum mechanics. Typical examples are the singular point interactions associated with a Dirac delta potential or its derivative in one dimension. When the potential is regular, the interaction term in the Schr\"odinger (or Dirac) equation is usually given by the product between the potential function and the wave function. However, when the potential is singular, this product sometimes is not well defined, and the interaction term may not make sense. Mathematically this problem can be solved by using the theory of self-adjoint extensions of symmetric operators (SAE), from which one finds a well defined self-adjoint hamiltonian. In one dimension, the self-adjoint extensions of the hamiltonian for a point interaction are members of a 4-parameter family, and are completely characterized by the boundary conditions the wave function satisfies at the singular point. One disadvantage of this approach, from a physicist's point of view, is that the self-adjoint hamiltonian is not given as a sum of two well defined operators, corresponding to the kinectic and the potential energies; the hamiltonian is given ``as a whole", and one lacks intuition about the specific properties of the ``potential". In this seminar I will present a formal approach to this problem based on the theory of distributions. In this approach the ill-defined product forming the interaction term in the Schr\"odinger equation is replaced by a well defined distribution concentrated in a single point. By imposing on this distribution some simple mathematical requirements, besides the probability conservation across the singular point, one finds that the allowable interaction terms are described by a family of 4-parameters, which are related to the boundary conditions at the singular point in exactly the same way as we find by the theory of SAE. I intend to discuss the relationships between the theory of SAE and this distributional approach, as well as to discuss some possibilities to formulate the latter (still formal) in a mathematical rigorous way.

**Continuum Mechanics and Industrial Mathematics**

26th September: Dr Alex Wray (University of Strathclyde)Title: The evaporative behaviour of asymmetric drops

Date: 1.00pm Tuesday 26th September

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

The evaporation of liquid drops has received extensive attention over time due to its fundamental significance in a variety of industrial contexts, not to mention the widespread consideration given to the so-called `coffee-stain effect’. Of particular interest are drops that are in some way asymmetric: it is known that the flow inside such drops is itself asymmetric as a result of non-uniformities in the evaporative flux, but the exact mechanism was not previously understood. Unfortunately the system is not amenable to the standard method described in the seminal 1997 paper of Deegan et al., but I discuss how the system may nonetheless be modelled. The finer details, especially in situations where the drop is non-slender, prove to be rather challenging, and much remains as yet unknown. I discuss what progress has been made so far, and discuss promising avenues.

Title: Phase change at the nanoscale

Date: 4.00pm Thursday 28th September

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

Nanotechnology has been a very important research topic due to the wide range of applications it has to offer

in multiple fields such as industry or medicine. Many of these applications involve high temperatures which can

even lead to a phase change and therefore it is crucial to understand how these processes occur at small length

scales.

It is widely known that heat transport at the nanoscale cannot be described in the same manner as for

macroscopic objects. There exists a large number of experimental observations which show that many thermodynamic

properties, such as the melt temperature or the thermal conductivity, become highly size-dependent at

the nanoscale and thus developing mathematical models which are able to describe this dependence accurately

is very important. In addition, most of the mathematical models describing heat transfer processes are based

on Fourier’s law, which states that the heat flux is proportional to the temperature gradient. However, it has

been shown that the classical equations break down at the nanoscale and thus other approaches are necessary

to describe heat conduction at small length or short time scales correctly. The Guyer-Krumhansl equation is a

very popular extension to the classical Fourier law that incorporates memory and non-localities, which become

significant at the nanoscale.

In this talk we will discuss the mathematical modelling of phase change and how nanoscale effects have been

incorporated into the mathematical description. We will show that the widely accepted equations are incorrect

and we will provide a new system. A mathematical model for the size-dependent melt temperature will also be

presented and we will show that there is an excellent agreement with experimental observations. In the end we

will discuss how the Guyer-Krumhansl equation affects a solidification process in a simple geometry.

Title: Analysis of a Fractal Ultrasonic Transducer

Date: 1.00pm Tuesday 10th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

Ultrasonic transducers are an essential tool in medical imaging, in imaging cracks in nuclear plants, and in a wide range of inverse problems.This talk will provide some theorems which can be used to predict the dynamics of a fractal ultrasound transducer whose piezoelectric components span a range of length scales. As far as we know this is the first to study waves in the complement to the Sierpinski gasket. This is an important mathematical development as the complement is formed from a broad distribution of length scales whereas the Sierpinski gasket is formed from triangles of equal size. A finite element method is used to discretise the model and a renormalisation approach is then used to develop a recursion scheme that analytically describes the key components from the discrete matrices that arise. It transpires that the fractal device has a significantly higher reception sensitivity and a significantly wider bandwidth than an equivalent Euclidean (standard) device. So much so that our engineering colleagues have built the world’s first fractal ultrasonic transducer which I will try and bring along !

Title: On the dependence of solutions of pdes on the coefficients

Date: 1.00pm Tuesday 24th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: In the setting of so-called evolutionary equations invented by Rainer Picard in 2009 we study a certain type of a continuity property of solution operators. We will describe homogenisation theory in the framework of this continuity property. In fact, it can be shown that $G$-convergence of matrix-coefficients is equivalent to convergence of certain inverses in the weak operator topology. With this, one can show various homogenisation results for a wide class of standard linear equations in mathematical physics. Furthermore, the genericity of memory effects to arise due to the homogenisation process in the context Maxwell's equations can be explained by operator-theoretic means.

Title: Marchenko Methods for Seismics: Improving images without a detailed model

Date: 1.00pm Tuesday 7th November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Seismic methods which rely on emitting, recording and processing seismic waves, are widely used to locate subsurface resources and monitor known reservoirs. They are part of any hydrocarbon exploration or geological carbon capture and storage project. One of the most powerful tools used in seismics is **migration**, a method of imaging which provides high-resolution details of the subsurface. First-order Born methods which have been traditionally used for most migration algorithms fail to accurately map subsurface interfaces, and create a number of artifacts, the most pernicious of which are "phantom" reflectors. These "phantom" reflectors are coherent forms of noise which are caused by the presence of higher-order scattering in the data (multiples). Recently, **Marchenko methods** have been developed which, among other uses, can provide images almost devoid of any multiple-related artifacts. This is possible because, even without a detailed model of the subsurface, Marchenko methods can obtain estimates of these multiples, something conventional methods lack. This talk will introduce Marchenko methods, contextualized from a geophysical and mathematical point of view, and show some of its recent applications which have been developed at the University of Edinburgh.

Title: The Jellycopter: Stable Levitation using a Magnetic Stirrer

Date: 1.00pm Tuesday 21st November 2017

Venue: Livingstone Tower, 9th floor, room LT907

Title: In laboratories around the world, scientists use magnetic stirrers to mix solutions and dissolve powders. It is well known that at high drive rates the stir bar jumps around erratically with poor mixing, leading to its nick-name 'flea'. Investigating this behaviour, we discovered a state in which the flea levitates stably above the base of the vessel, supported by magnetic repulsion between flea and drive magnet. The vertical motion is oscillatory and the angular motion a superposition of rotation and oscillation. By solving the coupled vertical and angular equations of motion, we characterised the flea’s behaviour in terms of two dimensionless quantities: (i) the normalized drive speed and (ii) the ratio of magnetic to viscous forces. However, Earnshaw’s theorem states that levitation via any arrangement of static magnets is only possible with additional stabilising forces. In our system, we find that these forces arise from the flea’s oscillations which pump fluid radially outwards, and are only present for a narrow range of Reynold's numbers. At slower, creeping flow speeds, only viscous forces are present, whereas at higher speeds, the flow reverses direction and the flea is no longer stable. We also use both the levitating and non-levitating states to measure rheological properties of the system.

Title: **Watching Sessile Droplets Evaporate: Beautiful (and never boring) phenomena!**

Date: 1.00pm Tuesday 28th November 2017

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The evaporation of a liquid drop on a solid substrate is a remarkably common phenomenon. Yet, the complexity of the underlying mechanisms has constrained previous studies to spherically-symmetric configurations. We recently demonstrated [1] detailed evolution of thermocapillary instabilities during evaporation of hemispherical and non-hemispherical sessile droplets and iii) non-hemispherical sessile droplets. Rigorous DNS (using our in house TPLS2 solver [2]) showed for the first time, breakage of symmetry and the consequent development of a preferential direction for thermocapillary convection. This results in counter-rotating whirling currents in the drop playing a critical role in regulating the interface thermal and fluid dynamics.

We will also present our recent-most investigations of well-defined, non-spherical evaporating drops of pure liquids and binary mixtures. We recently deduced a new universal scaling law for the evaporation rate valid for any shape and demonstrated that more curved regions lead to preferential localized depositions in particle-laden drops [3]. Furthermore, geometry induces well-defined flow structures within the drop that change according to the driving mechanism and spatially-dependent thresholds for thermocapillary instabilities. In the case of binary mixtures, geometry dictates the spatial segregation of the more volatile component as it is depleted. In the light of our results, we believe that the drop geometry can be exploited to facilitate precise local control over the particle deposition and evaporative dynamics of pure drops and the mixing characteristics of multicomponent drops.

Title: Fracture phenomena in foams: upscaling to PDE models

Date: 1.00pm Tuesday 5th December 2017

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Injection of a gas into a gas/liquid foam is known to give rise to instability phenomena on a variety of time and length scales. Macroscopically, one observes a propagating gas-filled structure that can display properties of liquid finger propagation as well as of fracture in solids. Using a discrete network model, which incorporates the underlying film instability as well as viscous resistance from the moving liquid structures, we describe both large-scale ductile finger-like cracks and brittle cleavage phenomena in line with experimental observations. Based on this discrete model, we then derive a continuum limit PDE description of both the ductile and brittle modes and draw analogy with Saffman--Taylor fingering in non-Newtonian continuum fluids and molecular dynamics simulations of fracture in crystalline atomic solids.

Title: More with less for seismic imaging

Date: 1.00pm Tuesday 23rd January 2018

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: In seismic exploration, a medium is excited and the medium response is measured at the receivers. The medium properties and measurements are related by the wave equation. Given the medium, computation of the measurements is referred to as the forward problem. Consequently, the inverse problem is estimation of medium properties from the given measurements. Advances in the microprocessor, computer memory and storage technologies, miniaturization and improved accuracy of sensors combined with operational advancements enabled exponential growth of measurement channels in seismic surveys since 1970s. With the current systems we easily collect 10-20Tb/day which leads to Petabytes or more data per survey. The challenge is to design acquisition systems with reduced number of sensors and measurements providing comparable data information or inversion quality with existing acquisition systems. We formulate this sampling problem in the form of an inverse problem. This talk discusses two ways we formulated the problem and the necessary ingredients in the formulation. An efficient way to address this problem is still under question and will be open to discussion.

**Numerical Analysis and Scientific Computing**

10th October: Dr Prashanth Nadukandi (University of Manchester)**Numerical Analysis and Scientific Computing**

Title: Stable computation of the trigonometric matrix functions: cos(sqrt(A)) and sinc(sqrt(A))

Date: 4.00pm Tuesday 10th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

Title: Perron-Frobenius theorem for multi-homogeneous maps and some applications

Date: 4.00pm Tuesday 24th October

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The nonlinear Perron-Frobenius theory addresses existence, uniqueness and maximality of positive eigenpairs for order-preserving homogeneous functions. This is an important and relatively recent generalization of the famous results for nonnegative matrices. In this talk I present a further generalization of this theory to "multi-dimensional" order-preserving and homogeneous maps, which we briefly call multi-homogeneous maps. The results presented are then used to discuss some nonlinear matrix and tensor eigenvalue problems and some of their applications.

Title: An introduction to multitrace formulations and associated domain decomposition solvers

Date: 4.00pm Tuesday 7th November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Multitrace formulations (MTFs) are based on a decomposition of the problem domain into subdomains, and thus domain decomposition solvers are of interest. The fully rigorous mathematical MTF can however be daunting for the non-specialist. We introduce in this work MTFs on a simple model problem using concepts familiar to researchers in domain decomposition. This allows us to get a new understanding of MTFs and a natural block Jacobi iteration, for which we determine optimal relaxation parameters. We then show how iterative multitrace formulation solvers are related to a well known domain decomposition method called optimal Schwarz method: a method which used Dirichlet to Neumann maps in the transmission condition. We finally show that the insight gained from the simple model problem leads to remarkable identities for Calder ́on projectors and related operators, and the convergence results and optimal choice of the relaxation parameter we obtained is independent of the geometry, the space dimension of the problem, and the precise form of the spatial elliptic operator, like for optimal Schwarz methods. We illustrate our analysis with numerical experiments. This is a joint work with X. Claeys and M.J. Gander

Title: Direct Solution of Sparse Linear Equations on Parallel Computers

Date: 4.00pm Tuesday 14th November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: As part of the H2020 FET-HPC Project NLAFET (http://www.nlafet.eu/), we are studying the scalability of algorithms and software for using direct methods for solving large sparse equations. In this talk we briefly discuss the structure of NLAFET and the scope of the Project. We then focus on algorithmic approaches for solving sparse systems: positive definite, symmetric indefinite, and unsymmetric. An important aspect of most of our algorithms is that although we are solving sparse equations most of the kernels are for dense linear algebra. We show why this is the case with a simple example before illustrating the various levels of parallelism available in the sparse case. The work described in this talk has been conducted by the STFC NLAFET Team who comprise: Florent Lopez, Stojce Nakov, and Philippe Gambron.

Title: The p-Laplacian on a segment. Spectral and time evolution problem

Date: 4.00pm Tuesday 21st November

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The non-linear spectral and time evolution problems associated to the p-Laplacian have attracted significant attention in recent years. In this talk we will examine various analytical properties of these two problems, when posed on a segment of finite length and subject to homogeneous Dirichlet boundary conditions at the end points. An explicit expression for the eigenfunctions can be found in terms of special functions. These eigenfunctions are naturally called p-sine functions, a terminology introduced by Elbert, Otani and others in the 1980s. The p-sine functions play a fundamental role in the theory of Sobolev embeddings, yet many questions about them remain open. During the talk we will discuss partial answers and challenges associated to some of these open questions. We only known, for example, that the p-sine functions form a Riesz basis of the Hilbert space L^2(0,1) for all p larger than or equal to a threshold p_1, where p_1 is the solution of a transcendental equation and is approximately equal to 1.043817. The confirmation of this threshold relies on the Beurling representation of the change of coordinate operator in terms of Dirichlet series and the answer to the basis question remains completely open for 1

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Date: 4.00pm Tuesday 30 January 2018

Venue: Livingstone Tower, 9th floor, room LT907

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Date: 4.00pm Tuesday 20 February 2018

Venue: Livingstone Tower, 9th floor, room LT907

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Date: 4.00pm Tuesday 27 February 2018

Venue: Livingstone Tower, 9th floor, room LT907

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Date: 4.00pm Tuesday 6th March 2018

Venue: Livingstone Tower, 9th floor, room LT907

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Date: 4.00pm Tuesday 27th March 2018

Venue: Livingstone Tower, 9th floor, room LT907

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**Population Modelling and Epidemiology**

30th March: Dr Robert Wilson (Mathematics and Statistics, University of Strathclyde)Title: Zooplankton Diapause in a Warmer World: Modelling the Impact of 21st Century Climate Change on Calanus Finmarchicus

Date: 1pm Wednesday 30th March 2016

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** To avoid starving in winter, many zooplankton species spend over six months dormant in deep waters. The time animals can remain dormant will likely be reduced by global warming. We therefore modelled changes in potential dormancy duration in the key species Calanus finmarchicus under 21st century climate change. Climate change impacts varied markedly. Western Atlantic populations see large reductions in potential dormancy duration, but the Norwegian Sea experiences only marginal change. The reductions in the Western Atlantic will likely cause important changes to the populations of C. finmarchicus and species that prey on it.

Title: TBA

Date: 1pm Wednesday 6th April 2016

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: TBA

Title: A General Methodological Framework for Identifying Disease Risk Spatial Clusters Based Upon Mixtures of Temporal Trends

Date: 1pm Wednesday 26th October 2016

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** We present a novel general Bayesian hierarchical mixture model for clustering areas based on their temporal trends. Our approach is general in that it allows the user to choose the shape of the temporal trends to include in the model, and examples include linear, general monotonic, and changepoint trends. Inference from the model is based on Metropolis coupled Markov chain Monte Carlo (MC)^3 techniques in order to prevent issues pertaining to multimodality often associated with mixture models. The effectiveness of (MC)^3 is demonstrated in a simulation study, before applying the model to hospital admission rates due to respiratory disease in the city of Glasgow between 2002 and 2011. Software for implementing this model will be made freely available as part of the R package CARBayesST.

**Title:** The Risk of Dengue for Non-Immune Foreign Visitors to the 2016 Summer Olympic Games in Rio de Janeiro, Brazil

**Date:** 1.00pm, Wednesday 25th May 2016

**Venue:** Livingstone Tower, 9th Floor, LT9.07

**Abstract:** Dengue is a viral infection caused by 4 dengue serotypes transmitted by mosquitoes that is an increasing problem in Brazil and other countries in the tropics and subtropics. As Brazil is the country with the highest number of dengue cases worldwide. Rio de Janeiro, the venue for the 2016 Olympic Games, has been of major importance for the epidemiology of dengue in Brazil. After the DENV 1–4 introductions in 1986, 1990, 2000 and 2011, respectively, the city has suffered explosive outbreaks. Properly quantifying the risk of dengue for foreign visitors to the Olympics is important. A mathematical model to calculate the risk of developing dengue for foreign tourists attending the Olympic Games in Rio de Janeiro in 2016 is proposed. A system of differential equation models the spread of dengue amongst the resident population and a stochastic approximation is used to assess the risk to tourists.

Title: Optimal Vaccination Age for Dengue in Brazil with a Tetravalent Dengue Vaccine

Date: 1pm Wednesday 2nd November 2016

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** With the first vaccine against Dengue being licensed in several endemic countries an important aspect that needs to be considered is the age at which it should be administered. If vaccination is done too early it is ineffective as individuals are protected by maternal antibodies, but if it is done later the infection may spread in the younger age groups, also the risks of hospitalisation and mortality change with age of infection, which is influenced by vaccination. However, to find the optimal vaccination age the possible coexistence of up to four distinct Dengue serotypes and the cross-reactions between these serotypes and Dengue antibodies need to be taken into account. We adapt a method previously applied to other infectious diseases and define the lifetime expected risk due to Dengue with respect to two different risk measures (hospitalization and lethality) which we then seek to minimize for a given three-dose vaccination strategy. Our results show that the optimal vaccination age not only depends on the risk measure but also on the number and combination of serotypes in circulation, as well as on underlying assumptions about cross-immunity and antibody dependent enhancement (ADE).

Title: Dancing in the Moonlight: Vertical Migration of Arctic Zooplankton during the Polar Night

Date: 1pm Wednesday 16th November 2016

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** This talk will focus on the results from my PhD, which I completed this year at the Scottish Association for Marine Science before starting here at Strathclyde. In recent years, evidence has been found of Diel Vertical Migration (DVM) in zooplankton during the
Polar Night in the Arctic Ocean. However, the drivers of this light mediated behaviour during an apparent lack of
illumination and food are poorly understood. A novel
dataset comprising 58 deployments of moored Acoustic Doppler Current Profilers is used in this study
to observe the vertical migratory behaviour of zooplankton on a pan-Arctic scale. Methods of circadian rhythm analysis are applied to detect synchronous activity. During the Polar Night, the moon is seen to control the vertical positioning of zooplankton, and a new type of migratory behaviour is described: Lunar Vertical Migration (LVM). This exists as LVM-day (24.8 hour periodicity) and LVM-month (29.5 day periodicity), and is observed throughout the Arctic Ocean. The results presented here show continuous activity throughout winter, and
challenge assumptions of a quiescent Polar Night.

Title: Disease Mapping and Visualization using Data from Spatio-Temporally Referenced Prevalence Surveys

Date: 1pm Wednesday 3rd May 2017

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:** We set out general principles and develop statistical tools for the analysis of data from spatio-temporally referenced prevalence surveys. Our objective is to provide a tutorial guide that can be used in order to identify parsimonious geostatistical models for prevalence mapping. A general variogram-based Monte Carlo procedure is proposed to check the validity of the modelling assumptions. We describe and contrast likelihood-based and Bayesian methods of inference, showing how to account for parameter uncertainty under each of the two paradigms. We also describe extensions of the standard model for disease prevalence that can be used when stationarity of the spatio-temporal covariance function is not supported by the data. We discuss how to define predictive targets and argue that exceedance probabilities provide one of the most effective ways to convey uncertainty in prevalence estimates. We describe statistical software for the visualization of spatio-temporal predictive summaries of prevalence through interactive animations. Finally, we illustrate an application to historical malaria prevalence data from 1334 surveys conducted in Senegal between 1905 and 2014.

Title: Including biology in spatial statistical models. Examples from vector-borne disease studies.

Date: 12.30pm, Thursday 1st June 2017

Venue: Livingstone Tower, 9th floor, room LT907

**Abstract:**

Vector borne diseases (e.g. Malaria, Dengue, Leishmaniasis) account for 20% of all infectious diseases, causing several million of infections and more than 1 million deaths annually. The majority of the vectors are insects (e.g. mosquitoes, midges and flies) and ticks, which biology and epidemiology are not often fully understood.

Biological and statistical models are used for mapping and modelling vector-borne diseases, however, rarely these methods are combined to produce maps and tools for disease surveillance and control (e.g. vector hot spots). In this talk I will present some techniques that can make data biologically meaningful; and the use of geo-bio-statistical models for tsetse flies (sleeping sickness) surveillance and control in Zambia. We show how mapping tsetse flies immigration, emigration, mortality and fertility can be the key element for successful disease eradication.

Title: Non-local Schrodinger Operators and Related Jump Processes

Date: 3pm Wednesday 15th March 2017

**Abstract: **Classical Schrödinger operators have been the object of much research involving functional analysis, probability and mathematical physics in the past decades. The recent interest in non-local Schrödinger operators consisting of the sum of a pseudo-differential operator and a multiplication operator greatly extended the range of applications, and inspired much new research in pure mathematics too. I will discuss how Feynman-Kac-type representations can be derived for the non-local cases and which random processes they give rise to. Then I will consider various sample path properties of these jump processes in terms of spectral properties of the generating non-local operators, and will contrast them with diffusions and classical Schrödinger operators.

Title: Stochastic Lotka-Volterra Food Chains

Date: 3.30pm Wednesday 19th April 2017

**Abstract: **We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka-Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka-Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and $n-1$ predators for $n\in\{2,3,4,\dots\}$. The $j$th predator eats the $j-1$th species and is eaten by the $j+1$th predator; this way each species only interacts with at most two other species - the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam '79 to a stochastic setting. A novelty of our analysis is the fact that we can describe the behavior the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species. This is joint work with Dang H. Nguyen.

Title: Time-Varying Feedback and its Control Ability

Date: 3.00pm Friday 19th May 2017

**Abstract: **Comparison to pure feedback control, time-varying feedback control has distinct advantages, e.g., in handling system nonlinearities, counteracting system uncertainties and achieving prescribed performance. But due to the time-variations, time-varying feedback always keeps most people away, and its potential has been investigated far from enough. Here I shall illustrate some good and ability of time-varying feedback, and introduce some applications in SDEs, as well as several problems to be further investigated.

Title: Last Passage Percolation Models in a Bernoulli Environment

Date: 3.00pm Wednesday 14th June 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **We will discuss two different last passage percolation models in an i.i.d. Bernoulli random environment. In particular, I will show explicit laws of large numbers and order of fluctuations for the last passage time - the maximum number of Bernoulli points one can collect by following a sequence of admissible steps that ends in a predetermined lattice site. I will show how the behaviour of these models change depending on the set of admissible steps (e.g. the LLN changes, directions that belong in a "percolation cluster” change) and also show how the order of fluctuations change if the direction of the path endpoint changes. This is joint work with Janosch Ortmann and Federico Ciech (Univ. of Sussex).

Title: Quantized Feedback Control for Control Systems with Saturation Nonlinearity

Date: 3.30pm Friday 16th June 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **In control systems, every physical actuator or sensor is subject to saturation owing to its maximum and minimum limits. Common examples of such limits are the deflection limits in aircraft actuators, the voltage limits in electrical actuators. Saturation nonlinearities are also purposely introduced into engineering systems such as control systems and neural network systems. In addition, one of the most important research areas in control theory is quantized control. Quantized feedback is found in many engineering systems including mechanical systems and networked systems. Since communication that need to transmit the feedback information from

the sensor to the controller may become less reliable as the bandwidth is limited. Here, I shall investigate quantized feedback control problems for systems subject to saturation nonlinearity.

Title: Stability of Two Kinds of Stochastic Runge-Kutta Methods for Stochastic Differential Equations

Date: 3.30pm Wednesday 5th July 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **We present two kinds of explicit Runge–Kutta methods for solving stochastic differential equations by using the three–term recurrence relations of Chebyshev and Legendre polynomials. The almost sure stability and mean-square stability of the numerical solutions generated by the two kinds of methods are investigated respectively. Numerical examples are provided to confirm theoretical results.

Title: Bayes' Rule and the Law

Date: 3.00pm Thursday 24th August 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **Bayesian inference is an approach in mathematical statistics where the probability of a hypothesis is updated as more evidence and data become available. It has wide applications in many areas such as machine learning, evolutionary biology, medicine and even in the judicial system. This talk will explore how Bayesian inference can be used in a specific court case to assist jurors in the process of legal decision making, demonstrating the power of mathematics in the court room.

Title: MLMC for Value-At-Risk

Date: 4.00pm Tuesday 19th September 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **In this talk, I explore Monte Carlo methods to estimate the Value-At-Risk (VaR) of a portfolio, which is a measure of the risk of the portfolio in some short time horizon. It turns out that estimating VaR involves approximating a nested expectation where the outer expectation is taken with respect to stock values at the risk horizon and the inner expectation is taken with respect to the option index and stock values at some final time. Following (Giles, 2015), our approach is to use MLMC to approximate the outer expectation where deeper levels use more samples in the Monte Carlo estimate of the inner expectation. We look at various control variates to reduce the variance of such an estimate. We also explore using an adaptive strategy (Broadie et al, 2011) to determine the number of samples used in estimating the inner expectation. Finally, we discuss using unbiased MLMC (Rhee et al., 2015) when simulating stocks requires time discretization. Our results show that using MLMC to approximate a probability of large-loss with an error tolerance of order $\epsilon$, we are able to get an optimal complexity of order $\epsilon^{-2}(\log(\epsilon^{-1})^2$ that is independent of the number of options, for a large enough number of options.

Title: On Uniqueness and Blowup Properties for a Class of Second Order SDEs

Date: 2.30pm Wednesday 18th October 2017

Venue: Livingstone Tower, LT9.07

**Abstract: **As the first step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multiplicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (X_t; Y_t) of the equations dX_t = Y_tdt, dY_t =|X_t|^\alpha dB_t, (X_0; Y_0) = (x_0; y_0). In particular, we prove that solutions are nonunique if 0 < \apha < 1 and (x_0; y_0) = (0; 0) and unique if 1=2 < \alpha and(x_0; y_0) \not= (0; 0). We also show that blowup in finite time holds if \alpha > 1 and (x_0; y_0) \not= (0; 0).