**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: Geometry and Continua

Date: 3.30pm Wednesday 14th March

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: This presentation will survey some applications of Differential Geometry to Continuum Mechanics. The aim is to motivate the use of the geometric terminology and apparatus rather than to present technical details. Accordingly, the style will be informal and the scope as comprehensive as possible.

Title: Applied and computational topology – theory, algorithms, applications and beyond

Date: 3.30pm Wednesday 18th April

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: In this talk I will present a few applications of topological data analysis I am working on: starting from computational neuroscience and medical image analysis, all the way to material sciences and analysis of dynamical systems. In all the cases, we will be searching for a hidden geometrical or topological structure that explains considered phenomena. The light-weight presentation of theory will be accompanied by a demonstration of effective algorithms implementing it. I believe that this general analytical framework can be useful in many other branches of science and I hope you may find it useful in your research.

Title: Playing with pendulums

Date: 3.30pm Wednesday 25th April

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: The (nonlinear) pendulum equation is one of the most basic differential equations modelling physical phenomena, and it turns up in a wide variety of cases. In this talk we start by presenting some examples of its appearance in models of liquid crystal cells. In this context, the pendulum equation is coupled with several types of boundary conditions, and for the mathematical analysis of the solutions of the associated boundary value problems the concept of "time map" is particularly helpful. We shall present that concept, and its implementation for the study of existence of bifurcating solutions to some boundary value problems arising from modelling liquid crystal cells. We also briefly present ongoing work on a boundary value problem where a seemingly minor change in the boundary condition leads to an unexpectedly more difficult analysis, thus illustrating some subtleties of this well-known equation.

This talk will be based on joint works with: M. Grinfeld, N. Mottram, J. Pinto (2009), E.C. Gartland Jr., M. Grinfeld, J. Pinto (2009), M.I. Mendez, J. Pinto (2017), and J. Pinto, K. Xayxanadasy (ongoing)

Supported by Sir David Anderson Bequest (University of Strathclyde), and Centre of Mathematical Analysis, Geometry, and Dynamical Systems (IST-Univ. Lisbon, Portugal)

Title: Fjord-shelf heat exchange in Kangerdlugssuaq Fjord, Southeast Greenland.

Date: 3.30pm Wednesday 2nd May

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: No region has felt the effects of global climate change more acutely than the cryosphere, where changes are driven largely by increasing ocean heat content at high latitudes. In southeast Greenland, acceleration and retreat of the marine-terminating glaciers contributes significantly towards global sea level rise. In Svalbard, increasing water temperature has decimated sea ice cover in many of the fjords, and had substantial implications for the local ecosystem. Understanding the dynamics of the fjords which accommodate these glaciers is therefore critical to our understanding the rapid changes occurring at high latitudes.

A realistic numerical model was constructed of Kangerdlugssuaq Fjord and the adjacent continental shelf. The Earth's rotation played a crucial role in the nature of the circulation and exchange in the fjord, and coastal winds were found to excite coastally-trapped internal waves which propagated into the fjord along the right-hand side. This process was capable of doubling the heat delivery toward the glacier terminus. The process also enhanced the background circulation via Stokes' Drift. Long periods of moderate wind stress were found to induce greatly enhanced heat flux towards the ice sheet, while short, strong gusts were found to have little influence, suggesting that the timescale over which the shelf wind field varies is a key parameter in dictating wintertime heat delivery from the ocean to the Greenland Ice Sheet.

This talk will be part of NBDES (North British Differential Equations Seminar)

Title: Energy dissipation at maximal rate

Date: 3.30pm Tuesday 29th May

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: The lecture will consider the situation of evolution equations endowed with a "free energy", in which the initial value problem possesses multiple solutions, and will discuss whether the particular solution that maximizes the rate of energy dissipation enjoys a special status.

Title: Some electromagnetic wave propagation models for moving media: an operator theoretical perspective

Date: 3.30pm Wednesday 30th May

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: The study of Maxwell's equations in moving media is of long standing interest, beginning with J.C. Maxwell himself. We consider Maxwell's equations with a drift term (Maxwell-Hertz-Cohn model) as our starting model for such situations and inspect its connection to the Maxwell-Minkowski model, which has replaced it, in an operator-theoretical framework. By discussing these models in a common normal form, following the ideas of [1], [2], we are able to conveniently compare these approaches. The Maxwell–Hertz–Cohn model suggests a velocity constraint by the speed of light, which was a strong motivation for the Maxwell-Minkowski model, where this constraint is actually built into the Minkowski structure of space and time. In the light of this historical link it may be interesting that – as we will show – the constraint in the Maxwell–Hertz–Cohn model to media moving slower than the speed of light is a mathematical artefact.

[1] R. Picard, *A structural observation for linear material laws in classical mathematical physics.* Math. Methods Appl. Sci., 32(14):1768–1803, 2009.

[2] R. Picard and D.F. McGhee, *Partial differential equations: a unified Hilbert space approach, vol. 55 of De Gruyter Expositions in Mathematics.* De Gruyter, Berlin, New York, 2011.

**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.

Title: Initial Values for Differential Algebraic Equations

Date: 3pm Tuesday 6th February

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: In this talk I will present a functional analytic (Hilbert space) perspective towards initial value problems for differential-algebraic-equations. These differential equations form a subclass of implicitly defined differential equations. We shall discuss the biggest subspace of the (inifinite-dimensional) state space for which there still exist classical solutions. Furthermore, we present connections to certain distributional solutions, define and relate certain aspects of (exponential) stability. This is joint work with Sascha Trostorff from TU Dresden.

Title: Remarks on Communicability in Networks

Date: 3pm Tuesday 13th February

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The concept of communicability has proved to be useful in the analysis of complex networks. But it is not clear what properties a communicability function must or should have. Trying the define a communicability function leads to many interesting questions. In this talk I will discuss some of them

Title: Just keep walking

Date: 3pm Tuesday 13th March

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Complex networks can model interactions in a host of applications, ranging from the WWW to neuroscience. Several very popular centrality measures for nodes in networks build on the concept of information flow, or "walks" round the graph. Centrality measures based on the combinatorics of walks are particularly suited to be analysed via matrix theoretical tools, since powers of the adjacency matrix encode in their entries the number of walks taking place between any two nodes in the graph. However, not all walks are created equal. In this talk I will describe how to extend some popular walk-based centrality measures to the non-backtracking framework, where information is not allowed to travel back and forth between two nodes, but can only move forward.

This talk is based on joint work with Prof. D. J. Higham, Prof. P. Grindrod, and Dr. V. Noferini.

Title: A one-dimensional model for self-propelled diffusions

Date: 3pm Tuesday 20th March

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: One of the new challenges of statistical mechanics arises from the study of interacting particle systems of self-propelled particles. Such models are at the root of many biological phenomena, such as bacterial migration, flocking of birds etc. In this talk we will consider a non-linear PDE for a Viksek-type model (the PDE being non-linear in the sense of McKean). The PDE at hand is i) not in gradient form and ii) it is non-uniformly elliptic (but hypoelliptic instead). Moreover, as typical in this framework, the dynamics exhibits multiple equilibria (stationary states). This is a joint work with P.Butta (La Sapienza, Rome), F. Flandoli (Scuola Normale, Pisa) and B. Zegarlinski (Imperial College).

Title: Determinants of tail risk in emerging and developed markets

Date: 2pm Tuesday 29th May

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: We study the distribution of extreme events risk across emerging and developed stock markets and empirically identify the determinants of tail risk across countries. A recent literature has shown that rare disasters can explain some of the most important puzzles in finance and that tail risk is priced in the cross section of asset returns. We find a strong empirical relationship between tail risk and the quality of institutions even after economic and financial variables have been accounted for. Better governance substantially reduces the probability of extreme events. In addition, we find that what differentiates developed and developing countries concerning extreme stock market risk is the quality of their institutions, not the depth of their financial markets, nor the degree of financial and trade openness. We also outline a theoretical model arising from physics that captures many of the features of our analysis and suggests a mechanism for the developments of tail risks.

Please note that the seminar time is changed to 2PM to avoid scheduling conflict with the department colloquium.

Title: Stochastic unfolding and homogenization

Date: 2pm Tuesday 26th June

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: The notion of two-scale convergence and the periodic unfolding method are prominent and useful tools in multiscale modeling and analysis of PDEs with rapidly oscillating periodic coe?cients. In this talk we are interested in the theory of stochastic homogenization for continuum mechanical models in form of PDEs with random coefficients. In particular, we introduce a stochastic unfolding method that features many similarities to periodic unfolding. We discuss the relations of stochastic unfolding to previously introduced notions of stochastic two-scale convergence. We apply the stochastic unfolding procedure to homogenization of a non-convex evolutionary gradient system of Allen-Cahn type. This talk is based on a joint work with Martin Heida and Stefan Neukamm.

Please note that the seminar time is changed to 2PM to avoid scheduling conflict with the department colloquium.

**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.

Title: Reconstructing Spatially Heterogeneous Thermal Maps using Light-based Metrology Sensors

Date: 1.00pm Tuesday 20th February 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: As the world enters the fourth industrial revolution, the automated age, robots are being used more ubiquitously. Industries are interested in autonomous manufacturing as it reduces costs and increases productivity. A vital aspect of autonomous precision manufacturing is large volume metrology, with metrology being the science of measurement. In such settings the robots need a sensing system to help them position themselves and the component they are working on. One popular sensing modality uses light rays, which travel through the volume of air, to undertake this. These optical-based metrology systems such as photogrammetry (using digital photography to calculate angles between the camera and a reflector) and laser trackers (using laser beams to measure distances from the laser tracker to the reflector) are crucial in improving the accuracy and quality associated with robotic assembly. In an industrial setting these positional measurement systems give rise to uncertainties which can in many instances be greater than the required tolerances. One source of uncertainty that arises when considering large scale industrial settings is light refraction, the bending of the light ray’s path, due (in part) to temperature fluctuations in the air. We will report on our recent work in using light-based sensor data to reconstruct the heterogeneous spatial map of the refractive index in the air. Such knowledge can then be used to discount the refractive effects and thereby reduce the uncertainty of this positioning problem. We will look at solving this inverse problem using Voronoi tessellations to spatially parameterise the refractive index map. A Bayesian approach, namely the reversible jump Markov Chain Monte Carlo method (rj-MCMC), is then used as the optimisation method in the inversion. A simulation tool has been developed in COMSOL so that the methodology can be tested on a broad range of problems.

Title: Nematic liquid crystal director structures in rectangular regions

Date: 1.00pm Tuesday 20th February 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: We consider a shallow rectangular well of nematic liquid crystal subject to weak

anchoring on the sides of the well. By considering weak anchoring instead of innitely

strong anchoring, we are able to analyse nematic equilibria in the well without the

need to exclude point defects at the corners, as done in previous work in the area. For

relatively weak anchoring, we are able to derive analytic expressions for the director

alignment angle in terms of an innite series of modes, involving roots of a transcen-

dental equation. The analytic forms of the director conguration are then used to

calculate critical anchoring strengths at which uniform and distorted director struc-

tures exchange stability. We also consider the asymptotic behaviour of the director

structure and energy for very strong anchoring. We show that in both cases - for the

transitions from uniform to distorted states and the limit of innitely strong anchoring

- the approximate analytic expansions agree very well with corresponding numerical

calculations of the full model.

Title: Asymptotic and numerical study of the planar stick-slip flow for viscoelastic fluids

Date: 1.00pm Tuesday 27th February 2018

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

The stick-slip flow can be considered as an important challenging viscoelastic benchmark due to the presence of the stick-slip transition point, where a sudden change happens in the boundary conditions of the flow. In this work, the stress singularity of the Oldroyd-B, PTT and Giesekus viscoelastic models are verified for transient and steady planar stick-slip flow. We have carried out numerical simulations of the steady stick-slip flow along streamlines in the presence of a solvent viscosity, considering a given Newtonian velocity field and a simplified version of the constitutive equations.

These results were published in a recent paper of Evans et al. (2017).

In addition, we performed a full numerical simulations of the complete governing equations system for the transient planar stick-slip flow confirming the asymptotic results presented in Evans et al. (2017).

In order to improve the numerical knowledge about this viscoelastic benchmark problem, the Cartesian stress formulation has been here originally assessed for the PTT and Giesekus models considering the solvent viscosity case. The latter has been for the first time used for solving the transient planar stick-slip flow.

Title: The motion of droplets and contact lines including multiscale density functional theory approaches

Date: 1.00pm Tuesday 20th March 2018

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

The moving contact line problem occurs when attempting to model the movement of the location where two fluid phases and a solid meet, as occurs when droplets spread (e.g. in inkjet printing), capillaries fill, insects walk on water, or in many other natural or technological instances. The problem exists when using the classical, macroscopic, equations of fluid motion as a singularity occurs in the predicted stresses and thus forces at the contact line, and the velocity is multi-valued. In this talk, we will look at contact line motion in different situations through multiscale methods that alleviate this singularity. Common to all situations in the talk will be the inclusion of a binding potential between fluid-fluid and fluid-solid interfaces, either explicitly or effectively by modelling variable density of fluid. We will discuss density functional theory (DFT) results to obtain binding potentials, and also dynamic DFT results for contact line motion in their own right. A preliminary study of a three phase (solid, liquid, vapour) system where phase-change can occur between all phases will also be mentioned. At various points, joint work with Andreas Nold, Ben Goddard, Serafim Kalliadasis, Luis G. MacDowell, Han Yu Yin and Andrew Archer will be presented.

**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

Title:

Date: 4.00pm Tuesday 30 January 2018

Venue: Livingstone Tower, 9th floor, room LT907

Abstract:

Title: Stochastic Separation Theorems and One-trial corrections of Legacy AI systems

Date: 3.00pm Tuesday 20 February 2018

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: In this talk we shall consider the problem of mistakes in Artificial Intelligence (AI) systems and motivate a technology for simple, real-time, computationally-efficient, and non-iterative improvements of the systems. The improvements are, in essence, shallow networks constructed on top of the existing AI computational architectures. Theoretical foundation of the technology is based on Stochastic Separation Theorems and the ideas of measure concentration. We show that, subject to mild technical assumptions on statistical properties of internal signals in the original AI, with probability close to one the technology enables instantaneous ''learning away'' of spurious and systematic errors. The method is illustrated with applications in image processing and face/human shape recognition/detection.

Title: Computational research on blood flow dynamics and the initiation of atherosclerosis

Date: 4.00pm Tuesday 27 February 2018

Venue: Livingstone Tower, 9th floor, room LT907

Abstract: Atherosclerosis, the leading underlying cause of heart failure and stroke, is an inflammatory disease affecting primarily the larger systemic arteries. Atherosclerotic lesions are non-uniformly distributed along the arterial network, suggesting an important role for hemodynamic factors, particularly wall shear stress, in their development. In this talk, I will present computationally fluid dynamic (CFD)-obtained results on the role of hemodynamics on the distribution of these stresses, around idealized and anatomically correct arterial geometries. The value and sensitivity of this work will be discussed on the basis of a reverse-engineering approach in finding direct correlations with biology. A short description of other ongoing and future group research activities will also be presented.

Title: Generalized locally Toeplitz sequences and some applications to fractional diffusion equations

Date: 4.00pm Tuesday 6th March 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: When discretizing a linear PDE by a linear numerical method, the computation of the numerical solution reduces to solving a linear system. The size of this system grows when the discretization parameter n increases, i.e., when we refine the discretization mesh. We are then in the presence of a sequence of linear systems {Anxn = bn} with increasing size.

It is usually observed in practice that the corresponding sequence of discretization matrices {An} inherits a structure from the continuous problem and enjoys an asymptotic spectral distribution, which is compactly described by a function, known as symbol. The knowledge of the symbol and of its properties has a very crucial role, since it can be used to perform a convergence analysis and to predict the behavior of preconditioned Krylov and multigrid methods applied to An, as well as to design effective preconditioners and multigrid solvers for the associated linear systems.

The main tool for computing the spectral symbol of a PDE discretization matrix is the theory of Generalized locally Toeplitz (GLT) matrix-sequences introduced in [5] as a generalization of both classical Toeplitz sequences and variable coefficient differential operators. In this talk, we give an overview of such a theory and we outline its application to the fractional diffusion equations (FDEs), an extension of classical diffusion equations used to model anomalous diffusion phenomena. Exploiting the GLT machinery, we provide a spectral analysis of some FDE discretization matrices and we use the retrieved spectral information either to discuss the convergence of recently proposed techniques [2, 4] or to design new preconditioning and multigrid strategies for both 1D and 2D FDE problems [1, 3].

References

1. M.Donatelli,M.Mazza,S.Serra-Capizzano,Spectral analysis and structure preserving preconditioners for fractional diffusion equations, J. Comput. Phys., 307 (2016) 262–279.

2. S. L. Lei, H. W. Sun, A circulant preconditioner for fractional diffusion equations, J. Comput. Phys., 242 (2013) 715–725.

3. H. Moghaderi, M. Dehghan, M. Donatelli, M. Mazza, Spectral analysis and multigrid preconditioners for two- dimensional space-fractional diffusion equations, J. Comput. Phys., 350 (2017) 992–1011.

4. H. Pang, H. W. Sun, Multigrid iterative methods for fractional diffusion equations, J. Comput. Phys., 231 (2012) 693–703.

5. S. Serra-Capizzano, Generalized locally Toeplitz sequences: spectral analysis and applications to discretized differential equations, Linear Algebra Appl., 366 (2003) 371–402.

Title: Computing the operator norm of nonnegative matrices

Date: 4.00pm Tuesday 22nd May 2018

Venue: Livingstone Tower, 9th floor, room LT908

Abstract: Computing the subordinate operator norm of a rectangular matrix is known to be NP-hard in general. However, by restricting ourselves to matrices with nonnegative entries, we show that under suitable conditions on the norms, the operator norm of such matrices can be computed efficiently with a nonlinear power method. A linear convergence rate to the global maximizer is discussed. In particular, our conditions include the hypercontractive case for which no similar results were previously known. As an application, we discuss how these results can be used to produce lower bounds on the log-Sobolev constant of finite Markov chains.

Joint work with Francesco Tudisco and Matthias Hein.

**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: 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: 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: Why Primary Production Peaks at Surface During Summer in the Subtropica, Oligotrophic Open Ocean?

Date: 1pm Tuesday 20th March 2018

Venue: Livingstone Tower, 9th floor, room LT908

**Abstract:** Classic understanding on environmental controls on primary production (PP) in the subtropical, oligotrophic ocean gyre is that phytoplankton growth and PP are limited by the upward supply of inorganic nutrients delivered largely by diffusion. Since nutrient supply mostly comes from below and light attenuates from surface to the depth, phytoplankton growth rate and PP should peak at some intermediate depth, coinciding with the deep chlorophyll maximum (DCM) layer. However, examination on the PP data measured at three stations (ALOHA, S1, and BATS) in the subtropical North Pacific and Atlantic reveals that PP peaks within the surface mixed layer despite the negligible nutrient concentration at surface and a pronounced deep chlorophyll maximum (DCM) around 100 m. While the formation of DCM can be largely explained by phytoplankton photo-acclimation (adjustments of chlorophyll-to-carbon ratios), the surface peak of phytoplankton growth rate is difficult to explain. I evaluate several hypotheses that try to explain the surface peak of phytoplankton growth rate. The preliminary finding is that the coexistence of high- and low-light adapted ecotypes can best explain the observed patterns of nutrient, chlorophyll, and PP. This is highly consistent with many biodiversity–ecosystem functioning (BEF) studies that biodiversity can enhance productivity and nutrient utilization via niche complementarity. Thus, evaluation of the effects of climate change on ocean productivity is hard to be reliable without considering biodiversity.

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).

Title: Randomized Numerical Schemes for (S)ODEs/SPDEs

Date: 4.00pm Tuesday 12th June 2018

Venue: Livingstone Tower, LT9.07

**Abstract: **A wide range of applications, for instance, in the engineering and physical sciences as well as in computational finance is still spurring the demand for the development of more efficient algorithms and their theoretical justification. In particular, the current focus lies on the approximation of ODEs/S(P)DEs which cannot be treated by standard methods found in textbook. We, therefore, first developed two randomized explicit Runge–Kutta schemes for ordinary differential equations (ODEs) with time-irregular coeffcient functions. In particular, the methods are applicable to ODEs of Carathéodory type, whose coeffcient functions are only integrable with respect to the time variable but are not assumed to be continuous. An important ingredient in the analysis are corresponding error bounds for the randomized Riemann sum quadrature rule. It is demanding to approximate numerical solutions of non-autonomous SDEs where the standard smoothness and growth requirements of standard Milstein-type methods are not fulfilled. In the case of a non-differentiable drift coefficient function f, we proposed a drift-randomized Milstein method to achieve a higher order approximation and discussed the optimality of our convergence rates. We also pushed the idea to the numerical solution of non-autonomous semilinear stochastic evolution equations (SEEs) driven by an additive Wiener noise. Usually quite restrictive smoothness requirements are imposed in order to achieve high order of convergence rate. It turns out that the resulting method converges with a higher rate with respect to the temporal discretization parameter without requiring any differentiability of the nonlinearity. Our approach also relaxes the smoothness requirements of the coefficients with respect to the time variable considerably.