Seminar Series 2020-21
SPAAM Seminar Series
The Statistics, Probability, Analysis and Applied Mathematics (SPAAM) seminar series will take place virtually at 3pm on Thursdays during term time on the SPAAM Microsoft Teams channel. It will host a variety of talks from PhD students involved in applied mathematics research at Warwick (see the bottom of this page for the talk abstracts!).
Each seminar will usually host two speakers (unless otherwise stated) with each talk taking around 15-20 minutes with 5-10 minutes of questions afterwards. Speakers and committee members will hang around for some time after the talks for coffee and further questions. Please do contact one of the committee if you would like to join and be added to the MS Teams channel. Note that these talks may be recorded for later viewing on our Youtube channel so do join with audio and video off if you don't wish to feature!
If you would like to give a talk this term, please contact Diogo Caetano (Diogo.Caetano@warwick.ac.uk) or Haoran Ni (Haoran.Ni@warwick.ac.uk) and we will find you a slot!
Term 3
Date | Talk 1 | Talk 2 |
29th April 2021 (Week 1) |
||
6th May 2021 (Week 2) |
Industry careers day | |
13th May 2021 (Week 3) | Felicitas Schmitz (Univ. of Regensburg) | Suzie Brown (Statistics) |
20th May 2021 (Week 4) | Lorenzo Quarisa (Maths) | |
27th May 2021 (Week 5) | ||
3rd June 2021 (Week 6) | Ryan Acosta Babb (Maths) | |
10th June 2021 (Week 7) | Oliver Wright (Maths) | |
17th June 2021 (Week 8) | ||
24th June 2021 (Week 9) | ||
1st July 2021 (Week 10) |
Abstracts
Week 3 (Talk 1) - Interaction of mean curvature flow and diffusion - Felicitas Schmitz (Univ. of Regensburg)
I present a geometric evolution problem, i.e., a PDE on an evolving hypersurface where the evolution of the geometry is not given but also part of the problem. The particular problem combines a type of mean curvature flow equation and a diffusion equation. After motivating the system of equations from a physical point of view, we discuss the evolution of several properties of the hypersurface's shape during the flow. My aim is to explain everything in a descriptive way instead of demonstrating technical details.
Week 3 (Talk 2) - Kingman genealogies for non-neutral population models - Suzie Brown (Statistics)
Kingman’s coalescent is known to be the limiting genealogical process for a large class of population genetic models (Kingman 1982) and that class is fully specified by the necessary and sufficient conditions of Moehle & Sagitov (2001). However, these results apply only to population models that are neutral, that is, without natural selection. I will present a class of non-neutral models, motivated by sequential Monte Carlo, that also admit Kingman genealogies in the large-population limit, under generic conditions analogous to those of Moehle & Sagitov. Joint work with Adam Johansen, Paul Jenkins and Jere Koskela.
Week 4 (Talk 1) - The vanishing viscosity limit for the Navier-Stokes equations - Lorenzo Quarisa (Maths)
A central problem in mathematical fluid dynamics is whether inviscid fluids can be asymptotically approximated by viscous fluids as the viscosity tends to zero. The answer strongly depends on whether a boundary is present, and on the boundary conditions satisfied by the viscous flows. In particular, the problem is still unsolved when the viscous flow is assumed to be stationary at the boundary (no-slip condition). This setting has been considered since at least the early 20th century, for instance in the work of Prandtl. In this case, a so-called 'boundary layer' is observed where the viscous flow undergoes a sharp transition.
Week 6 (Talk 1) - What's the Difference Between a Circle and a Square? Truncations of Fourier Series and the Multiplier Problem - Ryan Acosta Babb (Maths)
In one dimension, there is only one way to truncate a partial sum: count up to a certain N. For series eigenfunctions labelled by pairs of indices, as is the case of the Fourier series on Z^2, we may truncate in several ways. For example, do we count pairs of indices (n, m) with |n|,|m| ≤ N, or instead count them with |n|^2 + |m|^2 ≤ N? It is a curious fact that L^p convergence can be obtained in the former case (for all p) but "never" the latter: it fails for all p ≠ 2. We discuss the context and history of this problem.
Week 7 (Talk 1) - What does gas have to do with elections? A brief introduction to Interacting Multi-agent Systems - Oliver Wright (Maths)
The idea of using the kinetic theory of gases to describe multi-agent socio-economic systems was proposed by Galam et al. in 1982. These range from models of economies to traffic models to election polling. In this talk, I wish to discuss the motivation and derivation of such models, as well as show some recent results of numerical simulations in the area of pre-election polling.
Term 2
Date | Talk 1 | Talk 2 |
14th January 2021 |
Helicity for volume-preserving Anosov flows - Solly Coles (Maths) |
Stochastic parareal: a novel application of probabilistic methods to time-parallelisation - Kamran Pentland (MathSys) |
21th January 2021 |
Hierarchical structure in a condensed zero-range process - Simon Gabriel (Maths CDT) | Infinite Horizon Stochastic Differential Utility - Joe Jerome (Statistics) |
28th January 2021 | Mathematically Modelling Metal Sheet Spinning - Hanson Bharth (MASDOC) | Minimax rates in change point detection problems - Mengchu Li (Statistics) |
4th February 2021 | Instantons for rare events in heavy-tailed distributions - Mnerh Alqahtani (Maths) | Using deep learning to infer house prices from online images - Bhavan Chahal (MathSys) |
11th February 2021 | Monte Carlo methods for Fredholm integral equations - Francesca Crucinio (Statistics) | Outbreak detection using Bayesian hierarchical modelling and Gaussian random fields - Laura Guzman-Rincon (MathSys) |
18th February 2021 | The Bethe Ansatz and Sticky Brownian Motions - Dom Brockington (MASDOC) | Accuracy of the Balanced Truncation Method of Model Order Reduction - Matt Coates (MASDOC) |
25th February 2021 | Symmetry, supersymmetry and deformations in physics - Andrew Beckett (Edinburgh) | (Andrew is covering both slots) |
4th March 2021 | Asymptotic length of the concave majorant of a Lévy process - David Bang (Statistics) | Wave Propagation in Randomly Layered Heterogeneous Media - Alistair Ferguson (Univ. of Strathclyde) |
11th March 2021 | Polynomial Interpolation: An interactive introduction - Jack Thomas (MASDOC) | Ferronematics in One-Dimensional Channels - James Dalby (Strathclyde) |
18th March 2021 | Waves in free surface swirling flows - Emanuele Zuccoli (Maths) | Directed polymers, stochastic heat equation and KPZ - Dimitris Lygkonis (MASDOC) |
Abstracts
Week 1 (Talk 1) - Helicity for volume-preserving Anosov flows - Solly Coles (Maths)
The helicity of a volume-preserving flow measures the extent to which flow lines tangle around each other. Helicity is a useful invariant which helps to answer certain variational problems in magnetohydrodynamics. Due to Arnold, one can characterise helicity in terms of the linking numbers of knots constructed by closing up trajectories of the flow with geodesic arcs. In this talk we will describe Arnold's characterisation, followed by a new characterisation for the case of Anosov flows, in terms of the linking of periodic trajectories.
Week 1 (Talk 2) - Stochastic parareal: a novel application of probabilistic methods to time-parallelisation - Kamran Pentland (MathSys)
Most modern techniques used to numerically integrate partial differential equations rely on parallelising aspects of the spatial dimension and, whilst highly efficient, are reaching scale up limits. Further numerical speed-up is often limited by the sequential time stepping method used. Introducing parallelism into the temporal dimension is not an obvious route one chooses due to the inherent sequential nature of time (i.e. future solutions states depend upon previous states). Having only been developed over the last few decades, time-parallel methods enable us to do just this and are gaining popularity in a number of applications. In this talk we will introduce the parareal algorithm (an existing robust parallel-in-time numerical integrator) that solves initial value problems in parallel. It converges deterministically at a given rate (yielding parallel speed-up) which we aim to improve upon by incorporating probabilistic methods - thus we develop a stochastic parareal algorithm.
Week 2 (Talk 1) - Hierarchical structure in a condensed zero-range process - Simon Gabriel (Maths CDT)
Motivated by the physics of condensed matter, we study the limiting dynamics of stochastic particle systems on a microscopic and macroscopic scale. A particular simple but interesting toy model is the zero-range process. Most classical results show that the condensate is a single macroscopic cluster, however, more sophisticated formations are possible.
In this talk we will introduce the zero-range process, define condensation and discuss conditions to see a Poisson-Dirichlet distributed condensate in the thermodynamic limit.
Week 2 (Talk 2) - Infinite Horizon Stochastic Differential Utility - Joe Jerome (Statistics)
Stochastic differential utility has been widely studied since its formulation by Duffie and Epstein in 1992. It allows modelling of a much wider range of risk and intertemporal preferences and therefore provides a natural extension to the Merton problem for time-additive utility. However, whilst the finite time horizon problem is now fairly well understood, few have investigated the infinite horizon `lifetime' problem. In our paper we provide a novel formulation of the lifetime problem, highlighting and explaining the role of the transversality condition. We then discuss the parameters governing the agent's preferences, and show that certain parameter combinations considered in the literature are ill-posed over the infinite horizon.
We prove existence of a finite valued utility process for a large class of consumption streams and then show that, by considering a natural generalisation, we may assign a meaningful utility to any non-negative progressively measurable process. This means that, regardless of the choice of financial market, self-financing consumption streams are always evaluable.
Finally, we show existence and uniqueness of an optimal strategy in a Black-Merton-Scholes market.
Week 3 (Talk 1) - Mathematically Modelling Metal Sheet Spinning - Hanson Bharth (MASDOC)
Metal is a material which is used in abundance and the efficient use of it has been imperative to the advancement of civilization since the Bronze Age. New technology and engineering processes have been developed for metal spinning, but progress has been foiled by our understanding of the physics. Metal forming is an extremely energy-intensive process so it is vital that we iron this out. In this talk, I shall present an approximate model for metal sheet spinning which is then formulated into a matrix Wiener-Hopf problem. To alloy us to find a solution we borrow some tools from orthogonal polynomials to compute the Wiener-Hopf decompositions. Finally, I shall discuss a hidden free-boundary problem and some simulations to demonstrate.
Week 3 (Talk 2) - Minimax rates in change point detection problems - Mengchu Li (Statistics)
In this talk, I will introduce the change point detection problem in general and focus on the simple univariate mean change point problem. Some known minimax results in change point detection and localisation will be presented as well as the celebrated wild binary segmentation framework, which has been shown to be minimax optimal (up to log factor). The focus will then shift to the robust mean change point detection problem. Some preliminary results on the minimax rates of the problem will be presented, complemented by a simple heuristic algorithm.
Week 4 (Talk 1) - Instantons for rare events in heavy-tailed distributions - Mnerh Alqahtani (Maths)
Estimating the probability of a rare trajectory of a stochastic dynamical system lies at the centre of large deviation theory. In the principle of large deviation, which is analogous to the least action principle in quantum mechanics; this probability is dominated by one single path of a set of all possible other paths (which are also unlikely events). This path is the minimizer of the rate function or the action, which is called ‘instanton’. The difficulty of answering such question can emerge from the complexity of the rate function landscape, which is what this talk addressing. More precisely, the case of heavy-tailed distributions associate with non-convex rate functions. By reviewing Gärtner-Ellis theorem, and by the means of convex analysis tools, we will demonstrate the failure mode that can be encountered while minimizing such rate functions. Then, a modification is suggested, which is a nonlinear reparametrization of rare events of interest, followed by some examples of systems exhibit large deviation behaviour. This talk is based on a recent preprint, with Dr. Tobias Grafke (can be found here).
Week 4 (Talk 2) - Using deep learning to infer house prices from online images - Bhavan Chahal (MathSys)
House prices are of interest at both a micro- and macroeconomic level due to issues such as housing inequality. The variation in amenities, air pollution, and other important factors in a particular location can determine where individuals choose to buy a property. By simply walking through a neighbourhood, we can understand a large amount of information about its socio-economic status and likely house prices. In recent times, vast quantities of online images have become available from sources such as Google and Zoopla. Simultaneously, huge leaps in what can automatically be inferred from an image have occurred, drawing on recent advances in deep learning. Can we use deep learning and this vast corpus of images to automatically infer house prices of neighbourhoods across London?
Week 5 (Talk 1) - Monte Carlo methods for Fredholm integral equations - Francesca Crucinio (Statistics)
Fredholm integral equations of the first kind are the prototypical example of inverse ill-posed problem. They model, among other things, density deconvolution, image reconstruction and find applications in epidemiology, medical imaging, nonlinear regression settings. However, their numerical solution remains a challenging problem. Many techniques currently available require a preliminary discretisation of the domain of the solution or make strong assumptions about its regularity. We propose a novel Monte Carlo method that circumvents these two issues and performs an adaptive stochastic discretisation of the domain without requiring strong assumptions on the solution of the integral equation. We study the theoretical properties of this algorithm and compare it with alternatives both on simulated data and realistic systems.
Week 5 (Talk 2) - Outbreak detection using Bayesian hierarchical modelling and Gaussian random fields - Laura Guzman-Rincon (MathSys)
Identification and investigation of outbreaks remain a high priority for health authorities. Early detection of infectious diseases facilitates interventions and prevents further transmission. For that purpose, surveillance systems collect epidemiological data from patients, including location, onset time of infection and, with the recent improvements in genotyping techniques, the whole-genome sequencing of bacteria. For outbreak detection, several statistical methods have been proposed using spatial, temporal, or genetic data. However, techniques mixing genomics and epidemiological factors are still underdeveloped.
This talk will describe a new approach that aims to combine diverse sources of data for outbreak detection of infections caused by Campylobacter. The methodology tackles the problem as a classification task using Bayesian hierarchical models, based on an existing spatial-temporal model proposed by Spencer et al. [Spat. Spatio-temporal Epidemiol., 2(3), 173–183 (2011)]. The talk will show that the latent parameters of the model are Gaussian random fields adapted to different types of data. Moreover, the model will be applied to study reported cases of Campylobacter infections in two regions in the UK. Two scenarios will be explained, where a temporal-genetic and a spatial-genetic model are trained using the framework proposed. The method allows us to find potential diffuse outbreaks that could not be captured using spatial-temporal methods.
Week 6 (Talk 1) - The Bethe Ansatz and Sticky Brownian Motions - Dom Brockington (MASDOC)
We introduce a family of interacting Brownian motions called sticky Brownian motions, and show that for a special choice of interaction we can find the transition kernel in exact form using the Bethe Ansatz. We then study the fluctuations of the size of atoms in the infinite particle system, called the Howitt-Warren process.
Week 6 (Talk 2) - Accuracy of the Balanced Truncation Method of Model Order Reduction - Matt Coates (MASDOC)
Linear systems are used to model a range of physical processes, but in order to apply them numerically approximate systems of low or finite dimension must be constructed. A common method of doing so is the method of "Balanced Truncation" as well as related methods. Error bounds for this method exist, however the bounds we wish to work with are given in terms of the spectral properties of an integral operator associated to the complete linear system, called the Hankel operator. We discuss Schatten class bounds, which describe the summability and decay properties of the singular values of the Hankel operator in terms of certain integrability properties of its associated integration Kernel. In particular we discuss the application of this to the Heat Equation on the unit disk, and conclude that the singular values for the associated Hankel operator decay better than any polynomial.
Week 7 (Talk 1) - Symmetry, supersymmetry and deformations in physics - Andrew Beckett (Edinburgh)
Symmetry principles play a central role in modern physics, and much of the progress in theoretical physics in the 20th century was driven by the discovery and exploration of these principles. I will discuss some of these symmetries, how they have helped us to understand the world around us and how a hypothetical symmetry, supersymmetry, might help to resolve some of the biggest outstanding issues in physics today. I will also discuss how spacetime curvature is used to describe gravity in general relativity, and how this curvature deforms the algebraic structures we use to describe symmetry. Finally, I will talk about how all of these ideas come together in my own research on understanding supergravity via deformations of Lie superalgebras.
Week 8 (Talk 1) - Asymptotic length of the concave majorant of a Lévy process - David Bang (Statistics)
TBC.
Week 8 (Talk 2) - Wave Propagation in Randomly Layered Heterogeneous Media - Alistair Ferguson (Univ. of Strathclyde)
We consider the propagation of high frequency elastic waves in a layered crystalline material. We are particularly interested in the regime where multiple scattering is dominant and where an effective medium approach or homogenisation cannot be used. Each layer is locally anisotropic and the layer thicknesses and crystal orientations follow a stochastic (Markovian) process. We model monochromatic shear waves propagating in this two-dimensional heterogeneous media. Expressions for the amplitude of the local and global coefficients for the reflected and transmitted waves are derived and shown to satisfy energy conservation. The resulting stochastic differential equations lead to a self-adjoint infinitesimal generator and a Fokker-Planck equation via limit theorems. Explicit expressions for the moments of the probability distributions of the power transmission and reflection coefficients, are then derived. This work is important in deepening our understanding of the ultrasonic non-destructive testing of composites and polycrystalline components in the engineering sciences.
Week 9 (Talk 1) - Polynomial Interpolation: An interactive introduction - Jack Thomas (MASDOC)
We give a brief tour of polynomial approximation theory highlighting the main classical results and illustrating the key ideas with pictures. Along the way, I'll discuss the connection between this problem and (logarithmic) potential theory and the theory of Schwarz-Christoffel conformal maps. Time permitting, I will explain the link between polynomial approximation and my own research.
Week 9 (Talk 2) - Ferronematics in One-Dimensional Channels - James Dalby (Strathclyde)
We study a model system with nematic and magnetic orders, within a channel geometry modelled by an interval, [−D, D]. The system is characterised by a tensor-valued nematic order parameter Q and a vector-valued magnetisation M, and the observable states are modelled as stable critical points of an appropriately defined free energy. In particular, the full energy includes a nemato-magnetic coupling term characterised by a parameter c. We (i) derive L∞ bounds for Q and M; (ii) prove a uniqueness result in parameter regimes defined by c, D and material- and temperature-dependent correlation lengths; (iii) analyse order reconstruction solutions, possessing domain walls, and their stabilities as a function of D and c and (iv) perform numerical studies that elucidate the interplay of c and D for multistability.
Week 10 (Talk 1) - Waves in free surface swirling flows - Emanuele Zuccoli (Maths)
Free surface flows are of major importance in many fields of Fluid Dynamics, both from a theoretical and experimental point of view. It is known that a small perturbation to a base free surface flow leads to the formation of waves propagating on the surface of the fluid. In this talk I will first show how to construct a complete mathematical model describing the dynamics of such waves in an unbounded domain when the base flow is a free surface vortex. Then, I will analyze a simpler model able to describe a trapping/scattering behaviour which may occur in waves spinning around a free surface swirling flow.
Week 10 (Talk 2) - Directed polymers, stochastic heat equation and KPZ - Dimitris Lygkonis (MASDOC)
During recent years a lot of attention has been drawn to the study of singular SPDE. In this talk we will demonstrate a connection between the stochastic heat equation and KPZ equation with directed polymers and present some recent developments in the field.
Term 1
Date | Talk 1 | Talk 2 |
15th October 2020 |
Dislocations and Grain Boundaries: A Short Account - Anvar Atayev (MASDOC) |
Predicting Asymptotic Behaviour of Matched Solutions - Matthew King (MASDOC) |
22nd October 2020 | A phase field model for raft formation on biological membranes - Luke Hatcher (MASDOC) | N/A |
29th October 2020 | Social event | |
5th November 2020 | Meta stability in atmospheric jets - Nayef Shkeir (MathSys) | Point particle interactions on surface biomembranes - Philip Herbert (MASDOC) |
12th November 2020 | Recombination detection for viral genetic data - Ana Ignatieva (Statistics) | Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection - Phil Hanson (MASDOC) |
19th November 2020 | A family of continuous-time dynamical systems with a trivial CLT - Nicolò Paviato (MASDOC) | Kesten Processes and Wealth Generation - Samuel Forbes (MathSys) |
26th November 2020 | The fully nonconforming virtual element method for fourth order perturbation problems - Alice Hodson (MASDOC) | Why are living systems cellular? Modelling reaction-diffusion systems in biochemical systems - Connah Johnson (MathSys) |
3rd December 2020 | Transaction tax in a general equilibrium model - Osian Shelley (MASDOC) | Fixing Bias in Zipf Estimators using approximate Bayesian computation - Charlie Pilgrim (MathSys) |
10th December 2020 | Differentiating random and deterministic uncertainty in Reinforcement Learning - Jake Thomas (MathSys) | Exact MCMC inference for Wright-Fisher diffusions - Jaro Sant (MASDOC) |
Abstracts
Week 2 (Talk 1) - Dislocations and Grain Boundaries: A Short Account - Anvar Atayev (MASDOC)
First described by Vito Volterra in 1907 and formally discovered independently by Egon Orowan, Michael Polanyi and Geoffrey Taylor in 1934, dislocations, a type of crystal defect and their derivative structures, such as grain boundaries, have been studied extensively by the mathematical and engineering communities since their inception. In this seminar, we provide a short and non-technical overview of dislocations and grain boundaries, describe how they're formed, how they're modelled and why they are interesting to study. Furthermore, a short discussion on the problem of equilibrium configurations of small angle grain boundaries from a view of discrete dislocation dynamics will be presented.
Week 2 (Talk 2) - Predicting Asymptotic Behaviour of Matched Solutions - Matthew King (MASDOC)
When solving complex ODE's, a solution may be limited by a radius of convergence. In these instances, a second solution may be proposed that is valid across the remainder of the domain. A Full solution can then be offered by performing matching between these solutions. Under certain asymptotic limits it may not be the case that these solutions follow the same behaviour, and to be able to make predictions for the full solution in these limits we must understand the matching process. Motivated by an example from aeroacoustics this talk will look at when these complications may occur, why the matching may not behave as might be initially expected, and how correct predictions may still be made.
Week 3 - A phase field model for raft formation on biological membranes - Luke Hatcher (MASDOC)
In this talk we introduce and develop a model for phase separation on biological membranes. Motivated by observations of subdomains (rafts) on biological membranes which compartmentalise cellular processes we propose a model which couples the Helfrich energy to a Cahn-Hilliard energy. Using a perturbation method, we describe the geometry of the membrane as a graph over a sphere. The resulting energy is a small deformation functional that is coupled to the interface.
We first explore the energy for a diffuse interface. We begin by focusing on the equilibria of the energy functional and use a gradient flow to numerically verify that the model predicts the formation of stable raft-like structures. We will discuss the parameter dependence of these subdomains. Subsequently, by calculating a Γ-limit, we relate the diffuse interface approach to a sharp interface approach.
Finally, we use Onsager's variational principle to address the non-equilibrium dynamics of a membrane. We obtain a Cahn-Hilliard equation with degenerate mobility coupled to a small deformation equation. We will discuss the challenges in proving an existence result for this problem and how they can be overcome. Again, we will consider the free boundary problem for the corresponding sharp interface approach.
Week 4 - Social event
This week's talks have been postponed due to speaker availability. Instead we will be meeting for a coffee and a catch-up to discuss our plans for events this academic year and to ask you what sort of things you would be interested taking part or joining in with! Please do come along, all are welcome!
Week 5 (Talk 1) - Meta stability in atmospheric jets - Nayef Shkeir (MathSys)
Turbulence in atmospheres, oceans and plasma flows leads to coherent large-scale jets that persist for long-times. These jets may be steady or transition between several meta-stable jet configurations. The main question is: Under what conditions, and from what mechanisms, can the system switch, and with what probability, i.e. how likely is it that Jupiter looses one of its jets. In this talk, we present a study of the dynamics of these atmospheric jets on large rotating Jovian planets where we can apply the stochastically forced two-dimensional barotropic equation and its various approximations.
Week 5 (Talk 2) - Point particle interactions on surface biomembranes - Philip Herbert (MASDOC)
In this talk, we discuss a model for protein interactions on a near spherical biomembrane. This is motivated by the fact that proteins are responsible for many vital task and, at large distances, any interactions are believed to be predominantly membrane mediated. We begin by discussing a simplified model for the membrane and the proteins. This is followed by outlining a method for numerically approximating the membrane-protein system. We conclude with the differentiability of the energy of the membrane-protein system, which may be utilised for a gradient descent algorithm.
Week 6 (Talk 1) - Recombination detection for viral genetic data - Ana Ignatieva (Statistics)
The processes of genetic mutation and recombination are fundamental drivers of viral evolution. Mutation events produce small changes within the genome during replication, and are generally visible in sequencing data. Recombination, on the other hand, occurs when genetic material from two parent particles is mixed together before being passed on to the offspring -- with the potential to rapidly and drastically change its pathogenic properties. The detection of recombination events from a sample of genetic data is a very challenging problem. I will give a brief overview of the combinatorial, algorithmic and statistical aspects of recombination detection, and discuss recent work combining these approaches with an application to viral sequencing data.
Week 6 (Talk 2) - Diffusion Limits at Small Times for Coalescent Processes with Mutation and Selection - Phil Hanson (MASDOC)
We introduce several backwards in time models present in population genetics that model lines of ancestry when a population is subject to random mutation and natural selection. When considering the ancestry of an infinite population we find that the number of ancestors instantaneously becomes finite as soon as we look into the past. We consider how quickly this happens (and what "quickly" means in this context) and consider these processes close to zero, characterising their asymptotic mean and second order fluctuations.
Week 7 (Talk 1) - A family of continuous-time dynamical systems with a trivial CLT - Nicolò Paviato (MASDOC)
Great interest has been shown in proving limit laws, such as the central limit theorem and Donsker's invariance principle, for a large class of discrete and continuous-time systems. To study their rates of convergence we consider a martingale method introduced by Gordin, which in some cases leads to an unforeseen result. In this talk we will show some of the difficulties that arise in dealing with semiflows that have a contracting family of transfer operators. The work presented here was done in collaboration with Prof. Ian Melbourne.
Week 7 (Talk 2) - Kesten Processes and Wealth Generation - Samuel Forbes (MathSys)
Kesten processes are discrete stochastic multiplicative processes which have wide applicability in fields such as economics, social science and neuroscience. Results on the convergent case were studied rigorously in the 1970s by Harry Kesten (1931-2019). However it has only been fairly recently that the non-convergent case has been studied. I will introduce the theory of Kesten processes as well as show simulations and discuss possible applications to wealth generation.
Week 8 (Talk 1) - The fully nonconforming virtual element method for fourth order perturbation problems - Alice Hodson (MASDOC)
In recent years, the discretisation of partial differential equations via the virtual element method (VEM) has seen a rapid increase. The virtual element method was first introduced to solve second order elliptic problems and is a generalisation of both finite element and mimetic finite difference methods. VEM spaces can easily be constructed to enforce desirable properties of the discrete functions even on general polygonal meshes which makes the approach very interesting for a wide range of problems. In this talk, we present a class of nonconforming virtual element methods for a general fourth order PDE in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. We analyse these nonconforming VEMs and look at their application to fourth order perturbation problems. Finally, we showcase the flexibility of our method by demonstrating the ease with which our approach can handle nonlinear fourth order problems via some additional numerical experiments..
Week 8 (Talk 2) - Why are living systems cellular? Modelling reaction-diffusion systems in biochemical systems - Connah Johnson (MathSys)
From the most complex higher organisms to the simplest bacterium, cells are a basic structure of living systems. Cells act as a container for cell bound chemicals, utilise their own internal chemical reaction systems, and dynamically evolve through growth and division events. But what impact does the introduction of cell entities have on the local and global chemical concentration fields? To model the role cells play in shaping their environment we develop a hybrid discrete-continuous software suite. In the simulations cells are modelled as discrete agents coupled to a continuous parabolic PDE domain with a focus on implementing realistic chemical reactions. We show the local perturbation of chemical fields due to cell agents. These perturbations may lead to altered local chemistry and reaction specification in living systems, potential drivers of biological complexity.
Week 9 (Talk 1) - Transaction tax in a general equilibrium model - Osian Shelley (MASDOC)
In this talk, we consider the effects of a quadratic tax rate levied against two agents with heterogeneous risk aversions in a continuous-time, risk-sharing equilibrium model. The goal of each agent is to choose a trading strategy which maximises the expected changes in her wealth, for which an optimal strategy exists in closed form, as the solution to an FBSDE.
This tractable set-up allows us to analyse the utility loss incurred from taxation. In particular, we show why in some cases an agent can benefit from the taxation before redistribution. Moreover, when agents have heterogeneous beliefs about the traded asset, we discuss if taxation and redistribution can dampen speculative trading and benefit the agents, respectively.
No knowledge of stochastic calculus required.
Week 9 (Talk 2) - Fixing Bias in Zipf Estimators using approximate Bayesian computation - Charlie Pilgrim (MathSys)
Zipf’s law describes a relationship between the number of occurrences of a word in a book and that word's occurrence ranking. I will show how the most popular estimator to fit Zipf models is biased. I will then go on to describe a population Monte Carlo algorithm combined with approximate Bayesian computation that can fit Zipf models with much less bias.
Week 10 (Talk 1) - Differentiating random and deterministic uncertainty in Reinforcement Learning - Jake Thomas (MathSys)
It is common practice in statistics to model all uncertainty using the structure of probability theory and random phenomena. However, in many circumstances uncertainty is not inherently random, instead it is due to a lack of information about a deterministic quantity. In this talk I will describe how possibility theory can be used to give a more accurate picture of deterministic uncertainty and outline how this can be applied in the context of reinforcement learning.
Week 10 (Talk 2) - Exact MCMC inference for Wright-Fisher diffusions - Jaro Sant (MASDOC)
Inferring genetically relevant features such as selection, mutation and effective population size from population-wide data has been a perennial problem for geneticists. Most of the traditional methods used are based solely on present day genetic information which greatly impairs the inference as the data used is essentially a static snapshot of the population being considered. Recent advances in gene sequencing as well as improvements in the technologies required to retrieve DNA from old remains such as fossils (called ancient DNA (aDNA)), have allowed for the creation of genetic time series datasets spanning several centuries. Such datasets potentially hold a wealth of information with regards to how several genetic factors and phenomena have influenced and helped shape the population upon which they act. However, eliciting such information from the data requires the development of more intricate statistical procedures. To make matters worse, the transition density of the Wright-Fisher diffusion (which is somewhat the "standard" diffusion of choice to model changes in allele frequencies) is analytically unavailable, leading to an intractable likelihood. In this talk, we propose a Markov Chain Monte Carlo (MCMC) setup which takes into account both the temporal nature of the observations as well as the intractability of the likelihood, whilst allowing for an exact inferential treatment of the parameters of interest (i.e. without the need to resort to any approximations).