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Student Seminars 2013

This page provides details of the seminars given by MASDOC associates. There are additional weekly seminars offered by both departments and details of these can be found in the Student Handbook.

The MASDOC seminars, unless notified otherwise, currently take place on Thursday 4pm-5pm in A1.01. If you are interested in giving a talk or inviting a speaker please contact Owen Daniel or Barnaby Garrod.

Term 2

List of Speakers:

09/01/14 Iain Rice (Aston) - Topographic Visualisation of multi-beam SONAR

23/01/14 Maria Veretennikova - On well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations

13/02/14 Jere Koskela - A tour of Approximate Bayesian Computation algorithms

27/02/14 Dominic Yeo (Oxford) - Mean-field Forest Fires

6/03/14 Clare Perryman (Exeter) - Rate-Induced Bifurcations: Can We Adapt to a Changing Environment?

13/03/14 Daniel Sanz Alonso - Filtering Partially Observed Deterministic Dynamical Systems



Filtering Partially Observed Deterministic Dynamical Systems

Daniel Sanz Alonso

The aim of filtering is to estimate, in an on-line fashion, the value of a stochastic process, the signal, as noisy observations become available. In this talk I will study discrete-time, randomly initialized signals that evolve according to a deterministic map $\Psi$ and assume that only a low-dimensional projection of the signal, given by an observation operator $P,$ can be observed. I will focus on the situation where the noise in the observations is small and we determine conditions on $P,$ and its relation to the map $\Psi,$ which ensure that the signal can be accurately tracked in the long-time asymptotic regime. We thus address the question of what observations are sufficient to reconstruct the signal accurately if the initial state of the dynamical system is uncertain.


Rate-Induced Bifurcations: Can We Adapt to a Changing Environment?

Clare Perryman (Exeter)

Rate-induced bifurcations occur in forced systems where there is a stable state for every fixed level of forcing. When the forcing varies too fast, the system fails to adiabatically follow the continuously
changing stable state and destabilises. The instability defines often non-obvious thresholds at which real-world systems fail to adapt to changing external conditions. We report on a novel threshold type
whose intricate band structure is organised by composite canard trajectories due to a folded saddle-node singularity. The results are obtained for slow-fast dynamical systems with one fast and two slow variables, akin to simple climate and neuron models, using modern concepts from geometric singular perturbation theory.


Mean-field Forest Fires

Dominic Yeo (Oxford)

We consider various adjustments to the standard random graph process, where additionally edges or vertices are removed when large components appear. In contrast to the original model, sometimes such processes display critical phenomena at all times. We consider existing methods, challenges and lots of open problems.


A tour of Approximate Bayesian Computation algorithms

Jere Koskela

Approximate Bayesian Computation (ABC) is an approximate method for conducting Bayesian inference without evaluating likelihoods. Despite having no guarantee of consistency, it has gained popularity due to being tractable for very complex models and large data sets. I will present the basic ABC rejection algorithm (Pritchard et al., 1999) and its various optimisations based on ideas from MCMC and importance sampling. The utility of these algorithms will be illustrated by considering parameter inference in population genetics under the spatial Lambda-coalescent (Etheridge, 2008).


On well-posedness and regularity of the Cauchy problem for nonlinear fractional in time and space equations

Maria Veretennikova

I will discuss the established theory for fractional differential equations and present our new results concerning fractional Hamilton-Jacobi-Bellman-type equations.


Topographic Visualisation of multi-beam SONAR

Iain Rice (Aston)

Large SONAR systems can produce hundreds to thousands of beams worth of data, the display of which can easily result in data overload for SONAR operators. The signals observed in the underwater domain are characterised by high frequency sampling rates, low signal to noise ratios and an intrinsic level of uncertainty. I will be discussing methods for representing these high-dimensional signals in a low-dimensional, topographic visualisation space. A topographic projection preserves the geometric structures of the data and neighbourhood relationships by mainly retaining the relative similarities between data in the high and low dimensional spaces. The data space is considered both in terms of the high-dimensional points and as the generating functions of the signals. Finally the use of prototypes in the dissimilarity space is demonstrated on a simulated dataset.

Term 1

Phase transitions and the random-cluster representation for Delaunay Potts models

William Nollett

We investigate the existence of phase transitions for a class of continuum multi-type particle systems. The interactions act on hyperedges between the particles, allowing us to define a class of models with geometry-dependent interactions. We establish the existence of stationary Gibbsian point processes for this class of models. A phase transition is defined with respect to the existence of multiple Gibbs measures, and we establish the existence of phase transitions in our models by proving that multiple Gibbs measures exist.


The Omega arcsine law

Yuchen Pei

In this talk I am going to talk about the "Omega arcsine law", which is defined piecewisely, as an indefinite integral of the arcsine distribution function on [ - 2 , 2 ] and as the modulus of the x coordinate outside of this interval. I'll show with a large deviation argument that the Omega arcsine law is the limit shape of the Young diagrams distributed according Plancherel measure. I'll also talk about the Krein-Markov correspondence, which shows that the asymptotic transition measure for the Plancherel growth is the semicircle law. Then if there's still time I may also talk about the Omega arcsine law as asymptotics of separation of roots of some orthogonal polynomials and its relation to the Burgers euqtion.

The talk is largely based on S. V. Kerov's monograph "Asymptotic representation theory of the symmetric group and its application in analysis".

Existence of weak solutions to parabolic PDEs via time discretisation

Andrew Lam

Partial differential equations have become a standard mathematical tool to model phenomena on a continuum level. Among them are parabolic equations that models the evolution of a quantity through time. Some well-known examples are the heat equation, Navier-Stokes equations, Porous-media equation and Mean curvature flow equation. A desirable property we would like to have is the existence of solutions (defined in a suitable sense) to these parabolic equations. So that, even if an exact formula for the solution is not available, numerical simulations of the PDE are still meaningful.

In this talk I will talk about a method to prove existence of weak solutions to a diffusion equation with nonlinear reaction terms. The main idea is to discretise the PDE in time, which leads to a sequence of elliptic problems. However, since there are nonlinear terms present, the standard Lax-Milgram theorem does not apply and so we further discretise in space and reduce it to a finite dimensional problem. We show that these discrete solutions exist and then show that they converge weakly to a weak solution when we pass to the limit. If time permits, I will outline how to deal with other types of nonlinearities.


Generating pseudorandom numbers

Chris Pettitt

It is impossible for deterministic computers to generate truly random samples for stochastic simulations. The aim of a pseudorandom number generator is to generate numbers that are indistinguishable from truly random samples in the sense that they are not rejected by hypothesis tests. I will give examples of pseudorandom generators (good and bad) and explain some important properties and theorems associated with the construction of a realistic and efficient generator. Elementary number theory will make a surprise(?) appearance!


Density Functional Theory: The Classical Hard-Core Gas

Simon Bignold

This talk will cover some of the material used for my MSc dissertation, supervised by Christoph Ortner. The topic of the dissertation was density functional theory, specifically a classical consideration of the one-dimensional hard-core gas in the canonical ensemble.

The talk begins by introducing some basic techniques required for density functional theory, after which we develop the ideal gas as a motivating example. We conclude the talk with a brief look at one of the methods used to consider the one-dimensional hard-core gas.

Density functional theory is a topic of interest in mathematics, physics and in chemistry. This talk reviews material required for a specific problem in the hope of demonstrating some of the techniques and problems encountered in density functional theory. This talk should hopefully be accessible to anyone with a reasonable degree of mathematical literacy.