Abstracts

The abstracts for the talks and posters are given below.

Abstracts of Talks

Andrew Fenn - The Face-Polynomial of Nestohedra

I will present an introduction to polytopes and an overview of some of their uses. I will then move on to one of the most interesting aspects of polyhedra and present calculations of the face-polynomials of many important series of nestohedra. A nestohedron is a simple polytope which arises from a connected graph. A series of nestohedra is a set of nestohedra arising from a series of graphs which is defined so that each successive graph is obtained from the previous one by the addition of a node that is connected to some subgraph or by replacing a node with two nodes connected by an arc. This raises the problem of how the face polynomials are related to each other both within and beween series. Examples of such series of nestohedron include the Permutohedron and the Associohedron or Stasheff polytope.

The problem of how to describe the faces of a nestohedron in terms of its graph is well known. It is usually attacked using f-vectors. We show that an approach based on the face-polynomial of a polytope is a finer tool, since in the case of the face-polynomial we can transform operations on the graph into analytical techniques, which we cannot do with the f-vector.

Chris Johnson - Granular flows and avalanches

Granular materials are common in everyday life, but the mechanisms governing their behaviour are still poorly understood. An important class of granular flows are avalanches, which occur in both large-scale geological and environmental situations, and as part of a range of smaller-scale flows which are useful in industry. I will present some current theories of avalanche propagation and of granular flows in general, and show how these can be used to model a number of real-world problems.

Matthew Hunt - The Shape of a Mach Stem

Unlike light, when a shock wave reflects from a solid wall, the angle of incidence is not the same as the angle of reflection. Indeed given a particular incident angle, the reflection point may not be at the wall with a "stem" leading from the reflection point to the wall!! If this is the case, then we say that there is a Mach reflection. The aim of this talk will be to introduce some basic concepts from shock dynamics and to calculate the actual shape of the stem via a asymptotic expansion of the Euler equations.

Chaofeng Kou - Joint Mean-Covariance Modeling and Variable Selection for High Dimensional Longitudinal Data

The defining characteristic of a longitudinal study is that individuals are measured repeatedly through time. In many fields, such as multilevel studies and time series, people are interested in finding out how both the average value and the dispersion of the responses vary over time and how they are affected by different experimental treatments. This is the main objective of modelling of mean and covariance structures. One of the challenges of covariance modeling is that the estimation of covariance matrix should be positive definite. Also, technological innovations allow us to collect massive amount of longitudinal data with relatively low cost. Most of these variables are not significant and need to be removed form the final models. In this talk, I will introduce the main idea about how difficulty caused by positive definite constraint can be overcome, how covariance structures are modelled together with mean and then discuss how to select the most important covariates variables for the mean and covariance structures.

Gemma Lloyd - Knitting Mathematics

"We have sailed many months, we have sailed many weeks,
(Four weeks to the month you may mark),
But never as yet ('tis your Captain who speaks)
Have we caught the least glimpse of a Snark!"

Fit the Second - The Bellman's Speech

The Hunting Of The Snark (An agony in eight fits) by Lewis Carroll

We introduce the Snark and then use knitting to give a different way of representing these and indeed other mathematical objects. No prior knowledge of knitting is assumed (or required).

Stephen Miller - Framed Cobordism and the Homotopy Groups of Spheres: the Pontryagin Construction

Higher homotopy groups are a natural extension of the fundamental group, one of the basic tools of algebraic topology. They carry a lot of information about the topology of a space. For example, a map between "nice" spaces that induces an isomorphism on all homotopy groups must be a homotopy equivalence. Unfortunately, homotopy groups are not as well behaved as the fundamental group or the well known homology groups. Even calculating the homotopy groups of spheres, seemingly the simplest of topological spaces, is a major outstanding problem in topology. The Pontyagin construction gives some insight by translating the problem into another setting. Calculating the (n+i)-th homotopy group of the n-sphere turns out to be equivalent to classifying framed i-dimensional manifolds in co-dimension n, up to an equivalence relation known as framed cobordism. In this talk I plan to give an overview of homotopy groups and framed manifolds, and explain how the Pontryagin construction allows us to pass from one to the other. I will then show how to classify one- and two-dimensional framed manifolds in any co-dimension, which gives us some of the homotopy groups of spheres. Unfortunately this is only a small part of the problem, and in higher dimensions working with framed manifolds becomes difficult very quickly. For that reason, more results have been produced using algebraic methods such as spectral sequences. One goal of current research is to bridge the gap between the algebraic and geometric approaches by interpreting the algebraic sequences in terms of manifolds and related geometric objects.

Simon Perera - Grothendieck Rings of First Order Structures

In Model Theory we study objects from other areas of mathematics using a suitable first order language. Then we can often use the machinery of this branch of logic to learn something about the original area of mathematics. I will explain how for a given mathematical structure and with a chosen F.O.L. we can construct an algebraic object called the Grothenieck Ring from the collection of definable sets and maps. This object has similarities with the Grothendieck Rings in other areas of maths such as algebraic geoemetry and homology. I will show how to construct the Grothendieck Ring of a vector space using the standard F.O.L. for modules.

Mladen Savov - Lévy Process - Brownian Motion with Jumps ?

The Brownian motion has been named after the botanist Robert Brown, who observed the random movement of small particles suspended in a liquid or gas, which is due to numerous collisions with small water molecules. The first traces to such observations date back to Lucretius's poem "The nature of things". The mathematics and physics behind this movement have been thoroughly investigated. Common sense suggests that the movement is continuous and it turns out that the Brownian motion is the only process that can describe such a system. Imagine now that the small particle can teleport (may be too artificial but if you think of insurance company that receives an insurance claim it makes sense) from time to time but the key features of the Brownian motion are preserved. Such motion is described by the Lévy process. In this talk we discuss the basic features of a Lévy process process.

Hadi Zare - A Combinatorial Approach to Mirror Symmetry

I will try to explain what we mean by an invariant in algebraic topology, and how some of these are related to counting. I hope to give examples of some famous invariants, and state one of famous conjecture in theoretical physics known as Mirror Symmetry.

Abstracts of Posters

Edwin Broni-Mensah - A Simple and Generic Methodology to Suppress 'Non-Linearity' Error of Option Pricing

The application of lattice, trees, and quadrature methods to solve multi-dimensional problems are, usually, limited to five dimensions due to the exponential increase in computational time with each increment in dimensions. This is often referred to as the curse of dimensionality. Given that the curse of dimensionality is inherent in most numerical schemes, Monte Carlo methods are usually preferred when solving high dimensional problems. This work proposes and develops a generic methodology which is simple and applicable to calculate multidimensional options for a given computational effort and furthermore can easily be incorporated into several numerical schemes. The methodology is powerful as it eliminates the non-monotonic behaviour that is present due to the mistreatment of regions where the solution is highly non-linear. This novel approach allows for extrapolation techniques to be employed on data sets which are anything but monotonic.

Kwan Yee Chan - Asymptotic Theory of Separated Flow from an Airfoil

The theory of flow separation from a solid body surface has been developing since the middle of the 19th century. The importance of which was proved when it was realised that boundary layer separation at the leading edge of an airfoil is the main contributor to the limit of lift force acting on the airfoil in a fluid stream. It is with the method of matched asymptotic expansions, found throughout mathematical physics, that the most productive advances have been made. I will present a brief introduction to asymptotic theory of separated flow on airfoils; charting the beginnings up to the present, stating the principal ideas of the theory and it's applications.

Ali Everett - Commuting Involution Graphs of Symplectic Groups

Let G be a finite group, and X a subset of G. A commuting graph on (G,X) has the elements of X as its vertices, and edges exist between vertices x and y, if and only if x and y commute in G. A commuting involution graph of G is one where X is a conjugacy class of elements of order 2. Notions relating to commuting graphs (with a simple, yet pretty example), and a brief definition of a symplectic group will be given. Then a theorem describing the size of two graphs of a certain family of symplectic group will be given, with a (very, very) compact explanation of how one would prove the theorem. The actual graphs themselves will also be presented.

Elinor Jones - An Introduction to Lévy Processes

The theory of stochastic (random) processes is one of the most important mathematical developments of the twentieth century. These processes aim to model the interaction between 'chance' and 'time', and are not only mathematically rich objects in themselves, but have an extensive range of applications in Physics, Engineering, Ecology and Economics.

We take a look at a particular class of stochastic processes, known as Lévy Processes, named in honour of the French Mathematician Paul Lévy. Typically, a Lévy Process will consist of continuous motion, possibly interspersed with jump discontinuities of random size, appearing at random times along its sample path. Familiar examples include Brownian Motion, the Poisson Process and Stable Processes.

We aim to give a rigorous definition of a Lévy Process, coupled with some simple examples.

Robert Logue - Instability of Supersonic Compression Ramp Flow

The main objective of the current work is to investigate the stability of the supersonic flow past a compression ramp. For increasing ramp angles a recirculation region develops in the vicinity of the corner and for very large angles secondary separation is also observed. The work presented uses two distinct approaches to investigate the stability of the flow. First, a global linear stability analysis, taking disturbances proportional to eμt, is preformed. Secondly, numerical simulations have been carried out with the linearised unsteady equations, linearised about a steady state, using forced disturbances. The computations are carried out using a hybrid high-order finite difference and spectral method. The numerical technique is robust and can compute the basic flow accurately for large ramp angles. The stability analysis yields some surprising results.

Chris Munro - Polynomial Eigenvalue Problems

The Polynomial Eigenvalue Problem (PEP) arises in a number of applications, in this poster we will describe how PEPs with small to medium dimension coefficients can be solved by linearization. We will describe a number of applications that yield PEPs, and also mention the frequently occurring case of degree 2 or Quadratic Eigenvalue Problems. In addition, we will describe recent work that has focused on deflation methods for quadratic matrix polynomials.

Mladen Savov - Laws of Iterated Logarithm for Lévy Processes with Unbounded Variation at Small Times

The Law of the Iterated Logarithm (LIL) is one of the deepest results in the classical theory of probability. Historically, any stochastic process whose maximal rate of growth is described by the deterministic function (2t ln | ln t|)1/2 is said to exhibit LIL behaviour. The importance of this law is so big that nowadays we refer to LIL for a stochastic process even though the rate of growth of the process may have nothing to do with iterated logarithm ( (2t ln ln t)1/2 ). In this poster some new results on the small time behaviour of Lévy processes will be presented. Let X be a Lévy process (if you have doubts what a Lévy process is, please see Elinor's poster) and X*t = sups<t |Xs| , Xt = sups<t Xs. In this poster a comparatively systematic way of describing the deterministic functions, such that limsupt → 0 Xt / b1(t) = limsupt → 0 |Xt| / b1(t) = 1 and limsupt → 0 Xt / b1(t) = 1, is proposed. It is shown that there cannot be a universal link between the characteristics of the Lévy process and b1(t), and b2(t). Integral criteria are derived for checking whether a given function describes the maximal rate of growth for any Lévy process.

Paul Taylor - Monte Carlo Algorithms in Black-Box Group Computation

A Monte Carlo algorithm is one which can be guaranteed to return a correct answer with probability no less than (1-ε), where the value of ε is specified in advance by the user. We discuss the implementation of some randomized Monte Carlo algorithms for computing within black-box groups.

Andreas Vrahimis - Bias Corrected Bootstrap Method to Estimate the Optimal Smoothing Parameter in Density Estimation

We present the methodology behind the use of the bootstrap resampling method to estimate the optimal smoothing parameter in univariate kernel density estimation, along with the bias introduced with this method. The source of the bias is then identified and explained along with a correction that deals with the bias problem effectively. Results of a simulation study are presented that show how the correction manages to improve the method for various samples, simulated from distributions with different features. Also most of the already established methods are included in the simulation study to show that the corrected bootstrap method provides comparable results with these methods.