Wall's D(2) problem: algebraic topology and topological algebra

Wall's dimension 2, or D(2), problem is a conjecture involving three-dimensional spaces defined in a certain way, and algebraic objects associated to these spaces. I will try to briefly describe the problem and the algebraic objects involved, and give an idea of how connections between the spaces and the algebraic objects arise.

El Niño initialization

Understanding and predicting the El Nino phenomenon is of high priority in climate science. Simple models help to understand the dynamics while initialization improves predictability. I will explain a little about why El Nino occurs, talk about the concept of initialization and then describe my efforts to initialize a simple model.

Almost Intersecting Families That Are Almost Everything

A family of sets is called intersecting if any pair of sets in the family have a non-empty intersection. A natural question to ask is how large such a family of sets can be. The answer is quite surprising, and its proof is both beautiful and elegant. I will be talking on weakening the condition that sets must intersect, and seeing what we can say about the size of the family.

Baroclinic Instability and Equilibration of an unstable jet

Baroclinic instabilities are responsible for a lot of the weather systems seen in the mid-latitudes of the Earth and so their study is of great interest. The mechanism of these flows will be discussed as well as a brief introduction to initialisation problems concerning baroclinic jets.

The covering group of SL_2(R)

I will try to explain what a covering group is and show how to construct a non-trivial covering group for the matrix group SL_2(R). If there is time, I will show how the covering group is trivial when we choose a congruence subgroup of SL_2(R).

Finite area vortex dynamics on the surface of a sphere

I'll give a brief introduction to 2-D vortex dynamics and show plenty of pictures of vortices arising in nature. I'll then introduce a couple of problems I've looked at and give a quick summary of the methods used to solve them and then show some videos of the results and look at some examples of where these models can be applied in geophysics.

Good behaviour is rare

Suppose that you are asked to draw the graph of a continuous function. Almost surely your graph is smooth except, perhaps, at some sharp points. Unfortunately such a graph is too tame to give you a typical picture of a continuous function because almost all continuous functions are much wilder. In this talk, I will explain how badly most continuous functions (or other mathematical objects) behave.

Computational Stokesian Dynamics

The area of suspension mechanics is a highly applicable to many industries, from pharmaceutical processing plants to injection moulding of filled plastics. Suspensions are widely used but difficult to understand especially in large domains. The computational technique of Stokesian Dynamics considers a suspension of spherical particles within a viscous fluid undergoing a linear flow. There are exact integral equations to describe the physical system, which use the Oseen tensor to represent particles by a series of point forces. This integral form can be truncated (the essence of Stokesian Dynamics) to achieve a computational speedup -- however, it is still time-consuming when considering a large domain where far-field interactions converge only slowly in space. If the suspension is spatially periodic (over a lattice) great improvements can be made. Ewald summation has been previously applied to three dimensional domains to great effect. That work, however, cannot be applied to monolayers or confined suspensions. The aim of our work is to develop a method that will take advantage of the speedup offered by Ewald summation and apply it to a two-dimensional domain.

Why you need to know about Fractal Dimensions

This talk will try to communicate what fractal dimensions are, why they are interesting, and why other things are sometimes more interesting (like getting root on department computers ;) )

A New Look For The Nuetrino

Weyl's equation is a system of 2 homogenous linear partial differential equations for 2 complex unkowns. It's formulation requires spinors, Pauli matrices and covariant differentiation. Our new formulation only requires much simplier tools and becomes more intuitive and elegant.

Sound Propagation in Slowly Varying Flow Ducts

I'll be giving a brief introduction into duct acoustics with the the main application being to understand and therefore attempt to reduce the level of noise produced by a turbofan engine. The analysis begins with looking at the Navier Stokes equations. By performing a simple trick based upon our understanding of the physical situation we can deduce a set of equations that govern the acoustic field. Once these equations have been established, we can then apply various techniques to deduce solutions to these equations, and I am going to describe two of the main techniques we use, namely 'The Method of Multiple Scales' and the WKB method.

Longest convex chains

Let T be a triangle in the plane, and take n independent random points with uniform distribution in T. What is the maximal number of points among the chosen one which are in convex position with two fixed vertices of the triangle? We call such a collection of points a convex chain. The main result is to give the order of asymptotics for the expectation of this maximal length; we also prove a strong concentration result for the location of these convex chains, where the 'mystic' affine perimeter gets into sight as well. This work is joint with prof. Imre Barany.

TBD

Not available

Computational Complex Hyperbolic Geometry

Something about modeling Complex Hyperbolic space and it's automorphisms using C++ and using these models to compute fundamental domains for certain arithmetic subgroups.