The Polar Night Vortex during Stratospheric Sudden Warmings: Vertical Structure and Coherence

Stratospheric Sudden Warmings (SSWs) are events which occur in the Polar winter stratosphere approximately once every two years. The polar winter stratosphere is dominated by the Polar Vortex, a region of high potential vorticity in comparison with the background flow. During SSW events, the Polar Vortex is observed to weaken and eventually break down. Important questions arise from the nature of this breakdown. For example, what is the influence of planetary waves from the troposphere, and what implications does the variation with height hold for the underlying mechanisms?

The talk will start with a brief introduction to the middle atmosphere and it's characteristic properties, principally the concept of potential vorticity. Using observational data we then construct three-dimensional visualizations of the Polar Vortex. These visualizations are used to discuss the possible mechanisms responsible for the generation and instigation of SSWs. See here for further details http://www.appmath.columbia.edu/ssws/.

Hopf Algebras In Galois Module Theory

Given a finite Galois extension of fields L/K, classical Galois module theory seeks to describe the ring OL of algebraic integers of L as a module over the integral group ring OK[G]. In Hopf-Galois Module theory, we replace the Galois group by a (possibly non-unique) Hopf Algebra, giving a so-called Hopf-Galois Structure on the extension. We then investigate the behaviour of the ring of integers OL in this Hopf-Galois Structure. In the case that several Hopf Algebras can be attached to an extension in this way, we can compare the behaviour of OL in the different Hopf-Galois Structures. I shall give an overview of this theory and talk about my own work on certain elementary abelian extensions.

An introduction to Stokes Flow

In Stokes flow we consider the limit of the Navier-Stokes equations where in the battle for dominance between viscous terms and the inertia terms, the viscous terms win. This allows us to simplify our system considerably, however as we live in a world with inertia this simplified world can lead to some non intuitive results. In this talk we shall look at the Stoke's equations, their properties and a couple of interesting examples.

The paradox of the face

Higher mathematics in relation to the paradox of the face.

On the Cusp

The group SU(2,1;C) acts on complex hyperbolic space (a space with two complex dimensions represented by the negative lines in C^3 with respect to some hermitian form). On restricting from C to the integers of some imaginary quadratic numberfield the quotient of the action gives a non-compact fundamental domain (FD) with cusps (like SL(2,Z) acting on the upper half plane). There is an idea of a good approximation to a FD called a Siegel set (SS). SS's have the property that only a finite number of transformations can act non-trivially in them, so in that sense it is a matter of technique to derive a FD from one. In 2003 Francsics and Lax published an SS for SU(2,1;Z[i]) and later that year completed this to a FD. A FD is also known for SU(2,1;Z[(1+i\sqrt{3})/2]), this method is very pretty, but also very specific to the ring. I have extended the work of Fransics and Lax to deal with the class number 1 case. I will be talking about the contribution I have made to the proof and how the pari number theory library has been instrumental in computing these SS's.

Please note that I have made this sound much more complicated than it actually is in order to make it sound like I do real maths. This is a talk about basic linear algebra.

The eight 3-dimensional geometries

The eight 3-dimensional geometries arose from work of William Thurston in the late 1970's. By defining a 'geometry' in a certain way he was able to show that there exist eight distinct 'geometries' in 3 dimensions, which he proposed are the basic building blocks of all 3-dimensional manifolds. In this talk, I will try to explain the definition of a 'model geometry', describe the eight geometries and outline Thurston's proof that these are the only geometries that exist in three dimensions.

Real integration and random Riemann sums

Probably everyone in the world is at least vaguely familiar with the construction of the Riemann integral using Riemann sums. Riemann integrability is a condition on a function which assures that all Riemann sums using small enough intervals will be close to one value, the Riemann integral. My work last year examines Riemann sums on functions which are not Riemann integral. We treat Riemann sums as random variables, which gives a probabilistic method of integrating a function, by sampling its values at randomly chosen points. If this 'random Riemann sum' is progressively closer one value with higher and higher probablilities as we use smaller and smaller intervals, we can define the integral of our function to be equal to this value. The main result I will be discussing is that this construction is equivalent to the Lebesgue integral for Lebesgue measurable functions.

Presentations of Groups

I will present an open problem which has motivated much of my research (as well as the research of the other other topologists in this department). I will also give a geometric interpretation of the problem (with the aid of a balloon). Knowing what a group is would be ideal, but I will go through an example of a group carefully at the start.

Freestyle

Due to various technical difficulties it has been a while since we have had a PReSS talk. So to get some momentum Chris has proposed a PReSS talk freestyle. This is going to consist of a number of short talks (up to 10 minutes) on any topic. Talks by Shingo, Wajid, Chris and Brian.

Vortex Dynamics

Applied topology - looking at physical systems in a shapey manner

Summary: With My tutor we have created a new method for measuring the twisted nature of a 3-d curve be it open or closed, I will introduce this method, in an everyone friendly way (lots of pictures) and then discuss a few of the ways I intend to apply it.

Modeling biomedical flows

Summary: I'll talk about the neurosurgical problem that I am studying, how this can be modeled mathematically and how the equations that arise from the problem can be solved.

*Unplugged*

Beating the hypercube in certain connectedness properties.

Price game

On certain aspects of inter-dimensional lizards and stability in their breeding.

Nowhere differentiable continuous functions

After explaining the history of nowhere differentiable continuous functions, I shall give a precise statement of a theorem saying that most continuous functions are nowhere differentiable. If time permits, I shall state a theorem that I recently proved.

Some aspects of a continuum model for a chute flow of grains

Will give a brief introduction to my PhD topic and then go on to discuss some of the work done in the second year of my PhD. The work covers some solutions to a hyperbolic pde by the method of characteristics, a little bit of asymptotics and some numerical work (mainly finite difference methods). However, I will pitch the talk at a 'general interest' level and really only give the rudiments of the above-mentioned solutions.

From Calculus to Colouring in

I will explain how notions of vector fields, Lie algebras and topology may be translated to arbitrary sets with associative product. I will give an example of a simple combinatorial problem which is easily solved by this way of thinking, yet remained unsolved for a long time.