The caloron transform is a `fake' dimensional reduction which transforms a G-bundle over certain manifolds to a loop group of G bundle over a manifold of one lower dimension. This talk will review the caloron transform and show how it can be best understood using the language of pseudo-isomorphisms from category theory as well as considering its application to Bogomolny monopoles and string structures.

In the first part of the talk I will illustrate some basic concepts of differential geometry that lead to the notion of a holonomy group. Then I will explain Berger's classification of Riemannian holonomy groups and discuss questions that arose from it. Finally, I will focus on holonomy groups of Lorentzian manifolds and indicate briefly why all this is of relevance to present-day theoretical physics.

I will discuss two problems in porous media flow.

On a tropical island, fresh water may sit in the soil beneath the ground, floating on the ocean's salt water. This water is a valuable resource for the inhabitants, but requires sufficient rainfall to recharge the lens. In this paper, Green's functions are used to derive an integral equation to satisfy all of the conditions except those on the interfaces, which are then solved for numerically. Conditions under which the lens can be maintained will be described. This is work I did with an Honours student, Sue Chen, who is now at U. Melbourne.

In the second problem, I will discuss an "exact" solution to a problem in withdrawal from an unconfined aquifer. The problem formulation gives rise to a singular integral equation that can be solved using a nice orthogonality result I first met in airfoil theory. This is work with Hong Zhang from Griffith University.

This talk is a continuation of the talk on 30 April. The same abstract applies: In this talk, I shall briefly review the notion of stability for holomorphic vector bundles on compact complex manifolds as discussed in the first part of this talk (28 August 2009). Then I shall attempt to compute some explicit examples in simple situations, illustrating the use of basic algebraic-geometric tools. The level of the talk will be appropriate for graduate students, particularly those who have been taking part in the algebraic geometry reading group meetings.

Meshless methods such as vortex methods (VMs) and smoothed particle hydrodynamics (SPH) schemes offer many advantages in fluid flow computations. Particle-based computations naturally adapt to complex flow geometries and so provide a high degree of computational efficiency. Also, particle based methods avoid CFL conditions because flow quantities are integrated along characteristics. There are many approaches to improving numerical methods, but one of the most effective routes is quantifying the error through the direct estimate of residual quantities. Understanding the residual for particle schemes requires a different approach than for meshless schemes but the rewards are significant. In this seminar, I will outline a general approach to understanding convergence that has been effective in creating high spatial accuracy vortex methods, and then I will discuss some recent investigations in the accuracy of diffusion operators used in SPH computations. Finally, I will provide some sample Navier-Stokes computations of high Reynolds number flows using BlobFlow, an open source implementation of the high precision vortex method.

High on a mathematician's to-do list is classifying objects and structures that arise in mathematics. We see patterns in things and want to know what other sorts of things behave similarly. This poses several problems. How can you tell when two seemingly different mathematical objects are the same? Can you even tell when two seemingly similar mathematical objects are the same? In fact, what does "the same" even mean? How can you tell if two things are the same when you can't even see them! In this talk, we will take a walk through some areas of maths known as algebraic topology and category theory and I will show you some of the ways mathematicians have devised to tell when two things are "the same".

The Whitehead tower of a topological space seeks to resolve that space by successively removing homotopy groups from the 'bottom up'. For a path-connected space with no 1-dimensional local pathologies the first stage in the tower can be chosen to be the universal (=1-connected) covering space. This construction also works in the category Diff of manifolds. However, further stages in the two known constructions of the Whitehead tower do not work in Diff, being purely topological - and one of these is non-functorial, depending on a large number of choices. This talk will survey results from my thesis which constructs a new, functorial model for the 2-connected cover which will lift to a generalised (2-)category of smooth objects. This talk contains joint work with Andrew Stacey of the Norwegian University of Science and Technology.

The interspersed repeat content of mammalian genomes has been best characterized in human, mouse and cow. We carried out de novo identification of repeated elements in the equine genome and identified previously unknown elements present at low copy number. The equine genome contains typical eutherian mammal repeats. We analysed both interspersed and simple sequence repeats (SSR) genome-wide, finding that some repeat classes are spatially correlated with each other as well as with G+C content and gene density. Based on these spatial correlations, we have confirmed recently-described ancestral vs clade-specific genome territories defined by repeat content. Territories enriched for ancestral repeats tended to be contiguous domains. To determine if these territories were evolutionarily conserved, we compared these results with a similar analysis of the human genome, and observed similar ancestral repeat enriched domains. These results indicate that ancestral, evolutionarily conserved mammalian genome territories can be identified on the basis of repeat content alone. Interspersed repeats of different ages appear to be analogous to geologic strata, allowing identification of ancient vs newly remodelled regions of mammalian genomes.

Many complex datasets suffer from missing data, and interpolating these missing elements is a key task in data analysis. Moreover, it is often the case that we see only a linear combination of the desired measurements, not the measurements themselves. For instance, in network management, it is easy to count the traffic on a link, but harder to measure the end-to-end flows. Additionally, typical interpolation algorithms treat either the spatial, or the temporal components of data separately, but in many real datasets have strong spatio-temporal structure that we would like to exploit in reconstructing the missing data. In this talk I will describe a novel reconstruction algorithm that exploits concepts from the growing area of compressive sensing to solve all of these problems and more. The approach works so well on Internet traffic matrices that we can obtain a reasonable reconstruction with as much as 98% of the original data missing.

We show uniqueness up to sign of positive, orthogonal almost-Kahler structures on any non-scalar flat Kahler-Einstein surface. This is joint work with A. J. di Scala.

Data assimilation aims to solve one of the fundamental problems ofnumerical weather prediction - estimating the optimal state of the atmosphere given a numerical model of the dynamics, and sparse, noisy observations of the system. A standard tool in attacking this filtering problem is the Kalman filter.

We consider the problem when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. We derive the corresponding Kalman filter in a variational setting.

We analyze the variance constraining Kalman filter (VCKF) filter for a simple linear toy model and determine its range of optimal performance. We explore the variance constraining Kalman filter in an ensemble transform setting for the Lorenz-96 system, and show that incorporating the information on the variance on some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.

Using methods from dynamical systems theory we then systems where the un-observed variables evolve deterministically but chaotically on a fast time scale.

This is joint work with Lewis Mitchell and Sebastian Reich.