Modelling evolution of different genetic types in spatially structured populations.
Professor Alison Etheridge, University of Oxford, UK.
Since the pioneering work of Fisher, Haldane and Wright at the beginning of the 20th Century, mathematics has played a central role in theoretical population genetics. One of the outstanding successes is Kingman’s coalescent. This process provides a simple and elegant description of the way in which individuals in a population are related to one another. However, it only really applies to very idealised `unstructured’ populations in which every individual experiences identical conditions. Spurred on by the need to interpret the recent flood of DNA sequence data, an enormous industry has developed that seeks to extend Kingman’s coalescent to incorporate things like variable population size, natural selection and spatial and genetic structure. But, until recently, a satisfactory approach to populations evolving in a spatial continuum has proved surprisingly elusive. In this talk we describe a framework for modelling spatially distributed populations that was introduced in joint work with Nick Barton (IST Austria). As time permits we’ll not only describe the application to genetics, but also some of the intriguing mathematical properties of some of the resulting models.
Modeling the bio-fluid dynamics of reproduction: successes and challenges
Professor Lisa Fauci, Tulane University in New Orleans, Louisiana.
The process of fertilization in mammalian reproduction provides a rich example of fluid-structure interactions. Spermatozoa encounter complex, non-Newtonian fluid environments as they make their way through the cilia-lined, contracting conduits of the female reproductive tract. The beat form realized by the flagellum varies tremendously along this journey due to mechanics and biochemical signaling. We will present recent progress on integrative computational models of pumping and swimming in both Newtonian and complex fluids that capture elements of this complex dynamical system.
Mathematics, Astronomy and Physics – a Three-Body Problem
Professor Douglas C. Heggie, University of Edinburgh, Scotland.
From the time of the ancient Greeks until the death of Poincare just over a hundred years ago, applied mathematics and theoretical astronomy were much the same thing. But the astronomy was celestial mechanics: the motion of the planets. A hundred years ago this was the one part of astronomy which was left almost untouched by the spectacular growth of astrophysics, and the link between mainstream mathematics and mainstream astronomy was severed. The position has been transformed in the last decade or so by the discovery of numerous planetary systems beyond the solar system. It is now a major growth area in astronomy, and has brought the mathematical study of planetary systems once again to centre stage.
In this talk I will focus on the role which the classical gravitational three-body problem is playing in these developments, with particular focus on the averaging technique. It is a rich subject which sheds light on much besides extrasolar planets, including the evolution of stellar clusters around black holes, the death of comets, and even the evolution of the Moon.
Slippery issues in micro and nanoscale flows
Professor Shaun Hendy, MacDiarmid Institute for Advanced Materials and Nanotechnology, University of Auckland.
Nano and microscale fluid flows are very important in the natural world and are becoming increasingly useful in a variety of technological settings. Nevertheless there are still many aspects of nanoscale fluid dynamics that are poorly understood. The no-slip boundary condition for the flow of simple liquids over solid surfaces was considered to have been experimentally established in the 1920s, yet new measurement techniques have recently demonstrated that there can be large violations of this boundary condition at the nano and microscale. Here we consider the role of slip boundary conditions in fluid flows using a theoretical approach, complemented by molecular dynamics simulations, and experimental evidence where available. Firstly, we consider nanoscale flows in small capillaries, including carbon nanotubes, where we have observed a breakdown of the Lucas-Washburn equations for capillary uptake. We then consider the general problem of relating macroscopic boundary conditions for fluid flow to microscopic surface chemistry, and discuss several cases where we have been able to solve this problem analytically. Finally, we look at applications of these results to carbon nanotube growth, catalysis, and the best way to put insulation in your roof.
Discovering the geometry of chaos
Professor Bernd Krauskopf, Mathematics Department, The University of Auckland.
The fact that even simple systems can display chaotic dynamics, has been quite a revelation. The Lorenz system of just three ordinary differential equations is arguably the most famous chaotic dynamical system: its chaotic `butterfly attractor’ appears on numerous book covers, T-shirts and coffee mugs. But how does the chaos arise from simple dynamics when parameters are changed? This question can be answered by investigating how the three-dimensional phase space is organised by surfaces known as global invariant manifolds, and how this organisation changes on the route to chaotic dynamics. This type of global study is made possible by recently developed numerical methods for finding invariant manifolds that are based on the solution of two-point boundary value problems. On the route to chaos there are intriguing topological and geometrical structures. Did you know, for example, that the chaotic dynamics itself is organised by a space filling pancake?
Disease modelling and its impact on policy decisions
Professor Geoff Mercer, National Centre for Epidemiology and Population Health, Australian National University.
Disease modelling is a powerful tool that can influence government policy and operational decisions but it is currently underutilised. There are three natural time frames of interest to both the modeller and the decision maker: the past, the present and the future. The modelling of past disease outbreaks when coupled with data can provide valuable insights into the mechanisms of disease transmission and inform future interventions. Real time modelling for operational decision support is rarely undertaken but potentially has a huge benefit. The most commonly used disease modelling is future scenario modelling to inform policy decisions such as vaccine rollout. In this talk I will discuss the three different time frames of disease modelling and give case studies of each.
The statistical dynamics of geophysical flows with application to ensemble prediction and data assimilation
Dr Terry O’Kane, CSIRO Marine & Atmospheric Research.
Geophysical flows relevant to atmosphere-ocean large-scale circulations are complex, involving the interaction of inhomogenities (mean flows and topography) with turbulent eddies. An understanding of the statistics of turbulent geophysical flows is difficult due to the range of scales encompassed and the complexity of the interactions involved. The number of ensemble members required for direct numerical simulations to adequately sample the probability distribution function is in general prohibitive and until recently the development of a computationally tractable statistical dynamical theory for inhomogeneous turbulent flows has proved elusive. In this talk I will describe recent advances in the non-equilibrium statistical theory and numerics for geophysical flows and their application to problems in data assimilation and weather prediction.
Real-Time Control of Ambulance Fleets through Statistics, Simulation and Optimization.
Professor Shane Henderson, School of Operations Research and Information Engineering, Cornell University
Ambulance organizations everywhere face increasing call volumes, increasing traffic congestion, and shrinking budgets. To keep response times small, many employ some kind of system-status management (SSM). SSM is the practice of real-time control of the ambulance fleet, using GPS units on the ambulances to track location, and information from the ambulance crews to track status. Available ambulances are carefully stationed in real time, while not requiring too many moves of the ambulance crews. I’ll describe recent work in the area, highlighting the use of mathematical tools including statistics, approximate dynamic programming, simulation, and optimization. This work has motivated us to develop simulation optimization algorithms for high-performance computing environments, and I’ll discuss what we view as the key complexity there.