MSI Weekly Bulletin - Week starting Monday 14 August, 2006

The Shallow Water Wave equations are used to model many different systems such as the propagation and runup of tsunamis and the flow resulting from a dam break. This talk will review the main techniques used to solve non-linear hyperbolic conservation laws such as the shallow water equations. Methods to be discussed include the streamline diffusion and discontinuous Galerkin finite element methods, naive difference upwind and Godunov-type finite volume methods. The remainder of the talk will discuss in detail the application of a central-upwind Godunov finite volume method to the one-dimensional shallow water wave equations and its performance when solving certain exact analytical solutions.

I consider robust methods for the construction of sampling designs in spatial studies. The designs are robust against misspecified regression responses, and are tailored for possible use with predictors which are minimax robust against misspecified variance/covariance structures. The loss function is based on the mean squared error of the predicted values. This is maximized, analytically, over a neighbourhood quantifying the departures from the fitted linear regression response. This maximum is then minimized numerically - by simulated annealing, or sequentially - in order to obtain the optimal designs.

In 1904, Henri Poincaré conjectured that (in today's English) "A closed 3-dimensional, simply connected manifold is homeomorphic to the 3-sphere". After some missteps in the next fifty years, this conjecture seemed to be lying dormant for nearly 99 years. You will hear final news on this conjecture from Prof. Gerhard Huisken at 4:00pm 24 Ausgust MSI colloquium. In this talk, I merely want to prepare the audience for Gerhard Huisken's talk. I will review the development of low dimensional topology in the last 100 years, revisit some of the history and legends surrounding the Poincaré conjecture, such as the proof of Stephen Smale on the beach of Rio for high dimensional (greater than 4) case, Michael Freedman's proof of the 4-dimensional topological Poincaré conjecture, and their interactions with quantum field theory.