17th May 2009, 1400 MC206
Prof. Fátima Silva Leite
University of Coimbra
will speak on
Geometry of Rolling Maps
Rolling maps describe how one smooth manifold rolls on another, without twist or slip. We will focus on the geometry of rolling a Riemannian manifold on its aﬃne tangent space at a point. Both manifolds are considered to be equipped with the metric induced by the Euclidean metric of some embedding space. The Kinematic equations of a rolling motion can be described by a control system with constraints on velocities, evolving on a subgroup of the Euclidean group of rigid motions, describing simultaneously rotations and translations in space. Choosing the controls is equivalent to choosing one of the curves along which the two manifolds touch. Issues like controllability and optimal control of rolling motions will be addressed and illustrated for the most well studied of these nonholonomic mechanical systems, the rolling sphere. Other interesting geometric features of rolling motions will be highlighted.
16th April 2009, 1100 MC206
Prof. Wilhelm Kaup
will speak on
Local Tube Realizations of CR-Manifolds and Maximal Abelian Subalgebras
9th April 2009, 1325 MC206
Dr. Marcus Hegland
Centre Mathematics and its Applications ANU
will speak on
Approximating the Solution of the Chemical Master Equation
Chemical reactions are stochastic processes. While in many applications the kinetic rate equations provide an adequate model for the dynamics of the expected concentrations this is not the case when the numbers of molecules involved in the reactions are small. This situation occurs in many molecular biological processes including signalling and gene regulation. In order to understand the eﬀect of the “reaction noise” one needs stochastic models.
The chemical master equation is a continuous time discrete state Markov model which describes how the probabilities of the states evolve over time due to the chemical reactions. The states are vectors of integers. Instead of the concentrations of the kinetic rate equations here the copy numbers of the chemical species form the components of the states. There values are typically between zero and a few hundred. The state space grows exponentially with the number of diﬀerent chemical species and, as the probability of each state is recorded one faces the “curse of dimensionality”. This is one reason why essentially all computational approaches in this area use stochastic simulation methods.
In this talk I will discuss methods to determine solutions of the chemical master equations numerically. The approach we adopt uses a variant of the sparse grid technique based on state space aggregation, which is a ﬁnite volume type approach. The approximation order can be controlled by a method introduced by Per Loetstedt and his collaborators which uses a piecewise polynomial approximation combined with aggregation. A new bound for the approximation error using this approach will be given. The sparse grid method will be illustrated for a very simple model of a signalling cascade. I will report on some work of an ANU student who was able to solve the master equation for a simple signalling cascade with 100 proteins species.