MSI Weekly Bulletin - Week starting Monday 9 October, 2006

A new form of `Constant Term' method for extracting coefficients from rational generating functions is introduced. We have recently used this method to solve an open problem from the 1970's, that of enumerating directed lattice paths in a slit, subject to a pair of independent weights on each wall. This talk generalizes that problem and solution. The techniques utilized are a blend of combinatorial, analytic and algebraic. In particular we use the combinatorics of pavings to obtain closed form expressions for some orthogonal polynomials that occur in the generating functions.

Computers often give the wrong answers. This can be caused by human errors ("garbage in, garbage out"), by software errors (programming "bugs" or incorrect program specifications), or even by hardware errors. As an example of a hardware error we describe the Intel Pentium divide bug that is estimated to have cost Intel $US 500 million in 1994. Similarly expensive was the software error that caused the crash of an Ariane 5 rocket in 1996. In simulation we often want computers to generate "random" numbers that are independent and have a known distribution. This is not so easy as it seems. We give some examples where the "random" numbers turned out to be not so random as expected, with potentially disastrous results. To summarise, it is easy to produce random numbers when you don't want to, but not so easy when you do.

I will describe methods for computing in reductive algebraic groups. My emphasis will be on efficient algorithms for two basic conjugacy results: every element can be conjugated into the Borel subgroup and every semisimple element can be conjugated into the maximal torus. As an application, I will discuss a constructive version of Sylow's theorem for finite groups of Lie type.

The Smoluchowski equation is a partial differential equation that describes many diffusion based processes in physics and chemistry. This talk will begin by deriving the Smoluchowski equation for diffusion in colloids, and then discuss the finite element solution of the Smoluchowski equation. The results will be compared with those of stochastic simulation methods

Gene regulatory networks describe the interaction of genes and its products the proteins. Because relatively small numbers of copies of each substance are involved the dynamics of these networks are mainly driven by noise generated by the translation processes involving the genes and their products. Therefore these systems are best described by stochastic models. With these models, the stochastic master equations, one can follow the time development of the probability distribution for the states defined by copy numbers of each substance. As for each substance involved, the state space grows exponentially the challenges lie in the large discrete state spaces due to high dimensionality.

The method of layer potentials was introduced by C. Neumann to solve elliptic boundary value problems. His idea was based on the well known Green formula. This approach eventually reduces the solvability of the Drichlet problem to the problem of solving a certain singular integral equation. In this talk, we progress on the solvability of these integral equations, both for the Laplacian and for variable coefficient operators and systems.

I will introduce several holomorphically invariant (pseudo)distances and (pseudo)metrics on complex manifolds and give some applications to problems in complex analysis and geometry.