MSI Weekly Bulletin - Week starting Monday 8 May, 2006

Fast and reliable pseudo-random number generators are required for simulation and other applications in Scientific Computing. Because of Moore's law, random number generators that were satisfactory in the past may be inadequate today. We outline some requirements for good uniform random number generators, and describe a class of generators having very fast vector/parallel implementations. These generators are based on primitive or almost primitive polynomials, and the degrees of the polynomials correspond to the exponents of certain Mersenne primes. We consider how to combine two generators to give a generator with better statistical and/or cryptographic properties, and also discuss the problem of initialization.

Of much interest to physicists and mathematicians is the relationship between the geometry of a manifold and the spectrum of the Laplace-Beltrami operator on the manifold. I will consider one of the most basic characterisations of the spectrum, namely whether it is discrete or whether it has some continuous component, and the relation to the geometry of the underlying manifold. In this talk I will discuss some of the well known (more classical) results in this area along with some more recent work.