MSI Weekly Bulletin - Week starting Monday 20 November, 2006

RSA is currently the most widely used cryptosystem used in today's digital world, for which its three inventors received the prestigious Turing Award in 2003. In this talk, I will give some mathematical analysis of the RSA, particularly some cryptanalytic attacks such as the IFP/DLP attacks, Dionphantine attacks and lattice attacks on RSA.

According to McCann, the length of the aforementioned gradient is less than \pi; we improve that bound by constructing a maximum principle for the gradient squared. We will discuss this, a bit unusual, construction.

This is the first of a few informal talks, initially on pressure, entropy, thermodynamic formalism and particularly applications to fractals (2 chapters in Falconers book \"techniques in fractal geometry\" are particularly relevant). Then I hope to develop a little symbolic dynamics and use this to extend the previous material. The intention is to aid my own learning of the material as much as to aid anyone else\'s understanding. If others would like to contribute later, that would be helpful. The material connects with many of the things in fractal geometry that people here are developing.

I'll discuss two families of Laurent polynomials with hyperoctahedral symmetry, both indexed by partitions. The first, Koornwinder's orthogonal polynomials, includes all of the classical (and quantum classical) spherical functions and characters as special and limiting cases, as well as the Jack and Macdonald polynomials. The second, Okounkov's interpolation polynomials, is defined (and overdetermined) by specifying a large collection of zeros. Despite the significant differences in definitions, these two families are in fact closely related. I'll discuss this connection, and show how it leads to new proofs of the known properties of Koornwinder polynomials, as well as proofs of properties not previously discovered.

You may recall a fortnight ago Daniel described to you a homology theory, called singular homology, which he used to ``count holes" in topological spaces. It was decided that it would be a good idea if I introduced another gadget called de Rham cohomology that, while defined very differently, turns out to be isomorphic to singular homology on suitable nice spaces. In hindsight, this was perhaps too ambitious a task for a casual one hour talk, so thought I'd give an introduction to smooth manifolds and fibre bundles first, which is a necessary step to define de Rham cohomology. As a treat for the differential geometry savvy, I also thought I'd throw in a little material about Milnor's ``exotic spheres", which are spaces homeomorphic to n-spheres but not diffeomorphic to them.

We will describe two ways (Moser-Ebin-Marsden and Brenier-McCann) of writing uniquely a diffeomorphism as a composed of two such, one of which (the right one, say) preserving the measure. At the infinitesimal level, the related Helmholtz decomposition of vector fields will be also presented.