MSI Weekly Bulletin - Week starting Monday 27 November, 2006

For many years mathematicians and computer scientists have searched for a fast and reliable primality test. This is especially relevant nowadays, because the popular RSA public-key cryptosystem requires very large primes in order to generate secure keys. I will describe some efficient randomised algorithms that are useful, but have the defect of occasionally giving the wrong answer, or taking a very long time to give an answer. In 2002, Agrawal, Kayal and Saxena (AKS) found a deterministic polynomial-time algorithm for primality testing. I will describe the original AKS algorithm and some improvements by Bernstein and Lenstra. As far as theory is concerned, we now know that "PRIMES is in P", and this appears to be the end of the story. However, I will explain why it is preferable to use randomised algorithms in practice.

This is the second of a few informal talks on pressure, entropy, thermodynamic formalism and applications to fractals (2 chapters in Falconers book "techniques in fractal geometry" are particularly relevant). I will review the definition of pressure and show how to construct a Gibbs measure on certain fractals which then gives their dimension. I will also discuss entropy, the variational principle, equilibrium measures, and analogies with statistical mechanics.

(Talk 1) There are several generalisations of the notion of amenability for Banach algebras. Two of them are approximate amenability, where derivations must only be 'approximately inner' and weak amenability, which involves only examining the dual module. Here, we elucidate a Banach algebra which is approximately weakly amenable, but is neither approximately amenable nor weakly amenable.

In 1993 Deligne made a conjecture that the Hochschild complex of an associative algebra (a purely algebraic object) has a natural action of the chain of the little disks operad (a purely geometric object). The first correct proof due to Tamarkin had appeared in 1998. After it many other proofs were found but the story of the Deligne's conjecture is still far from being over. In my lecture I will explain the conjecture, its importance in noncommutative differential calculus and why it is so inspiring. I also will give a brief sketch of one of the most recent proofs, based on a combination of the resuslts due to Tamarkin and myself.