SMS årsmöte , 4-5 juni 2009

M. Jacobsson
The Temperley-Lieb algebra, the Jones Polynomial and Khovanov homology
I will give an overview of the Jones polynomial and Khovanov's homology theory of links, with an eye toward the origins in statistical mechanics such as state models and the Temperley-Lieb algebra.

P. Kurlberg
On the period of some pseudo-random number generators and "number-theoretical turbulence"
Given coprime integers b and n, let ord(b,n) be the multiplicative order of b modulo n. The length of the periods of some popular pseudorandom number generators (e.g., the linear congruential generator, and the Blum-Blum-Shub generator) turns out to be related to ord(b,n) for appropriately chosen b and n. We will investigate some conclusions by V.I. Arnold (based on numerics by F. Aicardy as well as analogies with the physical principle of turbulence) on the average of ord(b,n), as n ranges over integers. We will also give lower bounds on ord(b,n) for b fixed and n ranging over certain subsets of the integers, e.g., the set of primes, the set of "RSA moduli" (i.e., products of two primes), the full set of integers, and the images of these sets under the Carmichael lambda function. (The lower bounds in the case of RSA moduli shows that certain "cycling attacks" on the RSA crypto system are ineffective.)

A. Strömbergsson
The Boltzmann-Grad limit of the periodic Lorentz gas
The Lorentz gas describes an ensemble of non-interacting point particles in an infinite array of spherical scatterers. It was originally developed by Lorentz in 1905 to model, in the limit of low scatterer density (Boltzmann-Grad limit), the stochastic properties of the motion of electrons in a metal. In my talk I will consider the case of a periodic array of scatterers, and describe a stochastic process that governs the macroscopic dynamics of a particle cloud in the Boltzmann-Grad limit. The corresponding result has been known for some time in the case of a Poisson-distributed (rather than periodic) configuration of scatterers. Here the limiting process corresponds to a solution of the linear Boltzmann equation. However, the linear Boltzmann equation does not hold in the periodic set-up, and the random flight process that emerges in the Boltzmann-Grad limit is substantially more complicated.
The main tool in our approach is measure rigidity, a part of ergodic theory which has recently found important applications in several other problems in number theory and mathematical physics.
(This lecture is based on joint work with Jens Marklof, Bristol.)

B. Wennberg
Many particle systems and propagation of chaos
The Boltzmann equation was published in 1872, but it had essentially been derived in weak form by Maxwell. The derivation relies on a very strong hypothesis known as "the propagation of chaos", which says that if the particles in the gas initially are uncorreleted, then they remain uncorrelated as time evolves. There is still no satisfactory mathematical proof of this for the case of a real gas, but there are many partial results, and full proofs for other many particle systems. In the talk I will give a precise formulation of the notion of propagation of chaos, and say something of what can and cannot be proven rigorously.