Abstrakter
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.