University of Minnesota

The accuracy of the Nos-Hoover thermostat to sample the Gibbs measure depends on the

dynamics being ergodic. It has been observed for a long time that this dynamics is

actually not ergodic for some simple systems, such as the harmonic oscillator.

In this talk, we rigourously prove the non-ergodicity of the Nos-Hoover thermostat, for

the one-dimensional harmonic oscillator.

We will also show that, for some multidimensional systems, the averaged dynamics for the limit

of infinite thermostat "mass" has many invariants, thus giving

theoretical support for either non-ergodicity or slow ergodization.

Our numerical experiments for a two-dimensional central force problem

and the one-dimensional pendulum problem give evidence for

non-ergodicity.

We also present numerical experiments for the Nose-Hoover chain with

two thermostats applied to the one-dimensional harmonic

oscillator. These experiments seem to support the non-ergodicity of the

dynamics if the masses of the reservoirs are large enough and are

consistent with ergodicity for more moderate masses.

Joint work with Frederic Legoll and Richard Moeckel

I will introduce the moving frame approach to the analysis of invariant variational problems and the evolution of differential invariants under invariant submanifold flows. Applications will include differential geometric flows, integrable systems, and image processing.

New techniques in cell and molecular biology have produced huge advances in our understanding of signal transduction and cellular response in many systems, and this has led to better cell-level models for problems ranging from biofilm formation to embryonic development. However, many problems involve very large numbers of cells, and detailed cell-based descriptions are computationally prohibitive at present. Thus rational techniques for incorporating cell-level knowledge into macroscopic equations are needed for these problems. In this talk we discuss several examples that arise in the context of cell motility and pattern formation. We will discuss systems in which the micro-to-macro transition can be made more or less completely, and also describe other systems that will require new insights and techniques.