Georgia Institute of Technology College of Sciences

The main aim of this talk is to design efficient and novel numerical algorithms for highly oscillatory dynamical systems with multiple time scales. Classical numerical methods for such problems need temporal resolution to resolve the finest scale and become very inefficient when the longer time intervals are of interest. In order to accelerate computations and improve the long time accuracy of numerical schemes, we take advantage of various multiscale structures established from a separation of time scales. The framework of the heterogeneous multiscale method (HMM) will be considered as a general strategy both for the design and for the analysis of multiscale methods.(Keywords: Multiscale oscillatory dynamical systems, numerical averaging methods.)

We show that on any Riemannian manifold with the Ricci curvature non-negative we can construct a coupling of two Brownian motions which are staying fixed distance for all times. We show a more general version of this for the case of Ricci bounded from below uniformly by a constant k. In the terminology of Burdzy, Kendall and others, a shy coupling is a coupling in which the Brownian motions do not couple in finite time with positive probability. What we construct here is a strong version of shy couplings on Riemannian manifolds. On the other hand, this can be put in contrast with some results of von Renesse and K. T. Sturm which give a characterization of the lower bound on the Ricci curvature in terms of couplings of Brownian motions and our construction optimizes this choice in a way which will be explained. This is joint work with Mihai N. Pascu.

During the 17th Century the French priest and physicist Edme Mariotte observed that objects floating on a liquid surface can attract or repel each other, and he attempted (without success!) to develop physical laws describing the phenomenon. Initial steps toward a consistent theory came later with Laplace, who in 1806 examined the configuration of two infinite vertical parallel plates of possibly differing materials, partially immersed in an infinite liquid bath and rigidly constrained. This can be viewed as an instantaneous snapshot of an idealized special case of the Mariotte observations. Using the then novel concept of surface tension, Laplace identified particular choices of materials and of plate separation, for which the plates would either attract or repel each other. The present work returns to that two‐plate configuration from a more geometrical point of view, leading to characterization of all modes of behavior that can occur. The results lead to algorithms for evaluating the forces with arbitrary precision subject to the physical hypotheses, and embrace also some surprises, notably the remarkable variety of occurring behavior patterns despite the relatively few available parameters. A striking limiting discontinuity appears as the plates approach each other. A message is conveyed, that small configurational changes can have large observational consequences, and thus easy answers in less restrictive circumstances should not be expected.