Georgia Institute of Technology College of Sciences

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.

The study of mechanical linkages is a very classical one, dating back to the Industrial Revolution. In this talk we will discuss the geometry of the configuration spaces in some simple idealized examples and, in particular, their curvature and geometry. This leads to an interesting quantitative description of their dynamical behaviour.

Front propagation in fluid flows arise in power generation of automobile engines, forest fire spreading, and material interfaces of solidification to name a few. In this talk, we introduce the level set formulation and the resulting Hamilton-Jacobi equation, known as G-equation in turbulent combustion. When the fluid flow has enough intensity, G-equation becomes non-coercive and non-linearity no longer dominates. When front curvature and flow stretching effects are included, the extended G-equation is also non-convex. We discuss recent progress in analysis and computation of homogenization and large time front speeds in cellular flows (two dimensional Hamiltonian flows) from both Lagrangian and Eulerian perspectives, and the recovery of experimental observations from the G-equations.

The classical Weyl equidistribution theorem says that if v is a non-resonant vector then the sequence v, 2v, 3v... is uniformly distributed on a torus. In this talk we discuss the rate of convergence to the uniform distribution. This is a joint work with Bassam Fayad.