What happens when we decrease the length of a closed curve in
the plane as fast as possible? This seemingly simple question has a very
nice answer which involves a beautiful combination of partial differential
equations and planar geometry. Come and get a glimpse of the amazing
subject of geometric flows!

A key phenomenon in the study of cell-to-cell communication and
protein regulation is the all-or-none, ultrasensitive dose response, which
transforms a continuous input into a digital output. Multisite systems are
often used in conjunction with allosteric effects to create such a behavior.
In this talk I describe a non-allosteric mechanism for a multisite system to
present strongly ultrasensitive behavior. Applications are given to protein
activation through multisite phosphorylation, clusters of receptors and DNA
regulation through histone modifications.

The theory of soliton equations has been an active research area for the past forty-five years, with applications to algebra, geometry, mathematical physics, and applied mathematics. In this talk, I will explain how many of these equations arise as geometric evolution equations for curves and as the governing equations for surfaces in 3-space. In particular, I will use Quicktime movies and pictures produced in Palais' 3D-XplorMath mathematical visualization program to demonstrate properties of soliton equations and their associated geometric objects.