In this talk, I will use the Korteweg-de
Vries equation, non-linear Schrodinger equation, and
the sine-Gordon equation as models to explain some
remarkable properties of a certain class of non-linear
wave equations, the so called "soliton equations".
Some relations to differential geometry will be
discussed. I will also use Richard Palais'
3D-XplorMath Visualization Computer Program to help us
"see" some of these properties.
I will give an overview of my work on some problems founds at the
interface between mathematical and life sciences. This includes
modeling of initiation and development of cancer (viewed as an
evolutionary problem), some problems in biophysics (studying the
dynamics of RNA transcription); learning theory, and the evolution of
natural languages.
Among the nicest spaces in topology and geometry are
manifolds, i.e. spaces which locally look like an open ball in R^n. If X
is such a manifold and G is a finite group acting on it, the usual
quotient X/G in general will not be a manifold anymore (if the G-action
has stabilizers). The theory of orbifolds is a different approach to
taking quotients, leading to objects which behave as if they were
manifolds, but also have some surprising properties defying our
intuition.
