Paul Wiegmann (Chicago University)
``Mathematical aspects of Laplacian Growth: evolution of conformal
maps and singularities of growing patterns"
ABSTRACT:
A broad class of non-equilibrium growth processes in two dimensions
have a common law: the velocity of the growing interface is determined
by the gradient of a harmonic field (Laplacian growth). This kind of
growth is unstable, giving rise to fractal singular patterns.
Recently it has been recognized that the theory of Laplacian growth is
deeply related to fundamental aspects of evolution of Riemann surfaces
in the moduli space, and is described by a universal Whitham
hierarchy. Among them are integrable systems, 2D quantum gravity and
random matrices. In the talk I plan to review some of these
connections emphasizing a hydrodynamic interpretation of abstract
objects of Whitham theory and asymptotes of orthogonal polynomials.