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.