Physics (Internal)

"Laplacian Growth"

This event is part of the Condensed Matter Theory Seminar Series.

Numerous unstable non-equilibrium physical processes produce a multitude of rich complex patterns, both compact and (more often) fractal. Shapes of patterns in a long time limit include fingers in porous media, dendritic trees in crystallization, fractals in bacterial colonies and malignant tissues, river networks, and other bio- and geo- systems. The patterns are often universal and reproducible. Geometry and dynamics of these forms present great challenges, because mathematical treatment of nonlinear, non-equilibrium, dissipative, and unstable dynamical systems is, as a rule, very difficult, if possible at all. While many impressive experimental and computational results were accumulated, powerful analytic methods for these systems were very limited until recently. Remarkably, many of these growth processes were reduced (after some idealization) to a mathematical formulation, which owns rich, powerful and beautiful integrable structure, and reveals deep connections with other exact disciplines, lying far from non-equilibrium growth. In physics the examples include quantum gravity, quantum Hall effect, and phase transitions. In classical mathematics, this structure was found to be deeply interconnected with such fields as the inverse potential problem, classical moments, orthogonal polynomials, complex analysis, and algebraic geometry. In modern mathematical physics we established tight relations of nonlinear growth to integrable hierarchies and deformations, normal random matrices, stochastic growth, and conformal theory. This mathematical structure made possible to see many outstanding problems in a new light and solve several long-standing challenges in pattern formation and in mathematical physics.

During the talk I will provide brief history with key experiments and paradigms in the field, make short surveys of mathematics mentioned above, expose the integrable structure hidden behind the interface dynamics, and will present major results up to date in this rapidly growing field.