Professor Nicholas Ercolani, Department of Mathematics, The University of Arizona, and Professor Shankar Venkataramani, Department of Mathematics, The University of Arizona.

In this research tutorial group, we will explore the connections between the dynamics of planar domains, conformal maps and integrable systems, specifically the Toda lattice. Many practically relevant growth processes occur in two dimensional domains, and are approximately conformally invariant. Exactly conformal processes are scale-invariant, whereas "real" physical phenomena have a small scale cutoff, making the processes only approximately conformally invariant. Examples of such processes include Diffusion Limited Aggregation (DLA) and Hele-Shaw fingering.

A useful tool in studying exactly conformally invariant growth processes is the Riemann mapping theorem. The theorem asserts the existence of a conformal map that maps a given simply connected domain onto a "standard" planar region such as the unit disk. We can study the dynamics of the domain's growth entirely through the evolution of the Riemann map to the unit disk. One finds that this class of conformally invariant growth models has a dynamics which is completely integrable. In particular this means that these processes possess infinitely many constants of motion which turn out to be the harmonic moments of the domain. The dynamics of the domain's boundary is described by the so-called Toda Lattice system of differential equations which is one of the most fundamental modern models of a completely integrable system.

The computation of the Riemann map in this setting is equivalent to solving for the electrostatic equilibrium of a continuous charge distribution on the domain. One way to discretize this problem is to consider the electrostatic equilibrium of $N$ discrete charges of size $1/N$ in the limit $N \rightarrow \infty$. Although this problem is exactly conformally invariant in the limit, it is only approximately so for finite $N$, because of a small scale cutoff of size $1/N$. Understanding the aysmptotics of this problem will potentially help in understanding the domain dynamics of two dimensional growth processes that break conformal invariance at small scales.