The topic named in the first part of the title of this lecture is familiar to every student. My ultimate goal is to show how the theory of integrable equations, which has been extensively developed during the past twenty years, and the Whitham theory, which already has a ten-year history, are related to the classical problem of complex analysis. The Riemann theorem asserts that, if a domain in the complex plane has a boundary containing more than two points, then there exists a conformal mapping of this domain onto the unit disk. This is an existence theorem. Many applied sciences are engaged in constructing such conformal mappings in particular situations; moreover, these problems are related to applications in hydrodynamics, in the theory of oil-fields, and in aerodynamics. The necessity of constructing conformal mappings of special domains emerges very often.

I want to present a recent remarkable observation of Zabrodin and Wiegmann, who discovered a relation between the classical problem on conformal mappings of domains and the dispersionless Toda lattice a couple of months ago. I shall tell about the development of this observation in our joint paper (not yet published), namely, about its generalization to nonsimply connected domains and about the role which the methods of algebraic geometry play in it.

Before proceeding to the problem proper, I want to give a brief overview of the entire context in which it has arisen, in order to clarify what the Whitham equations are. Surprisingly, the same structures related to the Whitham equations arise in various fields of mathematics, not only in the theory of conformal mappings.