Quadrature Domains and Laplacian Growth in Modern Physics (07w5008)

Arriving in Banff, Alberta Sunday, July 15 and departing Friday July 20, 2007


(Imperial College London)

(Royal Institute of Technology, Stockholm)

Mark Mineev-Weinstein (Federal University of Rio Grande do Norte, Brazil)

(University of California at Santa Barbara)


This proposal seeks support for a workshop to bring together, for the first time, a number of investigators from a broad range of disciplines (pure and applied mathematicians, physicists and engineers) who have all recently become aware of surprising and exciting areas of commonality in their research areas. Within the last five years, a very promising theoretical link has been unveiled between areas of modern physics (ranging from integrable systems theory and condensed matter physics through to string theory and quantum gravity) and more classical subjects such as potential theory, complex analysis and free boundary problems in fluid mechanics. At the heart of this correspondence lies the mathematical concept of a quadrature domain and its physical realization broadly known as ``Laplacian growth''. With the same fundamental mathematical ideas currently impinging on so many disparate areas of research within the physical sciences, a workshop aimed at uniting workers in these various fields seems both timely and of broad potential impact.

The following list summarizes some of the many topics, linked in various ways described above, that are to be discussed at the workshop. Particular emphasis will be given to the links between the traditional mathematical school of quadrature domains and Laplacian growth (1)-(5) and areas of modern physics (6)-(10) where new applications have recently been noticed.

(1) classical quadrature domains (algebraic domains) and the Schwarz function;

(2) geometric function theory, Riemann surfaces and algebraic curves;

(3) Laplacian growth, deformation of quadrature domains, Hele Shaw flow, Loewner theory;

(4) questions of balayage and gravi-equivalence in potential theory;

(5) operator theory, sums of squares and the exponential transform;

(6) integrable systems theory: the dispersionless Toda hierarchy, the universal Whitham hierarchy and the Benney hierarchy;

(7) random matrix theory;

(8) diffusion-limited aggregation (DLA);

(9) 2D quantum gravity;

(10) string theory.