## Final Activity Report Summary - INTERFACE (Integrable Structures in Theory of Multiphase Flows)

In recent years, integrable structures were found in a class of hydrodynamics' problems, leading to a pattern formation in a regime far from equilibrium. Growth problems of this type are unified by the name Laplacian growth (LG). Also known as the Hele-Shaw problem (HSP), it refers to dynamics of a moving front, i.e. an interface, between two distinct phases driven by a harmonic scalar field, which is a potential for the growth velocity field.

The HSP appears in different physical and mathematical contexts and has a number of practical applications, e.g. in oil industry. The most interesting and most studied dynamics occurs in the two-dimensional problem. In experiments, the two-dimensional 2d geometry is realised in a Hele-Shaw cell, i.e. a narrow gap between two parallel plates. In this version the problem is also known as the Saffman-Taylor problem or viscous fingering. The problem recently acquired a new facet, due to its connection with modern soliton theory and matrix models. Further development in this direction led to the observation that the equations describing HSP turned out to be constraints, known as 'string' equations in soliton theory, imposed on dispersionless limits of solutions of integrable hierarchies.

During the initial stage of our project we dealt with finite-dimensional reductions of LG or related solutions of dispesionless hierarchies. In particular, we established the Hamiltonian structure of such reductions, describing, for instance, multi-finger patterns in Hele-Shaw cell.

During the later stages we studied important generalisations of the LG describing free-boundary flows in nonhomogeneous media, such as oil flow in stratified medium, and encountered new exciting links with the theory of quantum integrable systems of the Calogero-Moser type. Our observations led to new developments in fluid mechanics, theory of quadrature domains and analysis of partial differential equations, such as the classical Hadamard's problem.

The above outlined directions led to understanding the mathematical aspects of the Hele-Shaw type problems and advanced the theory of partial differential equations.

The HSP appears in different physical and mathematical contexts and has a number of practical applications, e.g. in oil industry. The most interesting and most studied dynamics occurs in the two-dimensional problem. In experiments, the two-dimensional 2d geometry is realised in a Hele-Shaw cell, i.e. a narrow gap between two parallel plates. In this version the problem is also known as the Saffman-Taylor problem or viscous fingering. The problem recently acquired a new facet, due to its connection with modern soliton theory and matrix models. Further development in this direction led to the observation that the equations describing HSP turned out to be constraints, known as 'string' equations in soliton theory, imposed on dispersionless limits of solutions of integrable hierarchies.

During the initial stage of our project we dealt with finite-dimensional reductions of LG or related solutions of dispesionless hierarchies. In particular, we established the Hamiltonian structure of such reductions, describing, for instance, multi-finger patterns in Hele-Shaw cell.

During the later stages we studied important generalisations of the LG describing free-boundary flows in nonhomogeneous media, such as oil flow in stratified medium, and encountered new exciting links with the theory of quantum integrable systems of the Calogero-Moser type. Our observations led to new developments in fluid mechanics, theory of quadrature domains and analysis of partial differential equations, such as the classical Hadamard's problem.

The above outlined directions led to understanding the mathematical aspects of the Hele-Shaw type problems and advanced the theory of partial differential equations.