Geometry and Fluctuations of Surfaces

Surfaces are described as a two-dimensional random froth. the dual of the "fisherman's net". Its geometry and fluctuations are seen as local, elementary topological transformations (dissociation and motion of dislocations and disclinations). These transformations are the "collisions" responsible for statistical equilibrium of the structure, characterized by observable relations (Aboav. Lewis). The surface and its dynamics can be represented as a many-body problem with short-ranged interactions (Telley). The discrete bodies are paraboloids attached to the cells of the froth, with one additional degree of freedom beside their position in physical space. Elementary topological transformations are caused by simple and orthogonal motions of the bodies. Telley's model has remarkable syrnmetries, notably stereology (it is identical to its cut). It also describes quantitatively the evolution of many natural froths (soaps, sintering of polycrystalline mosaics, growth of biological tissues, deposition of amorphous films). As a Hamiltonian, it allows for local curvature fluctuations, leading to change of genus.