The study of random surfaces and random interfaces has long held the interest of physicists and mathematicians. Only recently, however, have there arisen mathematical techniques for understanding simple interfaces of more than one dimension. The next simplest case,that of two dimensional interfaces in three space,is already quite difficult. The two-dimensional interfaces we study are called 'stepped surfaces'. Under the simplest choice of measure on these surfaces, the uniform measure for a fixed boundary, the large-scale shapes taken by these surfaces has begun to be worked out by the PI and Okounkov, using techniques from PDEs, analysis and tropical geometry. This model is essentially the only mathematically 'solved' model of random interfaces. Moreover it contains a great deal of mathematical connections with other areas: to random matrix theory, integrable systems, string theory and Gromov-Witten theory. For these reasons it is worth understanding this model better, and also worth looking for generalizations.
We are studying mathematical models of crystal surfaces. On an atomic scale, the surface of a crystal, such as a salt crystal or diamond, is rough and 'random', but at large scales it is typically smooth and facetted. How these large scale features arise from the microscopic interactions of the atoms comprising the crystal is, to a large extent, still mysterious. However we can make models of crystal surfaces which are computationally tractable in a mathematical sense, and display the same behavior as real crystals: in particular they display facetting and large-scale shape formation. By studying these models we hope to gain understanding not just of crystal surfaces but of the general phenomenon of how local interactions among a large number of constituents can develop into large-scale behavior.
In statistical mechanics we study the average behavior of large systems of particles in interaction. There are, unfortunately, few general methods to get exact results in this area, so we focus on "solvable" systems and attempt to extend the range of solvability. In this project we studied a specific random surface model called the dimer model: in its simplest form it is a random tiling of a planar region with 2X1 and 1X2 rectangles ("dominos"), but is more generally related to models of 2-dimensional magnetism (the Ising model) and crystal surfaces (Potts model, lozenge dimer model). This basic model has a unique feature that there exist combinatorial techniques (determinants and other linear algebra tools) to compute "physically meaningful" quantities. One of the goals of the project, worked out with grad student Z. Li, was to apply a new technique called holographic reduction, developed in computer science by L. Valiant, to generalize these combinatorial methods to other models. Sinceholographic reduction is at its roots a linear algebraic technique, we were able to find the equations governing the complete applicable range of this method. This allowed us to "solve" some new models related to random surfaces with various types of singularities. Another important technique in statistical mechanics is so-called integrability. This is a concept from dynamical systems theory: in essence it states that some potentially highly complicated nonlinear systems can be represented in coordinates in which they are just rotations in some high-dimensional space. It was known that certain statistical mechanical models have some form of integrability. We discovered, working with A. B. Goncharov, that integrability holds in the strongest sense in the dimer model. This fact allowed us to discover a natural geometric structure in the parameter spaces in the dimer model, and thus to give an essentially complete description of the physical behavior of all possible dimer models. It is remarkable that the dimer model fits into the framework of integrable systems: this fact provides a heretofore unknown link between the field of dynamical systems and statistical mechanics. I believe that this link is quite important and the future will see many applications working across these two distinct fields.
