Thomas Wolff Memorial Lecture in Mathematics

There exist deep and perhaps surprising connections between certain random surface models of direct probabilistic interest and algebraic geometry. The random surface models in question are rather simple: these are the so-called stepped surfaces, also known as 3D Ising interfaces at zero temperature, and their close relatives. Various matrix models may be also viewed as more distant members of the same family. The connections between these models and algebraic geometry go both ways. On the one hand, enumerative problems related to algebraic curves may be answered in terms of partition functions of random surface models with specific boundary conditions. On the other hand, the analysis of random surfaces models, including such basic questions as the law of large numbers and the fluctuations around it is controlled by some different geometry, which is, in a sense, the "probabilistic mirror" to the original enumerative problem. I'll try to explain both directions of this correspondence in my lectures."