We describe some results concerning the number of connected components of nodal lines of high frequency Maass forms on the modular surface. Based on heuristics connecting these to a critical percolation model, Bogomolny and Schmit have conjectured, and numerics confirm, that this number follows an asymptotic law. While proving this appears to be very difficult, some approximations to it can be proved by developing number theoretic and analytic methods. The work report on is joint with A. Ghosh and A. Reznikov.

Peter Sarnak, Professor, School of Mathematics. Through the works of Fermat, Gauss, and Lagrange, we understand which positive integers can be represented as sums of two, three, or four squares. Hilbert's 11th problem, from 1900, extends this question to more general quadratic equations.