Nodal Lines of Maass Forms and Critical Percolation

by Prof. Peter Sarnak (Princeton University and Institute for Advanced Study)

Thursday 24 May 2012, 11:30 → 12:30 Asia/Kolkata

AG-66 (Colaba Campus)

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 an exactly solvable 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 (Joint with A. Ghosh and A. Reznikov).