# Peter Sarnak

## Nodal Lines of Maass Forms and Critical Percolation

Peter Sarnak
Institute for Advanced Study
March 20, 2012

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.

## Pseudorandomness - Substitution sequences at primes

Peter Sarnak
Institute for Advanced Study
December 3, 2008

## School of Mathematics 75th - Number Theory, Symmetry and Zeta Functions

Peter Sarnak
Institute for Advanced Study
March 11, 2005

## Automorphic Forms - Concluding Remarks

Peter Sarnak
Institute for Advanced Study
April 7, 2001