# Oleksiy Klurman: Boundary-adapted arithmetic random waves

**Time: **
Tue 2019-05-07 15.15

**Lecturer: **
Oleksiy Klurman

**Location: ** Room F11, KTH

Abstract: In this talk, we test M. Berry's ansatz on nodal deficiency in presence of boundary. The square billiard is studied, where the high spectral degeneracies allow for the introduction of a Gaussian ensemble of random Laplace eigenfunctions ("boundary-adapted arithmetic random waves"). As a result of a precise asymptotic analysis, two terms in the asymptotic expansion of the expected nodal length are derived, in the high energy limit along a generic sequence of energy levels.

In particular, we shall focus on a number-theoretic aspect of this problem, describing the techniques introduced by E. Bombieri and J. Bourgain to study additive equation for integer points on the circles.