Published on *Max Planck Institute for Mathematics* (https://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Oleksiy Klurman
Affiliation:

KTH Stockholm/MPIM
Date:

Tue, 2019-11-12 14:00 - 15:00 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 equations for integral points on the circles. This is based on a joint work with V. Cammarota and I. Wigman.

**Links:**

[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] https://www.mpim-bonn.mpg.de/node/3444

[3] https://www.mpim-bonn.mpg.de/node/5312