# Emerging Topics Working Group: Nodal sets of Eigenfunctions

**Agenda**

All talks will take place in Simonyi Hall

**Monday, February 13**

10:30am-11:30am Alexander Logunov, Tel Aviv University, *Upper estimate for the volume of zero sets of Laplace eigenfunctions: in between real-analytic and smooth cases*

Abstract: Let D be a domain in the n-dimensional Euclidean space. Assume that the boundary of D is sufficiently smooth, but not real-analytic. Consider the sequence of Laplace eigenfunctions in D with the Dirichlet boundary condition. How large can the (n-1)-dimensional volume of the zero set of Laplace eigenfunction in D be?

We will explain the upper bound by the square root of the eigenvalue times the logarithm of the eigenvalue. Based on a joint work (in progress) with E. Malinnikova and N. Nadirashvili.

**Tuesday, February 14 **

10:30am-11:20am Dan Mangoubi, Hebrew University, *Harmonic functions - positivity and convexity*

Abstract: We present a sequence of positive quadratic forms associated with harmonic functions on Abelian groups. We show how the positivity property recovers the polynomial Liouville property and we prove a three spheres theorem in terms of random walks.

The talk is based on a joint work with Gabor Lippner.

11:40am - 12.30pm Eugenia Malinnikova, Norwegian University of Science and Technology, *An **improvement of Liouville theorem for discrete harmonic functions*

Abstract: The classical Liouville theorem says that if a harmonic function on the plane is bounded then it is a constant. At the same time for any angle on the plane, there exist non-constant harmonic functions that are bounded outside the angle.

The situation is different for discrete harmonic functions on Z^2. We show that the following improved version of the Liouville theorem holds. If a discrete harmonic function is bounded on 99% of the plane then it is constant. It is a report on a joint work (in progress) with L. Buhovsky, A. Logunov and M. Sodin.

**Wednesday, February 15**

10:30am -11:20am John Toth, McGill University, *The nodal intersection problem for Laplace eigenfunctions*

Abstract: I will discuss recent results regarding the problem of counting intersections of eigenfunction nodal sets with real analytic curves H and the connection to eigenfunction restriction bounds over H.

11:40am - 12:30pm Lev Buhovski, Tel Aviv University, *Critical points of eigenfunctions*

Abstract: On a closed Riemannian manifold, the Courant nodal domain theorem gives an upper bound on the number of nodal domains of n-th eigenfunction of the Laplacian. In contrast to that, there does not exist such bound on the number of isolated critical points of an eigenfunction. I will try to sketch a proof of the existence of a Riemannian metric on the 2-dimensional torus, whose Laplacian has infinitely many eigenfunctions, each of which has infinitely many isolated critical points. Based on a joint work with A. Logunov, F. Nazarov, and M. Sodin.

**Thursday, February 16**

10:30am -11:20am Hamid Hezari, University of California, Irvine, *Applications of small scale quantum ergodicity in nodal sets*

Abstract: We present some applications of small scale QE in obtaining improvements on various quantities associated to eigenfunctions such as L^p norms, size of nodal sets, order of vanishing of eigenfunctions, growth estimates of eigenfunctions, and inner radius of nodal domains.

11:40am - 12:30pm Steven Zelditch, Northwestern University, *Log lower bound on the number of nodal domains on some surfaces of negative curvature*

Abstract: An open problem is to prove that for any (or at least any generic) Riemannian metric, there is some sequence of eigenfunctions of the Laplacian for which the number of nodal domains tends to infinity. It sounds easy but as yet there are almost no examples of such metrics except for separation of variables situations, and the results of Ghosh-Reznikov-Sarnak and of Junehyuk Jung and myself. This talk is about a quantitative improvement in which we give a log lower bound for the number of nodal domains for negatively curved `real Riemann surfaces'. The same result should hold at least for any non-positively curved surface with concave boundary, but there are several technical obstructions.

**Friday, February 17**

10:30am -11: 20am Mikhail Sodin, Tel Aviv University, *Nodal sets of random spherical harmonics*

Abstract: In the talk I will describe what is known and (mostly) unknown about asymptotic statistical topology and geometry of zero sets of random spherical harmonics of large degree. I plan to discuss (a) several provoking open questions and (b) the first non-trivial lower bound recently obtained with Fedor Nazarov (work in progress) for the variance of the number of connected components of the zero set.

11:40am -12:30pm Melissa Tacy, Australian National University, *Equidistribution of random waves on shrinking balls*

Abstract: In the 1970s Berry conjectured that the behavior of high energy, quantum-chaotic billiard systems could be well modeled by random waves. That is random combinations of the plane waves e^{ik ·x}. On manifolds it is more natural to randomize over the eigenfunctions of the Laplace-Beltrami operator. In this talk I will present results showing that such random waves equidistribute on balls that shrink with the eigenvalue. This is joint work with Xiaolong Han.