It was unclear until recently whether the ergodicity of the geodesic flow on a given Riemannian manifold $$extract_itex$$M$$/extract_itex$$ has any significant impact on the growth of the number of nodal domains of eigenfunctions of Laplace-Beltrami operator $$extract_itex$$\Delta_M$$/extract_itex$$, as the eigenvalue $$extract_itex$$\lambda \to \infty$$/extract_itex$$. In this talk, I'm going to explain my recent work with Steve Zelditch, where we prove that, when $$extract_itex$$M$$/extract_itex$$ is a principle $$extract_itex$$S^1$$/extract_itex$$-bundle equipped with a generic Kaluza-Klein metric, the nodal counting of eigenfunctions is typically $$extract_itex$$2$$/extract_itex$$, independent of the eigenvalues. Note that principle $$extract_itex$$S^1$$/extract_itex$$-bundle equipped with a Kaluza-Klein metric never admits ergodic geodesic flow. This, for instance, contrasts the case when $$extract_itex$$(M,g)$$/extract_itex$$ is a surface with non-empty boundary with ergodic geodesic flow (billiard flow), in which case the number of nodal domains of typical eigenfunctions tends to $$extract_itex$$+\infty$$/extract_itex$$ (proven in my paper with Seung Uk Jang).

I will also present an orthonormal eigenbasis of Laplacian on a flat 3-torus, where every non-constant eigenfunction has exactly two nodal domains. This provides a negative answer to the question raised by Thomas Hoffmann-Ostenhof:
For any given orthonormal eigenbasis of the Laplace--Beltrami operator, can we always find a subsequence where the number of nodal domains tends to $$extract_itex$$+\infty$$/extract_itex$$?''