Some Estimates on the Geometry of Eigenfunctions in the High-Energy Limit

 Given a closed manifold $M$, the Laplace operator $\Delta$ is known to
posses a discrete spectrum of eigenvalues $\lambda$ converging to infinity.
We are interested in properties of the corresponding eigenfunctions
$\phi_\lambda$ as $\lambda$ becomes large (i.e. the high-energy limit).
From a physical point of view, $\{ \phi_\lambda \}$ represent stationary states
of a free quantum particle on $M$ - when $\phi_\lambda$ is $L^2$-normalized,
it may be interpreted as the probability density of a particle in $M$, having energy $\lambda$.

 Various questions about the geometry of $\phi_\lambda$ have been studied thoroughly:
for example, what is the size of the vanishing (nodal) set of $\phi_\lambda$;
how is the nodal set distributed; how big can the $L^p$ norms of $\phi_\lambda$ be
and how are they localized;  how many nodal domains are there; how large are the nodal domains, etc.

 We focus on some classical, as well as recent results along these lines -
this includes joint work with M. Mukherjee and S. Steinerberger.