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.

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