  Can you hear the shape of Liouville quantum gravity? We obtain a Weyl law for
the eigenvalues of Liouville Brownian motion: the $n$-th eigenvalue grows
linearly with $n$, with the proportionality constant given by the Liouville
area of the domain and a certain deterministic constant $c_\gamma$ depending on
$\gamma \in (0, 2)$. The constant $c_\gamma$, initially a complicated function
of Sheffield's quantum cone, can be evaluated explicitly and is strictly
greater than the equivalent Riemannian constant.
  At the heart of the proof we obtain sharp asymptotics of independent interest
for the small-time behaviour of the on-diagonal heat kernel. Interestingly, we
show that the scaled heat kernel displays nontrivial pointwise fluctuations.
Fortunately, at the level of the heat trace these pointwise fluctuations cancel
each other, which leads to the result.
  We complement these results with a number of conjectures on the spectral
geometry of Liouville quantum gravity, notably suggesting a connection with
quantum chaos.
