By highly excited quantum states, I mean the eigenfunctions
of a Laplacian or Schrodinger operator correspoinding to a large eigenvalue
. It is when a physical system is in an excited state
that the (semi-) classical approximation is valid. My main purpose is to
survey a series of new results (joint work with B. Shiffman, C. Sogge and
J.Toth) about the geometry of such states: how their size (sup-norms and
-norms) reflects the underlying geometry, and how the zeros and critical
points are distributed in typical cases. I will also present some slides
(made by physicist) of actual excited states
in various systems of physical and mathematical interest.