# Mark AgranovskyBar Ilan University, Israel

## Global geometry of stationary sets

The subject of the talk is characterizing of
stationary sets -- i.e. sets where a non-trivial solution is zero at any moment of time --
for evolution (heat and wave) equations in R^n or in bounded domains.
For time-harmonic (i.e., single mode)
solutions stationary sets coincide with nodal (zero) sets of
eigenfunctions of the Laplace operator.
Understanding of the geometry of nodal sets is a difficult
problem, and mostly information about only the local structure of
nodal sets is available. Nevertheless, when the solution has sufficiently
rich spectrum (i.e., involves infinitely many time harmonic modes) and correspondinlgly
the stationary sets are intersections of infinite families of nodal sets
of the eigenfunctions in the spectrum, the
global geometric structure of stationary sets can be well understood
and in certain situations completely described.
That is the case, for instance, when the initial
velocity vanishes near the boundary of a bounded domain, or in the case
of the whole space has compact support or decays sufficiently fast
at infinity.

Results of recent work on the subject will be presented.