Unboundedness of the first eigenvalue of the Laplacian in symplectic   category
  Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure \omega. For each such metric, we look at the first eigenvalue \lambda_1 of the Laplacian associated with it. We show that \lambda_1 can be made arbitrarily large, when we vary the metric. This generalizes previous results of Polterovich, and of Mangoubi. 
Lev Buhovsky
math.SP math.SG