An isometric action of a connected Lie group on a Riemannian manifold is called polar if there exists a connected closed submanifold that meets each orbit of the action and intersects it orthogonally. Dadok established in 1985 a remarkable, and mysterious, relation between polar actions on Euclidean spaces and Riemannian symmetric spaces. Soon afterwards an attempt was made to classify polar actions on symmetric spaces. For irreducible symmetric spaces of compact type the final step of the classification has just been achieved by Kollross and Lytchak. In the talk I want to focus on symmetric spaces of noncompact type. For actions of reductive groups one can use the concept of duality between symmetric spaces of compact type and of noncompact type. However, new examples and phenomena arise from the geometry induced by actions of parabolic subgroups, for which there is no analogon in the compact case. I plan to discuss the main difficulties one encounters here and some partial solutions.

Eigenfunctions of the Laplacian on a bounded domain represent the modes of vibration of a vibrating drum. The behavior of these eigenfunctions is closely related to the behavior of the underlying dynamical system of the billiard table. In this talk I first give a brief exposition on this relation and then I talk about the boundary traces of eigenfunctions and a recent joint work with Han, Hassell and Zelditch.

Finite subgroup of Cremona group is a classical topic in algebraic geometry since the 19th century.  In this talk we explain an extension of this problem to the symplectic category.  In particular, we will explain the symplectic counterparts of two classical theorems.  The first one due to Noether, says a plane Cremona map is decomposed into a sequence of quadratic transformations, which is generalized to the symplectic category on the homological level.  The second one is due to Castelnuovo and Kantor, which says a minimal G-surface either has a conic bundle structure or is a Del Pezzo surface.  The latter theorem lies the ground of classifications of finite Cremona subgroups due to Dolgachev and Iskovskikh.  This is an ongoing program joint with Weimin Chen and Tian-Jun Li