In this talk, I will report a recent joint work with Mu-Tao Wang. We construct examples of shrinkers and expanders for Lagrangian mean curvature flows. These examples are Hamiltonian stationary and asymptotic to the union of two Hamiltonian stationary cones found by Schoen and Wolfson. The Schoen-Wolfson cones are obstructions to the existence problems of special Lagragians or Lagrangian minimal surfaces in the variational approach. It is known that these cone singularities cannot be resolved by any smooth Lagrangian submanifolds. The shrinkers and expanders that we found can be glued together to yield solutions of the Brakke motion-a weak formulation of the mean curvature flow, and thus provide a canonical way to resolve the union of two such cone singularities. Our theorem is analogus to the Feldman-Ilmanen-Knopf gluing construction for the K\"ahler-Ricci flows.

An isometric action of a Lie group is called polar if it admits
sections, i.e. submanifolds which meet all orbits and always
perpendicularly. Polarity is a very restrictive condition. For example,
in case of linear actions on *R^*n polarity characterizes the isotropy
representations of symmetric spaces (Dadok).

The aim of this talk is to report on work in progress to prove an
infinite dimensional analogue of Dadok's theorem. C.-L. Terng has
constructed interesting examples of polar actions on Hilbert spaces by
affine isometries, the so called P(G,H) actions. Here G is a compact Lie
group, H a closed subgroup of G \times G, and P(G,H) consists of all
paths in G with end points in H. The action of P(G,H) on the Hilbert
space of L^2-curves in the Lie algebra of G is by gauge transformations.
Surprisingly the actions correspond also to isotropy actions of
symmetric spaces which are now infinite dimension and quotients of a
Kac-Moody group by the fixed point set of an involution. We conjecture
that the P(G,H) actions exhaust all polar actions on a Hilbert space.