Abstract: Loop homology of a moment-angle complex is a subalgebra in the loop homology of Davis-Januszkiewicz space, which is isomorphic to the Yoneda algebra $Ext_{k[K]}(k,k)$. (Here $k[K]$ is the Stanley-Reisner ring for the simplicial complex $K$). If $K$ is a flag complex, this Yoneda algebra is known; this allows to give a presentation for loop homology for the moment-angle complex and to describe homotopy groups of these spaces in terms of homotopy groups of spheres (using recent results of L. Stanton). If time permits, we will also consider the case of "almost flag" simplicial complexes.

Abstract: In the first part, we will discuss some geometric properties of the group of hamiltonian diffeomorphisms on M, Ham(M), associated to a closed symplectic manifold (M,\om) with respect to the Hofer metric. This group, although infinite dimensional, exhibits properties similar to compact Lie groups. Pushing this philosophy, it has been observed, classically, that when the symplectic manifold is endowed with a toric action, the centralizer of this action plays the role of a maximal torus in Ham(M). In this talk, we present results that support the Hofer geometric arguments supporting this philosophy. We also present some results w.r.t the intrinsic hofer geometry on the centraliser.

In the second part of the talk, we will study the embedding space of two disjoint standard symplectic balls of capacities (sizes) c1 and c2 in $S^2\times S^2$ with respect to any symplectic form. The set of admissible capacities for such embeddings is subdivided into polygonal regions in which the homotopy type of the embedding space is constant. We present the set of all stability chambers and also present the homotopy type of the relevant embedding spaces in some of these chambers.