Mid-Term Report Summary 2 - STEIN (TOPOLOGY OF STEIN MANIFOLDS)
The "STEIN" research project aims to study the topology of Stein, or Weinstein manifolds using invariants based on holomorphic curves. It has two main parts: the study of Weinstein manifolds from the point of view of their skeleta with the goal of obtaining a description of a Weinstein manifold as the cotangent bundle of its skeleton; and the study of symplectic invariants constructed from holomorphic curves, in relationship with the topology of free loop spaces. The main achievement for the first reporting period is the proof of a functorial isomorphism between linearized contact homology of a Stein fillable contact manifold (and more generally of a Liouville fillable contact manifold), and the S^1-equivariant symplectic homology of the filling (truncated in positive values of the action). This opens up new perspectives regarding the rigorous definition of linearized contact homology, as well as regarding the topological interpretation of linearized contact homology. The current active topics of investigation within the project touch upon string topology, microlocal analysis, pseudo-holomorphic curves in symplectic manifolds and Hamiltonian dynamics.