In this paper we study a version of the Dirichlet problem for the homogeneous complex Monge-Amperè equation.

This note discusses some geometrically defined seminorms on the group Ham (M,ω) of Hamiltonian diffeomorphisms of a closed symplectic manifold (M,ω), giving conditions under which they are nondegenerate and explaining their relation to the Hofer norm. As a consequence we show that if an element in Ham (M,ω) is sufficiently close to identity in the C²-topology then it may be joined to the identity by a path whose Hofer length is minimal among all paths, not just among paths in the same homotopy class relative to endpoints. Thus, true geodesics always exist for the Hofer norm. The main step in the proof is to show that a "weighted" version of the nonsqueezing theorem holds for all fibrations over S² generated by sufficiently short loops. Further, an example is given showing that the Hofer norm may differ from the sum of one sided seminorms.

We construct a new aperiodic symplectic plug and hence new smooth counterexamples to the Hamiltonian Seifert conjecture in ℝ²ⁿ for n ≥ 3. In other words, we describe an alternative procedure, to those of V.L. Ginzburg [Gi1, Gi2] and M. Herman [Her], for producing smooth Hamiltonian flows, on symplectic manifolds of dimension at least six, which have compact regular level sets that contain no periodic orbits. The plug described here is a modification of those built by Ginzburg. In particular, we use a different "trap" which makes the necessary embeddings of this plug much easier to construct.

This work is a contribution to the area of Strict Quantization (in the sense of Rieffel) in the presence of curvature and non-Abelian group actions. More precisely, we use geometry to obtain explicit oscillatory integral formulae for strongly invariant strict deformation quantizations of a class of solvable symplectic symmetric spaces. Each of these quantizations gives rise to a field of (pre)-C^*-algebras whose fibers are function algebras which are closed under the deformed product. The symmetry group of the symmetric space acts on each fiber by C^*-algebra automorphisms.

We provide a translation between Chekanov's combinatorial theory for invariants of Legendrian knows in the standard contact ℝ³ and Eliashberg and Hofer's contact homology. We use this translation to transport the idea of "coherent orientations" from the contact homology world to Chekanov's combinatorial setting. As a result, we obtain a lifting of Chekanov's differential graded algebra invariant to an algebra over ℤ[t,t^-1] with a full ℤ grading.

This paper presents a natural extension to foliated spaces of the following result due to Gromov: the h-principle for open, invariant differential relations is valid on open manifolds. The definition of openness for foliated spaces adopted here involves a certain type of Morse functions. Consequences concerning the problem of existence of regular Poisson structures, the original motivation for this work, are presented.

Due to a publisher error, the correct version of sections 5.4, 5.5, and 5.6 was not printed. The correct version is includedhere.