We consider Legendrian knots and links in the standard 3-dimensional contact space. In 1997 Chekanov [Ch] introduced a new invariant for these knots. At the same time, a similar construction was suggested by Eliashberg [E1] within the framework of his joing work with Hofer and Givernthal on Symplectic Field Theory ([E2],[EGH]). To a knot diagram, they associated a differential algebra A. Its stable isomorphism type is invariant under Legendrian isotopy of the knot.

In this paper, we introduce an additional structure on this algebra in the case of a Legendrian link. For a link of N components, we show that its algebra splits A = ⊕_{g ∈ G} A_g Here G is a free group on (N - 1) variables. The splitting is determined by the order of the knots and is preserved by the differential. It gives a tool to show that some permutations of link components are impossible to produce by Legendrian isotopy.

We present an equivariant Liapunov stability criterion for dynamical systems with symmetry. This result yields a simple proof of the energy-momentum-Casimir stability analysis of relative equilibria of equivariant Hamiltonian systems.

We extend the theorems concerning the equivariant symplectic reduction of the cotangent bundle to contact geometry. The role of the cotangent bundle is tken by the cosphere bundle. We use Albert's method for reduction at zero and Willett's method for non-zero reduction.

Using sympectic Floer homology, Seidel associated a module to each mapping class of a compact connected oriented two-manifold of genus bigger than one. We compute this module for mapping classes which do not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphisms. The Nielsen-Thurston representative of such a class is shown to be monotone. The formula for the Floer homology is obtained for a topological separation of fixed points and a separation mechanism for Floer connecting orbits. As examples, we consider the geometric monodroy of isolated plane curve singularities. In this case, the formula for the Floer homology is particularly simple.

We discuss the construction of toric Kähler metrics on symplectic 2n-manifolds with hamiltonian n-torus action and present a simple derivation of the Guillemin formula for a distinguished Kähler metric on any such manifold. the results also apply to orbifolds.

We provide a complete and self-contained classification of (compact connected) contact toric manifolds thereby finishing the work initiated by Banyaga and Molino and by Galicki and Boyer. Our motivation comes from the conjectures of Toth and Zelditch on the uniqueness of toric integrable actions on the punctured cotangent bundles on n-toru 𝕋ⁿ and of the two-sphere S². The conjectures are equivalent to the uniqueness, up to conjugation, of maximal tori in the contactomorphism groups of the cosphere bundles of 𝕋ⁿ and S² respectively.

Let M be a compact oriented simply-connected manifold of dimension at least 8. Assume M is equipped with a torsion-free semi-free circle action with isolated fixed points. We prove M has a perfect invariant Morse-Smale function. The major ingredient in the proof is a new cancellation theorem for invariant Morse theory.