Multi-oriented Symplectic Geometry and the Extension of Path Intersection Indices

Abstract: Symplectic geometry can be traced back to Lagrange and his work on celestial mechanics and has since then been a very active field in mathematics, partly because of the applications it offers but also because of the beauty of the objects it deals with.I this thesis we begin by the simplest fact of symplectic geometry. We give the definition of a symplectic space and of the symplectic group, Sp(n). A symplectic space is the data of an even-dimensional space and of a form which satisfies a number of properties. Having done this we give a definition of the Lagrangian Grassmannian Lag(n) which consists of all n-dimensional subspaces of the symplectic space on which the symplectic form vanishes. We carefully study the topology of these spaces and their universal coverings.It is of great interest to know how the elements of the Lagrangian Grassmannian intersect each other. A lot of efforts have therefore been made to construct intersection indices for elements of Lag(n). They have gone under many names but have had a sole purpose, namely to give us a way to determine how these elements intersect. We show how these elements are constructed and extend the definition to paths of elements of Lag(n) and Sp(n). We end this thesis by extending the definition of an index defined by Conley and Zehnder bu using the properties of the Leray index. Their index plays a significant role in the theory of periodic Hamiltonian orbit.