Legendrian intersections in the 1-jet bundle
In this thesis we construct a family of generating functions for a Legendrian embedding, into the I-jet bundle of a closed manifold, that can be obtained from the zero section through Legendrian embeddings, by discretising the action functional. We compute the second variation of a generating function obtained as above at a nondegenerate critical point and prove a formula relating the signature of the second variation to the Maslov index as the mesh goes to zero. We use this to prove a generalisation of the Morse inequalities thus refining a theorem of Chekanov. We also compute the spectral flow of the operator obtained by linearising the gradient equation of the action functional along a path connecting two nondegenerate critical points. We end by making a conjecture about the relation between the Floer connecting orbits and the gradient flow lines of the discrete action functional.