The role of the Legendre transform in the study of the Floer complex of cotangent bundles

Consider a classical Hamiltonian H on the cotangent bundle T*M of a closed orientable manifold M, and let L:TM -> R be its Legendre-dual Lagrangian. In a previous paper we constructed an isomorphism Phi from the Morse complex of the Lagrangian action functional which is associated to L to the Floer complex which is determined by H. In this paper we give an explicit construction of a homotopy inverse Psi of Phi. Contrary to other previously defined maps going in the same direction, Psi is an isomorphism at the chain level and preserves the action filtration. Its definition is based on counting Floer trajectories on the negative half-cylinder which on the boundary satisfy "half" of the Hamilton equations. Albeit not of Lagrangian type, such a boundary condition defines Fredholm operators with good compactness properties. We also present a heuristic argument which, independently on any Fredholm and compactness analysis, explains why the spaces of maps which are used in the definition of Phi and Psi are the natural ones. The Legendre transform plays a crucial role both in our rigorous and in our heuristic arguments. We treat with some detail the delicate issue of orientations and show that the homology of the Floer complex is isomorphic to the singular homology of the loop space of M with a system of local coefficients, which is defined by the pull-back of the second Stiefel-Whitney class of TM on 2-tori in M

The Physics Inside Topological Quantum Field Theories

We show that the equations of motion defined over a specific field space are realizable as operator conditions in the physical sector of a generalized Floer theory defined over that field space. The ghosts associated with such a construction are found not to be dynamical. This construction is applied to gravity on a four dimensional manifold, $M$; whereupon, we obtain Einstein's equations via surgery, along $M$, in a five-dimensional topological quantum field theory.Comment: LaTeX, 7 page

Morse homology for the heat flow

We use the heat flow on the loop space of a closed Riemannian manifold to construct an algebraic chain complex. The chain groups are generated by perturbed closed geodesics. The boundary operator is defined in the spirit of Floer theory by counting, modulo time shift, heat flow trajectories that converge asymptotically to nondegenerate closed geodesics of Morse index difference one.Comment: 89 pages, 3 figure

Unstable geodesics and topological field theory

A topological field theory is used to study the cohomology of mapping space. The cohomology is identified with the BRST cohomology realizing the physical Hilbert space and the coboundary operator given by the calculations of tunneling between the perturbative vacua. Our method is illustrated by a simple example.Comment: 28 pages, OCU-15

Squeezing in Floer theory and refined Hofer-Zehnder capacities of sets near symplectic submanifolds

We use Floer homology to study the Hofer-Zehnder capacity of neighborhoods near a closed symplectic submanifold M of a geometrically bounded and symplectically aspherical ambient manifold. We prove that, when the unit normal bundle of M is homologically trivial in degree dim(M) (for example, if codim(M) > dim(M)), a refined version of the Hofer-Zehnder capacity is finite for all open sets close enough to M. We compute this capacity for certain tubular neighborhoods of M by using a squeezing argument in which the algebraic framework of Floer theory is used to detect nontrivial periodic orbits. As an application, we partially recover some existence results of Arnold for Hamiltonian flows which describe a charged particle moving in a nondegenerate magnetic field on a torus. We also relate our refined capacity to the study of Hamiltonian paths with minimal Hofer length.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper40.abs.htm

Design and Implementation of a Software Defined Ionosonde. A contribution to the development of distributed arrays of small instruments

In order to make advances in studies of mesoscale ionospheric phenomena, a new type of ionosonde is needed. This ionosonde should be relatively inexpensive and small form factor. It should also be well suited for operation in a network of transmit and receiver sites that are operated cooperatively in order to measure vertical and oblique paths between multiple transmitters and receivers in the network. No such ionosonde implementation currently exists. This thesis describes the design and implementation of a coded continuous wave ionosonde, which utilizes long pseudo-random transmit waveforms. Such radar waveforms have several advantages: they can be used at low peak power, they can be used in multi-static cooperative radar networks, they can be used to measure range-Doppler overspread targets, they are relatively robust against external interference, and they produce relatively low interference to other users that share the same portion of the electromagnetic spectrum. The new ionosonde design is thus well suited for use in ionosonde networks. The technical design relies on the software defined radio paradigm and the hardware design is based on commercially available inexpensive hardware. The hardware and software implementation is shown to meet the technical and scientific requirements that were set for the instrument. The operation of the instrument is demonstrated in practice in Longyearbyen, Svalbard. With this new ionosonde design and proof of concept implementation, it has been possible to re-establish routine ionospheric soundings at Longyearbyen, Svalbard; to replace the Dynasonde instrument that was decommissioned several years ago. It is also possible to use this new design as a basis for larger networks of ionosondes. The software and hardware design is made publicly available as open source, so that anyone interested can reproduce the instrument and also contribute to the project in the future

Symplectic Floer homology of area-preserving surface diffeomorphisms

The symplectic Floer homology HF_*(f) of a symplectomorphism f:S->S encodes data about the fixed points of f using counts of holomorphic cylinders in R x M_f, where M_f is the mapping torus of f. We give an algorithm to compute HF_*(f) for f a surface symplectomorphism in a pseudo-Anosov or reducible mapping class, completing the computation of Seidel's HF_*(h) for h any orientation-preserving mapping class.Comment: 57 pages, 4 figures. Revision for publication, with various minor corrections. Adds results on the module structure and invariance thereo

A Morse complex for infinite dimensional manifolds - Part I

In this paper and in the forthcoming Part II we introduce a Morse complex for a class of functions f defined on an infinite dimensional Hilbert manifold M, possibly having critical points of infinite Morse index and coindex. The idea is to consider an infinite dimensional subbundle - or more generally an essential subbundle - of the tangent bundle of M, suitably related with the gradient flow of f. This Part I deals with the following questions about the intersection W of the unstable manifold of a critical point x and the stable manifold of another critical point y: finite dimensionality of W, possibility that different components of W have different dimension, orientatability of W and coherence in the choice of an orientation, compactness of the closure of W, classification, up to topological conjugacy, of the gradient flow on the closure of W, in the case dim W=2.Comment: LaTeX2e file, 62 pages, to appear in Advances in Mathematics. Revised version, substantial changes in the compactness par

Seiberg-Witten-Floer Homology and Gluing Formulae

This paper gives a detailed construction of Seiberg-Witten-Floer homology for a closed oriented 3-manifold with a non-torsion \spinc structure. Gluing formulae for certain 4-dimensional manifolds splitting along an embedded 3-manifold are obtained.Comment: 63 pages, LaTe

The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems

An equivariant version of Conley's homotopy index theory for flows is described and used to find periodic solutions of a Hamiltonian system locally near an equilibrium point which is at resonanc