Dietmar Salamon

The Maslov index for paths

Maslov’s famous index for a loop of Lagrangian subspaces was interpreted by Arnold [1] as an intersection number with an algebraic variety known as the Maslov cycle. Arnold’s general position arguments apply equally well to the case of a path of Lagrangian subspaces whose endpoints lie in the complement of the Maslov cycle. Our aim in this paper is to define a Maslov index for any path regardless of where its endpoints lie. Our index is invariant under homotopy with fixed endpoints and is additive for catenations. Duistermaat [4] has proposed a Maslov index for paths which is not additive for catenations but is independent of the choice of the Lagrangian subspace used to define the Maslov cycle. By contrast our Maslov index depends on this choice. We have been motivated by two applications in [10] and [12] as well as the index introduced by Conley and Zehnder in [2] and [3]. In [12] we show how to define a signature for a certain class of one dimensional first order differential operators whose index and coindex are infinite. In [10] we relate the Maslov index to Cauchy Riemann operators such as those that arise in

Infinite Dimensional Linear Systems with Unbounded Control and Observation: A Functional Analytic Approach.

Abstract : The object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators. On the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem. The implications of the theory for large classes of functional and partial differential equations are discussed in detail. Keywords: Representation of infinite dimensional systems, semigroups, boundary control, feedback, linear quadratic control.

Floer homology and Novikov rings

We prove the Arnold conjecture for compact symplectic manifolds under the assumption that either the first Chern class of the tangent bundle vanishes over π2(M) or the minimal Chern number is at least half the dimension of the manifold. This includes the important class of Calabi-Yau manifolds. The key observation is that the Floer homology groups of the loop space form a module over Novikov’s ring of generalized Laurent series. The main difficulties to overcome are the presence of holomorphic spheres and the fact that the action functional is only well defined on the universal cover of the loop space with a possibly dense set of critical levels.

Transversality in elliptic Morse theory for the symplectic action

Our goal in this paper is to settle some transversality question for the perturbed nonlinear Cauchy-Riemann equations on the cylinder. These results play a central role in the denition of symplectic Floer homology and hence in the proof of the Arnold conjecture. There is currently no other reference to these transversality results in the open literature. Our approach does not require Aronszajn’s theorem. Instead we derive the unique continuation theorem from a generalization of the Carleman similarity principle.

Connected Simple Systems and the Conley Index of Isolated Invariant Sets.

The object of this paper is to present new and simplified proofs for most of the basic results in the index theory for flows. Simple, explicit formulae are derived for the maps which play a central role in the theory. The presentation is self-contained.

Self-dual instantons and holomorphic curves

A gradient flow of a Morse function on a compact Riemannian manifold is said to be of Morse-Smale type if the stable and unstable manifolds of any two critical points intersect transversally. For such a Morse-Smale gradient flow there is a chain complex generated by the critical points and graded by the Morse index. The boundary operator has as its (x, y)-entry the number of gradient flow lines running from x to y counted with appropriate signs whenever the difference of the Morse indices is 1. The homology of this chain complex agrees with the homology of the underlying manifold M and this can be used to prove the Morse inequalities (cf. [33], [26]). Around 1986, Floer generalized this idea to infinite-dimensional variational problems in which every critical point has infinite Morse index but the moduli spaces of connecting orbits form finite-dimensional manifolds for every pair of critical points. The dimensions of these spaces give rise to a relative Morse index and the boundary operator is defined by counting connecting

Realization theory in Hilbert space

A representation theorem for infinite-dimensional, linear control systems is proved in the context of strongly continuous semigroups in Hilbert spaces. The result allows for unbounded input and output operators and is used to derive necessary and sufficient conditions for the realizability in a Hilbert space of a time-invariant, causal input-output operator ℐ. The relation between input-output stability and stability of the realization is discussed. In the case of finite-dimensional input and output spaces the boundedness of the output operator is related to the existence of a convolution kernel representing the operator ℐ.

Dynamical systems, shape theory and the Conley index

The Conley index of an isolated invariant set is defined only for flows; we construct an analogue called the ‘shape index’ for discrete dynamical systems. It is the shape of the one-point compactification of the unstable manifold of the isolated invariant set in a certain topology which we call its ‘intrinsic’ topology (to distinguish it from the ‘extrinsic’ topology which it inherits from the ambient space). Like the Conley index, it is invariant under continuation. A key point is the construction of a certain ‘index category’ associated with the isolated invariant set; this construction works equally well for flows or discrete time systems, and its properties imply the basic properties of both the Conley index and the shape index.

Floer Homology and the Heat Flow

Abstract.We study the heat flow in the loop space of a closed Riemannian manifold M as an adiabatic limit of the Floer equations in the cotangent bundle. Our main application is a proof that the Floer homology of the cotangent bundle, for the Hamiltonian function kinetic plus potential energy, is naturally isomorphic to the homology of the loop space.

KAM theory in configuration space

A new approach to the Kolmogorov-Arnold-Moser theory concerning the existence of invariant tori having prescribed frequencies is presented. It is based on the Lagrangian formalism in configuration space instead of the Hamiltonian formalism in phase space used in earlier approaches. In particular, the construction of the invariant tori avoids the composition of infinitely many coordinate transformations. The regularity results obtained are applied to invariant curves of monotone twist maps. The Lagrangian approach has been prompted by a recent study of minimal foliations for variational problems on a torus by J. Moser.