Joel W. Robbin

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

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.

Asymptotic behaviour of holomorphic strips

Abstract The asymptotic behaviour of a finite energy pseudoholomorphic strip with Lagrangian boundary conditions in a symplectic manifold is determined by an eigenfunction of the linearized operator at the (transverse) intersection.

A construction of the Deligne-Mumford orbifold

We define the Deligne Mumford orbifold axiomatically by a universal mapping property, show that this universal mapping property is equivalent to an infinitessimal universal mapping property, and use the latter to give an existence proof.

Lyapunov maps, simplicial complexes and the Stone functor

Let be an attractor network for a dynamical system f t : M → M , indexed by the lower sets of a partially ordered set P . Our main theorem asserts the existence of a Lyapunov map ψ: M → K ( P ) which defines the attractor network. This result is used to prove the existence of connection matrices for discrete-time dynamical systems.

Combinatorial Floer homology

Introduction Part I. The Viterbo-Maslov Index: Chains and traces The Maslov index The simply connected case The Non simply connected case Part II. Combinatorial Lunes: Lunes and traces Arcs Combinatorial lunes Part III. Floer Homology: Combinatorial Floer homology Hearts Invariance under isotopy Lunes and holomorphic strips Further developments Appendices: Appendix A. The space of paths Appendix B. Diffeomorphisms of the half disc Appendix C. Homological algebra Appendix D. Asymptotic behavior of holomorphic strips Bibliography Index

The moduli space of regular stable maps

We prove that the moduli space of regular stable maps in a complex manifold admits a natural complex orbifold structure. Our proof is based on Hardy decompositions and Fredholm intersection theory.

Phase functions and path integrals

This note is an introduction to our forthcoming paper [17]. There we show how to construct the metaplectic representation using Feynman path integrals. We were led to this by our attempts to understand Atiyah’s explanation of topological quantum field theory in [2]. Like Feynman’s original approach in [9] (see also [10]) an action integral plays the role of a phase function. Unlike Feynman, we use paths in phase space rather than configuration space and use the symplectic action integral rather than the (classical) Lagrangian integral. We eventually restrict to (inhomogeneous) quadratic Hamiltonians so that the finite dimensional approximation to the path integral is a Gaussian integral. In evaluating this Gaussian integral the signature of a quadratic form appears. This quadratic form is a discrete approximation to the second variation of the action integral. For Lagrangians of the form kinetic energy minus potential energy, evaluated on curves in configuaration space, the index of the second variation is well-defined and, via the Morse Index Theorem, related to the Maslov In-

Lie algebras of infinitesimal norm isometries

Abstract Given a norm on a finite dimensional vector space V , we may consider the group of all linear automorphisms which preserve it. The Lie algebra of this group is a Lie subalgebra of the endomorphism algebra of V having two properties: (1) it is the Lie algebra of a compact subgroup, and (2) it is “saturated” in a sence made precise below. We show that any Lie subalgebra satisfying these conditions is the Lie algebra of the group of linear automorphisms preserving some norm. There is an appendix on elementary Lie group theory.

The exponential Vandermonde matrix

is positive for 0 < x0 ≤ x1 ≤ x2 ≤ · · · ≤ xn. Proof: If we divide each row of the matrix W (x) by its leading entry we get another matrix of the same form with ai replaced by ai − a0. Hence we assume w.l.o.g. that a0 = 0. We prove the following stronger statement by induction on n: The function wm(x) := ∂w(x0, x1, . . . , xn) ∂xn∂xn−1 · · ·∂xn−m+1 is positive for m = 0, 1, 2, . . . , n and 0 < x0 ≤ x1 ≤ x2 ≤ · · · ≤ xn.