Abstract. We study some Riemannian metrics on the space of smooth regular curves in the plane, viewed as the orbit space of maps from S1 to the plane modulo the group of diffeomorphisms of S1, acting as reparameterizations. In particular we investigate the metric for a constant A> 0:

We present an extension of the lambda-calculus with differential constructions motivated by a model of linear logic discovered by the first author and presented in [Ehr01]. We state and prove some basic results (confluence, weak normalization in the typed case), and also a theorem relating the usual Taylor series of analysis to the linear head reduction of lambda-calculus.

Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2-metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two length-weighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudo-differential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudo-differential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.

Abstract. The L 2-metric or Fubini-Study metric on the non-linear Grassmannian of all submanifolds of type M in a Riemannian manifold (N, g) induces geodesic distance 0. We discuss another metric which involves the mean curvature and shows that its geodesic distance is a good topological metric. The vanishing phenomenon for the geodesic distance holds also for all diffeomorphism groups for the L 2-metric. 1.

A global theory of algebras of generalized functions

We present a category of locally convex topological vector spaces which is a model of propositional classical linear logic, based on the standard concept of Kothe sequence spaces. In this setting, the spaces interpreting the exponential have a quite simple structure of commutative Hopf algebra. The co-Kleisli category of this linear category is a cartesian closed category of entire mappings. This work provides a simple setting where typed -calculus and dierential calculus can be combined; we give a few examples of computations. 1

Abstract. In the present paper we study abelian extensions of connected Lie groups G modeled on locally convex spaces by smooth G-modules A. We parametrize the extension classes by a suitable cohomology group H 2 s (G,A) defined by locally smooth cochains and construct an exact sequence that describes the difference between H 2 s (G,A) and the corresponding continuous Lie algebra cohomology space H 2 c (g,a). The obstructions for the integrability of a Lie algebra extensions to a Lie group extension are described in terms of period and flux homomorphisms. We also characterize the extensions with global smooth sections resp. those given by global smooth cocycles. Finally we apply the general theory to extensions of several types of diffeomorphism groups.

Abstract. We consider a natural Riemannian metric on the infinite dimensional manifold of all embeddings from a manifold into a Riemannian manifold, and derive its geodesic equation in the case Emb(R, R) which turns out to be Burgers ’ equation. Then we derive the geodesic equation, the curvature, and the Jacobi equation of a right invariant Riemannian metric on an infinite dimensional Lie group, which we apply to Diff(R), Diff(S 1), and the Virasoro-Bott group. Many of these results are well known, the emphasis is on conciseness and clarity. Table of contents 1. Introduction..........................

Abstract. We clarify the question whether for a smooth curve of polynomials one can choose the roots smoothly and related questions. Applications to perturbation theory of operators are given. Table of contents 1. Introduction...........................

Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a compact manifold. 1.