Abstract. We explore the relationship between contact forms on S 3 defined by Finsler metrics on S 2 and the theory developed by H. Hofer, K. Wysocki and E. Zehnder in [9,10]. We show that a Finsler metric on S 2 with curvature K ≥ 1 and with all geodesic loops of length > π is dynamically convex and hence it has either two or infinitely many closed geodesics. We also explain how to explicitly construct J-holomorphic embeddings of cylinders asymptotic to Reeb orbits of contact structures arising from Finsler metrics on S 2 with K = 1 thus complementing the results obtained in [8].

We describe a model-theoretic setting for the study of Shimura varieties, and study the interaction between model theory and arithmetic geometry in this setting. In particular, we show that the modeltheoretic statement of a certain L ω 1 ,ω -sentence having a unique model of cardinality ℵ 1 is equivalent to a condition regarding certain Galois representations associated with Hodge-generic points. We then show that for modular and Shimura curves this L ω 1 ,ω -sentence has a unique model in every infinite cardinality. In the process, we prove a new characterisation of the special points on any Shimura variety.

We consider a closed orientable Riemannian 3-manifold (M, g) and a vector field X with unit norm whose integral curves are geodesics of g. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of g. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that X is the Reeb vector field of the 1-form λ obtained by contracting g with X. We apply these results to the case of great circle flows on the 3-sphere with two objectives in mind: one is to recover the result in [4] that a volume preserving great circle flow must be Hopf and the other is to characterize in a similar fashion great circle flows that are conformal relative to the almost complex structure in the kernel of λ given by rotation by π/2 according to the orientation of M .

Abstract. Let M be a three-dimensional contact manifold, andψ : D \ {0} → M × R a finite-energy pseudoholomorphic map from the punctured disc in C that is asymptotic to a periodic orbit of the contact form. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such thatψ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singular point. Examples of this behaviour, which are studied in some detail, include pseudoholomorphic maps into E p,q × R, where E p,q denotes a rational ellipsoid (contact structure induced by the standard complex structure on C 2 ), as well as contact structures arising from non-standard circle-fibrations of the three-sphere.

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