[en] We present some new notions of smoothness of geometrical boundaries. The relations between the classical notions of smoothness and ours are also investigated. Finally, the extension of locally Lipschitz continuous functions is studied in connection with geometrical properties of their demands of definition. (author). 17 refs, 2 figs

[en] For each 1 ≤ n ≤ 6 we present formulas for the number of n-nodal curves in an n-dimensional linear systems on a smooth, projective surface. The method yields in particular the numbers of rational curves in the system of hyperplane sections of a generic K3-surface imbedded in Pⁿ by a complete system of curves of genus n as well as the number 17,601,000 of rational (singular) plane quintic curves in a generic quintic threefold. (author). 23 refs

[en] D-branes in curved backgrounds can be treated with techniques of boundary conformal field theory. We discuss the influence of scalar condensates on such branes, i.e. perturbations of boundary conditions by marginal boundary operators. A general criterion is presented that guarantees a boundary perturbation to be truly marginal in all orders of perturbation theory. Our results on boundary deformations have several interesting applications which are sketched at the end of this note. (orig.)

[en] A notion of Cartan pairs as an analogy of vector fields in the realm of noncommutative geometry has been proposed previously. In this paper an outline is given of the construction of a noncommutative analogy of the algebra of differential operators as well as its (algebraic) Fock space realization. Co-universal vector fields and covariant derivatives will also be discussed

[en] Suppose a spacetime M is a quotient of a spacetime V by a discrete group of isometries. It is shown how causality conditions in the two spacetimes are related, and how one can learn about the future causal boundary on M by studying structures in V. The relations between the two are particularly simple (the boundary of the quotient is the quotient of the boundary) if both V and M have spacelike future boundaries and if it is known that the quotient of the future completion of V is past-distinguishing. (That last assumption is automatic in the case of M being multi-warped.)

[en] Motivated by the recent interest in dynamical properties of topologically non-trivial spacetimes, we study linear perturbations of spatially closed Bianchi III vacuum spacetimes, whose spatial topology is the direct product of a higher genus surface and the circle. We first develop necessary mode functions, vectors and tensors, and then perform separations of (perturbation) variables. The perturbation equations decouple in a way that is similar to but a generalization of those of the Regge-Wheeler spherically symmetric case. We further achieve a decoupling of each set of perturbation equations into gauge-dependent and gauge-independent parts, by which we obtain wave equations for the gauge-invariant variables. We then discuss choices of gauge and stability properties. Details of the compactification of Bianchi III manifolds and spacetimes are presented in an appendix. In the other appendices we study scalar field and electromagnetic equations on the same background to compare asymptotic properties

[en] Recently we have shown that for 2 + 1-dimensional thin-shell wormholes a non-circular throat may lead to a physical wormhole in the sense that the energy conditions are satisfied. By the same token, herein we consider an angular dependent throat geometry embedded in a 2 + 1-dimensional flat spacetime in polar coordinates. It is shown that, remarkably, a generic, natural example of the throat geometry is provided by a hypocycloid. That is, two flat 2 + 1 dimensions are glued together along a hypocycloid. The energy required in each hypocycloid increases with the frequency of the roller circle inside the large one. (orig.)

[en] Automorphic inflation is an application of the framework of automorphic scalar field theory, based on the theory of automorphic forms and representations. In this paper the general framework of automorphic and modular inflation is described in some detail, with emphasis on the resulting stratification of the space of scalar field theories in terms of the group theoretic data associated to the shift symmetry, as well as the automorphic data that specifies the potential. The class of theories based on Eisenstein series provides a natural generalization of the model of j-inflation considered previously.

[en] After results of the author (1980, 1981) and Vinberg (1981), the finiteness of the number of maximal arithmetic groups generated by reflections in Lobachevsky spaces remained unknown in dimensions 2≤n≤9 only. It was proved recently (2005) in dimension 2 by Long, Maclachlan and Reid and in dimension 3 by Agol. Here we use the results in dimensions 2 and 3 to prove the finiteness in all remaining dimensions 4≤n≤9. The methods of the author (1980, 1981) are more than sufficient for this using a very short and very simple argument

[en] We introduce one-dimensional sets to help describe and constrain the integral curves of an n-dimensional dynamical system. These curves provide more information about the system than zero-dimensional sets (fixed points). In fact, these curves pass through the fixed points. Connecting curves are introduced using two different but equivalent definitions, one from dynamical systems theory, the other from differential geometry. We describe how to compute these curves and illustrate their properties by showing the connecting curves for a number of dynamical systems.