Motivated by the work of Farber, Tabachnikov and Yuzvinsky on the motion planning problem for projective spaces, we give an estimate for the topological complexity (TC) of lens spaces in terms of certain generalized “skew” maps between spheres. This last concept turns out to be closely related to that for a generalized axial map developed by Astey, Davis and the author to characterize the smallest Euclidean dimension where (2-torsion) lens spaces can be immersed. As a result, this suggests an alternative simpler “TC-approach” to the classical immersion problem for real projective spaces, whose initial stages we settle by means of techniques in obstruction theory.

Let $M$ be a closed oriented surface of negative Gaussian curvature and let $\Omega$ be a non-exact 2-form. Let $\lambda$ be a small positive real number. We show that the longitudinal KAM-cocycle of the magnetic flow given by $\la\,\Omega$ is a coboundary if and only if the Gaussian curvature is constant and $\Omega$ is a constant multiple of the area form.

By investigating the relation between growth and value distribution A. Melas established a qualitative version of a Picard type theorem for universal Taylor series, that is: every universal Taylor series on the open unit disk $D$, assumes every complex value with at most one exception on infinite subsets of $D$ that approach the boundary of $D$ rather slowly. On the other hand, we show that there are universal Taylor series on $D$ such that the infinite subset of $D$ on which exactly one value is assumed, can approach the boundary of $D$ arbitrarily fast. Hence in view of Melas' work our result is the best possible. We also study the problem of interpolation by universal Taylor series.

We determine the Thurston–Bennequin invariant of graph divide links, which include all closed positive braids, all divide links and certain negative twist knots. As a corollary of this and a result of P. Lisca and A. I. Stipsicz, we prove that the 3-manifold obtained from $S^3$ by Dehn surgery along a non-trivial graph divide knot K with coefficient r carries positive, tight contact structures for every r except the Thurston-Bennequin invariant of K.

The concept of generalised (in the sense of Colombeau) connection on a principal fibre bundle is introduced. This definition is then used to extend results concerning the geometry of principal fibre bundles to those that only have a generalised connection. Some applications to singular solutions of Yang–Mills theory are given.

The main aim of this paper is to give an overview of work that started in the early 90's and contained non-linearly in a number of papers and preprints. The style will be informal since the author wants this paper to be accessible to non-experts. Formal definitions, such as “$\Omega$ is a Lipschitz domain” will not be given. They will be well known to experts and “intuitively” clear to non-experts. Technical statements or proofs will not be given either. The reader who is not happy with this may consult the references given. The reader who wants to read this overview and meets unknown words is advised in the first instance to ignore them.

We give a method to construct non-trivial $p$-harmonic morphisms via conformal change of the metric on the domain and/or the target manifold (Theorems 2.1, 2.5 and 2.8). As applications, we show the existence of higher dimensional harmonic spheres in general manifolds (Theorem 2.10) generalizing Sacks and Uhlenbeck's result on harmonic 2-spheres, prove some existence theorems for non-trivial $p$-harmonic morphisms between conformally flat spaces (Theorems 2.12, 3.1 and 3.2), give a method to construct minimal foliations via $p$-harmonic morphisms and show that many $\mathbf{R}^{m}$ with a conformally flat metric admit minimal foliations of codimension greater than two (Theorems 3.3, 3.4).

Different types of loss of stability of elastic structures are usually illustrated by simple models. This paper presents a family of structures which can demonstrate four cases of the double cusp catastrophe by the variation of a parameter. The path defined by this parameter is calculated in the diagram of the classes of the double cusp catastrophe. In one of the four cases the primary equilibrium path of the perfect structure intersects a secondary equilibrium surface at the critical point. In the other cases there are two secondary equilibrium paths, one of which belongs to critical state in all points. In all four cases the equilibrium paths of the imperfect structure are determined as well as the critical load in terms of the value of the simplest imperfections.

All $\cal {A}$-simple singularities of map-germs from $\bb {R}^n$ to $\bb {R}^p$, where $n\ge p$, of minimal corank (i.e. of corank $n-p+1$) have an M-deformation, that is a deformation in which the maximal numbers of isolated stable singular points are simultaneously present in the discriminant.

We introduce new concepts and properties of lightlike distributions and foliations (of dimension and co-dimension 1) in a space-time manifold of dimension $n$, from a purely geometric point of view. Given an observer and a lightlike distribution $\Omega $ of dimension or co-dimension 1, its lightlike direction is broken down into two vector fields: a timelike vector field $U$ representing the observer and a spacelike vector field $S$ representing the relative direction of propagation of $\Omega $ for this observer. A new distribution $\Omega_U^-$ is defined, with the opposite relative direction of propagation for the observer $U$. If both distributions $\Omega $ and $\Omega_U^-$ are integrable, the pair $\{\Omega,\Omega_U^-\}$ represents the wave fronts of a stationary wave for the observer $U$. However, we show in an example that the integrability of $\Omega $ does not imply the integrability of $\Omega_U^-$.

We modify arguments used in the study of Universal Taylor Series and extend them from simply connected domains to general domains in $\bb {C}$ to give a new simple and natural proof of the fact that all holomorphic functions are generically non extendable.

Multitype Moran sets are introduced in this paper. They appear naturally in the study of the structure of the quasi-crystal spectrum, and they generalize some known fractal structures such as self-similar sets, graph-direct sets and Moran sets. It is known that for any Moran set E with a bounded condition on contracting ratios, one has \[\dim_H E=s_*\le s^*=\dim_P E=\dimB E,\] where $s_*$ and $s^*$ are the lower and upper pre-dimension according to the natural coverings. For any multitype Moran set E with a bounded condition on contracting ratios, we prove $\dimB E{=}s^*$. With an additional assumption of primitivity, we prove that for multitype Moran sets, the above formula still holds. We also give some examples to show that if the condition of primitivity is not fulfilled, then it may happen that $\dim_H E<s_*$ or $\dim_P E<s^*$.

A sufficient condition is derived for a finite-time $L_2 $ singularity of the 3d incompressible Euler equations, making appropriate assumptions on eigenvalues of the Hessian of pressure. Under this condition $ \ \lim_{ t \uparrow T_*} \sup \|\frac{ D \omega} { Dt}\|_{L_2(\Omega)} = \infty $, where $\Omega \subset \mathbb{R}$ moves with the fluid. In particular, $| \omega | $, $| \S_{ij} | $, and $| \P_{ij} | $ all become unbounded at one point $(x_1, T_1) $, $T_1 $ being the first blow-up time in $L_2 $.

The structure of Dunford–Pettis sets, strong Dunford–Pettis sets and certain spaces of operators is studied in the context of the injective and projective tensor products of Banach spaces.

An analogue of the theory of integral closure and reductions is developed for a more general class of closures, called Nakayama closures. It is shown that tight closure is a Nakayama closure by proving a “Nakayama lemma for tight closure”. Then, after strengthening A. Vraciu's theory of *-independence and the special part of tight closure, it is shown that all minimal *-reductions of an ideal in an analytically irreducible excellent local ring of positive characteristic have the same minimal number of generators. This number is called the *-spread of the ideal, by analogy with the notion of analytic spread.