Given $r>1$, we consider convex bodies in $\E^n$ which contain a fixed unit ball, and whose extreme points are of distance at least $r$ from the centre of the unit ball, and we investigate how well these convex bodies approximate the unit ball in terms of volume, surface area and mean width. As $r$ tends to one, we prove asymptotic formulae for the error of the approximation, and provide good estimates on the involved constants depending on the dimension.

Let~$V$ be an analytic variety in some open set in~$\C^n$. For a real analytic curve~$\gamma$ with $ \gamma(0) = 0 $ and $ d \ge 1 $ define $ V_t = t^{-d}(V - \gamma(t)) $. It was shown in a previous paper that the currents of integration over~$V_t$ converge to a limit current whose support $ T_{\gamma,d} V $ is an algebraic variety as~$t$ tends to zero. Here, it is shown that the canonical defining function of the limit current is the suitably normalized limit of the canonical defining functions of the~$V_t$. As a corollary, it is shown that $ T_{\gamma,d} V $ is either inhomogeneous or coincides with $ T_{\gamma,\delta} V $ for all~$\delta$ in some neighborhood of~$d$. As another application it is shown that for surfaces only a finite number of curves lead to limit varieties that are interesting for the investigation of Phragm\'en--Lindel\"of conditions. Corresponding results for limit varieties $ T_{\sigma,\delta} W $ of algebraic varieties W along real analytic curves tending to infinity are derived by a reduction to the local case.

We classify linear weighted graphs up to the blowing-up and blowing-down operations which are relevant for the study of algebraic surfaces.

We will study the following question: Are nilpotent conjugacy classes of reductive Lie algebras over $p$-adic fields definable? By definable, we mean definable by a formula in Pas's language. In this language, there are no field extensions and no uniformisers. Using Waldspurger's parametrization, we answer in the affirmative in the case of special orthogonal Lie algebras $\mathfrak{so}(n)$ for $n$ odd, over $p$-adic fields.

A smooth affine surface $X$ defined over the complex field $\C$ is an $\ML_0$ surface if the Makar--Limanov invariant $\ML(X)$ is trivial. In this paper we study the topology and geometry of $\ML_0$ surfaces. Of particular interest is the question: Is every curve $C$ in $X$ which is isomorphic to the affine line a fiber component of an $\A^1$-fibration on $X$? We shall show that the answer is affirmative if the Picard number $\rho(X)=0$, but negative in case $\rho(X) \ge 1$. We shall also study the ascent and descent of the $\ML_0$ property under proper maps.

For $p$ a prime, a $p$-typical cover of a connected scheme on which $p=0$ is a finite \'etale cover whose monodromy group (i.e., the Galois group of its normal closure) is a $p$-group. The geometry of such covers exhibits some unexpectedly pleasant behaviors; building on work of Katz, we demonstrate some of these. These include a criterion for when a morphism induces an isomorphism of the $p$\nobreakdash-typi\-cal quotients of the \'etale fundamental groups, and a decomposition theorem for $p$-typical covers of polynomial rings over an algebraically closed field.

For a hyperbolic $3$-manifold $M$ with a torus boundary component, all but finitely many Dehn fillings yield hyperbolic $3$-manifolds. In this paper, we will focus on the situation where $M$ has two exceptional Dehn fillings: an annular filling and a toroidal filling. For such a situation, Gordon gave an upper bound of $5$ for the distance between such slopes. Furthermore, the distance $4$ is realized only by two specific manifolds, and $5$ is realized by a single manifold. These manifolds all have a union of two tori as their boundaries. Also, there is a manifold with three tori as its boundary which realizes the distance $3$. We show that if the distance is $3$ then the boundary of the manifold consists of at most three tori.

Let $\alpha$ and $\beta$ be two Furstenberg transformations on $2$-torus associated with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2$. It is shown that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable \CAs, it is shown that $\af$ and $\bt$ are approximately $K$-conjugate if (and only if) $\overline{\theta_1\pm \theta_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to the condition that the associated crossed product \CAs are isomorphic.

Starting with a 2-dimensional mod $p$ Galois representation, we construct a deformation to a power series ring in infinitely many variables over the $p$-adics. The image of this representation is full in the sense that it contains $\SL_2$ of this power series ring. Furthermore, all ${\mathbb Z}_p$ specializations of this deformation are potentially semistable at $p$.

L'objectif de cet article est d'\'etudier la notion d'amibe au sens de Favorov pour les syst\`emes finis de sommes d'exponentielles \`a fr\'equences r\'eelles et de montrer que, sous des hypoth\`eses de g\'en\'ericit\'e sur les fr\'equences, le compl\'ementaire de l'amibe d'un syst\`eme de~$(k+1)$ sommes d'exponentielles \`a fr\'equences r\'eelles est un sous-ensemble $k$-convexe au sens d'Henriques.