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 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.