We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting.
As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$ and maximal functions.

We study separable metric spaces with few types of countable dense
sets. We present a structure theorem for locally compact spaces
having precisely $n$ types of countable dense sets: such a space
contains a subset $S$ of size at most $n{-}1$ such that $S$ is
invariant under
all homeomorphisms of $X$ and $X\setminus S$ is countable dense
homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$
types of
countable dense sets is Polish. The natural question of whether every
Polish space has either countably many or $\mathfrak{c}$ many types of
countable
dense sets, is shown to be closely related to Topological Vaught's
Conjecture.

This paper provides an addendum to M. Hrušák
and J. van Mill ``Nearly countable dense homogeneous spaces.''
Canad. J. Math., published online 2013-03-08
http://dx.doi.org/10.4153/CJM-2013-006-8.

In this paper we regularize the Kepler problem on $S^3$ in several
different ways. First, we perform a Moser-type regularization. Then, we
adapt the Ligon-Schaaf regularization to our problem. Finally, we show
that the Moser regularization and the Ligon-Schaaf map we obtained can be
understood as the composition of the corresponding maps for the Kepler problem
in Euclidean space and the gnomonic transformation.

The paper is centered around a new proof of the infinitesimal rigidity
of convex polyhedra. The proof is based on studying derivatives of the
discrete Hilbert-Einstein functional on the space of "warped
polyhedra" with a fixed metric on the boundary.

The situation is in a sense dual to using derivatives of the volume in order to prove the Gauss infinitesimal rigidity of convex polyhedra. This latter kind of rigidity is related to the Minkowski theorem on the existence and uniqueness of a polyhedron with prescribed face normals and face areas.

In the spherical and in the hyperbolic-de Sitter space, there is a perfect duality between the Hilbert-Einstein functional and the volume, as well as between both kinds of rigidity.

We review some of the related work and discuss directions for future research.

Let $E$ be an elliptic curve over $\mathbb Q$ which has good supersingular
reduction at $p\gt 3$. We construct what we call the $\pm/\pm$-Selmer
groups of $E$ over the $\mathbb Z_p^2$-extension of an imaginary quadratic
field $K$ when the prime $p$ splits completely over $K/\mathbb Q$, and
prove they enjoy a property analogous to Mazur's control theorem.

Furthermore, we propose a conjectural connection between the
$\pm/\pm$-Selmer groups and Loeffler's two-variable $\pm/\pm$-$p$-adic
$L$-functions of elliptic curves.

Let $\mathbb{F}_q[t]$ denote the polynomial ring over the finite
field $\mathbb{F}_q$.
We employ Wooley's new efficient congruencing method to prove
certain multidimensional Vinogradov-type estimates in $\mathbb{F}_q[t]$.
These results allow us to apply a variant of the circle method
to obtain asymptotic formulas for a system connected to the problem
about linear spaces lying on hypersurfaces defined over $\mathbb{F}_q[t]$.

We consider finite groups acting on
quantum (or skew) polynomial rings. Deformations of the
semidirect product of the quantum polynomial ring with the acting group
extend symplectic reflection algebras and graded Hecke algebras
to the quantum setting over a field
of arbitrary characteristic.
We give necessary and sufficient conditions for such algebras to satisfy a
Poincaré-Birkhoff-Witt property using the theory of noncommutative
Gröbner bases.
We include applications to the case of abelian groups
and the case of groups acting on coordinate rings of quantum planes.
In addition, we classify graded automorphisms of the coordinate ring of quantum 3-space. In characteristic zero, Hochschild cohomology
gives an elegant description of the PBW conditions.

In this paper, we show that the failure of the unique branch
hypothesis (UBH) for tame trees
implies that in some homogenous generic extension of $V$ there is a
transitive model $M$ containing $Ord \cup \mathbb{R}$ such that
$M\vDash \mathsf{AD}^+ + \Theta \gt \theta_0$. In particular, this
implies the existence (in $V$) of a non-tame mouse. The results of
this paper significantly extend J. R. Steel's earlier results
for tame trees.

Consider a Shimura curve $X^D_0(N)$ over the rational
numbers. We determine criteria for the twist by an Atkin-Lehner
involution to have points over a local field. As a corollary we give a
new proof of the theorem of Jordan-Livné on $\mathbf{Q}_p$ points
when $p\mid D$ and for the first time give criteria for $\mathbf{Q}_p$
points when $p\mid N$. We also give congruence conditions for roots
modulo $p$ of Hilbert class polynomials.