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$BV$, 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$n$ types of countable dense sets: such a space contains a subset S$S$ of size at most n-1$n-1$ such that S$S$ is invariant under all homeomorphisms of X$X$ and X\,\backslash \,S$X\,\backslash \,S$ is countable dense homogeneous. We prove that every Borel space having fewer than \mathfrak{c}$\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or \mathfrak{c}$\mathfrak{c}$ many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

In this paper we regularize the Kepler problem on {{S}^{3}}${{S}^{3}}$ 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 presents 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 space 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$E$ be an elliptic curve over \mathbb{Q}$\mathbb{Q}$ that has good supersingular reduction at p\,>\,3$p\,>\,3$. We construct what we call the \pm /\pm $\pm /\pm$-Selmer groups of E$E$ over the \mathbb{Z}_{p}^{2}$\mathbb{Z}_{p}^{2}$-extension of an imaginary quadratic field K$K$ when the prime p$p$ splits completely over K/\mathbb{Q}$K/\mathbb{Q}$, and prove that they enjoy a property analogous to Mazur's control theorem.

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

Let {{\mathbb{F}}_{q}}\left[ t \right]${{\mathbb{F}}_{q}}\left[ t \right]$ denote the polynomial ring over the finite field {{\mathbb{F}}_{q}}${{\mathbb{F}}_{q}}$. We employ Wooley's new efficient congruencing method to prove certain multidimensional Vinogradov-type estimates in {{\mathbb{F}}_{q}}\left[ t \right]${{\mathbb{F}}_{q}}\left[ t \right]$. 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}}\left[ t \right]${{\mathbb{F}}_{q}}\left[ t \right]$.

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 Poincaré–Birkhoff–Witt conditions.

In this paper, we show that the failure of the unique branch hypothesis \left( \text{UBH} \right)$\left( \text{UBH} \right)$ for tame trees implies that in some homogenous generic extension of V$V$ there is a transitive model M$M$ containing Ord \cup \mathbb{R}$\cup \mathbb{R}$ such that M\,\vDash \,\text{A}{{\text{D}}^{+}}\,+\,\Theta \,>\,{{\theta }_{0}}$M\,\vDash \,\text{A}{{\text{D}}^{+}}\,+\,\Theta \,>\,{{\theta }_{0}}$. In particular, this implies the existence (in V$V$) 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_{0}^{D}\left( N \right)$X_{0}^{D}\left( N \right)$ over the rational numbers. We determine criteria for the twist by an Atkin–Lenher involution to have points over a local field. As a corollary we give a new proof of the theorem of Jordan and Livné on {{\mathbf{Q}}_{p}}${{\mathbf{Q}}_{p}}$ points when p|D$p|D$ and for the first time give criteria for {{\mathbf{Q}}_{p}}${{\mathbf{Q}}_{p}}$ points when p|N$p|N$. We also give congruence conditions for roots modulo p$p$ of Hilbert class polynomials.