Minhyong Kim defined the Selmer variety associated with a curve $X$ over a number field, which is a non-abelian analogue of the $\mathbb{Q}_p$-Selmer group of the Jacobian variety of $X$. In this paper, we define a torsion analogue of the Selmer variety. Recall that Mazur’s control theorem describes the behavior of the torsion Selmer groups of an abelian variety with good ordinary reduction at $p$ in the cyclotomic tower of number fields. We give a non-abelian analogue of Mazur’s control theorem by replacing the torsion Selmer group by a torsion analogue of the Selmer variety.

We study inclusions between primitive ideals in the universal enveloping algebra of general linear superalgebras. For classical simple Lie superalgebras, any primitive ideal is the annihilator of a simple highest weight module. It therefore suffices to study the quasi-order on highest weights determined by the relation of inclusion between primitive ideals. For the specific case of reductive Lie algebras, this quasi-order is essentially the left Kazhdan-Lusztig quasi-order. For Lie superalgebras, a description of the poset structure on the set primitive ideals is at the moment not known, apart from some low dimensional specific cases. We derive an alternative definition of the left Kazhdan-Lusztig quasi-order which extends to classical Lie superalgebras. We denote this quasi-order by $\unlhd$ and show that a relation in $\unlhd$ implies an inclusion between primitive ideals.

For $\mathfrak{gl}(m|n)$ the new quasi-order $\unlhd$ is defined explicitly in terms of Brundan’s Kazhdan-Lusztig theory. We prove that $\unlhd$ induces an actual partial order on the set of primitive ideals. We conjecture that this is the inclusion order. By the above paragraph one direction of this conjecture is true. We prove several consistency results concerning the conjecture and prove it for singly atypical and typical blocks of $\mathfrak{gl}(m|n)$ and in general for $\mathfrak{gl}(2|2)$. An important tool is a new translation principle for primitive ideals, based on the crystal structure underlying Brundan’s categorification on category $\mathcal{O}$. Finally we focus on an interesting explicit example; the poset of primitive ideals contained in the augmentation ideal for $\mathfrak{gl}(m|1)$.

We obtain a topological and weakly equivariant classification of closed three-dimensional Alexandrov spaces with an effective, isometric circle action. This generalizes the topological and equivariant classifications of Raymond [26] and Orlik and Raymond [23] of closed three-dimensional manifolds admitting an effective circle action. As an application, we prove a version of the Borel conjecture for closed three-dimensional Alexandrov spaces with circle symmetry.

Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n (x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

Let $\frak F_{\lambda}$, $\lambda\in \mathbb{C}$, be the space of tensor densities of degree $\lambda$ on the supercircle $S^{1|1}$. We consider the superspace $\mathfrak{D}_{\lambda_1,\lambda_2,\mu}$ of bilinear differential operators from $\frak F_{\lambda_1}\otimes\frak F_{\lambda_2}$ to $\frak F_{\mu}$ as a module over the orthosymplectic superalgebra $\mathfrak{osp}(1|2)$. We prove the existence and the uniqueness of a canonical conformally equivariant symbol map from $\mathfrak{D}_{\lambda_1,\lambda_2,\mu}^k$ to the corresponding space of symbols. An explicit expression of the associated quantization map is also given.