The purpose of this paper is to show that there is a natural connection between Teichmüller spaces and crystallographic groups. This connection was first discovered by the author when discussing the Deligne-Mumford compactification of moduli spaces (see \lq\lq {\it The Deligne-Mumford compactification and crystallographic groups}\rq\rq, preprint, 2018). In this paper, we will discuss the same connection from a slightly different viewpoint having in mind an analogy of the maximally degenerate frontier points of the augmented Teichmüller spaces and the cusp points of hyperbolic spaces.

For proper flat schemes over complete discrete valuation rings of mixed characteristic, we construct an isomorphism of certain subgroups of the Picard group and the first cohomology group of the structure sheaf. When the Picard functor is representable and smooth, our construction recovers and gives finer information to the isomorphism coming from its formal completion. An alternative proof of an old theorem of Mattuck is given.

For an isolated singularity which is Newton non-degenerate and also convenient, the Milnor number can be computed from the complement of its Newton diagram in the first quadrant by using Kouchnirenko's formula. In this paper, we consider tropical curves dual to subdivisions of this complement for a plane curve singularity, and show that there exists a tropical curve by which we can count the Milnor number. Our formula may be regarded as a tropical version of the well-known formula by the real morsification due to A'Campo and Gusein-Zade.

We give a formula of the connected component decomposition of the Alexander quandle: $\mathbb{Z}[t^{\pm1}]/(f_1(t),\ldots, f_k(t))=\bigsqcup^{a-1}_{i=0}\mathrm{Orb}(i)$, where $a=\gcd (f_1(1),\ldots, f_k(1))$. We show that the connected component $\mathrm{Orb}(i)$ is isomorphic to $\mathbb{Z}[t^{\pm1}]/J$ with an explicit ideal $J$. By using this, we see how a quandle is decomposed into connected components for some Alexander quandles. We introduce a decomposition of a quandle into the disjoint union of maximal connected subquandles. In some cases, this decomposition is obtained by iterating a connected component decomposition. We also discuss the maximal connected sub-multiple conjugation quandle decomposition.

We consider two families of subgroups of a group. Assume that each subgroup which belongs to one family is contained in some subgroup which belongs to the other family. We then discuss relations of relative hyperbolicity for the group with respect to the two families, respectively. If the group is hyperbolic relative to the two families, respectively, then we consider relations of relative quasiconvexity for a subgroup of the group with respect to the two families, respectively.

Let $G$ be a finite non-solvable group. We study homology of the complex $\mathcal{N}(G)$ of all non-trivial nilpotent subgroups of $G$. The determination of $H_{n}(\mathcal{N}(G))$ is reduced to that of homology of its subcomplex $\mathcal{N}_{\pi_{1}}(G)$ consisting of all nilpotent $\pi_{1}$-subgroups, where $\pi_{1}$ is the connected component of the prime graph of $G$ containing 2. Furthermore, $\mathcal{N}_{\pi_{1}}(G)$ is connected if $G$ possesses no strongly embedded subgroups.

We study the bifurcations of small amplitude time-periodic solutions and chaotic solutions of a two-component integro-differential reaction-diffusion system in one spatial dimension. The system has doubly degenerate points and triply degenerate points. The following results are obtained. (I) Around the doubly degenerate points, a reduced two-dimensional dynamical system on the center manifold is obtained. We find that the small amplitude stable time-periodic solutions can bifurcate from the non-uniform stationary solutions through the Hopf bifurcations for all $n$. (II) Around the triply degenerate point, a three-dimensional dynamical system on the center manifold is obtained. The reduced system can be transformed into normal form for the Hopf-Pitchfork bifurcation. The truncated normal form can possess the invariant tori and the heteroclinic loop. Furthermore, the system under the non $S^{1}$-symmetric perturbation may possess the Shil'nikov type homoclinic orbit. Numerical results for the integro-differential reaction-diffusion system are presented and found to be convincing.

In this work we consider constructions of genus three curves $Y$ such that $\text{End}(\text{Jac} (Y))\otimes \mathbb{Q}$ contains the totally real cubic number field $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$. We construct explicit three-dimensional families whose general member is a nonhyperelliptic genus 3 curve with this property. The case when $Y$ is hyperelliptic was studied in \textsc{J. W. Hoffman, H. Wang}, $7$-gons and genus $3$ hyperelliptic curves, Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales., Serie A. Matemàticas \textbf{107} (2013), 35--52, and some nonhyperelliptic curves were constructed in \textsc{J. W. Hoffman, Z. Liang, Y. Sakai, H. Wang}, Genus $3$ curves whose Jacobians have endomorphisms by $\mathbb{Q}(\zeta _7 +\bar{\zeta}_7 )$, J. Symb. Comp. \textbf{74} (2016), 561--577.

In this short note, we calculate the highest degree term of the hyperbolic torsion polynomial of a pretzel knot with three tangles. It gives a supporting evidence for a conjecture of Dunfield, Friedl and Jackson that the hyperbolic torsion polynomial determines the genus and fiberedness of a hyperbolic knot.

As a generalization of slant submanifolds and slant Riemannian maps, we introduce conformal slant Riemannian maps from Riemannian manifolds to almost Hermitian manifolds. We give non-trivial examples, investigate the geometry of foliations and obtain decomposition theorems by using the existence of conformal Riemannian maps. Moreover, we also investigate the harmonicity of such maps and find necessary and sufficient conditions for conformal slant Riemannian maps to be totally geodesic.

We give an upper bound for the rate of simultaneous recurrence for some commuting family of measure preserving transformations. This provides a generalization of the Boshernitzan quantitative recurrence theorem for single transformation. We also give a lower bound for the Hausdorff dimension of the invariant measures for such transformations and for countable group actions.

We construct examples of cohomogeneity one special Lagrangian submanifolds in the cotangent bundle over the complex projective space, whose Calabi-Yau structure was given by Stenzel. For each example, we describe the condition of special Lagrangian as an ordinary differential equation. Our method is based on a moment map technique and the classification of cohomogeneity one actions on the complex projective space classified by Takagi.

Let $\mathrm{F}$ be a non-Archimedean local field with finite residue field. In this paper, we generalize a classification of {\it generic} elements of the general linear group $\mathrm{GL}_n(\mathrm{F})$, $n \geqq 1$ that generate fields of degree $n$ over $\mathrm{F}$ and are {\it minimal} over $\mathrm{F}$, which was given by Hijikata, to an inner form $\mathrm{G}$ of $\mathrm{GL}_n(\mathrm{F})$, and by using the results of Dott [13], we classify supercuspidal representations of $\mathrm{G}$ that are induced from {\it very cuspidal} representations of maximal compact mod center, open subgroups, which are defined in terms of generic elements. This classification generalizes that of {\it epipelagic} supercuspidal representations of $\mathrm{G}$ which was given by Bushnell and Henniart for $\mathrm{GL}_n(\mathrm{F})$ and by Imai and Tsushima for $\mathrm{G}$.