Related papers: Square-tiled tori
One can describe an $n$-dimensional noncommutative torus by means of an antisymmetric $n\times n$-matrix $\theta$. We construct an action of the group $SO(n,n|\bf Z)$ on the space of antisymmetric matrices and show that, generically,…
We classify all edge-to-edge spherical isohedral 4-gonal tilings such that the skeletons are pseudo-double wheels. For this, we characterize these spherical tilings by a quadratic equation for the cosine of an edge-length. By the…
We study certain structural properties of fine zonotopal tilings, or cubillages, on cyclic zonotopes $Z(n,d)$ of an arbitrary dimension $d$ and their relations to $(d-1)$-separated collections of subsets of a set $\{1,2,\ldots,n\}$.…
In this paper, we classify up to Hamiltonian isotopy Lagrangian tori that split as a product of circles in $S^2 \times S^2$, when the latter is equipped with a non-monotone split symplectic form. We show that this classification is…
For each closed orientable surface we introduce a simplical complex with some additional structure which is a version of the complex of curves of this surface adjusted to investigation of its Torelli group. We call this complex the Torelli…
We prove the 2-torus $\mathbb T$, an abelian linear algebraic group, is a fine moduli space of labeled, oriented, possibly-degenerate inscribable similarity classes of triangles, where a triangle is {\it inscribable} if it can be inscribed…
We investigate conditions under which a two-dimensional complex semi-torus splits into a direct product of C^* and a one-dimensional compact complex torus.
We study normal compact K\"ahler spaces whose rational cohomology ring is isomorphic to that of a complex torus. We call them rational cohomology tori. We classify, up to dimension three, those with rational singularities. We then give…
In this paper, we prove the existence of full dimensional tori for $d$-dimensional nonlinear Schr$\ddot{\mbox{o}}$dinger equation with periodic boundary conditions \begin{equation*}\label{L1} \sqrt{-1}u_{t}+\Delta u+V*u\pm\epsilon…
We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides.
We classify orbifolds obtained by taking the quotient of a three tori by abelian extensions of Z/n x Z/n automorphisms, where each torus has a multiplicative Z/n action (n=3,4 or 6). This 'completes' the classification of orbifolds of the…
An anti-torus is a subgroup $<a,b>$ in the fundamental group of a compact non-positively curved space $X$, acting in a specific way on the universal covering space $\tilde{X}$ such that $a$ and $b$ do not have any commuting non-trivial…
We show that under a lower Ricci curvature bound and an upper diameter bound, a torus admits a finite-sheeted covering space with volume bounded from below and diameter bounded from above. This partially recovers a result of Kloeckner and…
We consider surface links in the 4-space which are presented by the form of simple branched coverings over the standard torus, which we call torus-covering links. In this paper, we study unknotting numbers of torus-covering links. In some…
We solve a problem posed by Cardinali and Sastry [2] about factorization of $2$-covers of finite classical generalized quadrangles. To that end, we develop a general theory of cover factorization for generalized quadrangles, and in…
We prove equivalent numerical conditions for a complete spherical variety to admit a toric structure, and for the smoothness of an arbitrary spherical variety along any given G-orbit. The conditions are in terms of spherical skeletons, a…
In this paper we give a classification of tilings of the sphere by congruent quadrilaterals with exactly two equal edges. The tilings are the earth map tilings, $(p,q)$-earth map tilings and their flip modifications, and quadrilateral…
We give an exact formula for the number of distinct square patterns of a given size that occur in the Squiral tiling.
Toroidal orbifolds and their resolutions are described within the framework of (2,2) Gauged Linear Sigma Models (GLSMs). Our procedure describes two-tori as hypersurfaces in (weighted) projective spaces. The description is chosen such that…
We prove a flat torus theorem for quadric complexes. In particular, we show that if a non-cyclic free abelian group $G$ acts metrically properly on a quadric complex $X$, then $G \cong \mathbb{Z}^2$ and $X$ contains a $G$-invariant…