Related papers: SL_2-Tilings Do Not Exist in Higher Dimensions (mo…

All varieties, extremal contractions, singularities are divided on exceptional and non-exceptional ones. Roughly speaking, there are the infinite families of non-exceptional varieties, extremal contractions or singularities and only the…

Quasicrystals allow for symmetries that are impossible in crystalline materials, such as eight-fold rotational symmetry, enabling the existence of novel higher-order topological insulators in two dimensions without crystalline counterparts.…

Currently only three spatial and one temporal dimensions are considered to be "physical". Recently, solutions to a plethora of questions have used the notion of extra-dimensions. The experimental verification of the existence of such extra…

A second-order topological insulator in three dimensions refers to a topological insulator with gapless states localized on the hinges, which is a generalization of a traditional topological insulator with gapless states localized on the…

We study the definability of maximal towers and of inextendible linearly ordered towers (ilt's), a notion that is more general than that of a maximal tower. We show that there is, in the constructible universe, a $\Pi^1_1$ definable maximal…

We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d=2 (the original…

The SL(2,Z) anomaly recently derived for type IIB supergravity in 10 dimensions is shown to be a consequence of T-duality with the type IIA string, after compactification to 2 dimensions on an 8-fold. This explains the identity of the…

We prove some conditions for the existence of higher dimensional algebraic fibering of group extensions. This leads to various corollaries on incoherence of groups and some geometric examples of algebraic fibers of type $F_n$ but not…

Starting with the light-cone Hamiltonian for gravity, we perform a field redefinition that reveals a hidden symmetry in four dimensions, namely the Ehlers $SL(2,R)$ symmetry. The field redefinition, which is non-local in space but local in…

We prove that 1) There exist infinitely many non-trivial codimension one "thick" knots in $\mathbb{R}^5$; 2) For each closed four-dimensional smooth manifold $M$ and for each sufficiently small positive $\epsilon$ the set of isometry…

Let n integer greater or equal to 4 and even and let T_n be the set of ribbon L-shaped n-ominoes. We study tiling problems for regions in a square lattice by T_n. Our main result shows a remarkable rigidity property: a tiling of the first…

We consider tilings of deficient rectangles by the set $\mathcal{T}_4$ of ribbon $L$-tetrominoes. A tiling exists iff the rectangle is a square of odd side. The missing cell is on the main NW--SE diagonal, in an odd position if the square…

We define iteration over a two dimensional manifold as analog of iteration over a path defined by Chen. We give several applications. Some of them include constructions of non-abelian modular symbol for $SL(3,\Z)$ and for $SL_{2/K}$, where…

We introduce a new class of $\mathfrak{sl}_2$-triples in a complex simple Lie algebra $\mathfrak{g}$, which we call magical. Such an $\mathfrak{sl}_2$-triple canonically defines a real form and various decompositions of $\mathfrak{g}$.…

We develop dimension theory for a large class of structures called espaliers, consisting of a set $L$ equipped with a partial order $\leq$, an orthogonality relation $\perp$, and an equivalence relation $\sim$, subject to certain axioms.…

We prove a lower bound on the rank of tensors constructed from families of linear maps that `expand' the dimension of every subspace. Such families, called {\em dimension expanders} have been studied for many years with several known…

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden non-spatial symmetries can occur…

We show that there exist constants $\alpha,\epsilon>0$ such that for every positive integer $n$ there is a continuous odd function $f:S^m\to S^n$, with $m\geq \alpha n$, such that the $\epsilon$-expansion of the image of $f$ does not…

We consider families of B\"or\"oczky tilings in hyperbolic space in arbitrary dimension, study some basic properties and classify all possible symmetries. In particular, it is shown that these tilings are non-crystallographic, and that…

Algebraic number theory relates SIC-POVMs in dimension $d>3$ to those in dimension $d(d-2)$. We define a SIC in dimension $d(d-2)$ to be aligned to a SIC in dimension $d$ if and only if the squares of the overlap phases in dimension $d$…