Related papers: Lattices generated by skeletons of reflexive polyt…
In this note, we discuss planar lattices generated by their atoms. We prove that if $L$ is a planar lattice generated by $n$ atoms, then both the left and the right boundaries of $L$ have at most $n+1$ elements. On the other hand, $L$ can…
We show that the number of conjugacy classes of maximal finite subgroups of a lattice in a semisimple Lie group is linearly bounded by the covolume of the lattice. Moreover, for higher rank groups, we show that this number grows sublinearly…
In this paper, we investigate pre-Calabi-Yau algebras and homotopy double Poisson algebras arising from homotopy Rota-Baxter structures. We introduce the notion of cyclic homotopy Rota-Baxter algebras, a class of homotopy Rota-Baxter…
We prove that, apart from some well-known low-dimensional examples, any compact hyperbolic Coxeter polytope has a pair of disjoint facets. This is one of very few known general results concerning combinatorics of compact hyperbolic Coxeter…
In this note we consider the motivic aspect of the middle cohomology of more than 200 classes of quasi-smooth Calabi--Yau threefolds inside weighted projective 4-space which come with an action of a cyclic group of even order. The action…
Given a finite set of lattice points, we compare its sumsets and lattice points in its dilated convex hulls. Both of these are known to grow as polynomials. Generally, the former are subsets of the latter. In this paper, we will see that…
We specify what is meant for a polytope to be reconstructible from its graph or dual graph. And we introduce the problem of class reconstructibility, i.e., the face lattice of the polytope can be determined from the (dual) graph within a…
Four dimensional reflexive polyhedra encode the data for smooth Calabi-Yau threefolds that are hypersurfaces in toric varieties, and have important applications both in perturbative and in non-perturbative string theory. We describe how we…
Based on the concept of dual cones introduced by J.Opgenorth we give an algorithm to compute a generating system of the group of automorphisms of a lattice endowed with a hyperbolic bilinear form.
Periodic lattices are natural generalizations of lattices, which arise naturally in diophantine approximations with rationals of bounded denominators. In this paper, we prove analogues of classical theorems in geometry of numbers for…
We present a universal normal algebra suitable for constructing and classifying Calabi-Yau spaces in arbitrary dimensions. This algebraic approach includes natural extensions of reflexive weight vectors to higher dimensions, related to…
Certain classical generating functions for elements of reflection groups can be expressed using fundamental invariants called exponents. We give new analogues of such generating functions that accommodate orbits of reflecting hyperplanes…
We introduce the notion of decomposable locally conformally product (LCP) manifolds and characterize those which are defined on quotients of Riemannian Lie groups by co-compact lattices.
We construct higher-dimensional Calabi-Yau varieties defined over a given number field with Zariski dense sets of rational points. We give two elementary constructions in arbitrary dimensions as well as another construction in dimension…
We prove that every finite lattice L can be embedded in a three-generated finite lattice K. We also prove that every algebraic lattice with accessible cardinality is a complete sublattice of an appropriate algebraic lattice K such that K is…
A renowned theorem of Blind and Mani, with a constructive proof by Kalai and an efficiency proof by Friedman, shows that the whole face lattice of a simple polytope can be determined from its graph. This is part of a broader story of…
It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure…
We affirm a conjecture of Sacks [1972] by showing that every countable distributive lattice is isomorphic to an initial segment of the hyperdegrees, $\mathcal{D}_{h}$. In fact, we prove that every sublattice of any hyperarithmetic lattice…
In his seminal 1951 paper "Extreme forms" Coxeter \cite{cox51} observed that for $n \ge 9$ one can add vectors to the perfect lattice $\sfA_9$ so that the resulting perfect lattice, called $\sfA_9^2$ by Coxeter, has exactly the same set of…
Geometrical constraints imposed on higher dimensional harmonic lattices generally lead to nonlinear dynamical lattice models. Helical lattices obtained by such a procedure are shown to be described by sine- plus linear-lattice equations.…