Related papers: The Honeycomb Conjecture
We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment…
A foliation is said to admit a foliated contact structure if there is a codimension 1 distribution in the tangent space of the foliation such that the restriction to any leaf is contact. We prove a version of the Weinstein conjecture in the…
As is remarked by B. Totaro, R. Thomas essentially proved that the Hodge conjecture is inductively equivalent to the existence of a hyperplane section, called a generalized Thomas hyperplane section, such that the restriction to it of a…
In their 2002 paper, Ciucu and Krattenthaler proved several product formulas for the number of lozenge tilings of various regions obtained from a centrally symmetric hexagon on the triangular lattice by removing maximal staircase regions…
In this paper we have given an algorithmic proof of an long standing Barnette's conjecture (1969) that every 3-connected bipartite cubic planar graph is hamiltonian. Our method is quite different than the known approaches and it rely on the…
We show that the hyperplane conjecture holds for the classes of $k$-intersection bodies with arbitrary measures in place of volume.
In 1989 H. Tverberg proposed a quite general conjecture in Discrete geometry, which could be considered as the common basis for many results in Combinatorial geometry and at the same time as a discrete analogue of the common transversal…
Rohatgi and the author recently proved a shuffling theorem for lozenge tilings of `doubly-dented hexagons' (arXiv:1905.08311). The theorem can be considered as a hybrid between two classical theorems in the enumeration of tilings:…
Denote by $M(P)$ the configuration space of a planar polygonal linkage, that is, the space of all possible planar configurations modulo congruences, including configurations with self-intersections. A particular interest attracts its subset…
It is well known that plane curves with the same endpoints are homotopic. An analogous claim for plane curves with the same endpoints and bounded curvature still remains open. In this work we find necessary and sufficient conditions for two…
Hexagonal polyominoes are polyominoes on the honeycomb lattice. We enumerate the symmetry classes of convex hexagonal polyominoes. Here convexity is to be understood as convexity along the three main column directions. We deduce the…
We show that every continuous map from one translationally finite tiling space to another can be approximated by a local map. If two local maps are homotopic, then the homotopy can be chosen so that every interpolating map is also local.
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was…
A typical decomposition question asks whether the edges of some graph $G$ can be partitioned into disjoint copies of another graph $H$. One of the oldest and best known conjectures in this area, posed by Ringel in 1963, concerns the…
Hadwiger's covering conjecture is that every $n$-dimensional convex body can be covered by at most $2^n$ of its smaller positive homothetic copies, with $2^n$ copies required only for affine images of $n$-cube. Convex hull of a ball and an…
Matt Blum conjectured that the number of tilings of a hexagonal dungeon with side-lengths $a,2a,b,a,2a,b$ (for $b\geq2a$) equals $13^{2a^2}14^{\lfloor a^2/2\rfloor}$. Ciucu and the author of the present paper proved the conjecture by using…
A honeycomb array is an analogue of a Costas array in the hexagonal grid; they were first studied by Golomb and Taylor in 1984. A recent result of Blackburn, Etzion, Martin and Paterson has shown that (in contrast to the situation for…
For a compact and convex window, Mecke described a process of tessellations which arise from cell divisions in discrete time. At each time step, one of the existing cells is selected according to an equally-likely law. Independently, a line…
We resolve a 25 year old problem by showing that The Paving Conjecture is equivalent to The Paving Conjecture for Triangular Matrices.
Classic mass partition results are about dividing the plane into regions that are equal with respect to one or more measures (masses). We introduce a new concept in which the notion of partition is replaced by that of a cover. In this case…