相关论文: Planar pseudo-triangulations, spherical pseudo-til…

We show that there is a constant $c$ such that any 3-uniform hypergraph $\mathcal H$ with $n$ vertices and at least $cn^{5/2}$ edges contains a triangulation of the real projective plane as a subgraph. This resolves a conjecture of…

We construct and study a bicategory of super 2-line bundles over graded Lie groupoids, providing a unified framework for geometric models of twistings of (Real) K-theory. The core of our work is to exhibit a wide range of models from the…

Part I: The two-dimensional Pascal Triangle will be generalized into a three-dimensional Pascal Pyramid and four-, five- or whatsoever-dimensional hyper-pyramids. Part II: The Bilateral Binomial Theorem will be generalised into a Bilateral…

We introduce a conjecture that we call the {\it Two Hyperplane Conjecture}, saying that an isoperimetric surface that divides a convex body in half by volume is trapped between parallel hyperplanes. The conjecture is motivated by an…

In this paper we study the problem of augmenting a planar graph such that it becomes 3-regular and remains planar. We show that it is NP-hard to decide whether such an augmentation exists. On the other hand, we give an efficient algorithm…

In this note, using some regular triangular tilings of the sphere, the Euclidean plane and the hyperbolic plane, we examine the potential relationship between their discrete Bakry - Emery curvatures and the smooth curvatures of their…

Counting Euclidean triangulations with vertices in a finite set $\C$ of the convex hull $\conv(\C)$ of $\C$ is difficult in general, both algorithmically and theoretically. The aim of this paper is to describe nearly convex polygons, a…

The main purpose of this paper is to summarize the basic ingredients, illustrated with examples, of a pseudoholomorphic curve theory for symplectic 4-orbifolds. These are extensions of relevant work of Gromov, McDuff and Taubes on…

The author has been interested in regions surrounded by cylinders of real algebraic hypersurfaces and their shapes and polynomials associated to them. Here, we formulate and investigate natural decompositions into such cylinders of real…

We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph problem---to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph---and…

We introduce orthogonal ring patterns in the 2-sphere and in the hyperbolic plane, consisting of pairs of concentric circles, which generalize circle patterns. We show that their radii are described by a discrete integrable system. This is…

We prove that the theory of the Farey graph is pseudofinite by constructing a sequence of finite structures that satisfy increasingly large subsets of its first-order axiomatization. This graph is an important object in the study of curve…

We elaborate on some important ideas contained in Lobachevsky's Pangeometry and in some of his other memoirs. The ideas include the following: (1) The trigonometric formulae, which express the dependence between angles and edges of…

We describe a promising approach to efficiently morph spherical graphs, extending earlier approaches of Awartani and Henderson [Trans. AMS 1987] and Kobourov and Landis [JGAA 2006]. Specifically, we describe two methods to morph…

We study a more general version of the gluings of hyperbolic orbifolds in the spirit of Gromov and Piatetski-Shapiro, where the gluing pieces, called the building blocks, are no longer assumed to be arithmetic or incommensurable. We prove…

We review an approach which aims at studying discrete (pseudo-)manifolds in dimension $d\geq 2$ and called random tensor models. More specifically, we insist on generalizing the two-dimensional notion of $p$-angulations to higher…

This paper contains theory on two related topics relevant to manifolds of normally hyperbolic singularities. First, theorems on the formal and $ C^k $ normal forms for these objects are proved. Then, the theorems are applied to give…

V. Arnold's problem 1987-14 asks whether there exist smooth hypersurfaces in $R^N$ (other than the conics in odd-dimensional spaces) for which the volume of the segment cut by any hyperplane from the body bounded by such a hypersurface is…

Partial generalizations of virtual polyhedra theory (sometimes under different names) appeared recently in the theory of torus manifolds. These generalizations look very different from the original virtual polyhedra theory. They are based…

We introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the later one…