Related papers: On topological and geometric $(19_4)$ configuratio…

We give restrictions on the weak combinatorics of line arrangements with singular points of odd multiplicity using topological arguments on locally-flat spheres in 4-manifolds. As a corollary, we show that there is no line arrangement…

A $configuration$ of a linkage $\Gamma$ is a possible positioning of $\Gamma$ in $\mathbb{R}^d$ and the collection of all such forms the configuration space $\mathcal{C}(\Gamma)$ of $\Gamma$. We here introduce the notion of the $symmetric…

We are dealing with the problem of space layout planning here. We present an architectural conceptual CAD approach. Starting with design specifications in terms of constraints over spaces, a specific enumeration heuristics leads to a…

Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao has shown recently that the maximum possible…

Let $V_1, V_2, V_3, \dots $ be a sequence of $\mathbb{Q}$-vector spaces where $V_n$ carries an action of $\mathfrak{S}_n$ for each $n$. {\em Representation stability} and {\em multiplicity stability} are two related notions of when the…

We classify the full set of convex central configurations in the Newtonian four-body problem. Particular attention is given to configurations possessing some type of symmetry or defining geometric property. Special cases considered include…

These expository notes are dedicated to the study of the topology of configuration spaces of manifolds. We give detailed computations of many invariants, including the fundamental group of the configuration spaces of $\mathbb{R}^2$, the…

The class of $\left(\binom{n+1}{2}_{n-1} \binom{n+1}{3}_3\right)$-configurations which contain at least $n-2$ $K_n$-graphs coincides with the class of so called systems of triangle perspectives i.e. of configurations which contain a bundle…

The classical no-three-in-line problem is to find the maximum number of points that can be placed in the $n \times n$ grid so that no three points lie on a line. Given a set $S$ of points in an Euclidean plane, the General Position Subset…

For a graph $G$ and a parameter $k$, we call a vertex $k$-enabling if it belongs both to a clique of size $k$ and to an independent set of size $k$, and we call it $k$-excluding otherwise. Motivated by issues that arise in secret sharing…

For $n \geq 2$, the $n$-th curvature set of a metric space $X$ is the set consisting of all $n$-by-$n$ distance matrices of $n$ points sampled from $X$. Curvature sets can be regarded as a geometric analogue of configuration spaces. In this…

Let $n$, $k$, and $t$ be integers satisfying $n>k>t\ge2$. A Steiner system with parameters $t$, $k$, and $n$ is a $k$-uniform hypergraph on $n$ vertices in which every set of $t$ distinct vertices is contained in exactly one edge. An…

A fundamental characteristic of computer networks is their topological structure. The question of the description of the structural characteristics of computer networks represents a problem that is not completely solved. Search methods for…

We study relations between $(n_4)$ incidence configurations and the classical Poncelet Porism. Poncelet's result studies two conics and a sequence of points and lines that inscribes one conic and circumscribes the other. Poncelet's Porism…

We present the concept of the topological symmetry group as a way to analyze the symmetries of non-rigid molecules. Then we characterize all of the groups which can occur as the topological symmetry group of an embedding of the complete…

Motivated by automated junction recognition in tracking data, we study a problem of placing a square or disc of fixed size in an arrangement of lines or line segments in the plane. We let distances among the intersection points of the lines…

Hass and Scott's example of a 4-valent graph on the 3-punctured sphere that cannot be realized by geodesics in any metric of negative curvature is generalized to impossible configurations filling surfaces of genus $n$ with $k$ punctures for…

We study supersolvable line arrangements in ${\mathbb P}^2$ over the reals and over the complex numbers, as the first step toward a combinatorial classification. Our main results show that a nontrivial (i.e., not a pencil or near pencil)…

We investigate an algebraic problem related to the determination of the fundamental group of a class of spaces of configurations on surfaces. The configuration spaces are spaces of points grouped into colors. Whether two points are allowed…

One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this…