Related papers: The $\Lambda$-property of a simple arc
In this paper we discuss some properties of fundamental groups and Alexander polynomials of plane curves. We discuss the relationship of the non-triviality of Alexander polynomials and the notion of (nearly) freeness for irreducible plane…
It is well-known since the time of the Greeks that two disjoint circles in the plane have four common tangent lines. Cappell et al. proved a generalization of this fact for properly separated strictly convex bodies in higher dimensions. We…
In this paper we collect the main properties of free curves in the complex projective plane and a lot of conjectures and open problems, both old and new. In the quest to understand the mystery of free curves, many tools were developed and…
We compute the cohomology ring of a generalised type of configuration space of points in $\mathbb{R}^r$. This configuration space is indexed by a graph. In the case the graph is complete the result is known and it is due to Arnold and…
This paper is a survey on arc spaces, a recent topic in algebraic geometry and singularity theory. The geometry of the arc space of an algebraic variety yields several new geometric invariants and brings new light to some classical…
We establish a structure theorem on the arc space of a $k$-scheme of finite type. More precisely, we show that the arc space is locally for the pro-smooth toplogy a product of an infinite dimensional affine space and of a non-noetherian…
Vogt's theorem, concerning boundary angles of a convex arc with monotonic curvature (spiral arc), is taken as a starting point to establish basic properties of spirals. The theorem is expanded by removing requirements of convexity and…
This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered…
This is an expository article detailing results concerning large arcs in finite projective spaces, which attempts to cover the most relevant results on arcs, simplifying and unifying proofs of known old and more recent theorems. The article…
We summarize the main ideas of General Relativity and Lorentzian geometry, leading to a proof of the simplest of the celebrated Hawking-Penrose singularity theorems. The reader is assumed to be familiar with Riemannian geometry and point…
We prove a conjecture of Tom Ilmanen's and Hubert Bray's regarding the existence of the outermost generalized apparent horizon in an initial data set and that it is outer area minimizing.
We investigate some combinatorial properties of convex polytopes simple in edges. For polytopes whose nonsimple vertices are located sufficiently far one from another, we prove an analog of the Hard Lefschetz theorem. It implies Stanley's…
For two non-congruent regular polygons of the same type, the method of finding the points in the plane at the equal distances to the vertices, is established. The existence of two points with this property is proved for two polygons with a…
A general method for constructing essential uniform algebras with prescribed properties is presented. Using the method, the following examples are constructed: an essential, natural, regular uniform algebra on the closed unit disc; an…
If $v$ is an eigenvector for eigenvalue $\lambda$ of a graph $X$ and $\alpha$ is an automorphism of $X$, then $\alpha(v)$ is also an eigenvector for $\lambda$. Thus it is rather exceptional for an eigenvalue of a vertex-transitive graph to…
Given a pair of real functions $(k,f)$, we study the conditions they must satisfy for $k+\lambda f$ to be the curvature in the arc-length of a closed planar curve for all real $\lambda$. Several equivalent conditions are pointed out,…
We present examples of metric spaces that are not Riemannian manifolds nor dimensionally homogeneous that satisfy the Tetrahedral Property. In spite of that, Euclidean cones over metric spaces with small diameter do not satisfy this…
This is an introduction to geometric algebra, an alternative to traditional vector algebra that expands on it in two ways: 1. In addition to scalars and vectors, it defines new objects representing subspaces of any dimension. 2. It defines…
The $\lambda$-property and the uniform $\lambda$-property were first introduced by R. Aron and R. Lohman in 1987 as geometric properties of Banach spaces. In 1989, Th. Shura and D. Trautman showed that the Schreier space possesses the…
In a series of papers in the 1960's, S. G\"ahler defined and investigated so-called m-metric spaces and their topological properties. An m-metric assigns to any tuple of m+1 elements a real value (more generally an element in a partially…