Related papers: Projective Cross-Ratio on Hypercomplex Numbers
It is understood now that all projective (and conformal) invariants of Riemannian metrics can be found by a transparent construction based on representation theory. So this article with a partial and quite cumbersome construction of…
Hypercontractive inequalities are a useful tool in dealing with extremal questions in the geometry of high-dimensional discrete and continuous spaces. In this survey we trace a few connections between different manifestations of…
Hyperbolic metric and different hyperbolic type metrics are studied in open sector domains of the complex plane. Several sharp inequalities are proven for them. Our main result describes the behavior of the triangular ratio metric under…
Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active…
This paper investigates the Castelnuovo-Mumford regularity of the generic hyperplane section of projective curves in positive characteristic case, and yields an application to a sharp bound on the regularity for nondegenerate projective…
We develop a complete theory of projective cross-ratios in n-dimensional Plane-Based Geometric Algebra (PGA), R(n,0,1), covering geometric objects of every grade: finite and ideal points, hyperplanes, and intermediate flats. For each object…
Using a quadratic version of the Bott residue theorem, we give a quadratic refinement of the count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the…
This paper introduces advances in the geometry of the transforms for cross ratio of four points in a line in the Desargues affine plane. The results given here have a clean, based Desargues affine plan axiomatic's and definitions of…
We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In…
We consider the ratio of two Gauss hypergeometric functions, in which the parameters of the numerator function differ from the respective parameters of the denominator function by integers. We derive explicit integral representations for…
We study properties of "hyperbolic directions" in groups acting cocompactly on properly convex domains in real projective space, from three different perspectives simultaneously: the (coarse) metric geometry of the Hilbert metric, the…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…
This article describes recent applications of algebraic geometry to noncommutative algebra. These techniques have been particularly successful in describing graded algebras of small dimension.
Geometric lower and upper estimates are obtained for invariant metrics on $\Bbb C$-convex domains containing no complex lines.
We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underly…
In this text, we explore the tools that Projective Differential Geometry can provide for the asymptotic analysis of classical fields on projectively compact manifolds. We emphasise on the case of order 2-compactifications and develop, in…
We study an isomorphism between the group of rigid body displacements and the group of dual quaternions modulo the dual number multiplicative group from the viewpoint of differential geometry in a projective space over the dual numbers.…
In recent years, several quantizations of real manifolds have been studied, in particular from the point of view of Connes' noncommutative geometry. Less is known for complex noncommutative spaces. In this paper, we review some recent…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…