Related papers: Notes on Bolyai's 'Appendix'
We use a combinatorial approximation of the hyperbolic plane to investigate properties of hyperbolic geometry such as exponential growth of perimeter and area of disks, and the linear isoperimetric inequality. This calculations give a…
There are three complete plane geometries of constant curvature: spherical, Euclidean and hyperbolic geometry. We explain how a closed oriented surface can carry a geometry which locally looks like one of these. Focussing on the hyperbolic…
This is the first in a series of papers showing that Haken manifolds have hyperbolic structures; this first was published, the second two have existed only in preprint form, and later preprints were never completed. This eprint is only an…
In this paper, we introduce an asymmetric metric on the space of marked Euclidean triangles, and we prove several properties of this metric, including two equivalent definitions of this metric, one of them comparing ratios of functions of…
The paper is a contribution to the conjecture of Kobayashi that the complement of a generic curve in the projective plane is hyperbolic, provided the degree is at least five. Previously the authors treated the cases of two quadrics and a…
Unexpected hypersurfaces arise when vanishing in points of a set $Z$ and higher-order vanishing along a general linear subspace fails to impose the expected number of independent conditions on forms of a fixed degree. The phenomenon was…
Reay's relaxed Tverberg conjecture and Conway's thrackle conjecture are open problems about the geometry of pairwise intersections. Reay asked for the minimum number of points in Euclidean d-space that guarantees any such point set admits a…
In the town of Saratov where he was prisonner, Poncelet, continuing the work of Euler and Steiner on polygons simultaneously inscribed in a circle and circumscribed around an other circle, proved the following generalization : "Let C and D…
We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude,…
There are multiple generalisations of the Pythagorean theorem to spherical and hyperbolic geometry. A natural one, involving areas of disks with radii equal to the sides of a proper triangle, was discovered in the hyperbolic case by Maria…
These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial…
Combining tools from category theory, model theory, and non-standard analysis we extend Baker-Beynon dualities to the classes of all Abelian $\ell$-groups and all Riesz spaces (also known as vector lattices). The extended dualities have a…
We discuss the most elementary properties of the hyperbolic trigonometry and show how they can be exploited to get a simple, albeit interesting, geometrical interpretation of the special relativity. It yields indeed a straightforword…
Four points ordered in the positive order on the unit circle determine the vertices of a quadrilateral, which is considered either as a euclidean or as a hyperbolic quadrilateral depending on whether the lines connecting the vertices are…
We define the simplest log-euclidean geometry. This geometry exposes a difficulty hidden in Hilbert's list of axioms presented in his "Grundlagen der Geometrie". The list of axioms appears to be incomplete if the foundations of geometry are…
``Can number and geometric spaces be reconstructed from their symmetries?'' This question, which is at the heart of anabelian geometry, a theory built on the collaborative efforts of an international community in many variants and with the…
The enduring legacy of Euclidean geometry underpins classical machine learning, which, for decades, has been primarily developed for data lying in Euclidean space. Yet, modern machine learning increasingly encounters richly structured data…
We deal with the quasi-symmetric algebra introduced by Paolo Aluffi, here named (embedded) Aluffi algebra. The algebra is a sort of "intermediate" algebra between the symmetric algebra and the Rees algebra of an ideal, which serves the…
This paper considers aspects of 4-manifold topology from the point of view of the null cone of a neutral metric, a point of view we call neutral causal topology. In particular, we construct and investigate neutral 4-manifolds with null…
Asymptotic symmetries are a general and important feature of theories with long-ranging fields, such as gravity, electromagnetism, and Yang-Mills. They appear in the formalism once the analytic behaviour of fields near infinity is specified…