Related papers: On the origin of Hilbert Geometry
We give a survey of our joint ongoing work with Ali Chamseddine, Slava Mukhanov and Walter van Suijlekom. We show how a problem purely motivated by "how geometry emerges from the quantum formalism" gives rise to a slightly noncommutative…
In this paper, we state and prove a simple geometric interpretation of the degrees of freedom of a chi-square distribution. The general geometric idea goes back at least to Fisher in the 1920s, but the exact result does not appear to have…
Herbert Gr{\"o}tzsch is the main founder of the theory of quasicon-formal mappings. We review five of his papers, written between 1928 and 1932, that show the progress of his work from conformal to quasiconformal geometry. This will give an…
Hilbert's paper on ``The Foundations of Physics (First Communication),'' is now primarily known for its parallel publication of essentially the same gravitational field equations of general relativity which Einstein published in a note on…
We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and…
Mostly aimed at an audience with backgrounds in geometry and homological algebra, these notes offer an introduction to derived geometry based on a lecture course given by the second author. The focus is on derived algebraic geometry, mainly…
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
This is largely a survey paper, dealing with Cartan geometries in the complex analytic category. We first remind some standard facts going back to the seminal works of F. Klein, E. Cartan and C. Ehresmann. Then we present the concept of a…
Metric spaces are a fundamental component of mathematics and have a paramount importance as a framework for measuring distance. They can be found in many different branches of mathematics, such as analysis and topology. This paper offers an…
In this paper, we provide an essentially self-contained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of three-manifolds. In…
Hanika, Schneider, and Stumme introduced geometric data set as a generalization of metric measure space for the computation of the observable diameter, and extended the observable distance between metric measure spaces to that between…
We describe the natural geometry of Hilbert schemes of curves in ${\mathbb P}^3$ and, in some cases, in ${\mathbb P}^n$ , $n\geq 4$.
The law of centripetal force governing the motion of celestial bodies in eccentric conic sections, has been established and thoroughly investigated by Sir Isaac Newton in his Principia Mathematica. Yet its profound implications on the…
One of the major developments of twentieth century physics has been the gradual recognition that a common feature of the known fundamental interactions is their gauge structure. In this article the authors review the early history of gauge…
Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a…
Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective…
E. C. Zeeman [1] has criticized the fact that in all articles and books until that moment (1967) the topology employed to work with the Minkowski space was the Euclidean one. He has proposed a new topology, which was generalized for more…
We develop a notion of rank one properly convex domains (or Hilbert geometries) in the real projective space. This is in the spirit of rank one non-positively curved Riemannian manifolds and CAT(0) spaces. We define rank one isometries for…
Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several…
Hilbert's 1917 discovery of reversed gravitational acceleration is discussed, and the connection to arXiv:1102.2870v2 [gr-qc] is explained.