Related papers: Geometrie der Zahlen. Ein \"Uberblick
In this chapter we take up the quantum Riemannian geometry of a spatial slice of spacetime. While researchers are still facing the challenge of observing quantum gravity, there is a geometrical core to loop quantum gravity that does much to…
A simple visual representation of Minkowski spacetime appropriate for a student with a background in geometry and algebra is presented. Minkowski spacetime can be modeled with a Euclidean 4-space to yield accurate visualizations as…
Every convex body K in R^n has a coordinate projection PK that contains at least vol(0.1 K) cells of the integer lattice PZ^n, provided this volume is at least one. Our proof of this counterpart of Minkowski's theorem is based on an…
Some examples and basic properties of ultrametric spaces are briefly discussed.
With the help of the recently introduced parametric geometry of numbers by W. M. Schmidt and L. Summerer, we prove a strong version of a conjecture of Schmidt concerning the successive minima of a lattice.
This article serves as an introduction to the special volume on Positive Geometry in the journal Le Matematiche. We attempt to answer the question in the title by describing the origins and objects of positive geometry at this early stage…
Minkowski's 2nd theorem in the Geometry of Numbers provides optimal upper and lower bounds for the volume of a $o$-symmetric convex body in terms of its successive minima. In this paper we study extensions of this theorem from two different…
The four dimensional spacetime continuum, as first conceived by Minkowski, has become the dominant framework within which to describe physical laws. In this paper, we show how this four-dimensional structure is a natural property of…
The geometry of Quantum Mechanics in the context of uncertainty and complementarity, and probability is explored. We extend the discussion of geometry of uncertainty relations in wider perspective. Also, we discuss the geometry of…
This paper provides an overview of modern digital geometry and topology through mathematical principles, algorithms, and measurements. It also covers recent developments in the applications of digital geometry and topology including image…
This paper is a survey of some of the developments in coarse extrinsic geometry since its inception in the work of Gromov. Distortion, as measured by comparing the diameter of balls relative to different metrics, can be regarded as one of…
We introduce orbifolds from the classical point of view, using charts, and present orbifold versions of elementary objects from Algebraic Topology, such as the fundamental group, coverings and Euler characteristic; Differential…
The ancient unsolved problem of congruent numbers has been reduced to one of the major questions of contemporary arithmetic: the finiteness of the number of curves over $\bf Q$ which become isomorphic at every place to a given curve. We…
In pregeometry a metric arises as a composite object at large distances. We investigate if its signature, which distinguishes between time and space, could be a result of the dynamics rather than being built in already in the formulation of…
The Minkowski problem in convex geometry concerns showing that a given Borel measure on the unit sphere is, up to perhaps a constant, some type of surface area measure of a convex body. Two types of Minkowski problems in particular are an…
The secret network of the universe: How quantum geometry might complete Einstein's dream. - An informal introduction to quantum geometry (loop quantum gravity), spin networks, quantum black holes, and the work of Abhay Ashtekar, Carlo…
These are lecture notes for the course "MATS4120 Geometry of geodesics" given at the University of Jyv\"askyl\"a in Spring 2020. Basic differential geometry or Riemannian geometry is useful background but is not strictly necessary. Exercise…
This text is a slightly edited version of lecture notes for a course I gave at ETH, during the Winter term 2000-2001, to undergraduate Mathematics and Physics students. Contents: Chapter 1 - Examples of Dynamical Systems Chapter 2 -…
We investigate the geometric properties of lightlike surfaces in the Minkowski space $\R^{2,1}$, using Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a…
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…