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Incidence: Projective Geometry
Projective geometry is the proper domain for the notion of incidence for straight lines and points; it knows no other basic notions. The basic ideas...
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Felix Klein, Sophus Lie, contact transformations, and connexes
Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups)...
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Absolute Plane Geometry
This chapter is an introduction to the absolute plane geometry.
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Ordered absolute geometry
Bachmann’s absolute geometry provides a common foundation of Euclidean and hyperbolic geometry without any assumptions about order or free mobility....
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Infinite-dimensional (dg) Lie algebras and factorization algebras in algebraic geometry
Infinite-dimensional Lie algebras (such as Kac-Moody, Virasoro etc.) govern, in many ways, various moduli spaces associated to algebraic curves. To...
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A Car as Parabolic Geometry
We show that a car, viewed as a nonholonomic system, provides an example of a flat parabolic geometry of type (SO(2, 3), P 12), where P 12 is a Borel...
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Parallelism: Affine Geometry
Straight line and incidence are the simplest geometric notions. In our opening chapter, however, we will add the notion of parallelism, which will...
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Inner Ideals of Real Simple Lie Algebras
A classification up to automorphism of the inner ideals of the real finite-dimensional simple Lie algebras is given, jointly with precise...
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The geometry of discrete asymptotic-geodesic 4-webs in isotropic 3-space
The geometry of webs has been investigated over more than a century driven by still open problems. In our paper we contribute to extending the...
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What Is Geometry?
This introductory chapter has rather a philosophical, or more precisely a metamathematical character: It does not do mathematics, but talks about...
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The Simplicity Degree of Tarski’s Euclidean Geometry of Ruler and Dividers is 5
We present an axiom system for the plane Euclidean geometry of ruler and dividers constructions, expressed in Tarski’s language, with points as the...
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Point line geometry in the tropical plane
We study the classical result by Bruijn and Erdős regarding the bound on the number of lines determined by a n -point configuration in the plane, and...
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Affine convex geometry – Part 2
The main objective of this chapter is to study some important (starlike or convex) bodies associated to a given convex body. These are the so-called...
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Pre-lie Algebras
Pre-Lie algebras, also called Vinberg algebras, have become an important tool in combinatorics, differential geometry, the theory of operads and in...
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Counting in Algebraic Geometry
In this introductory chapter we give a rather informal overview of the most prominent aspects of enumerative geometry, both in its classical and...
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Noncommutative Geometry of Random Surfaces
AbstractWe associate a noncommutative curve to a periodic, bipartite, planar dimer model with polygonal boundary. It determines the inverse...
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Interfacial phase-change and geometry modify nanoscale pattern formation in irradiated thin films
In this paper, we consider the linear stability of ion-irradiated thin films where the typical no-penetration boundary condition has been relaxed to...
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Distance: Euclidean Geometry
In the third century BC, EuclidEuclid compiled the mathematical knowledge of the time in his book “The Elements”. In his geometrical considerations,...
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Minkowski Geometry—Some Concepts and Recent Developments
The geometry of finite-dimensional normed spaces (= Minkowski geometry) is a research topic which is related to many other fields, such as convex...