Related papers: Basic coordinate-free non-Euclidean geometry
We show that Euclidean geometry in suitably high dimension can be expressed as a theory of orthogonality of subspaces with fixed dimensions and fixed dimension of their meet.
These lecture notes provide an introduction to logarithmic geometry with a view towards recent applications in the desingularization theory.
This Lecture Notes is meant to introduce noncommutative algebraic geometry tools (which were invented by M. Artin, W. Schelter, J. Tate, and M. Van den Bergh in the late 1980s) and also graded skew Clifford algebras (which were introduced…
We present a survey of the application of Cones' Non-Commutative Geometry to gravitation. Bases of the theory and Euclidian gravity models are reviewed. Then we discuss the problem of a Lorentzian generalization of the theory and review…
This paper presents a coordinate free pre-metric formulation of charge free Maxwell-Minkowski electrodynamics, and of the developed by the authors non-linear Extended Electrodynamics. First we introduce some formal relations from…
A new mathematical theory, non-associative geometry, providing a unified algebraic description of continuous and discrete spacetime, is introduced.
Non-Euclidean method of the generalized geometry construction is considered. According to this approach any generalized geometry is obtained as a result of deformation of the proper Euclidean geometry. The method may be applied for…
We trace the development of arguments for the consistency of non-Euclidean geometries and for the independence of the parallel postulate, showing how the arguments become more rigorous as a formal conception of geometry is introduced. We…
These lectures notes are an intoduction for physicists to several ideas and applications of noncommutative geometry. The necessary mathematical tools are presented in a way which we feel should be accessible to physicists. We illustrate…
This is an introduction to noncommutative geometry, from an affine viewpoint, that is, by using coordinates. The spaces $\mathbb R^N,\mathbb C^N$ have no free analogues in the operator algebra sense, but the corresponding unit spheres…
The first part of these notes is a self-contained introduction to generalized complex geometry. It is intended as a `user manual' for tools used in the study of supersymmetric backgrounds of supergravity. In the second part we review some…
An attempt is made to extend some of the basic paradigms of dynamics, from the viewpoint of (continuous) flows, to non-metric manifolds.
Just as for non-abelian gauge theories at strong coupling, discrete lattice methods are a natural tool in the study of non-perturbative quantum gravity. They have to reflect the fact that the geometric degrees of freedom are dynamical, and…
The concept of a physical space, which actualizes Euclidean geometry, is not confined to the statics of solids but extensible to the phenomena where Newtonian mechanics is valid, defining its concept of time. The laws of propagation of…
Non-perturbative quantum general relativity provides a possible framework to analyze issues related to black hole thermodynamics from a fundamental perspective. A pedagogical account of the recent developments in this area is given. The…
Foundations of the theory of vertex algebras are extended to the non-Archimedean setting.
This is a set of expository lecture notes created originally for a graduate course on holomorphic curves taught at ETH Zurich and the Humboldt University Berlin in 2009/2010. The notes are still incomplete, but due to recent requests from…
In a recent paper we have suggested that a formulation of quantum mechanics should exist, which does not require the concept of time, and that the appropriate mathematical language for such a formulation is noncommutative differential…
This survey article is an introduction to Diophantine Geometry at a basic undergraduate level. It focuses on Diophantine Equations and the qualitative description of their solutions rather than detailed proofs.
In this paper the model considered by Arkani-Hamed, Cohen and Georgi in the context of (de)constructing dimensions has been studied by making use of non-commutative geometry. The non-commutative geometry provides a natural framework to…