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We review aspects of our formalism for differential geometry on noncommutative and nonassociative spaces which arise from cochain twist deformation quantization of manifolds. We work in the simplest setting of trivial vector bundles and…
We consider the compactification of Matrix theory on tori with background antisymmetric tensor field. Douglas and Hull have recently discussed how noncommutative geometry appears on the tori. In this paper, we demonstrate the concrete…
Classical differential geometry can be encoded in spectral data, such as Connes' spectral triples, involving supersymmetry algebras. In this paper, we formulate non-commutative geometry in terms of supersymmetric spectral data. This leads…
This is a survey of some recent advances in the theory of singular traces in which the authors have played some part and which were inspired by questions raised by the book of Alain Connes (Noncommutative Geometry, Academic Press 1994).…
The theory of noncommutative geometry provides an interesting mathematical background for developing new physical models. In particular, it allows one to describe the classical Standard Model coupled to Euclidean gravity. However,…
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several…
Around 1980 Connes extended the notions of geometry to the non-commutative setting. Since then {\it non-commutative geometry} has turned into a very active area of mathematical research. As a first non-trivial example of a non-commutative…
A general question behind this paper is to explore a good notion for intrinsic curvature in the framework of noncommutative geometry started by Alain Connes in the 80s. It has only recently begun (2014) to be comprehended via the intensive…
Non commutative geometry is creating new possibilities for physics. Quantum spacetime geometry and post inflationary models of the universe with matter creation have an enormous range of scales of time, distance and energy in between. There…
We study a class of noncommutative geometries that give rise to dimensionally reduced Yang-Mills theories. The emerging geometries describe sets of copies of an even dimensional manifold. Similarities to the D-branes in string theory are…
The first part of these notes gives an introduction to noncommutative projective geometry after Artin--Zhang. The second part provides an overview of the work of Polishchuk that reconciles noncommutative two-tori having real multiplication…
Noncommutativity is an idea dating back to the early times of Quantum Mechanics and that string theory induced noncommutative (NC) geometry which provides an effective framework to study short distance spacetime dynamics. Also, string…
After a brief review of matrix theory compactification leading to noncommutative supersymmetric Yang-Mills gauge theory, we present solutions for the fundamental and adjoint sections on a two-dimensional twisted quantum torus in two…
We survey noncommutative spacetimes with coordinates being enveloping algebras of Lie algebras. We also explain how to do differential geometry on noncommutative spaces that are obtained from commutative ones via a Moyal-product type…
A review of the applications of noncommutative geometry to a systematic formulation of duality symmetries in string theory is presented. The spectral triples associated with a lattice vertex operator algebra and the corresponding…
This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September…
I give a summary of the progress made on using the elegant construction of Alain Connes noncommutaive geometry to explore the nature of space-time at very high energies. In particular I show that by making very few natural and weak…
After introducing a noncommutative counterpart of commutative algebraic geometry based on monoidal categories of quasi-coherent sheaves we show that various constructions in noncommutative geometry (e.g. Morita equivalences, Hopf-Galois…
The foundations of matrix geometry are discussed, which provides the basis for recent progress on the effective geometry and gravity in Yang-Mills matrix models. Basic examples lead to a notion of embedded noncommutative spaces (branes)…
In this pedagogical mini course the basics of the derivation of the noncommutative structures appearing in string theory are reviewed. First we discuss the well established appearance of the noncommutative Moyal-Weyl star-product in the…