相关论文: Noncommutative geometry through monoidal categorie…

We present a short introductory overview of the non-commutative extensions of several classical physical theories. After a general discussion of the reasons that suggest that the non-commutativity is a major issue that will eventually lead…

In this survey article we describe some geometric results in the theory of noncommutative rings and, more generally, in the theory of abelian categories. Roughly speaking and by analogy with the commutative situation, the category of graded…

These are expanded lecture notes of a mini-course whose objectives were to introduce the basic concepts, constructions and techniques of noncommutative geometry, as well as their uses as a framework for modelling quantum spacetime. Key…

We argue that there should exist a "noncommutative Fourier transform" which should identify functions of noncommutative variables (say, of matrices of indeterminate size) and ordinary functions or measures on the space of paths. Some…

Noncommutative geometry has seen remarkable applications for high energy physics, viz. the geometrical interpretation of the Standard Model. The question whether it also allows for supersymmetric theories has so far not been answered in a…

In this article, we will define non-commutative covering spaces using Hopf-Galois theory. We will look at basic properties of covering spaces that still hold for these non-commutative analogues. We will describe examples including coverings…

In this note, making use of noncommutative $l$-adic cohomology, we extend the generalized Riemann hypothesis from the realm of algebraic geometry to the broad setting of geometric noncommutative schemes in the sense of Orlov. As a first…

In this paper we introduce a notion of {\it generalized operad} containing as special cases various kinds of operad--like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories…

A generalization of the notion of a (pseudo-) Riemannian space is proposed in a framework of noncommutative geometry. In particular, there are parametrized families of generalized Riemannian spaces which are deformations of classical…

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,…

In the context of a noncommutative differential calculus on the algebra of real valued functions of an $n$-dimensional manifold $M$, a commutative and associative product of 1-forms is naturally defined. Ordinary differential calculus…

A natural extension of the standard model within non-commutative geometry is presented. The geometry determines its Higgs sector. This determination is fuzzy, but precise enough to be incompatible with experiment.

We show that every involutive Hopf monoid in a complete and finitely cocomplete symmetric monoidal category gives rise to invariants of oriented surfaces defined in terms of ribbon graphs. For every ribbon graph this yields an object in the…

A review of the relationships between matrix models and noncommutative gauge theory is presented. A lattice version of noncommutative Yang-Mills theory is constructed and used to examine some generic properties of noncommutative quantum…

A covariant functor from the category of generic complex algebraic curves to a category of the AF-algebras is constructed. The construction is based on a representation of the Teichmueller space of a curve by the measured foliations due to…

Together with collaborators, we introduced a noncommutative Riemannian geometry over Moyal algebras and systematically developed it for noncommutative spaces embedded in higher dimensions in the last few years. The theory was applied to…

We give a survey on higher invariants in noncommutative geometry and their applications to differential geometry and topology.

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…

We illustrate an isomorphic representation of the observable algebra for quantum mechanics in terms of the functions on the projective Hilbert space, and its Hilbert space analog, with a noncommutative product in terms of explicit…

After an introduction to some basic issues in non-commutative geometry (Gel'fand duality, spectral triples), we present a "panoramic view" of the status of our current research program on the use of categorical methods in the setting of…