We illustrate the thesis that if time did not exist, we would have to create it if space is noncommutative, and extend functions by something like Schroedinger’s equation. We propose that the phenomenon is a somewhat general mechanism within noncommutative geometry for ‘sponteneous time generation’. We show in detail how this works for the su2 algebra [xi, xj ] = 2ıλǫijxk as noncommutative spac...

We review some applications of noncommutative geometry to the study of transverse geometry of Riemannian foliations and discuss open problems.

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 Dirac-Ramond operators are constructed and shown to naturally incorporate target space and discrete worldsheet dualities as isometries of the noncommutative space. The...

This text is based on my lectures delivered at the School on Algebraic K-Theory and Applications which took place at the International Center for Theoretical Physics (ICTP) in Trieste during the last two weeks of May of 2007. It might be regarded as an introduction to some basic facts of noncommutative algebraic geometry and the related chapters of homological algebra and (as a part of it) a no...