Riemannian Quantum topologie

The concepts of classical differential geometry may be expanded in a variety of
ways to fit a noncommutative setting. Given that many of them have their own rich
pure mathematical theories, we believe that it is impossible to say with certainty
which of them is accurate. However, if we were to evaluate what noncommutative
differential geometry should ideally take into account, we may come up with the
following points. There should be a sizable selection of interesting examples from
various mathematical disciplines. Even if certain features of the theory may become
irrelevant in the classical case, noncommutative geometry should reduce to classical
geometry as a particular case. The majority of classical differential geometry
structures ought to have equivalents in noncommutative geometry. Finally, as
geometry was designed to be a practical discipline, there should be applications,
which traditionally have meant applications in physics and applied mathematics.
As a result, one of the guiding ideas of this work has been to cover both the
background in pure mathematics and its applications, from categories to
cosmology and from modules to Minkowski space. We’ll attempt to explain both
elements by beginning with a simple concept. A large portion of the work will be
drawn from our own research articles, which were motivated by the
aforementioned viewpoint. This includes, but is not limited to, our experience with
quantum groups (a “quantum groups approach to noncommutative geometry”). In
summary, we provide a noncommutative geometry style that is especially
constructive and computeable and, when appropriate, offer linkages to other
methods.