This is related to Anweshi's question about [theories of noncommutative geometry](https://mathoverflow.net/questions/10512/).


Let's start out by saying that I live, mostly, in a commutative universe. The only noncommutative rings I have much truck with are either supercommutative, almost commutative (filtered, with commutative associated graded), group algebras or matrix algebras, none of which really show many of the true difficulties of noncommutative things.


So, here's my (somewhat pithy) question: what's noncommutative geometry good for?


To be a bit more precise, I have a vague sense that $C^\*$ stuff is supposed to work well in quantum mechanics, but I'm somewhat more interested in more algebraic noncommutative geometry. What sorts of problems does it solve that we can't solve without leaving the commutative world? Why should, say, a complex algebraic geometer learn some noncommutative geometry?



Applications of noncommutative geometry
Charles,


a couple of reasons why a complex algebraic geometer (certainly someone who is 
interested in moduli spaces of vector bundles, as your profile tells me) might 
at least keep
an open verdict on the stuff NC-algebraic geometers (NCAGers from now on) are trying to do.


in recent years ,a lot of progress has been made towards understanding moduli spaces of
semi-stable representations of 'formally smooth' algebras (think 'smooth in the NC-world).
in particular when it comes to their etale local structure and their rationality.
for example, there is [this book](http://win.ua.ac.be/~lebruyn/LeBruyn2007a.pdf), by someone.


this may not seem terribly relevant to you until you realize that some of the more
interesting moduli spaces in algebraic geometry are among those studied. for example, the
moduli space of semi-stable rank n bundles of degree 0 over a curve of genus g is the moduli
space of representations of a certain dimension vector over a specific formally smooth algebra,
as Aidan Schofield showed. he also applied this to rationality results about these spaces.


likewise, the moduli space of semi-stable rank n vectorbundles on the projective plane with Chern 
classes c1=0 and c2=n is birational to that of semi-simple n-dimensional representations of the free algebra
in two variables. the corresponding rationality problem has been studied by NCAG-ers (aka 'ringtheorists'
at the time) since the early 70ties (work by S.A. Amitsur, Claudio Procesi and Ed Formanek). by their results, we NCAGers,
knew that the method of 'proof' by Maruyama of their stable rationality in the mid 80ties, couldn't possibly work.


it's rather ironic that the best rationality results on these moduli spaces (of bundles over the
projective plane) are not due to AGers but to NCAGers : Procesi for n=2, Formanek for n=3 and 4 and
Bessenrodt and some guy for n=5 and 7. together with a result by Aidan Schofield these results show
that this moduli space is stably rational for all divisors n of 420.


further, what a crepant resolution of a quotient singularity is to you, is to NCAGers the moduli space of certain
representations of a nice noncommutative algebra over the singularity.


likewise, when you AGers mumble 'Deligne-Mumford stack', we NCAGers say 'ah! a noncommutative algebra'.


