This is a question that has been winding around my head for a long time and I have not found a convincing answer. The title says everything, but I am going to enrich my question by little more explanations.
As a layman, I have started searching for expositories/more informal, rather intuitive, also original account of non-commutative geometry to get more sense of it, namely, I have looked through
The English translation of Review of non-commutative algebra by Alain Connes,
Surveys in non-commutative geometry, Clay mathematics proceedings, Volume 6,
Nevertheless, I am not satisfied with them at all. It seems to me, that even understanding a simple example, requires much more knowledge that is gained in grad school. Now for me, this field merely contains a lot of highly developed machineries which are more technical (somehow artificial) than that of other fields.
The following are my questions revolving around the significance of this field in Mathematics. Of course, they are absolutely related to my main question.
How can a grad student be motivated to specialize in this field? and
What is (are) the well-known result(s), found solely by non-commutative geometric techniques that could not be proven without them?
