# What is the significance of non-commutative geometry in mathematics?

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

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.

1. How can a grad student be motivated to specialize in this field? and

2. What is (are) the well-known result(s), found solely by non-commutative geometric techniques that could not be proven without them?

• I think Alain Valette's answer to your Q2 is a very good example. As for Q1, I have found that some combination of "there is some beautiful mathematics here" and "there might be a better chance of getting a grant or a postdoc if you do this, than if you do something idiotic like the cohomology of Banach algebras", quite forceful. – Yemon Choi Feb 11 '12 at 8:46