Application of Chern class I am taking a course about differentiable manifold and I need to prepare a representation about Chern class.
Although now I am familiar with the properties of Chern class, but I can not find good examples of the applications of Chern class. As a topological invariant of vector bundle, I really want to find some example to use Chern class to distinguish some different vector bundle. I am also looking other applications which can illustrate why Chern class is useful. Thanks!

The first time I really used Chern classes was when writing this blog post. The goal was to compute the cohomology ring of a hypersurface of degree $d$ in $\mathbb{CP}^3$, and I ended up computing its Chern classes, Pontryagin classes, Wu classes, and Stiefel-Whitney classes to do this. It was fun. 
