For my graduate (master) thesis I am studying the theory of Chern Classes. As a possible personal development the only sensible idea I have so far, and which I frankly think is impossible, is to work on the inverse problem, i.e., given a class in cohomology, is there a vector bundle which has that class as one of its Chern classes? The following question is to understand what it is already known about this and if maybe I might focus my attention on a small sub-problem and if it is an idea worth telling my supervisor at all or just forget about it.

1. Has this problem been analysed before? Is it a sensible matter to tackle?
2. I know how to find Chern Classes taking conjugate invariant symmetric polynomials of the curvature matrix of a connection on the manifold (so deRahm Cohomology). Is this approach better, worse, equivalent to studying line divisors, picard groups and related? (Is there any other approach to finding Chern classes?)