In August of 2013, I finally presented on Milnor’s Characteristic Classes for my final comprehensive exam. I am officially a PhD candidate now!
I focused on his proof of the polynomial ring structure of the cohomology ring of the complex Grassmann manifold. I assumed everyone already had working definitions of the Euler class and Stiefel-Whitney classes. I presented Milnor’s definition of Chern classes based on the Gysin sequence, and then powered through the notationally heavy proof.