I learned the difference between a "differentiable manifold" and a "differential manifold," today. It's a subtle one.
Thesis work has been going all right. I wouldn't rightly call it research, because it's retracing the work of others, in order to assess how "my" result [1] sits in the body of knowledge. It's hard work trying to understand this stuff, but it makes me uneasy that I haven't been contributing any ideas of my own into the mix.
On the plus side, I've been doing all sorts of fun differential topological stuff: reading Milnor's paper on exotic differential structures, and now learning about isotopies of diffeomorphisms and gluing manifolds. I might even learn about vector bundles, which is a topic I hadn't expected to learn unless I "defected" to geometry.
Sometimes it's nice to be able to bake your cake and eat it, too. q:
[1] More accurately, I should say that it's Fred Gehring's result. The good ideas are his, and at best I've proven a few corollaries.
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