August 2, 2009
I recently (well, it’s been a month or two) decided I wanted to do maths at a serious level. That meant diving again in books, and tryint to get my mind in shape again. The basics were still there of course : it’s like biking, you never really forget. But basics aren’t enough.
I decided I would follow the tracks I followed already before my Ph.D, so I dived into algebra. Then I read algebraic geometry ; Hartshorne’s “Algebraic geometry”, which didn’t really stick : I was lacking in both commutative algebra and category theory.
So I read a few things about commutative algebra (my old course notes, a dusty book), then Mac Lane’s “Categories for the working mathematician”. Ouch. Brain. Hurt. Much. But that was welcome pain : it made the bigger picture appear more clearly. I should still try to find something about the Goedel-Bernays axioms for categories : I’d like to have clean bases.
Then I tried read the EGA (Grothendieck and Dieudonné’s “Éléments de géométrie algébrique”) — that was probably a mistake : that series of books was obviously never meant to be read! The examples are sparse and undetailed, there is little in the way of explaining what the bare results mean (sometimes I would take a few pages back and say : “Oh, they’re proving that this construction is local in such and such a way… couldn’t they just say that to make it smoother!?”). A good reference, but not a learning tool. The zero-numbered chapters are probably the most helpful. If only I could lay my hand on Grothendieck’s Tohoku!
I’ll probably try to read Hartshorne’s once again : I’m sure I’ll be able to get the best of it this time. The ultimate goal of this algebraic geometry reading is to read Deligne’s work on the Weil conjectures. Then I’ll probably try to read Wiles’s proof of the Shimura-Taniyama-Weil conjecture (which implies the famous FLT — Fermat’s Last Theorem). I’m probably still a bit short on Galois theory, number fields and representation theory for those though.
And of course, the ultimate goal of this maths reading is to be able to read my Ph.D report, get up to date with the results in the field, then go further.
PS: for those who don’t know about that site yet : NUMDAM has a great number of very interesting things — including the EGA and Deligne’s work on Weil conjectures, Néron’s work on minimal models ; oldies and goodies, but also more recent stuff.