I don’t really anticipate doing all that much serious blogging for the next few weeks, but I might do a few posts like this one.

First, I learned from Qiaochu in a comment that the commutativity of the endomorphism monoid of the unital object in a monoidal category can be proved using the Eckmann-Hilton argument. Let be this object; then we can define two operations on as follows. The first is the tensor product: given , define , where is the isomorphism. Next, define . It follows that by the axioms for a monoidal category (in particular, the ones about the unital object), so the Eckmann-Hilton argument that these two operations are the same and commutative.

Second, I discovered that Serre’s GAGA paper is available online freely, thanks to Numdam. So is the Borel-Serre paper on the Riemann-Roch theorem, both of which I probably ought to read sometime, given their heavy citations in Hartshorne (and they both look fairly accessible, actually). Sadly, it appears that Grothendieck has wished for his own work to be taken down, and that mathematicians may comply. See this discussion over at the Secret Blogging Seminar.

Thirdly, I may as well say what I plan to talk about when I resume serious blogging. I have a series on one of the following in mind:

- Algebraic geometry: Various topics in Hartshorne II-III (mostly III). Say more about differentials and the conormal sequence, ampleness, the cohomology of affine and projective space, Serre duality, flatness/smoothness, cohomology along fibers.
- Algebraic geometry: Do some abstract nonsense, e.g. Grothendieck topologies and sheaves on them.
- (Edit) Commutative algebra: cover the homological theory of local rings, e.g. Koszul complexes, depth, Cohen-Macaulayness, properties of regular local rings (in particular, the highly important fact that they are UFDs). If I do any algebraic geometry, much of this will be necessary anyway.
- Analysis/PDE: Finish the proof of Malgrange-Ehrenpreis, cover Sobolev spaces on and on manifolds, estimates for the Laplacian and elliptic operators, Hodge theory
- Representation theory: talk about the center of where is a semisimple Lie algebra. The theorems of Harish-Chandra (on characters) and Chevalley (relating this to the enveloping algebra of a Cartan subalgebra).
- Differential geometry: The sphere theorem, Gromov-Hausdorff convergence, Ehresmann connections, characteristic classes (via Chern-Weil theory)

OK, that’s a long list. Ideally I’d get to all of them on this blog at some point, but hopefully I can say a fair bit on at least two or three of them by the end of the academic year.

February 21, 2010 at 1:52 pm

I vote for Grothendieck topologies etc. It seems this is the view most working algebraic geometers take, yet it is presented almost nowhere to my knowledge.

My other vote would be differential geometry, which I really like, but just don’t do much anymore.

February 27, 2010 at 1:10 pm

As a starry-eyed optimist I cannot help but dream that the blogosphere is a suitable setting for those wishing to initiate a systematic program to understand the functorial presentation of elementary algebraic geometry (as outlined by James Borger). As I understand it current students (future practitioners) — at least those residing in the blogosphere — have all expressed considerable interest in this point-of-view, and it seems unfortunate that while the resources (including those mathematicians found on that thread) are available, no centralized effort has been made to extract this information out of them.

Fortunately, I think the Stacks Project is doing remarkably well.

February 27, 2010 at 1:32 pm

Well, the new blog Effective Descent is apparently intended to fill that void. It’s too bad nothing seems to have been posted yet.

I don’t know where I’ll be whenever I do resume posting in a few more weeks. Most likely I’ll try to get somewhere in Fantechi’s FGA Explained. It’d be great if I could do cohomology on sites in general and start getting to the really fancy stuff generalizing Hartshorne chapter III, but it might not be possible (or realistic) for me to do this until college or whenever I can talk to an algebraic geometer on a daily basis.

February 27, 2010 at 7:41 pm

[…] Project blog Filed under: Uncategorized — by pmoduli @ 7:41 pm After recent discussion I found the recently-established Stacks blog. Here you can find many comments on the progress on […]