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 1 be this object; then we can define two operations on End(1) as follows.  The first is the tensor product: given a,b, define a.b := \phi^{-1} \circ a \otimes b \circ \phi, where \phi: 1 \to 1 \otimes 1 is the isomorphism.  Next, define a \ast b := a \circ b.  It follows that (a \ast b) . (c \ast d) = (a . c) \ast (b. d) 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 \mathbb{R}^n and on manifolds, estimates for the Laplacian and elliptic operators, Hodge theory
  • Representation theory: talk about the center of U(\mathfrak{g}) where \mathfrak{g} 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.

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