My Forget Me Not plug-in for Safari was reviewed on MacWorld!

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It’s great to read the comments and find out what users wish were different: a lot of people don’t just want to Unclose Window but also Unclose Tab.

For me, I really would have liked to have Unclose Window be underneath the Edit menu, but because the undo hierarchy is linked to the window (i.e., when you switch windows, the possible things you can undo changes) it isn’t possible to undo the very destruction of the window itself.

The whole idea of “undoing” something is very amusing, especially the modern sense; the old sense of “undoing” something by destroying it is quite dissimilar from the idea of restoring to a previous state.

In August, two of my friends from college have died: Daniel Bartlett and Michelle Knapp. I’m not sure what else I can say; I remember them so clearly…

From Recent Advances in the Langlands Program, quoted in This Week’s Finds:

First of all, it should be remarked that according to the Tannakian phylosophy, one can reconstruct a group from the category of its finite-dimensional representations, equipped with the structure of the tensor product.

I suppose one should think of this as the categorification of Pontrjagin duality?

For a long while, I had wondered how this goes; this Introduction to Tannaka Duality and Quantum Groups will probably answer my questions.

Here is a question that I haven’t been able to find very much about:

What are the finite subgroups of the rotation groups $SO(n)$?

For examples, I can take a Coxeter group, and choose elements corresponding to rotations (e.g., the subgroup generated by products of generators), but that’s not going to produce very many examples.

Here’s a very easy theorem.

Theorem. All closed orientable 3-manifolds are parallelizable. All closed orientable 3-manifolds are the boundary of a 4-manifold.

Proof: Let $M$ be an orientable $3$-manifold. Recall that the Wu class $v$ is the unique cohomology class such that $\langle v \cup x, [M] \rangle = \langle Sq(x), [M] \rangle$, and Wu’s theorem says that $w(M) = Sq(v)$. The up-shot is that Stiefel-Whitney classes are homotopy invariants, even though they are defined using the tangent bundle.

Since $M$ is orientable, we have $w_1(M) = 0$. Since $\dim M = 3$, the Steenrod squares $Sq^2$ and $Sq^3$ kill everything, so $v_2 = 0$ and $v_3 = 0$. By Wu’s theorem, $w_2(M) = Sq^1(v_1) + v_2 = 0$, and $w_3(M) = Sq^1(v_2) + v_3 = 0$. In other words, all the Stiefel-Whitney classes vanish. ∎

Orientability matters; after all, being orientable is the same thing as $w_1$ vanishing. For example, $RP^2 \times S^1$ is not parallelizable, since $w_1(RP^2 \times S^1) = w_1(RP^2) + w_1(S^1) \neq 0$.