## Sunday, December 9, 2007

### Putting in Time

Well, I spent most of the afternoon trying to learn things. Might as well have just watched movies though.

Suppose I let R_{k-1}^{k} be the poset of vector subspaces of real k-space whose dimensions are bigger than 0, and no more than k-1 (that is, a point in this poset is a vector space E that is a subspace of real k-space, and 0<dim E<k), ordered by inclusion. This is then a topological category (that is, a category internal in Top), with object space the disjoint union of so many Grassmannians and morphism space the disjoint union of flag manifolds (flags of length 2, I guess you'd say). I've been trying to sort out the realization of this category, because it comes up in a homotopy limit I'd like to calculate. The nerve is a simplicial set whose non-degenerate simplices are flag manifolds (the length gives the dimension of the simplex), and so the largest dimension in which there are non-degenerate simplices is k-1 (those chains E_1<E_2<...<E_{k-1} where dim E_i = i). Anybody have a reference for somebody else's work on this topic? Because I'm not getting anywhere, and it feels like the sort of thing people would have been interested in already. [Update: Found it]

Also finished reading E. T. Bell's "Men of Mathematics" today. I don't remember getting a whole lot out of it, but I guess that'll happen. So now I gotta figure out what to read next. Kinda feel like I should read "Women in Mathematics", which I feel like I've seen in the library. Also just heard about an author Pynchon, sounded like he might be fun.

Guess I should go prepare something to teach LaTeX to high schoolers.