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Ultrafilter! Waffle! Below are the 10 most recent journal entries recorded in the "Ultrafilter! Waffle!" journal:
May 7th, 2011
02:34 am

FLMPotD
Let K/k be the splitting field of an irreducible quintic over k. Show that K cannot contain a root of any irreducible cubic over k.

Current Mood: tired
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March 31st, 2011
01:10 pm

FLMPotD
Compute \sum_{k=0}^n (-1)^k \binom{n}{k}/(k+1).

Current Mood: moody
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March 26th, 2011
02:17 pm

FLMPotD
Find a commutative ring R, a multiplicative subset S \subset R, and finitely generated R-modules M and N such that the natural map S^{-1}Hom(M,N) \to Hom(S^{-1}M,S^{-1}N) is not an isomorphism.

Current Mood: busy
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March 7th, 2011
10:04 pm

FLMPotD
Show that a finite p-group G can be recovered from the mod p cochains on BG, as an E_\infty-algebra.

(This implies that a finite nilpotent group can be recovered from its integral cochains...I think the same should be true for any finite group, but I don't quite know enough about representation theory and group cohomology over Z to be sure about a certain step of the argument, and maybe my intuitions from how things work for p-groups really only apply to nilpotent groups. Note that this is not true for infinite groups, as there are nontrivial groups with trivial cohomology, a fact which is crucial to the proof of the Kan-Thurston theorem.)

Current Mood: okay
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February 23rd, 2011
05:32 pm

FLMPotD
Show that if the homology of a spectrum is free, then its Atiyah-Hirzebruch spectral sequence for any (co)homology theory with torsion-free coefficients degenerates at E_2.

Current Mood: okay
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10:02 am

On cleverness in LARPs
While I've never written a LARP, one of the subtler points about writing them that I've noticed from playing is that it is difficult to make plots that require people to be clever in character. The basic difficulty is that if I'm presented with a challenge to which I (the player) see an obvious solution (or even just first step towards a solution), but my character sheet doesn't even mention this obvious idea, I will assume that for some reason it is not obvious to my character. If my character sheet says "you hope to figure out a way to do X" and there's an obvious way to do X, in character it doesn't make sense for me to be "figuring out a way" if the obvious way was obvious to my character. So if you're writing a character and part of achieving that character's goals involves coming up with a solution to a problem, you need to make sure the solution is difficult enough (or inaccessible with the knowledge that the character starts out with at the beginning of the game) and that you don't conspicuously leave out any obvious ideas the character should have for going about the solution.

Anyways, the moral of the story is that character sheets need to be very thorough, because an omission of an idea in a character sheet is just as significant as an inclusion.

Current Mood: thoughtful

February 17th, 2011
05:34 pm

Bah
I don't understand why people ever say things in math without accompanying them with an explanation of what's actually going on in them. It's just so...why???

(Specifically, this was prompted by thinking about the fact that a simplicial group is automatically a Kan complex. Like two years ago I read about this and they just gave some unexplained formula for how you can use the group structure to fill horns. Then today, as I was bored in a seminar, I figured out that it's nothing but a souped-up version of the Eckmann-Hilton argument saying that the composition of paths in a group is canonically homotopic to their pointwise product using the group structure. Why didn't they explain this in the book where I read it before??)

Current Mood: ranting

February 4th, 2011
03:37 pm

FLMPotD
Prove Picard's big theorem (in a neighborhood of an essential singularity, a holomorphic function's image misses at most one point of C) using only the fact that the holomorphic universal cover of C-{0,1} is the unit disk, elementary complex analysis, and covering space theory.

(In my complex analysis class years ago, we learned the proof of Picard's little theorem along these lines, which I found very beautiful, and then proceeded to do a very analytic and unenlightening proof of Picard's big theorem. I had always assumed that there must not exist a similarly nice proof, and then recently I tried to come up with one and discovered that there was one.)

Current Mood: sick
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February 3rd, 2011
12:14 am

FLMPotD
Show that the ring Q[sin,cos] (say as a subring of the ring of functions on R) is spanned by the functions sin nx and cos nx as n varies.

(This one's from . There's an unenlightening trivial solution using obscure trig identities, and then a more interesting conceptual solution.)

Current Mood: merp
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February 2nd, 2011
10:34 pm