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May 7th, 2011
02:34 am

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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.

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March 31st, 2011
01:10 pm

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

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March 26th, 2011
02:17 pm

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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.

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March 7th, 2011
10:04 pm

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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.)

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February 23rd, 2011
05:32 pm

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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.

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10:02 am

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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.

I should mention that not all (perhaps even not most) people play characters this way, which sometimes bothers me (and causes my characters to be much less successful than others'). For instance, in a game I played in recently, one of the tasks of a group of characters was to design a budget, which involved raising some money and apportioning it in a way that would receive approval from a majority of the group. Everyone's character sheets just said things like, "you want to fund this, you don't want to fund that, you don't want to raise money this way". There was a very obvious solution to the budgeting process, which was to write down all the things you cared to fund and compare it to the total funds available and see how much you needed to raise (in fact, you didn't need to raise much money at all, even if you wanted to fund almost everything that anyone wanted to fund). Nothing was said about this (or the results of it) in the character sheets, so as I played my character, this is something that wouldn't happen until everyone sat down and started debating the budget (and at that point it would actually happen pretty quickly, because it is kind of the only way to do it). However, one of the players decided to do this ahead of time, and came into the game with a grand budget plan to present to the group, most points of which were fairly quickly accepted. While in some ways this is something that would have made some sense for their character, this is something that I would never do if my character sheet didn't say to do it. Seriously, if you have some budget priorities and you know how much money you have to spend and your character hasn't already done the obvious calculation and thought about it (with the outcome and your thoughts about it mentioned in the character sheet), that seems to me to be a nontrivial statement about your character.

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.

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February 17th, 2011
05:34 pm

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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??)

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February 4th, 2011
03:37 pm

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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.)

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February 3rd, 2011
12:14 am

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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 somthng2remembr. There's an unenlightening trivial solution using obscure trig identities, and then a more interesting conceptual solution.)

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February 2nd, 2011
10:34 pm

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FLMPotD
Show that x^4-16x^2+4 is reducible mod p for every prime p.

(Stolen from a comment by secret_panda on a post on somthng2remembr's facebook. Yay stalking!)

(PS: Actually, my argument should work for anything of the form x^4+bx^2+c^2 for b and c integers and p odd.)

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