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06:58 pm
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FLMPotD
Without using Bott periodicity in any way, compute the homotopy groups of BO up to \pi_7.
(In particular, this computation allows you to find all the homotopy groups from merely the fact that they are periodic, rather than the stronger statement which identifies all the intermediate deloopings.)
(ETA: Hrm, my computation at first glance seems inconsistent with p_1=w_2^2 mod 2...I tried working this out in my head on the train ride to New Jersey, but failed...)
Current Mood:  carsick (bussick?)
Tags: flmpotd
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07:00 pm
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FLMPotD
Show that for n=1,2,4,8, the Hopf map S^{2n-1}\to S^n is the image under the J-homomorphism of the map S^{n-1}\to O(n) sending a unit norm element of the n-dimensional division algebra to its multiplication action. (In the n=1 case, beware that the "J-homomorphism" is actually not even a homomorphism!)
Current Mood:  too pretty to do math
Tags: flmpotd
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02:36 am
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Fall party!
Yay HRSFA fall party! Awesome Mgame, scavenger hunt, race, and telephone pictionary. And YAY GIANT ROLL OF BUBBLEWRAP!
Odd observation: when I become a dog, five girls become cats. No boys become animals. Why do boys not want to be animals? Also, why is no one else a dog?
OK, I should sleep now. And um maybe actually start prepping my splash class tomorrow?
Current Mood:  tired
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05:17 pm
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FLMPotD
Fix some homology theory. Suppose a map f is an isomorphism on homology and has the right lifting property for cofibrations that are isomorphisms on homology. Show that f is a weak equivalence.
Current Mood:  okay
Tags: flmpotd
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08:52 pm
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:)
![[info]](http://l-stat.livejournal.com/img/userinfo.gif) convexbaka: Sigh
cannibalism is not good
but oooh it was sooo good
Current Mood:  victorious
Tags: quotes
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06:45 pm
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FLMPotD
Let X be a path-connected space. Show that the space of nonempty finite subsets of X, topologized naturally as a quotient of \bigcoprod_{n>0} X^n, is weakly contractible.
Current Mood: tired
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04:42 pm
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FLMPotD
Ordinary nth cohomology with coefficients in A is classified by maps to K(A,n). Find an analogous statement for cohomology with local coefficients.
Current Mood:  hungry
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10:04 am
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FLMPotD
Let X be a (pointed) co-H-space, i.e. a pointed space with a "comultiplication" X \to X \vee X which has a homotopy 2-sided counit. Show that for any multiplicative cohomology theory, all products in the reduced cohomology of X vanish.
Current Mood: okay
Tags: flmpotd
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03:10 pm
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FLMPotD
When people think about two spaces being homotopy equivalent, they usually think of it in terms of "deforming" one space into the other, even though this isn't really the definition (and requires an ambient space!). Here's a way of making this rigorous:
Show that two spaces are homotopy-equivalent iff you can realize them as different fibers of a quasifibration (a map whose fibers are equivalent to its homotopy fibers, even though the map may not be a fibration).
(Can you do it with an actual fibration? I feel like it ought to be possible, but I don't know how...)
Current Mood: sick
Tags: flmpotd
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09:05 am
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Is this about Math SIG?
I was teaching some kind of algebra, at Mathcamp I think. Then there were suddenly a whole lot of people in the audience. And there was like pizza and all these pizza advertisements. It was very disruptive. I was annoyed.
Current Mood:  lonely
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08:28 pm
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Math SIG!
HRSFA now officially has a Math SIG.
Current Mood: happy
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01:25 am
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Epiphany
These ideas have been swirling around in my head for a while, but somehow I never really formulated them quite right until now.
A (pointed) space is a chain complex of (pointed) sets. Homotopy groups are homology of chain complexes of pointed sets. Alternatively, they are Ext from the two point set to your space, in the category of pointed sets.
I love algebraic topology.
Current Mood:  sleepy
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08:31 pm
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FLMPotD
Another problem from Math Overflow: Let T be a linear endomorphism the space of nxn matrices over some field that preserves determinants. Show that there are invertible U and V such that T(A)=UAV or T(A)=UA^tV, where A^t is the transpose of A.
Current Mood: okay
Tags: flmpotd
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10:33 pm
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FLMPotD
Show that you can partition the plane into two sets, neither of which contains a (nonconstant) path.
(This problem, with the extra condition that you give an explicit construction, has been unsolved at the 20 Questions Seminar and Math Overflow for several days. Also, those two sites are awesome.)
Current Mood: tired
Tags: flmpotd
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03:25 pm
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Pretty combinatorics!
FLMPotD: Show that the number of (unordered) partitions of an integer into numbers of the form 2^n-1 is the same as the number of partitions as \sum a_i with a_i \geq 2a_{i+1}.
Current Mood: tired
Tags: flmpotd
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01:17 am
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FLMPotD
Let A be a finite-dimensional associative algebra over a field equipped with a linear map S:A \to A and a nondegenerate symmetric pairing satisfying (ab,c)=(b,(Sa)c). Show that S is an involutive antihomomorphism and that the category of finite dimensional A-modules is naturally equivalent to its own opposite category via a functor that sends A to itself.
(I came up with this actually just to show that A is injective as a module over itself.)
Current Mood: tired
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11:08 am
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FLMPotD
Assume that every homology sphere is homeomorphic to a sphere (this is false) and that whenever R \times A is a topological manifold, so is A (according to Wikipedia, this is also false, a counterexample being given by the suspension of a homology sphere). Describe an algorithm that decides whether a simplicial complex is a topological manifold.
(In general, there do exist topological manifolds with triangulations for which they are not PL manifolds, constructed using homology spheres. However, I don't know whether in general it is possible to decide whether a triangulation is a manifold--I suspect the answer is no because of fundamental group issues, but I really don't know.)
Current Mood: hungry
Tags: flmpotd
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07:41 pm
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FLMPotD
Show that every autohomeomorphism of R^n of prime order has a fixed point.
Current Mood: okay
Tags: flmpotd
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11:07 pm
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Hmmm...
It is odd to not be writing a puzzlehunt. I just discussed some ridiculously overconstrained but awesome puzzle ideas with HRSFA. Among them: Race of the Two-Headed Monsters, but with each individual half-line being grammatical and them forming their own puzzle.
Also, it is unnerving how much Jason L is like hahafaha.
Current Mood: ineffective
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01:59 am
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<3
There is now a group of 4 devoted Mao players in HRSFA.
(Also, random guy asked for directions as I walked back from Mather at 1:30 AM, and then was kind of over the top in thanking me. It was mildly creepy?)
Current Mood: sleepy
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