152,678 reputation
15329601
bio website math.berkeley.edu/~qchu
location Berkeley, CA
age 24
visits member for 4 years, 3 months
seen 1 hour ago

I'm a third-year graduate student in pure mathematics at UC Berkeley. You can contact me at qchu[at]math[dot]berkeley[dot]edu.


23h
answered Non-Galois number fields and complex embeddings
1d
answered Does the functor that preserves limit always have a left adjoint?
2d
awarded  Nice Answer
Oct
20
awarded  ring-theory
Oct
19
comment Why only two binary operations?
@isomorphismes: well, who knows? We could spend all day arguing about precisely what is and is not natural. What I'm willing to say is that categories have been a very fruitful point of view historically and I expect they'll continue to be in the future.
Oct
19
revised Why only two binary operations?
added 109 characters in body
Oct
19
answered Why only two binary operations?
Oct
19
comment Compactness of Lie groups
@Brenin: I mean two things by this, which is why I didn't want to be precise. First, $\text{GL}_n(\mathbb{C})$ deformation retracts onto $\text{U}(n)$, so the two are homotopy equivalent. But second, it turns out that the two have essentially the same representation theory, provided that you're careful to restrict your attention to algebraic representations of $\text{GL}_n(\mathbb{C})$.
Oct
19
answered automorphism of the projective space $\mathbb{P}_A^n$
Oct
19
comment Calculating $\pi_2$ of a certain free loop space
Okay, strictly speaking that first comment only shows that we get some semidirect product decomposition, not that we get the one associated to the usual action of $\pi_1$ on $\pi_2$. But this probably comes from thinking a bit harder about how the fibration $\Omega X \to LX \to X$ works (which I haven't, yet).
Oct
19
comment Calculating $\pi_2$ of a certain free loop space
Incidentally, the simplest example of a space I know with nontrivial action of $\pi_1$ on $\pi_2$ is $X = \mathbb{RP}^2$, where $\pi_1 = \mathbb{Z}_2$ acts on $\pi_2 = \mathbb{Z}$ by reversing signs. So there's your interesting semidirect product.
Oct
19
comment Calculating $\pi_2$ of a certain free loop space
Okay, I know how to prove that semidirect product decomposition now. First, with all the other $\pi_k$ computed, you can show that two important boundary maps in the long exact sequence in homotopy associated to the fibration $\Omega X \to LX \to X$ vanish (assume $X$ path-connected here or else I have to specify which basepoint I take based loops at), giving a short exact sequence $\pi_2(X) \to \pi_1(LX) \to \pi_1(X)$. Second, the inclusion $X \to LX$ of constant loops into all loops is a splitting of this sequence.
Oct
19
revised Calculating $\pi_2$ of a certain free loop space
deleted 16 characters in body
Oct
19
answered Calculating $\pi_2$ of a certain free loop space
Oct
18
comment When is the free loop space simply connected?
@Jason: Yes, maybe I should've mentioned that it's possible to be very explicit about what $\pi_0(LX)$ is. I am less sure about $\pi_1(LX)$ though. Perhaps it's the semidirect product of $\pi_1(X)$ and $\pi_2(X)$ in general?
Oct
18
comment Could the equivalence classes in the construction of quotient group be the orbits of some group action?
Yes. It's $S$ acting on $G$ by right multiplication.
Oct
18
answered Compactness of Lie groups
Oct
18
revised When is the free loop space simply connected?
deleted 39 characters in body
Oct
18
comment When is the free loop space simply connected?
@Mike: ah, you're right of course. The special fact about compact Lie groups I had in mind is a corresponding fact about Lie algebra cohomology.
Oct
18
comment When is the free loop space simply connected?
At least in the last paragraph you are conflating the free and based loop spaces.