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Qiaochu Yuan

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Name Qiaochu Yuan
Member for 3 years
Seen 3 hours ago
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Location Berkeley, CA
Age 22
I'm a second-year graduate student in pure mathematics at UC Berkeley. You can contact me at qchu[at]math[dot]berkeley[dot]edu.
11h
 comment Orbit structure of linear representations of complex Lie groups
In the case of the dual of the adjoint representation, this is intensively studied under the keyword "coadjoint orbit."
15h
 comment How do we express measurable spaces using type theory?
Haskell can at least model hereditarily finite sets: that's a fairly simple recursive data type (a hereditarily finite set is a finite list of hereditarily finite sets, more or less).
Jun
12
 comment Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
Yes, I think it's true in any case that $U(-)$ is the colimit over $\text{Hom}(A_n, -)$.
Jun
11
 awarded ● Nice Question
Jun
11
 awarded ● Popular Question
Jun
11
 comment Which categories are the categories of models of a Lawvere theory?
@Zhen: ah, I see. I would also guess that $G$ generated by a single object under finite coproducts is the correct condition.
Jun
10
 comment Which categories are the categories of models of a Lawvere theory?
Great! Am I correct in guessing that by "Lawvere theory" here you mean a multisorted Lawvere theory, and that if I was only interested in Lawvere theories with one sort the second condition should be "there exists an object in $C$ such that..."?
Jun
10
 asked Which categories are the categories of models of a Lawvere theory?
Jun
5
 comment Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
Cool. I think basically the same argument works for C*-algebras as well (using the C*-algebras of continuous functions on $D_n$ instead).
Jun
5
 comment Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
@Simon: this could work. No continuity or boundedness hypotheses should be necessary; this sort of thing is enforced by naturality (see the argument below).
Jun
5
 revised Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
added 123 characters in body
Jun
5
 asked Defining holomorphic functions in terms of Banach algebras, and similarly for C*-algebras
Jun
3
 awarded ● Nice Answer
Jun
1
 comment Is Gouvêa-Mazur’s “Infinite Fern” a fractal?
What does "is a fractal" mean here?
May
30
 comment Wrong-way Frobenius reciprocity for finite groups representations
@domenico: yes, but perhaps it's cleaner to start from the map in the other direction.
May
30
 accepted Wrong-way Frobenius reciprocity for finite groups representations
May
30
 answered Decomposition into irreducibles of symmetric powers of irreps.
May
30
 comment Inner automorphisms and $K$-theory
Crossposted to math.SE: math.stackexchange.com/questions/406088/…
May
29
 comment Wrong-way Frobenius reciprocity for finite groups representations
I added an explicit map. Something went wrong when I tried to write this map down abstractly and I'm not sure how to fix it.
May
29
 revised Wrong-way Frobenius reciprocity for finite groups representations
deleted 148 characters in body; deleted 2 characters in body
May
29
 revised Wrong-way Frobenius reciprocity for finite groups representations
added 550 characters in body; added 11 characters in body
May
29
 comment Is every (one dimensional) n-bud of total degree n also a formal group law?
Sage can symbolically manipulate multivariate polynomials (sagemath.org/doc/constructions/…) although SageMathCloud wasn't happy with the above example for some reason.
May
29
 revised Wrong-way Frobenius reciprocity for finite groups representations
added 56 characters in body
May
29
 revised Wrong-way Frobenius reciprocity for finite groups representations
added 618 characters in body; added 476 characters in body; deleted 31 characters in body
May
29
 accepted Is every (one dimensional) n-bud of total degree n also a formal group law?
May
29
 answered Wrong-way Frobenius reciprocity for finite groups representations
May
29
 comment Intersection graphs of 2-element subsets
What's unnatural about the adjacency matrix? Regarded as an operator $\mathbb{R}^V \to \mathbb{R}^V$ it's perfectly coordinate-free.
May
29
 comment Do there exist transcendental numbers which are not hypertranscendental?
The title and the body ask different questions. A simple example for the body question is $2 \pi i$.
May
29
 revised Is every (one dimensional) n-bud of total degree n also a formal group law?
edited body
May
29
 comment Are roots of transcendental elements transcendental?
@darij: Let $k$ contain two elements $a, b$ such that $a^2 = ab = b^2 = 0$ and let $A = k[t]/(at^3 - b)$. By construction, $t$ is algebraic. It looks like $t^2$ might be transcendental, although I'm not sure how to prove it.
May
29
 comment Are roots of transcendental elements transcendental?
@quid: $k$ is a commutative ring, not a field. You can't conclude that $k[t]$ is finitely generated as a $k$-module either (since we may have for example $k = \mathbb{Z}, t = \frac{1}{2}$).
May
29
 answered Is every (one dimensional) n-bud of total degree n also a formal group law?
May
27
 comment Can you prove the Fundamental Theorem of Algebra just using fixed point theory?
I played around once with proving FTA from the Banach fixed point theorem but I couldn't get it to work. You can prove FTA from the Lefschetz fixed point theorem, though. Does that still qualify as "fixed point theory"?
May
27
 awarded ● Enlightened
May
27
 accepted Is it true that Nature promotes products?
May
26
 comment Embedding Theorem for topological spaces, and in general
One theorem of the first form is "every second-countable Tychonoff space embeds into $[0, 1]^{\mathbb{N}}$."
May
26
 comment Embedding Theorem for topological spaces, and in general
Are you asking for theorems of the form "every nice topological space embeds into some even nicer topological space" or for theorems of the form "every nice subcategory of $\text{Top}$ embeds into some even nicer category"?
May
24
 comment Dijkgraaf-Witten TQFT vs. Representation Theory?
The question seems vague. Can you be more specific?
May
24
 awarded ● Nice Answer
May
24
 awarded ● Nice Answer
May
21
 comment objects which can’t be defined without making choices but which end up independent of the choice
Here's one possibility: consider the Grothendieck group of the category of finite groups, where $[A] = [B] + [C]$ whenever there is a short exact sequence $0 \to B \to A \to C \to 0$. Then the Grothendieck group should be free abelian on the finite simple groups, and the image of a finite group in the Grothendieck group should be precisely the simple groups in a composition series, with appropriate multiplicities.
May
21
 comment How much of character theory can be done without Schur’s lemma or the Artin-Wedderburn theorem?
I think these "ghost representations" do exist; the literature on fusion categories might be a place to find them?
May
21
 comment objects which can’t be defined without making choices but which end up independent of the choice
The fundamental group can be recovered from the category of covering spaces; it's the unique group $G$ such that the category of covering spaces is equivalent to $G\text{-Set}$. In that sense it doesn't depend on a choice of basepoint.
May
20
 answered objects which can’t be defined without making choices but which end up independent of the choice
May
19
 comment What are the main structure theorems on finitely generated commutative monoids?
This sounds quite hard. Isn't the category of finitely generated commutative idempotent monoids equivalent to the category of finite lattices?
May
17
 comment Is there any proof that you feel you do not “understand”?
I'm not sure (I meant to work this out sometime but haven't gotten around to it). Some Cartesian closed category where the morphisms are computable functions. The point would be that Godel numbering provides something like a surjection $\mathbb{N} \to \mathbb{N}^{\mathbb{N}}$ in such a category, so $\mathbb{N}$ must have the fixed point property.
May
17
 answered Is there any proof that you feel you do not “understand”?
May
17
 comment Is there any proof that you feel you do not “understand”?
Does this count as a proof at the undergraduate level?
May
17
 comment Is there any proof that you feel you do not “understand”?
The recursion theorem ought to be a corollary of Lawvere's fixed point theorem, right?
May
15
 comment Reference/quote request: “All of combinatorics is the representation theory of $S_n$”
I think Igor Pak (?) once said something like "gambling is the applied representation theory of the symmetric group," but I don't have a citation so I may have just imagined this.