52,057 reputation
16202463
bio website math.berkeley.edu/~qchu
location Berkeley, CA
age 24
visits member for 5 years, 9 months
seen 11 mins ago

I am a third-year graduate student at UC Berkeley. I'm interested in the interplay between homotopy theory, quantum field theory, manifold topology, and higher category theory.


5m
comment A particularly “natural” algebraic structure with three commutative, pairwise-distributive operations
I don't understand why what you've already written isn't an answer to most of these questions. As you already know, sending a sequence $a_1, a_2, \dots$ to the corresponding formal power series $\sum a_i x^i$ turns Cauchy convolution into "pointwise" multiplication, and sending it to the formal Dirichlet series $\sum \frac{a_i}{i^s}$ turns Dirichlet convolution into "pointwise" multiplication. The quotation marks can be removed if you impose growth conditions. In what way is this not an answer to #3, #4, and #8?
10h
comment Can the projective line be provided with a ring structure?
If $K$ doesn't contain a square root of $-1$, then your group is the quotient of the multiplicative group of $K[i]$ by the multiplicative group of $K$. The closest ring in sight is $K[i]$, which, being a field, has no nontrivial quotients.
10h
comment Can the projective line be provided with a ring structure?
Your proposed multiplication isn't well-defined if $K$ itself already contains a square root of $-1$: in this case, $K[i]$ has zero divisors, so it's possible for the "product" of two points to have both coordinates zero and hence to not be a well-defined point on the projective line.
11h
answered Can the projective line be provided with a ring structure?
1d
comment Virasoro-like algebras over the quaternions
@Sébastien: this already fails for $n = 1$. The bracket is $[A, B] = AB - BA$ which is not $\mathbb{H}$-bilinear in either variable since neither $A$ nor $B$ need be central. Did you understand my point earlier about $\mathbb{H}$-modules not being a symmetric monoidal category?
2d
comment SO$(4)$ (& SO$(n)$) characterization?
Added line breaks where your paragraphs were. Hope you don't mind.
2d
revised SO$(4)$ (& SO$(n)$) characterization?
added 4 characters in body
2d
comment Virasoro-like algebras over the quaternions
@Chanler: if you just write down the standard definition as applied to a left $\mathbb{H}$-module, you get a definition which doesn't apply to $\mathfrak{gl}_n(\mathbb{H})$ (the Lie bracket on $\mathfrak{gl}_n(\mathbb{H})$ fails to be $\mathbb{H}$-bilinear in the appropriate way). Are you sure that's what you want?
2d
comment Virasoro-like algebras over the quaternions
What is a Lie algebra over $\mathbb{H}$? (The Lie operad is a symmetric operad in, say, abelian groups, so I know how to define a Lie algebra in any symmetric monoidal $\text{Ab}$-enriched category. But $\mathbb{H}$-modules aren't such a category.)
2d
answered SO$(4)$ (& SO$(n)$) characterization?
Jul
30
comment Why only Normed Linear Spaces?
We do in fact do this. See, for example, en.wikipedia.org/wiki/Absolute_value_(algebra) and en.wikipedia.org/wiki/Banach_algebra.
Jul
29
revised Does the following characterize local presentability?
added 3 characters in body
Jul
29
answered Does the following characterize local presentability?
Jul
28
comment Does every Lawvere theory arise in this way?
@David: no, that already fails in the case of groups. The way you recover a Lawvere theory from the free algebra over it on one generator $X$ is by taking the opposite of the full subcategory on the finite coproducts of $X$, not by taking the full subcategory on the finite products of $X$.
Jul
26
comment Is there a generalization of homotopy groups to fractional dimensions
Is there an $E_{\infty}$ ring spectrum $E$ and an invertible $E$-module spectrum $F$ such that $F^{\otimes 2} \cong E[1]$? ($\otimes$ here denotes the $E$-module smash product.) This gives a notion of fractional $E$-homology and cohomology.
Jul
24
comment Loop space generalization
$S^1$ has two interesting kinds of extra structure as an object of the homotopy category of based spaces: first, it's a cogroup object, but second, it's a group object. The first kind of extra structure is the one related to loop spaces, while the second kind of extra structure is the one related to its description as $B \mathbb{Z}$. The Eilenberg-MacLane spaces $B^n \mathbb{Z}, n \ge 2$, on the other hand, are only group objects, and have no cogroup structure (since their cohomology has interesting cup products).
Jul
22
awarded  Good Question
Jul
22
awarded  Good Question
Jul
22
awarded  Nice Question
Jul
21
comment Three questions about modular forms frequently asked to me
Crossposted: math.stackexchange.com/questions/1369276/…