Qiaochu Yuan
Reputation
538/400 score
 10h comment Existence of a solution of a system of polynomial equations It's easy to write down lots of examples where there are no fixed points; for example, because we have $f_i(x) > x_i$ for all $i$ (e.g. we could take $f_i(x) = x_i + 1$, or $f_i(x) = x_i^2 + 1$). What examples are you interested in? 1d awarded polynomials May 2 comment Holomorphic contractibility of GL(H)? What do "holomorphically contractible" and "holomorphic $K(\mathbb{Z}, 2)$" mean? May 1 comment Time-Energy Uncertainty Relation in relativistic Quantum Mechanics Probably going to get better answers on physics.stackexchange.com. May 1 awarded Enlightened Apr 30 awarded Nice Answer Apr 30 comment Maps from $S^3$ to $S^3$ For what it's worth, I didn't downvote, but you're not doing yourself any favors by insulting people and questioning their expertise. Denis is correct that there is a notion of polynomial map in this setting, namely a map all of whose coordinates are given by polynomials (thinking of $S^3$ as embedded in $\mathbb{R}^4$ in the usual way); for example, the maps $z \mapsto z^n$ coming from the group structure on $SU(2)$ are polynomial maps in this sense. For a discussion of degree see, for example, en.wikipedia.org/wiki/Degree_of_a_continuous_mapping. Apr 30 comment Maps from $S^3$ to $S^3$ The space of maps has connected components labeled by degree, which is an integer. Is this the sort of thing you want to know? Right now your question is very vague. Apr 30 comment From Weyl groups to Weyl groupoids? It's easier to have this discussion on the Lie group rather than Lie algebra level. There for $G$ a compact connected Lie group you can consider the groupoid whose objects are maximal tori and whose morphisms are conjugations. Every object in this groupoid is isomorphic, and all of their automorphism groups are the Weyl group. Apr 30 comment “Small” simplicial complex with torsion trees So just to clarify, these spanning trees are not in fact trees? Apr 29 answered Representation viewpoint on Chern Weil (cohomology computations done with rep theory?) Apr 29 comment what is the universal cover of GL(2,R)? The OP also says "and therefore $G'$ was $\mathbb{C}^2$" so I think it is clear he meant $\mathbb{C}^{\ast} = \mathbb{C} \setminus \{ 0 \}$. (I prefer the notation $\mathbb{C}^{\times}$.) Apr 28 comment what is the universal cover of GL(2,R)? $GL_2^{+}(\mathbb{R})$, as a manifold, is $\mathbb{R}^3 \times SO(2)$. (More generally, any connected Lie group, as a manifold, is its maximal compact times some $\mathbb{R}^n$.) So its universal cover, as a manifold, is $\mathbb{R}^4$. Apr 28 comment Does it make sense to compare sets (polytopes) with different dimensions? @Dima: I don't know what that means. The OP claims to have proven a statement and then wants to know whether the statement makes sense. If the OP doesn't know whether it makes sense then how did they prove it? Apr 26 comment “Spatial (geometrical)” realization of Elementary topos? @Simon: yes, I required the tensor product to distribute over colimits above. Elementary topoi can be ind-completed and I think the result is now monoidal cocomplete although I haven't checked this. Apr 26 comment “Spatial (geometrical)” realization of Elementary topos? @Andrej: well, that's stronger than an analogy, right? One is even a special case of the other. I really just mean an analogy here. It's very naive: colimits are like addition. A more precise analogy would have been to commutative monoids since coproducts don't have inverses. For example, like in commutative monoids, there is a zero object, and biproducts. In this analogy presheaf categories are analogous to free abelian groups. Apr 26 revised “Spatial (geometrical)” realization of Elementary topos? added 89 characters in body Apr 26 answered “Spatial (geometrical)” realization of Elementary topos? Apr 26 answered Probability theory without deductive closure Apr 26 comment When are Morita classes represented by certain structured algebra objects? Also, depending on what you mean by "semisimple module category," I don't think your claim that these are all categories of modules over algebras is true either, since (with the definitions I have in mind) there may be infinitely many simple objects.