# Theo Johnson-Freyd

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 Name Theo Johnson-Freyd Member for 3 years Seen 16 hours ago Website Location Berkeley Age 27
Graduate student at UC Berkeley, studying quantum field theory.
 1d accepted Can distinct open knots correspond to the same closed knot? 1d comment Differential form on a compact manifold whose exterior derivative is nowhere zero?It might be helpful for you to draw a picture. A 2-form on $S^2$ can be thought of as a map $\mathrm T S^2 \to \mathrm T^* S^2$ from the tangent to the cotangent bundle, and as such it has, at each point, a kernel which is either $0,1,$ or $2$-dimensional. Since your 2-form is a restriction from $\mathbb R^3$, the kernel is the intersection of $\mathrm T S^2$ with the kernel of the map $\mathrm T\mathbb R^3 \to \mathrm T^*\mathbb R^3$, and you can draw this. The 2-form vanishes exactly when the kernel is everything, so work out at which points $\ker \mathrm d\omega$ is parallel to $S^2$. 1d comment What are the main structure theorems on finitely generated commutative monoids?@Noah S: I imagine that there are certain versions of the problem "classify numerical monoids" that really mean "understand the primes". Certainly there are "classification" problems among prime numbers that are not known. 1d comment Closed geodesic loops around points in compact manifolds(I take it you don't count the constant loop.) Of course, if $M$ has nontrivial $\pi_1$, then you can find a geodesic with fixed endpoints representing any nontrivial homotopy class by minimizing the energy. In the simply-connected case I don't see an immediate proof. May18 comment In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal?@Ryan: Thanks for the comment! I was unaware of their work, but I'll look it up. May18 asked In cell-decomposed manifolds, how easy is it to arrange for the tubular neighborhood of a diagonal to contract onto the diagonal? May17 answered Can distinct open knots correspond to the same closed knot? May16 comment Reasons to prefer one large prime over another to approximate characteristic zero... go into developing precisely this type of heuristic. May16 comment Reasons to prefer one large prime over another to approximate characteristic zeroI think this is an interesting question, to which I do not have an answer. I will point out that in some sense no prime is better than any other: for any particular finite set of primes, certainly there are sentences that fail exactly on that set. So your question presupposes something about "interesting" questions that can be answered by an algorithm, or about questions that are "likely" to come up in "research". I doubt that pure model theory and pure number theory can give an absolute answer to things about "interesting" questions and "likely research", but conversely much work does ... May14 comment reflection LIE groupsI don't have a useful answer to your question, but I vaguely recall at least one conference talk in which $O(n)$ was compared, with quite some success, to a finite reflection group, so this question is not entirely out of left field. May14 comment the category of right comodule of coalgebra is a monoidal category , why?Please look over mathoverflow.net/howtoask and revise this question: it is missing definitions, should have better capitalization and punctuation, and might as well have the mathematics typeset as actual TeX (see the box "How to write math" on the right-hand column). But probably no change will fix your question, as in general there is no good monoidal structure for the category of comodules of a general coalgebra. Two situations in which comodule categories are monoidal are when the coalgebra in question is cocommutative, or when the coalgebra is given a Hopf algebra structure. May5 asked When does an even-dimensional manifold fiber over an odd-dimensional manifold? May5 answered Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field? May5 asked Does a connected manifold with vanishing Euler characteristic admit a nowhere-vanishing vector field? May5 awarded ● Popular Question Apr24 awarded ● Nice Question Apr21 awarded ● Notable Question Apr20 comment Cocompleteness of the category of small $A_\infty$ categoriesThe 1-category of small $A_\infty$ categories is certainly cocomplete, as it is the category of representations of a Lawvere algebraic theory and hence presentable. But this is, of course, not the category that you care about. Presumably, you care instead about some version of an $(\infty,2)$-category, since $A_\infty$ categories are some version of $(\infty,1)$-categories. Apr20 comment What does a mathematician expect from mathematics education? I have voted to close this question as subjective and argumentative. If it is to be saved, at a minimum the first three sentences must go. I suppose that, being CW, I could just go in and change the question. Perhaps I will do so if the question is not closed, but even with the changes I see to make, I'm not sure I can make the question into an appropriate one for MO. Apr19 comment How to understand Chern-Simons actionI have no complaints about Urs's answer below, but if you want a less high-brow discussion, I recommend Freed's articles "Classical Chern-Simons Theory Part I" arxiv.org/abs/hep-th/9206021 and "Part II" ace1.ma.utexas.edu/users/dafr/cs2.pdf . Apr19 comment HIgher Homotopy Groups and Representation TheoryWell, I mean, there are many applications of the general statement that $\pi_2$ is trivial. But for no $G$ can elements of $\pi_2(G)$ be applied to some particular construction, the way that $\pi_1$ or $\pi_3$ can. Apr18 answered HIgher Homotopy Groups and Representation Theory Apr18 comment HIgher Homotopy Groups and Representation Theory"if G˜ is the simply connected universal cover of G then all representations of g can be integrated to representations of G". You should include "finite-dimensional" somewhere in that sentence. The Lie algebra $\mathbb R$ acts on $\mathcal C^\infty(I)$, where $I$ denotes the open unit interval $I = (0,1)$, by sending the basis vector to $\frac{\partial}{\partial x}$, but this representation is not integrable to a representation of $\mathbb R$ on $\mathcal C^\infty(I)$. Apr16 comment Felder Kazhdan classical MEWell, it has only been a few months... Apr13 comment Analogy between the exterior power and the power setTo make the algebraic structure on $P(X)$ closer to that on $\Lambda(X)$, you can use the disjoint union rather than the union. Then the last bullet looks a little better. But note that the $\Lambda$ side of the last bullet doesn't make sense in arbitrary categories --- rather, it has something special to do with usual modules over a ring in which $2$ is invertible. Apr7 comment On mentioning recommenders' names in cover letter for postdoctoral applications@Yemon: I entirely agree, but meant only to illustrate that questions of this type can be very discipline-specific. I should also emphasize that I have not spent much time at academia.SE — for all I know, it could be entirely populated with mathematicians. Apr7 comment “Augmenting” a category by an associative binary operationI am likely confused about some aspect of your construction, but I don't see how $\zeta \ast C$ can be a category unless $\zeta: \alpha\times\alpha \to \alpha$ is associative. If $\zeta$ is associative, then this is (almost, but not quite) the operation of "base change" of a category: if $C$ is any category, and $(\alpha,\zeta,a\in\alpha)$ any associative algebra, then "$C \otimes \alpha$" is the category with the same objects as $C$ and morphisms $\hom_{C\otimes \alpha}(x,y) = \hom_C(x,y)\times \alpha$. I have no useful comments about colimits. Apr7 comment uniqueness in $\infty$ categories.@Eric: It looks like your comments would make a good answer to this question. Mar31 comment On mentioning recommenders' names in cover letter for postdoctoral applications@Ben: At least in the US, the correct answer to this question is fairly discipline-specific. We have MathJobs, and as Xuhan expressed below, it already handles most of the "what letters to expect"-type stuff. Many disciplines and departments, especially in the sciences, have adopted academicjobsonline.org, which is based on MathJobs, but there is variance, and some disciplines are still in the transition. Some disciplines, especially in the humanities, don't use any standardization like this, and cover letters are vital. So I disagree that academia.SE is an obviously-better fit. Mar31 revised On mentioning recommenders' names in cover letter for postdoctoral applicationsadded request for responder's background Mar31 answered On mentioning recommenders' names in cover letter for postdoctoral applications Mar23 comment If a graph embeds in the projective plane or the torus is there a bound on the number of edge crossings it has in the plane?Please read mathoverflow.net/howtoask and revise this question. Mar18 awarded ● Popular Question Mar17 awarded ● Nice Question Mar17 comment Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds?(I fixed a formatting conflict.) Mar17 revised Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds?formatting fix Mar17 asked Are there results from gauge theory known or conjectured to distinguish smooth from PL manifolds? Mar13 comment How to triangulate a math reference?I think this is a great triple of questions. You might think about breaking it into three questions, with links between them, as they are part of a coherent conversation. Mar11 awarded ● Popular Question Mar8 comment Contraction of graded vector fields on de Rham complexSorry, that was a typo. I meant that to speak about $d_{dR}$, you need to say how it's supposed to have bidegree $(0,1)$. What I should have written was $d_{dR} : L_{(0,1)} \otimes \Omega^\bullet \to \Omega^\bullet$. But if you and I disagree about which the generating line in degree $(0,1)$ is (and in particular don't choose an isomorphism between them) then we will each have a $d_{dR}$, and no way to compare them, except that in any fixed skeletalization they'll differ by some scalar. Mar8 answered Contraction of graded vector fields on de Rham complex Mar8 asked How many flat connections has a line bundle in algebraic geometry? Mar6 comment Topological characterization of the closed interval $[0,1]$.My answer mathoverflow.net/questions/76134/… also discusses $[0,1]$ from the point of view of its coalgebraic structure, as you mention in your last paragraph. This is similar also to Liviu Nicolaescu's answer below. But I disagree with your comment that "in some sense $[0,1]$ is the best topological space". Mar5 awarded ● Popular Question Feb27 comment Number of graphs with a cycleI second Boris Bukh's comment: please provide more details, motivation, and so on. Most mathematicians don't like thinking about unmotivated and uncontextualized questions. Note that because certain graphs have extra symmetries, often exact formulas are much simpler or much more complicated (to the point of being unavailable) depending on whether you work with labeled or unlabeled graphs. It can also happen that what you care about are certain asymptotics, rather than exact formulas. So all of these concerns might be included in your question, and will guide answers. Feb18 comment On Tamarkin’s proof of Etingof-Kazhdan quantization of Lie bialgebraGreat question. I have some thoughts and ideas on this (actually, I have begun a project joint with O. Gwilliam related to these questions), but nothing completed or answer-worthy. I'd be happy to discuss some of our ideas offline. Conversely, if you come across any discussion in the literature (or write anything yourself), I'd be very grateful if you sent me a pointers. Feb12 comment Exactness is often an open condition. How often?... For abelian groups, I think I can do the infinitesimal case from a spectral-sequence argument, using the fact that the if the associated-graded of a filter abelian group is zero, then the filtered abelian group was already zero; but there, I don't even know "infinitesimal upper semicontinuity" except near exact complexes, and I really don't know how to say non-infinitesimal statements. Feb12 comment Exactness is often an open condition. How often?@Mariano: Right, for finite-dimensional vector spaces I know how to do everything I've asked. Already for infinite-dimensional vector spaces, though, I know how to prove semicontinuity for "infinitesimal" open neighborhoods, but I don't know enough about infinite-dimensional vector spaces to say things finitely... Feb10 asked Exactness is often an open condition. How often? Feb9 comment subrings with identityPlease read mathoverflow.net/howtoask and then revise this question. At present, it looks like homework. If it is homework, please read mathoverflow.net/faq and look over other sites where your question is more appropriate.