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    • Page created: bullet cluster by Urs Schreiber at 2018-03-31 00:53:25 UTC.

      Author comments:

    • Page created: proof of the prime number theorem by Todd Trimble at 2018-03-31 00:45:16 UTC.

      Author comments:

      New page which gives the details of the proof of the prime number theorem, as extracted from Zagier (after Newman), but less compressed.

    • On the orthogonal group page, there is a table of π i(SO(n))\pi_i(SO(n)) for 1≤i≤121 \leq i \leq 12 and 2≤n≤122 \leq n \leq 12. The reference given is the Encyclopedic Dictionary of Mathematics which actually has a table which goes up to i=15i = 15 and n=17n = 17. In a comment to my last MO question, David Roberts suggested that I add these additional groups to the table. I am happy to do this, but as the person who created the table used the same resource, they made a decision to stop at i=12i = 12 and n=12n = 12; maybe there was a good reason for this.

      Should I expand the table or is it fine as is?

    • Added to simple group an example I was given on MO: the simple group of cardinality κ\kappa, given by taking the smallest normal subgroup of Aut(κ)Aut(\kappa) containing the 3-cycles. This is essentially the ’even’ permutations for an infinite set.

      This is nice, because I was trying to think of a simple group (or one with only small normal subgroups) with inaccessible cardinality, and some obvious tricks weren’t working.

    • I decided to make my first contributions to the nLab, so am following the request, and saying what I did here.

      I fixed a typo in the definition of composition of correspondences. It is now a little unclear, so I also started a new page, internal tensor product (the link was already there, just no page). I’ll try to add some details tomorrow when I have time.

    • Has anyone developed models for the homotopy theory of HℚH \mathbb{Q}.module spectra over rational topological spaces a bit?

      I expect there should be a model on the opposite category of dg-modules over rational dg-algebras. Restricted to the trivial modules it should reduce to the standard Sullivan/Quillen model of rational homotopy theory. Restricted to the dg-modules over ℚ\mathbb{Q} it should reduce to the standard model for the homotopy theory of rational chain complexes, hence equivalently that of HℚH \mathbb{Q}-module spectra.

      Is there any work on this?

    • I added to endomorphism the observation that in a cartesian monoidal category, if an internal endomorphism monoid End(c)End(c) exists and is commutative, then cc is subterminal.

    • Added the following to the page on the Gray tensor product:

      The Gray tensor product as the left Kan extension of a tensor product on the full subcategory Cu of 2Cat is on page 16 of

      Since I’m really new at this, feel very free to give advice or corrections, thanks, Keith

    • I have corrected and expanded my note (at 4-sphere: here) of the result of Roig-Saralegi 00, p. 2 on minimal rational dg-models of the following maps over S 3S^3

      S 4 ↓ S 3,AAS 4//S 1 ↓ S 3,AAS 0 ↓ S 3 \array{ S^4 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^4//S^1 \\ \downarrow \\ S^3 } \,,\phantom{AA} \array{ S^0 \\ \downarrow \\ S^3 }

      induced from the “suspended Hopf action” of S 1S^1 on S 4S^4.

      My aim in extracting this is to rename the generators given in Roig-Saralegi 00, p. 2 such as to make their degrees and their pattern more manifest. I hope I got it right now:

      fibration vector space underlying minimal dg-model differential on minimal dg-model S 4 ↓ S 3 Sym •⟨h 3⟩⊗⟨ω 2p⏟deg=2p,f 2p+4⏟deg=2p+4|p∈ℕ⟩ d:{ω 0 ↦0 ω 2p+2 ↦h 3∧ω 2p f 4 ↦0 f 2p+6 ↦h 3∧f 2p+4 S 0 ↓ S 3 Sym •⟨h 3⟩⊗⟨ω 2p⏟deg=2p,f 2p⏟deg=2p|p∈ℕ⟩ d:{ω 0 ↦0 ω 2p+2 ↦h 3∧ω 2p f 0 ↦0 f 2p+2 ↦h 3∧f 2p S 4//S 1 ↓ S 3 Sym •⟨h 3,f 2⟩⊗⟨ω 2p⏟deg=2p,f 2p+4⏟deg=2p+4|p∈ℕ⟩ d:{ω 0 ↦0 ω 2p+2 ↦h 3∧ω 2p f 2 ↦0 f 2p+4 ↦h 3∧f 2p+2 \array{ \text{fibration} & \array{\text{vector space underlying} \\ \text{minimal dg-model}} & \array{ \text{differential on} \\ \text{minimal dg-model} } \\ \array{ S^4 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{deg = 2p + 4}{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_4 & \mapsto 0 \\ f_{2p+6} & \mapsto h_3 \wedge f_{2p + 4} \end{aligned} \right. \\ \array{ S^0 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3\rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg = 2p }{ \underbrace{ f_{2p} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_0 & \mapsto 0 \\ f_{2p+2} &\mapsto h_3 \wedge f_{2p} \end{aligned} \right. \\ \array{ S^4//S^1 \\ \downarrow \\ S^3 } & Sym^\bullet \langle h_3 , f_2 \rangle \otimes \left\langle \underset{deg = 2p}{ \underbrace{ \omega_{2p} }}, \underset{ deg =2p + 4 }{ \underbrace{ f_{2p + 4} }} \,\vert\, p \in \mathbb{N} \right\rangle & d \colon \left\{ \begin{aligned} \omega_0 & \mapsto 0 \\ \omega_{2p+2} &\mapsto h_3 \wedge \omega_{2p} \\ f_2 & \mapsto 0 \\ f_{2p+4} & \mapsto h_3 \wedge f_{2p + 2} \end{aligned} \right. }

    • added to Massey product a paragraph on their relation to A ∞A_\infty-algebras and a bunch of references on that relation

    • created Cahiers topos.

      Do I understand correctly that this gadget is named after the journal that Dubuc’s original article appeared in? What a strange idea.

    • Created the bare minimum at limit spaces. I was surprised it was missing.

    • Currently, the page Tools for the advancement of objective logic says that the described paper discusses the “concrete particular”, but this paper does not discuss “concrete particular”.

      [For context, Lawvere regards “concrete particular” as a category error, as he has stated elsewhere—basically to Lawvere, the abstract and the concrete are two aspects of a universal/general concept, and not to be discussed at the level of particulars. The passage from the particular to the universal/general is marked by choosing a subclass of observables about the particular as definitive—this generates the abstract general, and thence the concrete generals. The connection between the particulars and the concrete general comes from the fact that each particular can be observed in the ways specified by the abstract general, and that the abstract generals embed as representables into the concrete generals.]

      To be precise, the 1994 paper talks about the particular, and then the abstract and concrete general.

      However, I haven’t edited the page yet because I believe that in order to keep the nlab internally consistent, one may need to also update the page abstract general, concrete general and concrete particular; however, I am aware of a significant discussion elsewhere in the nforum, and I see that maybe some consensus was already reached there in favor of the notion of the “concrete particular”.

      The latter page contains the nlab’s synthesis of these ideas which does not quite match Lawvere’s—which is generally fine, but I think we should be careful about attributing this interpretation to Lawvere himself (which we are in danger of doing in the page on the 1994 paper).

      Do you all have any thoughts on how to proceed?

    • at sober space the only class of examples mentioned are Hausdorff spaces. What’s a good class of non-Hausdorff sober spaces to add to the list?

    • Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).

      In further reaction I have started a minimum of explanation for one more item in the list: bottom-up and top-down model building.

    • I have tweaked the Idea-section at naturalness, and I added a pointer to the first decent discussion that I have seen: Clarke 17

    • I am running into the following simple question and am wondering if there is anything useful to be said.

      Let

      𝒜∈dgcAlg ℚ \mathcal{A} \in dgcAlg_\mathbb{Q}

      be a differential graded-commutative algebra in characteristic zero, whose underlying graded algebra is free graded-commutative on some graded vector space VV:

      𝒜=(Sym(V),d). \mathcal{A} = (Sym(V), d) \,.

      Consider an odd-graded element

      c∈𝒜 odd, c \in \mathcal{A}_{odd} \,,

      and write (c)(c) for the ideal it generates.

      In this situation I’d like to determine whether it is true that

      1. there is an inclusion 𝒜/(c)↪𝒜\mathcal{A}/(c) \hookrightarrow \mathcal{A};

      2. for every element ω∈𝒜\omega \in \mathcal{A} there is a decomposition

        ω=ω 0+cω 1 \omega = \omega_0 + c \omega_1

        for unique ω 0,ω 1∈𝒜/(c)↪𝒜\omega_0, \omega_1 \in \mathcal{A}/(c) \hookrightarrow \mathcal{A}.

      For example if c≠0∈V odd↪𝒜 odd↪𝒜c \neq 0 \in V_{odd} \hookrightarrow \mathcal{A}_{odd} \hookrightarrow \mathcal{A} is a generator, then these conditions are trivially true.

      On the other extreme, if cc is the product of an odd number >1\gt 1 of odd generators, then it is not true. For example if c=c 1c 2c 3c = c_1 c_2 c_3, with c 1,c 2,c 3∈V odd↪𝒜 oddc_1, c_2, c_3 \in V_{odd} \hookrightarrow \mathcal{A}_{odd}, then for instance c(1+c 1)=c(1+c 2)=cc (1 + c_1) = c (1 + c_2) = c and so the coefficient ω 1\omega_1 is not unique.

      Is there anything useful that one can say in general?

    • For purposes of linking, I had given an entry to decomposable differential form.

      In more general ℕ\mathbb{N}-graded-commutative algebras than just that of differential forms, is there any established terminology for

      0.\;\;\;\;\;0. elements that are sums of decomposables, i.e. sums of monomials in elements of degree 1?

      What I’d really need is terminology for:

      1. elements HH of degree n+1n+1 which split off at least one factor of degree 11, hence H=α⏟deg=1⋅βH = \underset{deg = 1}{\underbrace{ \alpha}} \cdot \beta;

      2. elements which are finite sums of these.

      Is there anything?

    • Here is another stub: Albert algebra.

      It would be nice to get a reference to clear up the number of (real) Albert algebras. John Baez's octonion paper, among other literature (including our Jordan algebra), takes it for granted that there is only one (which is true, over the complex numbers, but people are usually working over the real numbers). But John himself points out on a Wikipedia talk page that there are two (and that's what I followed).

    • We had a paragraph on split ocotnions buried in the entry composition algebra.

      In order to be able to link to it, I have given that paragraph its own entry, now split octonions. But this deserves to be expanded of course.

    • Over in another thread, David Roberts asks for explanation of a bunch of terms in QFT (here).

      Here I started a minimum of explanation for one item in the list: protection from quantum corrections.

    • at Bockstein homomorphism in the examples-section where it says

      B nU(1)≃B n+1ℤ \mathbf{B}^n U(1) \simeq \mathbf{B}^{n+1}\mathbb{Z}

      I have added the parenthetical remark

      (which is true in ambient contexts such as ETop∞GrpdETop\infty Grpd or Smooth∞GrpdSmooth \infty Grpd)

      Just to safe the reader from a common trap. Because it is not true in Top≃∞GrpdTop \simeq \infty Grpd. The problem is that in all traditional literature the crucial distinction between TopTop and ETop∞GrpdETop \infty Grpd (or similar) is often appealed to implicitly, but rarely explicitly. In Top≃∞GrpdTop \simeq \infty Grpd we have instead B nU(1)≃K(U(1),n)\mathbf{B}^n U(1) \simeq K(U(1), n).

    • if you have been looking at the logs you will have seen me work on this for a few days already, so I should say what I am doing:

      I am working on creating an entry twisted smooth cohomology in string theory . This is supposed to eventually serve as the set of notes for my lectures at the ESI Program K-Theory and Quantum Fields in the next weeks.

      This should probably sit on my personal web, and I can move it there eventually. But for the moment I am developing it as an nnLab entry because that saves me from prefixing every single wiki-link with

       nLab:

    • Took a stab at a general formulation of Poisson summation formula, although the class of functions to which it is supposed to apply wasn’t nailed down (yet).

      (Some of the ingredients of Tate’s thesis are currently on my mind.)

    • I have added the Fierz identities that give the S 2S^2-valued supercocycle in 5d here.

      Added this briefly also at Fierz identity: here

    • The observation that the conditional expectation enjoys a universal property inspired me to write some ''random text''.

    • At field (physics) I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.

      This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.

      So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from scalar field in parallel with the abstract discussion.

    • At dichotomy between nice objects and nice categories I added a quote from Deligne about allowing awful schemes gave a nice category of schemes. I can’t find the page I was thinking of where this dichotomy is also mentioned along with attribution of the general idea to Grothendieck. I wanted to add it there as well.

    • I created a stub entry for Hörmander topology, just to record some references.

      The following seems to be waiting for somebody to answer it:

      Consider the deformed Minkowski metric

      η ε≔diag(−1+iε,1+iε,…,1+iε) \eta_\epsilon \coloneqq diag( -1 + i \epsilon, 1+ i \epsilon , \dots , 1 + i \epsilon )

      for ε>0∈ℝ\epsilon \gt 0 \in \mathbb{R}.Then consider the ε\epsilon-deformed Feynman propagator Δ F,ε,Λ\Delta_{F,\epsilon,\Lambda} with momentum cut off with scale Λ\Lambda.

      The question: does the limit satisfy

      Δ F=limε→0Λ→∞Δ F,ε,Λ \Delta_{F} \;=\; \underset{ {\epsilon \to 0} \atop {\Lambda \to \infty} }{\lim} \Delta_{F,\epsilon,\Lambda}

      in the Hörmander topology for tempered distributions with wave front set contained in that of the genuine Feynman propagator Δ F\Delta_F?

    • for ease of linking I have given antibracket its own little entry (it used to just redirect to BV-BRST complex).

      I had also given local antibracket an little entry of its own. Possibly these two should be merged…

    • have created extension of distributions with the statement of the characterization of the space of point-extensions of distributions of finite degree of divergence: here

      This space is what gets identified as the space of renormalization freedom (counter-terms) in the formalization of perturbative renormalization of QFT in the approach of “causal perturbation theory”. Accordingly, the references for the theorem, as far as I am aware, are from the mathematical physics literature, going back to Epstein-Glaser 73. But the statement as such stands independently of its application to QFT, is fairly elementary and clearly of interest in itself. If anyone knows reference in the pure mathematics literature (earlier or independent or with more general statements that easily reduce to this one), please let me know.

    • I created Bishop’s constructive mathematics by moving some material from Errett Bishop and adding some more discussion of what it is and isn’t. Comments and suggestions are very welcome; I’m still trying to figure out the best way to describe the relationship of this theory to other things like topos logic.

    • I added some references to convex space and began a new entry on homomorphism.

      It would be great to see the article on convex spaces continue... it sort of trails off now. I've tried to enlist Tobias Fritz.

    • This is probably a request for Todd!

      Over on colimits for categories of algebras there’s a corollary I really need right now, about Eilerberg-Moore categories being cocomplete, and the remark:

      The hypotheses of the preceding corollary hold when CC is a complete, cocomplete, cartesian closed category and CC is the monad corresponding to a finitary algebraic theory.

      That sounds like exactly what I want, but when I click on finitary algebraic theory I get taken to a page that doesn’t have the definition of “finitary algebraic theory”. I think I know what this means, so I could guess and stick it in, but I think I should let the expert do it.

      Oh, whoops! - as usual, I actually need a multi-sorted generalization. But still it would be nice to have this clarified.

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