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    • This page has been ‘mucked up’. The table has been destroyed giving just a block of text. I could roll back but thought it better for people to see the mess!

    • starting something – this should be the last of the types if light mesons in the list

      v1, current

    • I made some changes to bivector. While the idea section is correct (and should be strictly adhered to!) but the previous definition is wrong in general! The previous definition is consistent and used in wikipedia but it misses both the direct relation of bivectors, trivectors and general polyvectors to determinants as well as the standard nontrivial usage of bivectors in analytic geometry wher bivectors define equivalence classes of parallelograms and in particular with a point in space given define an affine plane. If we adhere to wikipedia and not to standard treatments in geometry (e.g. M M Postnikov, Analytic geometry) then we miss the nontriviality of the notion of bivector and its meaning which is more precise than that of a general element in the second exterior power.

      Bivector in a vector space VV is not any element in the second exterior power, but a DECOMPOSABLE vector in the second tensor power – in general dimension just such elements in Λ 2V\Lambda^2 V have the intended geometric meaning and define vector 2-subspaces and of course affine 2-subspaces if a point in the 2-subspace is given. It is true that every bivector in 2-d or in 3-d space is decomposable, but in dimension 4 this is already not true. Thus the bivectors form a vector space just in the dimensions up to 33. Similarly, trivectors form a vector space just in the dimensions up to 44. In the context of differential graded algebras, polyvector fields are usually taken as arbitrary elements in the exterior powers of vector fields.

    • Created page on Paul Fendley for linking from Fusion Categories

      v1, current

    • Hello, I added some words about an ambiguity I found in the literature about the name “dilatino”. If I am wrong, I’d be thankful to the one who makes me notice the mistake!

      diff, v2, current

    • need the link, but nothing here yet

      v1, current

    • stated the definition Δf=⋆d⋆df\Delta f = \star d \star d f and spelled out how this gives the usual component formula:

      ⋆d⋆df =⋆d⋆(∂ jf)dx j =⋆d(1(D−1)!|det((g ij))|g ij(∂ jf)ε ik 2⋯k Ddx k 2∧⋯∧dx k D) =⋆∂ k 1(1(D−1)!|det((g ij))|g ij(∂ jf)ε ik 2⋯k Ddx k 1∧dx k 2∧⋯∧dx k D) =|det((g ij))|1D!(D−1)!ε l 1l 2⋯l Dg l 1k 1g l 2k 2⋯g l Dk Dε ik 2⋯k D⏟=det((g ij) −1)δ i k 1∂ k 1(|det((g ij))|g ij(∂ jf)) =1|det((g ij))|δ i k 1∂ k 1(|det((g ij))|g ij(∂ jf)) =1|det((g ij))|∂ i(|det((g ij))|g ij(∂ jf)) \begin{aligned} \star d \star d f & = \star d \star (\partial_j f) d x^j \\ & = \star d \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{ i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \star \partial_{ \color{magenta} k_1} \left( \tfrac{1}{ \color{green} (D-1)! } \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \, g^{i j} (\partial_j f) \, \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } d x^{ \color{magenta} k_1 } \wedge d x^{ \color{green} k_2 } \wedge \cdots \wedge d x^{ \color{green} k_{D} } \right) \\ & = \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } \underset{ = \det\big( (g_{i j})^{-1} \big) \delta^{ \color{magenta} k_1 }_i }{ \underbrace{ \tfrac{1}{ { \color{orange} D! } { \color{green} (D-1)! } } \epsilon_{ \color{orange} l_1 l_2 \cdots l_D } g^{ { \color{orange} l_1 } { \color{magenta} k_1 } } g^{ { \color{orange} l_2 } { \color{green} k_2 } } \cdots g^{ { \color{orange} l_D} { \color{green} k_D } } \epsilon_{ i {\color{green} k_2 \cdots k_{D} } } } } \, \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \delta^{ \color{magenta} k_1 }_i \partial_{ \color{magenta} k_1 } \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \\ & = \frac{1}{ \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } } \partial_{i} \left( \sqrt{ \left\vert det\big( (g_{i j}) \big) \right\vert } g^{i j} (\partial_j f) \right) \end{aligned}

      diff, v7, current

    • expanded brane

      first a little remark on what D-branes are abstractly, in reply to an MO-question, then something on fundamental branes, going along with the discussion on the Café

    • brief category:people entry in order to make hyperlinks work

      v1, current

    • a stub, for the moment just so as to complete a pattern of entries, but I added pointer to

      v1, current

    • Added to BF-theory the reference that right now I am believing is the earliest one:

      Gary Horowitz, Exactly soluable diffeomorphism invariant theories Commun. Math. Phys. 125, 417-437 (1989)

      But maybe I am wrong. Does anyone have an earlier one? I saw pointers to A. Schwarz articles from the late 70s, but I am not sure if he really considered BF as such.

    • starting something – not done yet but need to save

      v1, current

    • I have split off ordinal sum from the entry on joins as it is needed in other entries as well. I have not revised the entry just doing a cut and paste, so it needs more work!

    • for completeness, to satisfy links

      v1, current

    • am finally completing this table (previously at quark) to be !include-ed into relevant entries, for ease of cross-linking

      v1, current

    • for completeness and to satisfy links – nothing much here yet

      v1, current

    • I have reorganized somewhat entry Maxim Kontsevich (with some new links and few bits of additional info) and created a related stub Vassiliev invariant, just to record a link to an impressive online bibliography maintained by Dror Bar-Natan and Sergei Duzhin, hosted at Duzhin's webpage at Russian Academy of Sciences.

    • 1) Change the composition symbol from “:” functors and “*\ast” for natural transformations to “∘\circ” 2) Some aligning and spacing

      Luidnel Maignan

      diff, v4, current

    • Requested clarification in the currently false/confusing entry universal+differential+envelope (sorry I couldn't do more: this will need someone more qualified than me). -- Benoit Jubin

    • added mentioning of imaginary numbers more generally in star algebras.

      Also re-arranged the existing text slightly, for clarity.

      diff, v5, current

    • Following his passing yesterday from Covid-19, I have added a bit more on his contributions that are relevant to the nPOV.

      diff, v3, current

    • This page should be merged with “essentially algebraic theory”, as the two notions are equivalent. Since the “cartesian theory” has little in it, I’m attempting to redirect to “essentially algebraic theory”. There is also relevant stuff at “cartesian logic”, though I’m not sure how best to organize that and link to it.

      Steve Vickers

      diff, v5, current

    • I assume “in” is what was meant.

      Joshua Schwartz

      diff, v7, current

    • I’m supposed to do this to be listed somewhere?

      Joshua Schwartz

      v1, current

    • Made a start. I linked to lenses (in computer science) since according to the penultimate entry there, Gödel provided the first instance. I guess the connection goes via games, linear logic, etc.

      Anyway, to be expanded by someone who knows about these things.

      v1, current

    • I added a bunch of material to nilpotent group, including an inductive definition of nilpotency and of central series (which is how proofs about nilpotent groups often actually go) and a mention of the dual coinductive notions.

    • I don’t have any grand plans for this page, but thought I’d add it as food for thought. Also added a new section ’Philosophical musings’ to foundations of mathematics, and linked to this page there.

      v1, current

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