Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limit limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes

Show all

12421 to 12480 of 14796

  1. <
  2. 1
  3. ...
  4. 204
  5. 205
  6. 206
  7. 207
  8. 208
  9. 209
  10. 210
  11. 211
  12. ...
  13. 247
  14. >

    • I was unsatisfied with the entry Eilenberg-MacLane object. So I changed the wording at the beginning. Maybe it's an improvement, maybe something better needs to be done.

    • created equivariant homotopy theory – table displaying the various cohesive ∞\infty-toposes and their bases ∞\infty-toposes (for inclusion in “Related entries” at the relevant entries)

    • created G-space, a glorified disambiguation page.

    • I have been adding some stuff to j-invariant, but it’s not really good yet (this here just in case you are watching the logs and are wondering what’s happening)

    • Here is a note to myself or anyone else to add the following new preprint to Galois group when the nLab is back online.

      • Akhil Mathew, The Galois group of a stable homotopy theory, http://arxiv.org/abs/1404.2156.

      I’ve only just read the introduction, but it looks pretty great…

    • started model structure on operads

      by the way: I noticed that the page operad has not a single reference. Maybe somebody feels like filling in his favorite ones...

    • I've created a page for Fell Bundles. It's only really a stub at the moment but I'll get around to expanding it eventually. The nLab POV of Fell bundles looks very different from the classical view but the two views can easily be reconciled (which I guess should form part of the expansion).

    • for some reason I created brief entry inversion involution, but there is not really much of a point, I have to admit. But now it exists.

    • I have expanded a bit the previous stub entry Goerss-Hopkins-Miller theorem. It’s still stubby, but less so.

      I have added

      • more of the pertinent references;

      • an actual Idea-section

      • the statement of the Hopkins-Miller theorem in the version as it appears in Charles Rezk’s notes.

      Maybe this feeble step forward inspires Aaron to add more… :-)

    • started a note at local-global principle.

      Need to interrupt now. This clearly can be extended indefinitely…

    • I added the word ‘helps’ to the entry at genus of a surface since its genus does not fully classify the surface, you need orientability (and then is it a surface with boundary or not).

    • started an entry cubic curve,

      For the moment I wanted to record (see the entry) a pointer to Akhil Mathew’s identification of that eight-fold cover of ℳ cub\mathcal{M}_{cub} (hence of ℳ ell\mathcal{M}_{ell}) which is analogous to the 2-fold cover of the “moduli stack of formal tori” Bℤ 2B \mathbb{Z}_2 that ends up being the reason for the ℤ 2\mathbb{Z}_2-action on KUKU.

      So here is the question that I am after: that cover is classified by a map ℳ ell→Bℤ/8ℤ\mathcal{M}_{ell} \to B \mathbb{Z}/8\mathbb{Z}, hence we get a double cover of the moduli space of elliptic curves d:ℳ ell→Bℤ/2ℤd \colon \mathcal{M}_{ell} \to B\mathbb{Z}/2\mathbb{Z}.

      Accordingly there is a spectrum Q≔d *(𝒪 top)Q \coloneqq d_\ast(\mathcal{O}^{top}) equipped with a ℤ 2\mathbb{Z}_2-action whose homotopy fixed points is tmftmf, I suppose: tmf≃Q ℤ 2tmf \simeq Q^{\mathbb{Z}_2}. (Hm, maybe I need to worry about the compactification…).

      I’d like to say that QQ is to tmftmf as KUKU is to KOKO. This is either subject to some confusion (wich one?) or else is an old hat. In the second case: what would be a reference?

    • have created geometric infinity-stack

      gave Toën’s definition in detail (quotient of a groupoid object in an (infinity,1)-category in TAlg ∞ op↪SpecSh ∞(C)T Alg_\infty^{op} \stackrel{Spec}{\hookrightarrow}Sh_\infty(C) ) and indicated the possibility of another definition, along the lines that we are discussing on the nnCafé

    • I made a new page called twisted form. Unfortunately, this stole the redirect from a sub-heading on differential form. The page is still pretty much a stub. I hope to enlarge it soon.

    • New references at symmetric function and new stub noncommutative symmetric function. An (unfinished?) discussion query from symmetric function moved here:

      David Corfield: Why does Hazewinkel in his description of the construction of Λ\Lambda on p. 129 of this use a graded projective limit construction in terms of projections of polynomial rings?

      John Baez: Hmm, it sounds like you’re telling me that there are ’projections’

      Λ n+1→Λ n \Lambda_{n+1} \to \Lambda_n

      given by setting the (n+1)(n+1)st variable to zero, and that Hazewinkel defines Λ\Lambda to be the limit (= projective limit)

      ⋯→Λ 2→Λ 1→Λ 0 \cdots \to \Lambda_2 \to \Lambda_1 \to \Lambda_0

      rather than the colimit

      Λ 0→Λ 1→Λ 2→⋯ \Lambda_0 \to \Lambda_1 \to \Lambda_2 \to \cdots

      Right now I don’t understand the difference between these two constructions well enough to tell which one is ’right’. Can someone explain the difference? Presumably there’s more stuff in the limit than the colimit.

      Mike Shulman: I think the difference is that the limit contains “polynomials” with infinitely many terms, and the colimit doesn’t. That’s often the way of these things.

      Actually, on second glance, I don’t understand the description of the maps in the colimit system; are you sure they actually exist? What exactly does it mean to “add in new terms with the new variable to make the result symmetric”?

      David Corfield: The two constructions are explained very well in section 2.1 of the Wikipedia article.

      Mike Shulman: Thanks! Here’s what I get from the Wikipedia article: the projections are easy to define. They are surjective and turn out to have sections (as ring homomorphisms). The ring of symmetric functions can be defined either as the colimit of the sections, or as the the limit of the projections in the category of graded rings. The limit in the category of all rings would contain too much stuff.

    • I have added to K-orientation pointers to the articles by Atiyah-Bott-Shapiro and to Joachim (2004), together with a brief paragraph.

    • Given the series of entries lately, I naturally came to the point that I started to want a “floating context” table of contents. So I started one and included it into relevant entries:

      But this needs more work still, clearly.

    • Missing from braid group was the precise geometric definition, so I put that in.

    • Created BICEP2, currently with the following text:

      BICEP2 is the name of an astrophysical experiment which released its data in March 2014. The experiment claims to have detected a pattern called the “B-mode” in the polarization of the cosmic microwave background (CMB).

      This data, if confirmed, is widely thought to be due to a gravitational wave mode created during the period of cosmic inflation by a quantum fluctuation in the field of gravity which then at the era of decoupling left the characteristic B-mode imprint on the CMB. This fact alone is regarded as further strong evidence for the already excellent experimental evidence for cosmic inflation as such (competing models did not predict such gravitational waves to be strong enough to be detectable in this way).

      What singles out the BICEP2 result over previous confirmations of cosmic inflation is that the data also gives a quantitative value for the energy scale at which cosmic inflation happened (the mass of the hypothetical inflaton), namely at around 10 1610^{16}GeV. This is ntoeworthy as being only two order of magnituded below the Planck scale, and hence 12 or so orders of magnitude above energies available in current accelerator experiments (the LHC). Also, it is at least a curious coincidence that this is precisely the hypothetical GUT scale.

      It is thought that this value rules out a large number of variant models of cosmic inflation and favors the model known as chaotic inflation.

    • just for completeness so that I don’t have gray links elsewhere, I have created some minimum (nothing exciting) at quantum fluctuation.

    • I did some editing at exponential function, to restore what I had believed to be a clear argument, which had been edited out by Colin Tan in favor of his own argument. His argument has been moved to a footnote.

    • Added a statement of the Schmidt decomposition to the page "tensor product of Banach spaces". It seemed the right place for it, since this is the page where the tensor product of Hilbert spaces is discussed.

    • added to chiral differential operator a paragraph briefly summarizing how the Witten genus of a complex manifolds is constructed by Gorbounov, Malikov and Schechtman. Copied the same paragraph also into the Properties-section at Witten genus

    • Wrote up more stuff at pi.

      Incidentally, there are some statements at irrational number that look a little peculiar to me. For example:

      In the early modern era, Latin mathematicians began work with imaginary numbers, which are necessarily irrational. They subsequently proved the irrationality of pi, (…)

      I suppose Legendre could qualify as a “Latin mathematician of the early modern era” if we take a sufficiently broad view (e.g., he spoke a language in the Latin “clade”), but somehow I feel this is not what the author really had in mind; there were those Renaissance-era Italians who began work with imaginaries IIRC. :-) Probably it would be good to rephrase slightly.

      Also this:

      There is an easy nonconstructive proof that there exist irrational numbers aa and bb such that a ba^b is rational; let bb be 2\sqrt{2} and let aa be either 2\sqrt{2} or 2 2\sqrt{2}^{\sqrt{2}}, depending on whether the latter is rational or irrational. A constructive proof is much harder

      Not that hard actually: take a=2a = \sqrt{2} and b=2log3log2b = 2\frac{\log 3}{\log 2}, where a b=3a^b = 3 if I did my arithmetic correctly. Pretty sure that can be made constructive. (Again, I think it’s probably just a case of several thoughts being smooshed together.)

    • Added the proof that every positive operator is self-adjoint.
    • For my first contribution to nLab, I've typed up my notes on effect algebras, with the definitions of a generalized effect algebra and morphism of effect algebras. The proofs here are more basic than most that I've seen on the wiki, but I've decided to include them in the spirit of this being a public lab book.

12421 to 12480 of 14796

  1. <
  2. 1
  3. ...
  4. 204
  5. 205
  6. 206
  7. 207
  8. 208
  9. 209
  10. 210
  11. 211
  12. ...
  13. 247
  14. >
Top of Page