adding another reference

Anonymous

]]>there had been no references at *Hilbert space*, I have added the following, focusing on the origin and application in quantum mechanics:

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John von Neumann,

Mathematische Grundlagen der Quantenmechanik. (German) Mathematical Foundations of Quantum Mechanics. Berlin, Germany: Springer Verlag, 1932.George Mackey,

The Mathematical Foundations of Quamtum MechanicsA Lecture-note Volume, ser. The mathematical physics monograph series. Princeton university, 1963E. Prugovecki,

Quantum mechanics in Hilbert Space. Academic Press, 1971.

Page created, but author did not leave any comments.

Egbert Rijke

]]>an essentially empty stub, for the moment just to satisfy a link long requested at *harmonic analysis*

I have added at *HomePage* in the section *Discussion* a new sentence with a new link:

If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page *Welcome to the nForum (nlabmeta)*. I re-doscivered it only after my recent related comment here.

Wanting to know if any progress has been made in this area I searched and found this old forum post by Urs comparing the IKKT matrix model of the type IIb string theory to Loop Quantum Gravity. He says that both display 'radical background independence'

https://www.physicsforums.com/threads/lqg-strings-and-the-ikkt-matrix-model.8391/

Have the prospects for this type of research changed since 2003? What are the prospects for the IKKT model and, more ambitiously, the unification of LQG and String Theory? ]]>

added to tmf a section that gives an outline of the proof strategy for how to compute the homotopy groups of the $tmf$-spectrum from global sections of the $E_\infty$-structure sheaf on the moduli stack of elliptic curves.

A point which I wanted to emphasize is that

The problem of constructing $tmf$ as global sections of an $\infty$-structure sheaf has a tautological solution: take the underlying space to be $Spec tmf$.

From this tautological but useless solution one gets to the one that is used for actual computations by one single crucial fact:

In the $\infty$-topos over the $\infty$-site of formal duals of $E_\infty$-rings, the dual $Spec M U$ of the Thom spectrum, is a well-supported object. the terminal morphism

$Spec M U \to *$in the $\infty$-topos is an effective epimorphism, hence a covering of the point.

Using this we can pull back the tautological solution of the problem to the cover and then compute there. This is what actually happens in practice: the decategorification of the pullback of $Spec tmf$ to $Spec M U$ is the moduli stack of elliptic curves. And it is a happy coincidence that despite this drastic decategorification, there is still enough information left to compute $\mathcal{O} Spec tmf$ on that.

]]>I note that Goncalo Tabuada has moved so we should edit his nLab page when editing is available: webpage

]]>Prompted by signs that the current server is reaching its limits with the existing software, I am going to attempt to carry out the migration to the cloud (Amazon Web Services, aka AWS), that has been intended since we raised funds for this purpose early this year. For those who do not remember, we will be using an AWS account owned by (and billed to) the Topos Institute. Brendan Fong is our principal contact at the Topos Institute.

The full migration will take a long time. I do not really have time for it myself at the moment, but the time has come that it is necessary to do something. I am going to try to gradually build functionality up; the first goal is simply for the nLab to be viewable (i.e. it will exist in read-only mode to begin with).

**I am now going to disable the current nLab server**, to prevent writes to the database whilst I am migrating existing content. My apologies for the inconvenience, but with the tiny amount of time that I have, this is the only simple way to do things cleanly.

I hope to be able to get read-only mode up this evening European time; I will keep you updated. The nForum will remain up.

PS - This is not just a question of moving the software over. The old/original Instiki will be completely gone once the migration is complete. The nLab should look the same/similar on the surface, but the only traces of the original code will be the CSS.

]]>A stub with a few references.

]]>for when the editing functionality is back, this here is a good textbook to record at *quantization*:

- Nik Weaver,
*Mathematical Quantization*, Chapman and Hall/CRC 2001 (ISBN:9781584880011)

an initial stub with distracting noise. I submit for your derision, Model theory

]]>finally created a stub for super Yang-Mills theory

]]>am finally giving this its own entry. Nothing much here yet, though, still busy fixing some legacy cross-linking…

]]>created [[fivebrane 6-group]] with the idea

]]>I have added to *Teichmüller theory* a mini-paragraph Complex structure on Teichmüller space with a minimum of pointers to the issue of constructing a complex structure on Teichmüller space itself.

Maybe somebody has an idea on the following: The Teichmüller orbifold itself should have a neat general abstract construction as the full subobject on the étale maps of the mapping stack formed in smooth $\infty$-groupoids/smooth $\infty$-stacks into the Haefliger stack for complex manifolds : via Carchedi 12, pages 37-38.

Might we have a refinement of this kind of construction that would produce the Teichmüller orbifold directly as on objects in $\infty$-stacks over the complex-analytic site?

]]>I started an idea section at transgression, but it could probably use some going over by an expert. I hope I didn’t mess things up too badly. I was reading Urs’ note on “integration without integration” on the train ride home and fooled myself into thinking I understood something.

By the way, this reminded me of a discussion we had a while back

]]>have started *closed cover*, for the moment mainly in order to record references.

I fixed a link to a pdf file that was giving a general page, and not the file!

]]>I have added pointer to the arXiv copy to the item

- Daniel Berwick-Evans, Pedro Boavida de Brito, Dmitri Pavlov,
*Classifying spaces of infinity-sheaves*(arXiv:1912.10544)

I have edited a bit at *Fredholm operator*. Also started a stubby *Fredholm module* in the process. But it remains very much unfinished. Have to interrupt now for a bit.

There is an obvious similarity between the four adjoints describing cohesion for the Sierpinski $(\infty, 1)$-topos in Example 6.1.2 of dcct and four of the seven adjoints (third to sixth) for arrow categories of pointed categories having pullbacks and pushouts MO answer+comment. So

$\Pi([P \to X]) = X$ ; $Disc(X) = [X \to X]$; $\Gamma([P \to X]) = P$ ; $coDisc(Q) = [Q \to \ast]$.

$[f: A \to B] \mapsto coker(f)$; $A \mapsto [0 \to A]$; $[A \to B] \mapsto B$; $A \mapsto [A \to A]$; $[A \to B] \mapsto A$; $A \mapsto [A \to 0]$; $[f: A \to B] \mapsto ker(f)$.

It seems there’s a linearization happening, and we might want to consider, as we did a while ago, whether there is a Freyd-like result for AT-$(\infty, 1)$-categories between stable ones and toposes.

Forming the (co)monads from the adjunction strings, the Sierpinski case gives us the expected three for cohesion, ʃ $\dashv\; \flat \;\dashv\; \sharp$. The other case gives us six. Now, are there traces of these extra three in the nonlinear case, trying to show themselves?

Consulting the cohesion - table, ʃ does have a left adjoint, that is, if one lifts it up to *infinitesimal* shape $\Im$. It’s the reduction modality, $\Re$. And by the same move, its lift to $\rightsquigarrow$ has a further left adjoint $\rightrightarrows$.

So is there a connection between $\Re$ and the modality $[A \to B] \to [0 \to B]$? Maybe we’re not so far from the tangent (infinity,1)-category construction as a halfway house, with an element above a space $X$, a parameterized spectrum, as an infinitesimal there.

And what of the other two modalities, $[A \to B] \mapsto [0 \to coker f]$ and $[A \to B] \mapsto [ker f \to 0]$?

]]>finally splitting this off, for ease of organizing references. Not much here yet…

]]>added references (also to *Burnside ring*):

Erkki Laitinen,

*On the Burnside ring and stable cohomotopy of a finite group*, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, Laitinen79.pdf:file)Wolfgang Lück,

*The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups*(arXiv:math/0504051)

Mention “adjoint chain” terminology.

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