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    • Page created, but author did not leave any comments.

      v1, current

    • A new stub recording few relevant references. Relevant for statistical mechanics, integrable models and so on. The conjecture has been proven in 2011 by Its and others in the Annals paper listed.

      v1, current

    • Late-night phone edit – more tomorrow

      v1, current

    • Deleted old queries about formatting that are no longer relevant.

      diff, v15, current

    • Deleted old queries about formatting that are no longer relevant.

      diff, v15, current

    • Add reference to “A category-theoretic approach to the semantics of programming languages”.

      diff, v2, current

    • Are there slides available of Stuart Presnell’s “12 Reasons to be Interested in Topos Theory”?

    • brief category:people-entry for hyperlinking references

      v1, current

    • I added a blurb from and link to Connes most recent lecture (Temps et aléa du quantique (english)) on the time page. (Added both for its innate interest and to help understand the transit between mathematics and philosophical intuition).

      When I was in preparatory school, my teacher asked me (…) “what is a variable?”. I reflected and reflected, and after a while, I said “time”. (…) The topic of my talk is that I believe we are all used, because of our constitution and so on, to attribute variability to the passing of time. The thesis which I will propose and try to back with mathematical results is the following: I believe that the true variability is quantum, and that the true variability is the fact that when you take a quantum observable it doesn’t have a single value, but it has many possible values which are given by the spectrum of the operator, plus the fact that discrete variables cannot coexist with continuous variables without the quantum formalism. I will explain how time emerges from these facts. I have never tried to explain this idea, I know it’s difficult, and its difficult because in my mind it is backed up by an intuition which comes from many years of work, and this is the most difficult thing to transmit. (…) How to explain this? (…) The answer I believe comes from Von Neumann (suitably implemented and very much ameliorated). (…) In the 40’s and 50’s Von Neumann was asking what does it mean to have a subsystem? What does it mean that somehow, the Hilbert space in which you work is a Hilbert space in which you have partial knowledge of things because the system is a composite system and there is a part of the system which you know and a part which you ignore? What Von Neumann was trying to understand was factorizations. (gives lecture on factorizations…) By the way, I should say that this is why I spent many years studying Noncommutative Geometry: the simplest geometric origin of Von Neumann factorization is foliations. If you take the simplest foliation (well, I don’t know if it’s the simplest), the [???] foliation of the sphere bundle of a Riemann surface, you get the most exotic factorization of Von Neumann? (type III1).

    • Some naive ramblings, just thinking out loud, in case anyone feels inspired to offer a comment:

      I am trying to see how close to an ordered configuration space of points one can get with mapping spaces, which a priori give un-ordered configuration spaces of points.

      The idea I have is – in words – the following:

      An ordered configuration of points in 3\mathbb{R}^3 (say) is, up to homotopy, the same as

      a) An un-ordered configuration of points in 3× 1\mathbb{R}^3 \times \mathbb{R}^1,

      such that this

      b) projects to an 1\mathbb{R}^1-labeled un-ordered configuration in 3\mathbb{R}^3;

      and

      c) projects to an 3\mathbb{R}^3-labeled un-ordered configuration in 1\mathbb{R}^1.

      Meaning that the points in the configuration are distinct not only as points in 3× 1\mathbb{R}^3 \times \mathbb{R}^1, but also after projection as points in 3\mathbb{R}^3 and as points in 1\mathbb{R}^1.

      Here condition c) is what imposes an ordering on the “labels” in 1\mathbb{R}^1, since an arrangement of distict points on the real line puts these points into linear order.

      The formal statement of this idea should be that

      • the ordered un-labeled configuration space nConf {1,,n}( 3)\underset{n \in \mathbb{N}}{\sqcup} \underset{ {}^{ \{1,\cdots, n \} } }{Conf}(\mathbb{R}^3)

      is a fiber product (in the 1-category of topological spaces) of

      • the unordered XX-labeled configuration spaces Conf( k,X)Conf( \mathbb{R}^k, X ) with points disappearing when labeled by the base-point of XX

      as follows:

      nConf {1,,n}( 3) homeoConf( 3,S 1)×Conf(𝔻 3+1) relConf( 1,S 3) \underset{n \in \mathbb{N}}{\sqcup} \underset{ {}^{ \{1,\cdots, n \} } }{Conf}(\mathbb{R}^3) \;\simeq_{homeo}\; Conf\big( \mathbb{R}^3, S^1 \big) \underset{ Conf\big( \mathbb{D}^{3+1} \big)_{rel} }{\times} Conf\big( \mathbb{R}^1, S^3 \big)

      That this is the case should essentially come down to observing that this fiber product encodes the above “in words” description.

      First I thought that this 1-categorical fiber product is a homotopy-retraction of the corresponding homotopy fiber product, but now I think this can’t be.

      This is because all items in the above are homotopy equivalent to based mapping spaces as

      Maps */(S 3,S 4)× hMaps */(S 0,S 4)Maps */(S 1,S 4) Maps^{\!\ast/\!}\big( S^3, S^4\big) \underset{ Maps^{\!\ast/\!}\big( S^0, S^4\big) }{\times^h} Maps^{\!\ast/\!}\big( S^1, S^4\big)

      and of these I know the rational models, and there is no way for any homotopy fiber product of these to be equivalent to the rational model for the ordered configuration space.

      So now I am thinking that maybe I should regard the configuration spaces above as smooth manifolds, and as such as smooth stacks, and then think of them as differential refinements of the homotopy types of these mapping spaces (which are Cohomotopy cocycle spaces), to find that the ordered configuration space, as a smooth manifold/stack, is a homotopy fiber product of differentially refined Cohomotopy cocycle spaces.

      But not sure if that’s a fruitful picture…

    • Stub for dressing action (also called dressing transformation) etc. used in the study of integrable systems, Poisson-Lie duality and so on. Starting with collecting references.

      v1, current

    • brief category:people-entry for hyperlinking references at M5-brane and elsewhere

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • I know that it’s pretty elementary, but sometimes teaching algebra makes you think of things, so I preserved some observations (mostly not my own) on the quadratic formula the other day.

    • just for ease of hyperlinking, I am giving this its own little entry

      v1, current

    • Some minimum (gravity simulation software)

      v1, current

    • σi=iσ\sigma i = i\sigma is a natural transformation from σσ\sigma\sigma to σ\sigma (if I am not mistaken).

      Anonymous

      diff, v9, current

    • At asymptotic series I have made explicit the proof that the Taylor series of a smooth function is always asymptotic (here)

    • Basic definition of Wasserstein spaces and references. More to come.

      v1, current

    • I am resuming my old unfinished (and unublished) work on universal noncommutative flag varieties and noncommutative Grassmannians. One of the motivations has some avatars in operator theoretic setting and in relation to integrable systems. Thus I started revising pages and (re)collecting references on infinite-dimensional Grassmann varieties and creating some new pages like this one for Sato Grassmannian.

      v1, current

    • Berry’s phase is the stub about one of the most common applications of parallel transport in quantum physics, with its own applications in molecular and atomic physics, quantum computing and so on.

    • I note that Zoran has started an entry pro-C-star-algebra. I was wondering if inverse limits in topological -algebras are exact. If not then taking the limit seems a strange thing to do. It would be better to handle the pro-object as such. I.e. within the pro-category. Zoran, can you enlighten me? :-) I suspect that if the C-algebras are finite dimensional as vecor space then there would be not much difference… any thoughts?

    • Hello. There is possible confusion between the notions of saturated set and saturated subset, which have different pages and mean different things. Any advice on how to handle this?

      Thanks!

    • Defined a complete flag so I can link here from another page in good conscience.

      Mike Pierce

      diff, v4, current

    • Page created, but author did not leave any comments.

      v1, current

    • For now created page, more to be added soon.

      v1, current

    • A student asked “What is a cobordism?” and I checked and realized that the nnLab entry cobordism was effectively empty.

      So I have now added some basic text in the Idea-section and added a bare minimum of references. Much more should be done of course, but at least now there are pointers.

    • a small contents entry, to be !include-ed into relevant entries, for ease of hyperlinking

      v1, current

    • Page created, but author did not leave any comments.

      v1, current

    • added pointer to the Encyclopedia.com entry which turns out to have more detailed scientific information than Wikipedia

      (happened to need to look up which work of Gauss came out of his occupation with geodesy; and Wikipedia is most brief about this, while Encyclopedia.com has thorough three paragraphs on it)

      will also cross-link our Gaussian entries with this one now…

      diff, v4, current

    • only now do I realize that we have parallel entries

      I suppose these refer to the same person and should be merged?

      Even after some googling I can only find this very likely.

      diff, v2, current