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Since there wasn’t a page octonionic Hopf fibration, I started one, copying over the start of quaternionic Hopf fibration. The weirdness of the octonions doesn’t prevent anything there, does it?
Corrected discussion of MLL and MILL, and removed redirects for MILL (those should go to linear logic.
brief category:people-entry for hyperlinking references at Dp-D(p+2)-brane bound states, M2-M5-brane bound state and at E-string
another item for the list KK-compactifications of M-theory – table
just a stubby minimum so far, for the moment just so as to record references.
starting something here. For the moment this is just the list of references which I had previously recorded at RR-field tadpole cancellation, now joined by a quick Idea-sentence and a minimal statement of the actual cancellation condition
Added a short entry on Jean-Pierre Marquis. He gave a lovely talk at the meeting in Toulouse on History, Philosophy and Homotopy The nLab was mentioned and several of the well known nLabbers and nCaféists as well. Ralf Kromer’s talk was also excellent.
brief category:people-entry for hyperlinking references at KU-local stable homotopy theory
brief category:people-entry for hyperlinking references at fuzzy sphere
brief category:people-entry for hyperlinking references at Dp-D(p+2)-brane bound states, Yang-Mills monopoles, knot invariants, chord diagrams and fuzzy spheres.
brief category:people-entry for hyperlinking references at Dp-D(p+2)-brane bound states, Yang-Mills monopoles, knot invariants, chord diagrams and fuzzy spheres.
I just noticed and noted that Gabriella Böhm wrote a book, on generalizations in Hopf world,
There is now an extremely stubby stub at stack semantics.
Created a page on Sanjeevi Krishnan, who works in directed homotopy and its applications.
Are there slides available of Stuart Presnell’s “12 Reasons to be Interested in Topos Theory”?
Created person page: is an author of the Handbook of Homotopy Theory
Created person page, is an author of the Handbook of Homotopy Theory
Is an author for the Handbook of Homotopy Theory
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 (say) is, up to homotopy, the same as
a) An un-ordered configuration of points in ,
such that this
b) projects to an -labeled un-ordered configuration in ;
and
c) projects to an -labeled un-ordered configuration in .
Meaning that the points in the configuration are distinct not only as points in , but also after projection as points in and as points in .
Here condition c) is what imposes an ordering on the “labels” in , 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
is a fiber product (in the 1-category of topological spaces) of
as follows:
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
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…
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.
brief category:people-entry for hyperlinking references at configuration space of points and Euclidean G-space
brief category:people-entry for hyperlinking references at configuration space of points and at Euclidean G-space
I am starting landscape of string theory vacua -- hah! :-)
Created hypervirtual double category.
At asymptotic series I have made explicit the proof that the Taylor series of a smooth function is always asymptotic (here)
Stub on a subject between integrable systems theory and (pseudo)differentialoperator theory.
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.
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.