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I fixed a link that was not working. (The brackets were interfering with the link address.) see here
I created Hoàng Xuân Sính as a result of recent G+ discussion, and David Eppstein creating an English Wikipedia page for her. There is now a link to that page at 2-group and a(n updated) link to her thesis.
for those who check the logs and are wondering: I went through a fairly long list of category:people-entries on people based in and around London, updating affiliation links, references and related nLab entries.
discovered that we already had a stub on Weyl quantization. Cross-linked a bit and added the following reference on Weyl quantization of Chern-Simons theory (also to quantization of 3d Chern-Simons theory):
Jørgen Andersen, Deformation quantization and geometric quantization of abelian moduli spaces, Commun. Math. Phys., 255 (2005), 727–745
Razvan Gelca, Alejandro Uribe, The Weyl quantization and the quantum group quantization of the moduli space of flat SU(2)-connections on the torus are the same, Commun.Math.Phys. 233 (2003) 493-512 (arXiv:math-ph/0201059)
Razvan Gelca, Alejandro Uribe, From classical theta functions to topological quantum field theory (arXiv:1006.3252, slides pdf)
Razvan Gelca, Alejandro Uribe, Quantum mechanics and non-abelian theta functions for the gauge group SU(2) (arXiv:1007.2010)
I have added a new link to the page on Grothendieck. There is a good new article on a CNRS site.
given the concept of Heisenberg Lie n-algebra, there is an evident definition of Weyl n-algebra: its universal enveloping E-n algebra.
I noted that down for reference at Weyl n-algebra. In the process I noticed that Markarian proposed a different definition just a few months back
Just some obvious stuff at maximal partial function to satisfy some links.
started a stub for moduli space of Calabi-Yau spaces. Nothing really there yet, except some references and some cross-links.
I tried to start an entry theta function, but it’s hard to tell for me if anything of it has been saved. The nLab is too busy doing something else than serving pages.
I have added the statement of lemmas 4.1, 4.2 of Menni-Lawvere to cohesive topos here and to points-to-pieces transform here.
I am starting a table of contents theta functions - contents and am including it as a “floating table of contents” into relevant entries
The entry entitled (x, y) ⊙ (u, v) = (xu + 2yv, xv + yu) has been started. Is it Spam? or does that definition mean something useful to the nPOV.
made explicit in the Idea section of functional equation the statement that the functional equation of a zeta function is the incarnation under Mellin transform of the automorphy of the automorphic form that it comes from
gave automorphic L-function a minimum of an Idea-section, presently it reads as follows:
An automorphic L-function Lπ is an L-function built from an automorphic representation π, in nonabelian generalization of how a Dirichlet L-function Lχ is associated to a Dirichlet character χ (which is an automorphic form on the (abelian) idele group).
In analogy to how Artin reciprocity implies that to every 1-dimensional Galois representation σ there is a Dirichlet character χ such that the Artin L-function Lσ equals the Dirichlet L-function Lχ, so the conjectured Langlands correspondence says that to every n-dimensional Galois representation σ there is an automorphic representation π such that the automorphic L-function Lπ equals the Artin L-function Lσ.
stub for holomorphic block, for the moment just to record two references
gave prequantum line bundle a little entry of its own
Wrote Engeler’s lemma, something I hadn’t heard of until recently.
I added a few comments to Hilbert basis theorem about related work by Gordon and Noether (chronologically, on either side of Hilbert’s work).
I created the article left-determined model category.
I had begun adding to prime ideal theorem (en route adding to compactness theorem), but have decided to stop midstream because it looks as though much more general results are known, which I’d need to read up on it before writing further.
One thing I’ll mention now is that the surmise (due to Toby?) that UF is equivalent to the prime ideal theorem for rigs seems to be known and subsumed under these general results. Banaschewski’s name comes up as one having a key lattice-theoretic insight into this topic: “Every nontrivial distributive complete lattice with a compact top element contains a prime element.”
The entry finite field was looking a little sad, so I added to it.
Created splitting field.
Do we have a page about natural transformations between (∞,1)-categories? I wanted to add a link to this paper (working today on catching up with the arXiv…) but I couldn’t find where to put it.
New entry binormal topological space.
We have a bit of a mess of closely related entries related to étale homotopy groups which existed more or less in parallel without seeming to know much of each other. I have tried to do some minimum of cross-linking and cleaning up, but this needs more attention.
There is more even, there is Grothendieck’s Galois theory and what not. (Maybe we need to wait until somebody gives a course on this and uses the occasion to clean it all up and harmonize it.)
Created exponent of a group.
I added more material to unique factorization domain.
I’ve slowly been trying to improve the article topological map since this thread. I just added a small note on embedded graphs versus abstract graphs, motivated by Bruce Bartlett’s interesting recent post at the n-café.
I created a stub for falling factorial, mainly to record the simple fact I learned yesterday that it counts the number of injections between two finite sets.
There doesn’t seem to be a discussion for this page differential cohesion and idelic structure. Is this to be the general page for ’inter-geometry’?
If so, it might be worth recording An Huang, On S-duality and Gauss reciprocity law, (Arxiv).
added to eta invariant a lightning section On manifolds with boundaries: as sections of the determinant line, essentially just a glorified pointer to Freed 95 for the time being.
Created intermediate model structure. Needs more cross-linking, but I have to go.
In two recent threads [1, 2] I had started to look into elementary formalization of the following obstruction problem in higher geometry:
given
a Klein geometry H→G,
a WZW term LWZW:G/H⟶Bp+1𝔾conn;
a Cartan geometry X modeled on G/H
then:
Here are first concrete observations, holding in any elementary ∞-topos (meaning: this may be proven using HoTT, not needing simplicial or other infinite diagrams):
First, a lemma that turns the datum of a global WZW term Fr(X)⟶Bp+1𝔾conn on the frame bundle of X (each of whose fibers looks like the formal disk 𝔻 around the base point, or any other point, in G/H) into something closer to cohomological data on X. In the following Fr(X) may be any fiber bundle E and Bp+1𝔾conn may be any coefficient object A.
Lemma. Let E→X be an F-fiber bundle associated to an Aut(F)-principal bundle P→X. Then A-valued functions on E are equivalent to sections of the [F,A]-fiber bundle canonically associated to P.
Proof. By the discussion at infinity-action, the universal [F,A]-fiber bundle [F,A]/Aut(F)→BAut(F) is simply the function space [F,A]BAut(F) formed in the slice over BAut(F), with F regarded with its canonical Aut(F)-action and A regarded with the trivial Aut(F)-action.
Now, by universality, sections of P×Aut(F)[F,A]→X are equivalently diagonal maps in
[F,A]/Aut(F)↗↓X⟶BAut(F)But by Cartesian closure in the slice and using the above, these are equivalent to horizontal maps in
E=P×Aut(F)F⟶A×BAut(F)↘↙BAut(F)Finally by (∑BAut(F)⊣BAut(F)*) this is equivalent to maps E→A. □
[ continued in next comment ]
am splitting off complex volume from hyperbolic manifold
stub for Bloch group
added details to Borel regulator, with discussion of Becker-Gottlieb transfer and the refinement to differential algebraic K-theory by the transfer index conjecture.
started some minimum at odd Chern character and cross-linked a little
I created Michal-Bastiani smooth map, and linked to it from diffeological space, Andree Ehresmann and locally convex topological vector space.
Very stubby beginning of BHK interpretation.
Started Witt-Burnside functor.
I split off coprojection from projection and added a remark about their monicity (or lack thereof).
I added a new section to Bayesian reasoning, Exchangeability, which outlines the de Finetti Representation theorem. As indicated, there’s a multivariate version. This was used to talk about Bose-Einstein statistics.
I wonder if anything interesting would happen with a HoTT rendition of statistical meachanics.
Someone has got the Euler-Lagrange equation page to redirect to hollymolly, and added that word at the bottom. How do we undo such vandalism?
during a talk on homotheties of Cartan geometries that I heard yersterday, it occurred to me that this concept has an immediate simple general abstract formulation in differential cohesive homotopy theory. Made a note on this now at homothety.
Added to F1 a section on Borger’s absolute geometry and then split it off as a stand-alone entry (minimal as it is) Borger’s absolute geometry.
I have been working on filling genuine content into
The first part Infinitesimal symmetries should be about readable, it starts out plenty expositionary, I hope, but towards the end it is still very terse. I wanted to get much further today, but it didn’t work out that way.
There was an ancient query box discussion sitting in the entry dg-Lie algebra which hereby I am moving from there to here.
begin of ancient discussion
+–{: .query}
Tim: I have changed the wording that Zoran suggested slightly. Of course, a dgla is an internal Lie algebra, a term that needs making precise in an entry, but then we must make precise the tensor product, and the symmetry. All that abstract baggage is, of course, in other entries, but I think it best to avoid the term ’simply’. I have heard it expressed that category theorists tend to use the term ’simply’ aand other similar terms too much from the point of view others working in neighbouring disciplines.
For instance, if someone knows de Rham theory from a geometric viewpoint, we know that in the long run it will be useful for them to understand the differential graded algebra from a categorical viewpoint as that is one of the most fruitful approaches for geometrically significant generalisations and applications BUT the debutant can get very put off by thinking that they have to understand lots of category theory before they can start understanding the de Rham complex. In fact coming from that direction they can understand the category theory via the de Rham theory. So I suggest that we simply avoid ’simply’!!
I know some researchers in other subject areas are looking with interest to the nLab as a quick means of entry into some interesting mathematics and a handy reference for definitions and background. That is great but it perhaps means that we have to be a bit careful about our natural feeling that the categorical approach is nearly always the ’best’. ’Simply’ is one problem, another is, I think, use of diagrams rather than formulae. My feeling is that both should be given (though the diagrams are more difficult to get looking nice).
Urs: these are all good points. In general I believe it will be good to offer different perspectives in an nLab entry, and explain what they are useful for, each. I take the point that the word “simply” for the categorical perspective may raise unintended feelings, so maybe it should be avoided or at least not left uncommented.
But we should also not hide the important point here, which is hinted at by the word simply: I think that the important point is that the abstract category-theoretic formulation which packages a long list of detailed definitions in a single statement such as “internal Lie algebra” allows us to recognize that that list of definitions is right.
There are many definitions that one can dream up. But some are better than others and category theory can explain why.
For instance I have seen experts who calucalted with differential graded algebra all day long be mystified by why exactly all the sign rules are as they are. The best explanation they had was: it works and yields interesting results. They were positively interested to learn that all these signs follow automatically and consistently by realizing that differential graded algebra is algebra internal to the category of chain complexes.
This doesn’t mean that it is best to introduce DGCA in this internal language. But it does mean that it is worthwhile pointting out that lots of nitty-gritty details of definitions can “simply” be derived by starting with an abstract internal definition and then turning the crank.
Tim: I could not have put it better myself. I was wondering if there might not be some way in which this viewpoint might not be expressed explicitly. Perhaps David C has some thoughts.. sort of ’the unreasonable effectiveness of categorical language’?
My intention for my own contribution (with help hopefully) is to gradually add glosses in the lexicon entries so as to help interpret in both directions, categorically,and geometrically.
For instance, in the construction of the cobar one take the tensor algebra of the suspension of the cokernel (is it?) of the augmentation. WHY?!!!!!!! How can one understand this? Magic? It works? In fact it is still a bit of a mystery to me and saying that it comes from such and such a categorical property still needs spelling out for me. I have asked rational homotopy theorists and have partially understood things from their point of view but there are still gaps in my understanding of it and some of them worry me!
Toby: One should be able to say something like, ’From a category-theoretic perpsective, a differential graded Lie algebra is simply an internal Lie algebra in an appropriate category of chain complexes.’. This advertises what Urs says, that definitions come automatically from the category-theoretic perspective, without pretending that this will be simple to anyone coming from outside that perpsective.
Zoran Škoda: Tim, your question about the intricacies of cobar construction in the category of chain complexes is an interesting one, which I can not fully answer, specially in a short answer. However, still the categorical picture simplifies the viewpoint and the definition at least,and gives a direction how to proceed there as well. Given a dgca C one looks at the functor Tw(C,A) assigning to an algebra A the set of solution of the Maurer-Cartan equation dt+t*t=0 where * is the convolution product. Cobar construction is the (co)representative of this covariant functor. If you take Tw(C,A) as a contravariant functor on the coalgebras, for fixed A, then its representative is the bar construction (this is said in different words in entry twisting cochain). So bar and cobar construction are simply representatives of very natural functors; accidents of the realization of these functors by formulas in Ch are a bit unfortunate as you pointed out.
=–
end of ancient discussion
I created a stub page for cartographic group, with definitions and a reference. The not-yet-existent article was already linked to from child’s drawing.
I wanted to add some references on WKB approximation, but found that we have two entries semiclassical approximation and WKB method. WKB or semiclassical expansion is one and the same thing: asymptotic expansion of quantum mechanical amplitudes in Planck constant. On the other hand, “WKB method” is often used to limit considerations just to the stationary phase approximation way of doing the expansion, rather than say to the path integral equivalent (the latter anyway used mainly in physics treatments of semiclassical expansion only).
One can “historically” limit to just one dimension and just to asymptotics of integral expressions in first order, so in some sense one can limit to some particular case as WKB approximation, but for a modern researcher, WKB and semiclassical method is one and the same thing. I can hardly split the discussion and references to the two entries, so I would rather have them merged into one entry and restrict any mention of the difference in scope to a historical subsection. What do you think about it (Urs, especially). (In fact it makes some sense to rename WKB method entry into 1-dimensional WKB method and to discuss just the old early theory there).
At semiclassical approximation, I added references on so called exact WKB method, very popular recently, stemming from Voros 1983, where one looks at WKB expansion to all orders and understands it in the sense of Borel summability.
A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’institut Henri Poincaré A39:3, 211-338 (1983) euclid
Alexander Getmanenko, Dmitry Tamarkin, Microlocal properties of sheaves and complex WKB, arxiv/1111.6325
Kohei Iwaki, Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) 474009 arxiv/1401.7094; Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv:1409.4641
arc spaces from Nash’s work reminded me of Nash equilibrium so I wrote the stub for mathematical economics. I added also a reference relating some model in that field to tropical geometry.
am working on a new chapter of the geometry of physics cluster:
Still a bit rough towards the end, but I need a break now.
In the course of this I started splitting off a further chapter geometry of physics – prequantum geometry, but that’s rudimentary at the moment.
sorry, first created it in David Corfields’ web, then in nLab. DId not keep track where I was.
I gave D-brane geometry a minimum of content and references. This is what in string theory was called “D-geometry” in the glory 90s and so I added a disambiguation line at the top of the latter entry.
We did not have the page on coalgebraic logic so I just created as a place for links for now, we could expand on it later.
started co-Kleisli category with a minimum of content. Even though its formally dual to Kleisli category, of course, it may be worthwhile to have a separate entry.
I have created random variable with some minimum context.
In addition I have added pointers to Kolmogorov’s original book and to some modern lecture notes to probability theory and some related entries.
I have briefly cross-linked probability space with possible worlds, indicating a similarity of concepts and an overlap of implementations.
I am working on a further chapter of geometry of physics titled geometry of physics – WZW terms.
So far there is just the introduction.
As usual, in the course of this I will be touching related entries. Right now I have copied the bulk of that introduction also to the entry WZW model in the section Topological term – WZW term – For the 2d WZW model, replacing the material that was there before (which I had had written, too, but the new version is better).