Mirror symmetry for correlation functions of tropical observables:

- Andrey Losev, Vyacheslav Lysov,
*Tropical mirror symmetry: correlation functions*, arXiv:2301.01687

Recently more references and some reorganization at tropical geometry.

]]>Continuation from 21:

Yes, I read some of those papers as you suggest, in fact I was doing research on the categorical/general nonsense origin of the cyclic cohomology coming from the Hopf algebras as in those papers. This is subject of my preprint on the arXiv. It went half a way: they have a cyclic structure on certain complex which is not (co)monadic but is obtained by using some functor on a (co)monadic complex. I figured out that the mechanism should involve distributive laws and (co)monadic simplicial objects, but left the additional functor for the time being out of the story. One of the reasons I believed in this was the existance of coefficients for that cyclic cohomology as discovered by Hajac, Rangipour, Khalkhali etc. and other was from some conversations from Jibladze who explained me some of his and Pirashvili’s insight. Bohm and Stefan have solved the question of the categorical origin of Hopf-cyclic cohomology independently few years later, by monads and distributive laws; my main theorem is special case of theirs. Kaygun had a different approach, with some similarity, he also did not know of my work, what he learned a couple of years later when we overlapped at MPI.

There were few more unfortunate twists why I did not pursue to finish the attack to the problem. When I told Alain Connes in 2004 that I had some general approach he seemed excited in a conversation, but when I have him first couple of pages of my manuscript he dismissed it saying “this is category theory!?” I had in fact another motivation for my preprint, related to orbifolds and groupoids, the work of Baranovsky and the theorem of Hinich on monoidal center (later used by Francis and Ben-Zvi who did not know of my contribution), which in a different form I discovered independently and somewhat before his paper was out (he also had the result earlier).

]]>Urs 20

Indeed, Urs, this at foliation *is* an unfinished list, I am sorry. When I have the access, nowdays, I usually add the link to MathSciNet review, so even if a reference is not in your language you can get the review or, if you have no access, the title etc. in English. Unfortunately, sometimes I do not have MR access or do not finish my record as it happened here. I added MathReviews links to hundreds of references in $n$Lab in a massive effort taking many days, including MR numbers to many of your references, improving the accessability and easening estimating if one needs to get hold of a reference. Still after all that and similar effort I am accused of not fostering accessability. Of course, this MR link may be less important for those links which have the arXiv number and link (I would still urge you and others to overtly write the arXiv number in the link name and link to the arXiv abstract rather than to the pdf at the arXiv, this saves many problems, e.g. when one accesses with Kindle, where it is difficult to change the address and pdfs do not download). This Russian original pdfs are free, thanks to mathnet.ru, one of the reasons I link them sometimes. Most scans of old literature we have and use are thanks to Russian enthusiasts, and it is fair to return the favor to them by giving the links also to the literature they can access freely and more understandable to them; this may also prompt some of the people from their math community to join the $n$Lab effort. Unfortunately, the Doklady are not there, as some bandit organization has took the copyright from the Soviet Academy of Arts and Sciences for something the state financed for decades (the Soviet Union had huge spending for science and education and now the pearl journal of all that effort is in private hands and one can not freely distribute that old research!! I will write about the issue elsewhere and I would appreciate any insider information one may have).

The whole series of articles is on characteristic classes of foliations and related issues on cohomology of formal vector fields: Godbillon-Vey class and Gelfand-Fuks series of works. I have opened an entry Gelfand-Fuks cohomology as well, but not all of these references seem to fit there (I thought at the beginning to transfer them there, but not all of the works Gelfand-Fuks are about the GF cohomology!! though there are connections).

Then comes a text titled “Modular Hecke algebras and their Hopf symmetry”. My immediate reaction here is. What on earth is this doing here?

“Reference” is in fact a longer record:

- A. Connes, H. Moscovici,
*Modular Hecke algebras and their Hopf symmetry*, Mosc. Math. J., 4:1 (2004), 67–109; math.QA/0301089, ams;*Hopf algebras, cyclic cohomology and the transverse index theory*, math.DG/9806109, Comm. Math. Phys.**198**, n.1, 1998 MR99m:58186 doi;*Rankin-Cohen brackets and the Hopf algebra of transverse geometry*, Mosc. Math. J., 4:1 (2004), 111–130

If one writes several papers together in a single reference paragraph, by the standard literature convention from physics, they are about the same topic. This time, as the titles suggest, the **transverse geometry** (of a foliation, of course, otherwise it were not there) and, clearly, about some Hopf algebras arising there. These guys introduced those Hopf algebras to describe the geometry of foliations, I think the construction has no earlier analogue in the subject which one could refer to. I do not think that saying that it is about a “Hopf algebra in the context of foliation” or saying that it is a “further development of foliation theory” will help that much, regarding it is seen from the title and the context: this article introduces a new Hopf algebra related to foliation theory. The useful information to add is however that the cyclic cohomology of those Hopf algebras is part of an effort by Connes and Moscovici to derive the local index formulas (those are at the cocycle level rather than at the cohomology level, so they are more precise) pertaining to the transverse geometry. This is the information which happened to be missing here, thanks for reminding me.

(Of course, ideally the article would have a longer paragraph on those which would refer and lead to this item, rather than mumbling half a way from within a literature list.)

]]>Zoran, here is an example of literature lists which I find unhelpful to the extent of being a distraction, as long as they are not further commented:

The References-section at *foliation* is a hybrid of a list which I think mainly you gave and some edits that I added. After the first bit to which I added comments to orient the reader follows a list of what I would call “random references”.

Here is what I find unhelpful:

The first is entirely in Russian, I can’t even read that. So I have no way to have any idea of what that reference might be good for. Maybe it’s the key reference that will remove the dead-lock in my research and maybe I should put all energy into getting hold of a translation, but I will never know.

A little bit below is the next Russion text. Then comes a text titled “Modular Hecke algebras and their Hopf symmetry”. My immediate reaction here is. What on earth is this doing here? Is that at all related to the topic of the entry even?! Probably it is, propbably there was a reason that somebody (I guess you, but didn’t check) added this reference here. But without the slightest comment it could just as well be a reference on the science of ice cream. I would not know why I should look at it.

I think this is not just a nuisance for me, the reader, but it is also sad for you (or whoever it was) who probably read that reference, saw that it contained some insights about foliations, and wanted to share that insight. This effort is all wasted if there is no comment accompanying the reference.

If you see what I mean.

]]>If you want to provide information for somebody new to a subject, there are usually the basic facts he should learn first.

I agree! Being basic *is* an information. Belonging to a specific class is an information. If you have a list with 10 items out of 25 labelled basic or important, then it is not needed to label each of the remaining as not important (or “further”).

everybody can collect as much random literature

Being in $n$Lab means the reference is liked, or author appreciated, by some $n$Lab contributor and not random.

]]>saying “further developments are in” is a VACUOUS information.

Usually not. If you want to provide information for somebody new to a subject, there are usually the basic facts he should learn first. And the current research questions that vary and expand on that should come later.

Today with Google, everybody can collect as much random literature on any topic as anyone else. The big question that Google cannot answer is: where to start? What’s the core, what’s the variation, where is the readable introduction in the haystack of chatter about the subject.

That’s what I think is the informaiton that a literature list on the nLab can try to provide. If the list just looks like a page of Google scholar output, then there is no extra value added.

]]>Although I agree with a lot of what you say here, Zoran, sometimes an obvious non-substantial line can help break up a block of text and make things easier to read. (I am meaning this as a general point and not specifically about this entry.)

]]>I erased the “modern” from “Modern textbooks…”, as the references in this section are not any more modern than the rest. The area is about a decade old so far. The subject is yet in *statu nascendi* and it is not possible to divide references into original and “developments”. Almost every reference should qualify as original in this period. Some of the newest papers have just different flavours of the theory, for example analytic as opposed to algebraic approach and so on. I am a bit uneasy with 580 put into the books as it is just a collection of articles and before it was *next* to one arXiv article which is in this collection, now they are separated.

I personally think that saying that some article is important historically or influential is useful (though opinionated and probably not shared by all $n$Lab writers) and that saying a content clarification (“an approach following Mumford”, “algebraic axiomatization” etc.) is much better and more useful, but saying “further developments are in” is a VACUOUS information. Similarly comments “it is also useful to look into” and alike which should be true for all references in $n$Lab (if not, let them out of here). Let us comment just with phrases which have non-obvious meaning. If something is already in the title of the paper or applies to all useful papers, it is not a piece of information.

]]>With a long unstructured list, I get lost!

I second that.

Let’s always try to add some kind of comment with references, and be it ever so brief or vague. Some of our entries have huge lists of references that are rather random or at least – and that’s what matters – must appear so to the reader, and that sometimes gets to the point of being just as useful as no list at all, because one ends up googling with more keywords added anyway.

I usually try to add some comments like (preferably in this order of appearance):

Original references include…

Surveys and reviews include…

Modern textbook accounts/lecture notes are…

Further developments include…

The first to correctly give credit (and often to explain why a concept is named the way it is), the next right afterwards to provide overviews for the hasty readers, then next to provide in-depth accounts for the reader who cares about them, and finally a list with all kinds of further developments.

I know that there is not always a clear-cut distinction between these cases, but usually one can come up with something that should be useful to the reader. Maybe good to think of how one would give references to a student who is asked to look into these things:

“You should know that these here are the original references, but first have a look at this review here to get some idea and then try to read this standard textbook here, which gives a comprehensive modern account. Finally to dig deeper notice this recent article here…”

]]>@Zoran: I understand that, and please do not take my comments as a criticism of your list. My comment was more general namely that we sometimes in an entry have an unstructured list which has ‘just grown’ as more titles are added, but without that much of an idea of the relevant importance of the papers / notes etc. in the list and that is less useful.

(Added later: I have restructured a little, and added a link to a book by Maclagen and Sturmfelds. I also fixed a dead link at Mark Gross.)

]]>Tim, the area is just emergent, it is still impossible to say which texts are important, unless you are one of the main experts in the field, when you will be, of course, biased. You are welcome to improve with comments, but I think it will be quite a difficult task. Simply there are many approaches, subareas and applications and the titles often tell you at least something about it.

]]>Personally, I agree with Zoran about keeping the reference list with the main text (in anticipation of expanding the main text), and I agree with Tim about the organisation of the reference list (which I wouldn't know how to do in this field).

]]>My own preferences in many of the reference lists would be to have a few key texts listed (say introductory summaries etc.), followed by ’historically’ important texts, then general references which are less structured. With a long unstructured list, I get lost!

]]>No, I would not separate. Of course, the text should be expanded. (BTW, it is still a small entry in comparison with average entry written by Urs.) It is practical to have the things together at this page size, especially as it is easier later to separate subtopics. For example, I think that the tropicalization may eventually have a separate page with the corresponding references moved along with the text.

]]>Perhaps we should have a separate sources for tropical geometry entry and expand the text in this one. The list of references is very useful but with the smallness of the text ….

]]>More references at tropical geometry.

]]>Thanks David, as I had missed that discussion, or rather I saw it at the time and meant to check back later ….. Well I suppose now is later!

I learnt of that stuff teaching operational research and computer science, (e.g. for any given alphabet, A, the set of languages over A is an idempotent semiring, and lots of the stuff relies on ideas from finite automata theory! Neat! We ran a seminar on it back in the early 2000s (before we were shut!))

]]>If you follow the links from matrix mechanics, you’ll see we discussed Litvinov’s paper. If path integration is a process of multiplying some quantity along a path and then adding across all paths, then if you use the tropical rig it becomes a matter of adding along a path, and then taking the minimum over all paths, which is a very classical mechanics thing to do, i.e., the least action principle.

]]>I do not really understand them but there are complete idempotent semirings which are used in analysis of certain Markov chain type processes, complete with a corresponding analysis. Gaubert’s notes mention these more in passing.

The name Maslov is one that I remember as being fruitful. There is a survey article by Litvinov

http://arxiv.org/PS_cache/math/pdf/0507/0507014v1.pdf

that may be useful. Maslov developed a measure theory for these tropical situations. Some of this looks right for ‘analysis’ in categorical situations (or so I have thought when I was teaching Max+. It is good for students to see an immediately useful bit of abstract algebra and they always seems to enjoy the strange sums: $2\oplus 3 = 3$ etc.

]]>I’m sure there’s interesting scope for taking algorithms and changing the rig, as in matrix mechanics. The Generalized Distributive Law by Srinivas M. Aji and Robert J. McEliece shows how many algorithms are rig-relatives of each other.

What, Tim, are those ’seemingly weird probablistic semirigs’?

]]>I have added some links to the algebra behind tropical geometry. There are numerous applications of min-plus and max-plus in Computer Science, and many of the methods, say, of linear tropical algebra, are generalisations of well-known algorithms that apply to very interesting stochastic situations. There is also the Floyd-Warshall algorithm which interprets in terms of Enriched category theory (old paper by Vaughan Pratt). At one time I wondered if the enriched POV could be used in the other CS applications and perhaps some of the recent tropical geometry ideas may be well worth following up in some seemingly weird probablistic semirigs.

]]>I added a stub for Mark Gross. By the way there is a conference **Tropical geometry and homological mirror symmetry** (similar to the title of Gross’s book) around July 9-15 in Croatia, the web site will be created soon. Katzarkov is the key organizer (Gross was initially supposed to participate, I am not sure now). I am trying to help a bit with local contacts, accomodation and so on in Split (well, Split is the location, if everything goes well).

New stubs tropical geometry and tropical semiring (with rig version included). Note the new book by Gross.

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