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What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory?

In my view, that means it should start off with unpunctured surfaces (and in fact, it would be fine with me if it never went further).

So far as I understand, this means that the results involved might well predate the invention of cluster algebras, but I still think that it would be nice to have an exposition of them from a cluster algebras perspective. I am hoping someone else agrees (and has consequently been inspired to write something along these lines).

My ideal answer (while I'm dreaming) would not assume familiarity with cluster algebras, and as little knowledge of Teichmuller theory as possible.

Hugh Thomas
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  • I am so looking for a nice readable introduction to precisely the other half of cluster algebra theory (the algebraic and combinatorial parts). But on the geometric side, I have been recommended Gekhtman, Shapiro, Vainstein (the AMS book, not the arXiv paper) -- or is it yet another geometric component of cluster algebra theory? (NB: I have no idea about geometry.) – darij grinberg Jan 19 '15 at 20:27
  • @darijgrinberg Gekhtman-Shapiro-Vainshtein takes a Poisson geometry approach, which is great, but I would like something more direct if possible. (Though maybe what I want can be found in there.) If you post a more specific question about "the algebraic and combinatorial parts", I may be able to help. Have you seen the notes from Fomin's Park City course, written up by Nathan Reading (arXiv:math/0505518)? – Hugh Thomas Jan 19 '15 at 21:04
  • I know of them, though I'd like something with proofs. Thanks for reminding me of them, though. – darij grinberg Jan 19 '15 at 21:09
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    http://www.sciencedirect.com/science/article/pii/S0001870809003387 ? – Tom Copeland Jan 20 '15 at 08:41
  • I don't know if this helps Hugh, but @darijgrinberg you might like Ralf Schiffler's new book, "Quiver Representations" (Springer). It's not explictly cluster-y but does cover the combinatorial realisation of types A and D by triangulations of polygons, for example, by relating it to the cluster category. Its style is "algebraic and combinatorial", for sure. – Jan Grabowski Jan 20 '15 at 09:57
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    Looking through my folder of cluster algebra notes, I did also spot some notes of Schiffler from a meeting in Sao Pedro (put "schiffler sao pedro" in your favourite search engine). Lecture 4 is an short intro to cluster algebras from surfaces. – Jan Grabowski Jan 20 '15 at 09:58
  • Also, it'd be helpful to know whether you're looking for something alternative to the canonical starting point of the Fomin-Shapiro-Thurston paper? [I'm certain you and the other contributors above know of this - I'm just naming it explicitly "for the record", for non-experts finding their way here.] – Jan Grabowski Jan 20 '15 at 10:03
  • All suggestions are welcome, but in the interests of clarifying my question, I'll explain why Tom and Jan's suggestions aren't quite what I'm looking for. Tom's suggestion (a paper by Schiffler) doesn't have any Teichmuller theory at all, and what I really want is a straightforward explanation of the link to Teichmuller theory (interpreting cluster variables as lambda-lengths, etc.). Jan's suggestion (plus its sequel by Fomin-Thurston) are canonical sources, but I was hoping for something easier for a student to read, and which devotes less of its energy to the punctured case. – Hugh Thomas Jan 20 '15 at 13:40
  • I see, thanks. But now I don't have any suggestions left to offer :-( – Jan Grabowski Jan 20 '15 at 13:45
  • @JanGrabowski, thanks for pointing out Schiffler's notes, which I was unfamiliar with, and which may come in handy (but for the same reason as for the Schiffler paper Tom suggested, are not really what this question is asking for). – Hugh Thomas Jan 20 '15 at 13:47
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    You've, of course, checked out the two intros by L. Williams and B. Keller. Have you reviewed these lectures https://math.berkeley.edu/~williams/CA.html ? – Tom Copeland Jan 20 '15 at 18:52
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    @TomCopeland Thank you for reminding me about Lauren's notes, http://arxiv.org/abs/1212.6263 ! They seem to cover exactly what I wanted. If you make that an answer, I will accept it. – Hugh Thomas Jan 22 '15 at 21:48
  • I'll leave it open, if you don't mind, to encourage someone to find another good paper. – Tom Copeland Jan 23 '15 at 08:37
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    I should point out the canonical source for sources on cluster algebras: Fomin's "Cluster algebras portal", http://www.math.lsa.umich.edu/~fomin/cluster.html – Hugh Thomas Jan 24 '15 at 03:10

1 Answers1

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Belatedly, per comments: Lauren Williams gave a survey course with recorded lectures, referencing her paper "Cluster algebras: an introduction."

More recently: Introduction to Cluster Algebras: Chapter 6 (2020)

Some background/motivational material:

"The Positive Grassmannian (from a mathematician’s perspective)" slides by Williams

"Combinatorics of KP solitons from the real grassmannian" by Kodama and Williams

KP solitons, total positivity, and cluster algebras" presentation slides by Williams

"A mathematician's unanticipated journey through the physical world," a popularizing article by Hartnett article in Quantamagazine on Williams and her work.

Tom Copeland
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    (Worth noting that this question is quite old and by now Hugh has probably learned what he wanted to about cluster algebras from surfaces.) These notes of Ralph Schiffler, about cluster algebras from surfaces in particular, look nice: https://webusers.imj-prg.fr/~patrick.le-meur/CIMPA/Course-Schiffler.pdf – Sam Hopkins Mar 17 '21 at 01:48
  • Motivational: See allusions to cluster algebras in the body and comments to https://mathoverflow.net/questions/184803/guises-of-the-stasheff-polytopes-associahedra-for-the-coxeter-a-n-root-system – Tom Copeland Mar 17 '21 at 04:14
  • Related MO-Q: https://mathoverflow.net/questions/108797/what-do-cluster-algebras-tell-us-about-grassmannians – Tom Copeland Mar 17 '21 at 05:01