What is a good introduction to cluster algebras from surfaces?

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.

• Jan 20, 2015 at 8:41
• 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 20, 2015 at 9:58
• You've, of course, checked out the two intros by L. Williams and B. Keller. Have you reviewed these lectures math.berkeley.edu/~williams/CA.html ? Jan 20, 2015 at 18:52
• @TomCopeland Thank you for reminding me about Lauren's notes, arxiv.org/abs/1212.6263 ! They seem to cover exactly what I wanted. If you make that an answer, I will accept it. Jan 22, 2015 at 21:48
• I should point out the canonical source for sources on cluster algebras: Fomin's "Cluster algebras portal", math.lsa.umich.edu/~fomin/cluster.html Jan 24, 2015 at 3:10

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.