Abstract
Publ.math.IHES 126, 247--318 (2017) We identify spaces of half-translation surfaces, equivalently complex curves
with quadratic differential, with spaces of stability structures on Fukaya-type
categories of punctured surfaces. This is achieved by new methods involving the
complete classification of objects in these categories, which are defined in an
elementary way. We also introduce a number of tools to deal with surfaces of
infinite area, where structures similar to those in cluster algebra appear.
