Flat surfaces and stability structures
  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. 
Fabian Haiden, Ludmil Katzarkov, Maxim Kontsevich
math.AG math.RT