Moduli spaces of punctured Poincar\'e disks
  The Tamari lattice and the associahedron provide methods of measuring associativity on a line. The real moduli space of marked curves captures the space of such associativity. We consider a natural generalization by considering the moduli space of marked particles on the Poincar\'{e} disk, extending Tamari's notion of associativity based on nesting. A geometric and combinatorial construction of this space is provided, which appears in Kontsevich's deformation quantization, Voronov's swiss-cheese operad, and Kajiura and Stasheff's open-closed string theory. 
Satyan L. Devadoss, Benjamin Fehrman, Timothy Heath, Aditi Vashist
math.AT math-ph math.CO math.MP