CATS3: Higher Categorical Structures in Algebraic Geometry Pisa, September 1--5, 2008 Speaker: Joachim Kock Title: Feynman graphs, and nerve theorem for modular operads Abstract: Leinster-Weber theory allows you to do the following. Start with a nice monad, extract a category of 'shapes', and characterise the algebras for the monad as those presheaves on the category of shapes that satisfy a certain Segal condition. I will start with the basic case of categories: the monad is the free-category monad on graphs, the shapes are the finite (non-empty) linear orders, and the classical nerve theorem results. Secondly I will explain a nerve theorem for polynomial monads in terms of finite rooted trees, and how this result relates to multicategories and operads. Finally I will explain how the polynomial viewpoints lead to similar theorems for coloured cyclic and modular operads (compact symmetric multicategories). This is work in progress with AndrŽ Joyal.