Robert Penner (University of Southern California)
rpenner@math.usc.edu
Recent joint work with Greg McShane has answered the following question: Which curves can be short in a given cell of the cell decomposition of Teichmueller space? The answer involves a new combinatorial structure called "screens on fatgraphs" as we shall describe. The techniques of proof involve path-ordered product expansions of holonomies, Ptolemy transformations, and the triangle inequality. This is a main step in giving a combinatorial description of the Deligne-Mumford compactification of moduli space, which we shall also discuss as time permits.