abstract: We will discuss the higher dimensional filling functions for mapping class groups of surfaces. We will establish bounds for these families of functions and show they exhibit phase transitions at the rank (as measured by 3 genus+number of punctures-3); this phase transition is analogous to a corresponding result for symmetric spaces which results from the combined work of Brady--Farb, Hindawi, Leuzinger, and Wenger. This is joint work with Cornelia Drutu.