The PI studies automorphism groups and moduli spaces, using techniques from homotopy theory and geometric group theory. The goals of the project are to determine the homotopy type of the classifying space of the stable automorphism group of free groups, to prove homological stability theorems for new families of groups, to adapt the tools of Waldhausen's K-theory of spaces to the mapping class groups and automorphism groups of free groups, and finally to obtain a better understanding of exotic infinite loop space operads.
To study a large collection of objects, it is useful to classify the objects, that is to arrange them into meaningful classes. This is very much like classifying books by subject in a library and corresponds in mathematics to finding invariants, or common properties of the objects considered. One can consider a finer type of classification which moreover takes into account the notion of neighborhood, also called the topology. Such a classification remembers which objects or classes of objects are closely related to each other. The collection of objects is then classified by a topological space, not just by a set. This space is called a moduli space for the objects. It encodes a lot of information about the objects. In particular, the symmetries of the objects, also called automorphisms, play an important role in the topology of the moduli space. The PI studies moduli spaces of geometric objects like graphs, surfaces and higher dimensional manifolds.
