Abstract
The last 20 years have seen tremendous progress in our understanding of the homology of diffeomorphism groups of even-dimensional manifolds after a suitable stabilization (adding many copies of Sn x Sn), due to work by Madsen--Weiss and Galatius--Randal-Williams. The odd-dimensional case is a mystery until now; the naive generalization of the aforementioned results is certainly false. I will report on some progress that has been made in this direction, in particular about the evaluation of the homology in an important special case, namely the connected sum of many copies of Sn x Sn+1. This is joint work with Jens Reinhold.