Title: On the homology of diffeomorphism groups made discrete.

Abstract: Let $G$ be a finite dimensional Lie group and $G^{delta}$ be the same group with the discrete topology. The classifying space $BG$ classifies principal $G$-bundles and the classifying space $BG^{delta}$ classifies flat principal $G$-bundles (i.e. those bundles that admit a connection whose curvature vanishes). The natural homomorphism from $G^{delta}$ to $G$ induces a continuous map from $BG^{delta}$ to $BG$. Milnor conjectured that this map induces an equivalence after the profinite completion. In this talk, we discuss the same map for infinite dimensional Lie groups, in particular for diffeomorphism groups and symplectomorphisms. In these cases, we use techniques from homotopy theory to show that the map from $BG^{delta}$ to $BG$ induces a split surjection on cohomology with finite coefficients in the stable range. If time permits, I will discuss applications of these results in foliation theory, in particular, characteristic classes of flat surface bundles.