Geometry of Chow Quotients of Grassmannians -- Eugene Tevelev

We consider Kapranov's and Lafforgue's compactication of the moduli space of ordered n-tuples of hyperplanes in the r-dimensional projective space in linearly general position. For r=1 this is $\bar M_{0,n}$. For r=2 this is the KSBA moduli space of certain stable surfaces that admit canonical resolution of singularities defined using membranes (also known as tropical linear subspaces) in the Bruhat-Tits building. Geometry of this moduli space is (universally) bad a-la Vakil. This is joint work with Keel. If time permits I will describe evolution of these ideas that allows to effectively construct moduli of stable del Pezzo surfaces (joint with Keel and Hacking).