Moduli spaces of cubic hypersurfaces

In this talk I will give an overview of some recent work, joint with Samuel Grushevsky, Klaus Hulek, and Radu Laza, on the geometry and topology of compactifications of the moduli spaces of cubic threefolds and cubic surfaces. A focus of the talk will be on some results regarding non-isomorphic smooth compactifications of the moduli space of cubic surfaces, showing that two natural desingularizations of the moduli space have the same cohomology, and are both blow-ups of the moduli space at the same point, but are nevertheless, not isomorphic, and in fact, not even K-equivalent.