A central problem that has guided much work in algebraic geometry over the past several decades has been to compactify the moduli space of smooth curves and understand its birational geometry. For the most part, this first goal was achieved in the landmark work of Deligne and Mumford who showed that we may compactify M_g by allowing our moduli space to parametrize (stable) nodal curves. However, this is far from the only way that we could've compactified and it is an interesting question to try and find alternative compactifications which are still modular, in the sense that they can be described as moduli spaces of curves satisfying certain properties. In this talk, I will give a highly informal guided tour through the Hassett-Keel Program, which has thus far produced several other modular compactifications of M_g, and related them to one another through explicit birational transformations. Time permitting, I will also describe some of my own work in this direction, and/or briefly introduce more recent work carrying out a similar program for moduli of K3 surfaces.

The Grothendieck ring of varieties encodes algebraic varieties through cut-and-paste relations. Two varieties are called L-equivalent if their classes become equal after multiplication by a power of the class of the affine line. How much geometry does this relation remember, and how does it relate to D-equivalence—the equivalence of derived categories? In this talk, I will explore these questions for cubic fourfolds, where L-equivalence forces very general members to be isomorphic, while special ones exhibit richer behaviour. I will also discuss related results for hyper-Kähler manifolds. This is joint work with Simone Billi.