K-stability and moduli of quartic K3 surfaces

We show that K-moduli spaces of $(\mathbb{P}^3, cS)$ where $S$ is a quartic surface interpolates between the GIT moduli space and the Baily-Borel compactification as $c$ varies in $(0,1)$. We completely describe the wall crossings of these K-moduli spaces, hence verifying Laza-O’Grady’s prediction on the Hassett-Keel-Looijenga program for quartic K3 surfaces. We also obtain the K-moduli compactification of quartic double solids, and classify all Gorenstein canonical Fano degenerations of $\mathbb{P}^3$. This is based on joint work with K. Ascher and K. DeVleming.