Moduli of curves and K-stability

The K-moduli theory provides us with an approach to study moduli of curves. In this talk, I will introduce the K-moduli of certain log Fano pairs and how it relates to moduli of curves. We will see that the K-moduli spaces interpolate between different compactifications of moduli of curves. In particular, the K-moduli gives the last several Hassett-Keel models of moduli of curves of genus six.