I will introduce the concept of K-moduli illustrated by some examples of moduli spaces and related comparisons with GIT and wall-crossing phenomenon.

I will introduce the concept of K-moduli illustrated by some examples of moduli spaces and related comparisons with GIT and wall-crossing phenomenon.

Determining a modular interpretation of birational models arising from the MMP on Mg, the moduli space of genus g curves, is known as the Hassett-Keel program. The first few steps are well-understood yet the program remains quite incomplete in general for any genus g > 3. In this talk, we will discuss the complete Hassett-Keel program in genus 4. This is joint work with Kenneth Ascher, Yuchen Liu, and Xiaowei Wang.