Understanding the singularities is a central topic in the study of mean curvature flow, and the models of the singularities are known to be the self-shrinkers. In 2012, Colding-Minicozzi proved the compactness of self-shrinkers in R^3, and it is one of the starting points of the structure theory of self-shrinkers. In this talk, I will discuss our recent result on the compactness of self-shrinkers with fixed genus. The result relies on the study of the topology of self-shrinkers, and some blow-up and blow-down arguments of mean curvature flow. This is a joint work with Zhichao Wang.
