The entropy is a natural geometric functional introduced by Colding-Minicozzi to study the singularities of mean curvature flow, and it roughly measures the complexity of a hypersurface of Euclidean space.

In this talk, I will survey some recent progress with Jacob Bernstein on understanding the geometry and topology of hypersurfaces with low entropy

We use a weak mean curvature flow together with a surgery procedure to show that all closed hypersurfaces in Euclidean four space with entropy less than or equal to that of the round cylinder, are diffeomorphic to the three-sphere. This is joint work with Jacob Bernstein