Explore the mathematical foundations of Brakke's mean curvature flow in this advanced lecture that forms part of a comprehensive course series on geometric analysis. Delve into the theoretical framework and analytical techniques used to study the evolution of surfaces under mean curvature, building upon concepts introduced in the first part of the series. Examine the rigorous mathematical treatment of varifolds and their role in understanding weak solutions to mean curvature flow, including the fundamental properties and regularity theory that make Brakke's approach particularly powerful for handling topological changes during the flow evolution. Learn about the key analytical tools and measure-theoretic methods essential for working with this generalized notion of mean curvature flow, with particular attention to how these techniques apply to free boundary problems and geometric evolution equations.