Explore recent advancements in self-expanders of mean curvature flow and their applications in geometric analysis. Delve into this hour-long lecture that examines the negative gradient flow for the area functional and its finite-time singularities. Gain insights into self-similarly expanding solutions, or self-expanders, which model the asymptotic behavior of the flow during cone-like singularity formation and its long-term evolution. Discover how these concepts contribute to solving complex problems in geometric analysis and expand your understanding of mean curvature flow dynamics.