Explore recent developments in constant mean curvature hypersurfaces in this advanced mathematics lecture. Delve into two min-max theorems for constructing prescribed mean curvature hypersurfaces in non-compact spaces, including Euclidean space and asymptotically flat manifolds. Learn about the half-volume spectrum of a manifold and its relation to the Allen-Cahn min-max theory. Discover how this theory is used to find hypersurfaces associated with the half-volume spectrum, consisting of constant mean curvature components enclosing half the manifold's volume and possible minimal components. Examine topics such as Yau's Problem, scalar curvature in asymptotically flat manifolds, and the Weyl law for the half-volume spectrum. Gain insights from Liam Mazurowski's research at Caltech, building upon previous work in the field of geometric analysis.