Isoperimetry by stretching

Isoperimetric boundaries minimise area for fixed an enclosed volume, with sharp regularity theory ensuring smoothness away from a closed singular set of codimension seven. I will discuss recent work, with G. Niu, which constructs isoperimetric regions from hypersurfaces in closed manifolds. As a direct application, we show that a wide variety of topological types and singular sets arise in isoperimetric boundaries.