Almgren-Pitts min-max theory considers the space of integral currents on a manifold with the associated mass functional. Minimal hypersurfaces arise as the critical points of the mass functional, and so can be constructed using min-max techniques applied to certain families of integral currents. A particular set of families is the Gromov-Guth p-sweepouts. The min-max masses associated with these families are the p-widths.
This thesis calculates several p-widths for p <= 13 in the case of the round three-sphere by explicit construction of p-sweepouts and Lusternik-Schnirelmann topological arguments. It follows from recent developments in min-max theory that there is a minimal surface with genus > 1, index <= 9 and area equal to the 9-width.