This thesis consists of four parts. In the first part, we introduce the results, and sketch the ideas behind the proofs. In the second part, we construct genus one singularities for mean curvature flow using an interpolation argument. In the third part, we bound some Almgren-Pitts min-max p-widths of the unit 3-sphere by proving a strong multiplicity one min-max theorem. In the fourth part, we present results concerning p-widths and minimal surfaces in the unit 3-ball.