[1] It has played roles in the solutions to a number of conjectures in geometry and topology found by Almgren and Pitts themselves and also by other mathematicians, such as Mikhail Gromov, Richard Schoen, Shing-Tung Yau, Fernando Codá Marques, André Neves, Ian Agol, among others.
[11] In his PhD thesis, Almgren proved that the m-th homotopy group of the space of flat k-dimensional cycles on a closed Riemannian manifold is isomorphic to the (m+k)-th dimensional homology group of M. This result is a generalization of the Dold–Thom theorem, which can be thought of as the k=0 case of Almgren's theorem.
Existence of non-trivial homotopy classes in the space of cycles suggests the possibility of constructing minimal submanifolds as saddle points of the volume function, as in Morse theory.
In his subsequent work Almgren used these ideas to prove that for every k=1,...,n-1 a closed n-dimensional Riemannian manifold contains a stationary integral k-dimensional varifold, a generalization of minimal submanifold that may have singularities.
He showed that when the dimension n of the manifold is between 3 and 6 the minimal hypersurface obtained using Almgren's min-max method is smooth.