While visiting the University of Munich, he met Constantin Carathéodory who pointed him towards two important texts, Van der Waerden on abstract algebra and Speiser on group theory.

During the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders Mac Lane, substantially advanced American teaching and research in abstract algebra.

Birkhoff's approach to this development of universal algebra and lattice theory acknowledged prior ideas of Charles Sanders Peirce, Ernst Schröder, and Alfred North Whitehead; in fact, Whitehead had written an 1898 monograph entitled Universal Algebra.

Birkhoff then worked with Richard S. Varga, a former student, who was employed at Bettis Atomic Power Laboratory of the Westinghouse Electronic Corporation in Pittsburgh and was helping to design nuclear reactors.

Extending the results of Young, the Birkhoff–Varga collaboration led to many publications on positive operators and iterative methods for p-cyclic matrices.