The son of the mathematician
George David Birkhoff, Garrett was born in
Princeton, New Jersey. He began the
Harvard University BA course in 1928 after less than seven years of prior formal education. Upon completing his Harvard BA in 1932, he went to
Cambridge University to study
mathematical physics but switched to studying
abstract algebra under
Philip Hall. 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. Birkhoff held no Ph.D., a qualification British higher education did not emphasize at that time, and did not obtain an M.A. Nevertheless, after being a member of Harvard's
Society of Fellows, 1933–36, he spent the rest of his career teaching at Harvard. During the 1930s, Birkhoff, along with his Harvard colleagues
Marshall Stone and
Saunders Mac Lane, substantially advanced American teaching and research in
abstract algebra. In 1941 he and Mac Lane published
A Survey of Modern Algebra, the second undergraduate textbook in English on the subject (
Cyrus Colton MacDuffee's
An Introduction to Abstract Algebra was published in 1940). Mac Lane and Birkhoff's
Algebra (1967) is a more advanced text on
abstract algebra. A number of papers he wrote in the 1930s, culminating in his monograph,
Lattice Theory (1940; the third edition remains in print), turned
lattice theory into a major branch of
abstract algebra. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics,
universal 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. During and after
World War II, Birkhoff's interests gravitated towards what he called "engineering" mathematics. During the war, he worked on radar aiming and ballistics, including the
bazooka. In the development of weapons, mathematical questions arose, some of which had not yet been addressed by the literature on
fluid dynamics. Birkhoff's research was presented in his texts on fluid dynamics,
Hydrodynamics (1950) and
Jets, Wakes and Cavities (1957). Birkhoff, a friend of
John von Neumann, took a close interest in the rise of the electronic computer. Birkhoff supervised the Ph.D. thesis of
David M. Young on the numerical solution of
the partial differential equation of Poisson, in which Young proposed the
successive over-relaxation (SOR) method. 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. Birkhoff's research and consulting work (notably for
General Motors) developed computational methods besides
numerical linear algebra, notably the representation of smooth curves via
cubic splines. Birkhoff published more than 200 papers and supervised more than 50 Ph.D.s. He was a member of the
National Academy of Sciences, the
American Philosophical Society, and the
American Academy of Arts and Sciences. He was a
Guggenheim Fellow for the academic year 1948–1949 and the president of the
Society for Industrial and Applied Mathematics for 1966–1968. He won a
Lester R. Ford Award in 1974. == Selected books ==