Garrett Birkhoff (January 19, 1911, Princeton, New Jersey, USA – November 22, 1996, Water Mill, New York, USA) was an American mathematician.

The son of the mathematician George David Birkhoff, Garrett 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 in England to study mathematical physics but switched to studying abstract algebra under Philip Hall. While visiting the University of Munich, he met 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 even bother obtaining an MA. Nevertheless, after being a member Harvard's Society of Fellows,1933-36, he spent the rest of his career teaching at Harvard. From these facts can be inferred the number and quality of Birkhoff's papers published by his 25th year.

During the 1930s, Birkhoff, along with his Harvard colleagues Marshall Stone and Saunders MacLane, substantially advanced American teaching and research in abstract algebra. In 1941 he and MacLane published A Survey of Modern Algebra, the first undergraduate textbook in English on the subject. Birkhoff and MacLane (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 subject owed little to Alfred North Whitehead's 1898 monograph bearing the same name.

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. This weapons-related work culminated 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. His research and consulting work (notably for General Motors) began to employ computational methods, such as numerical linear algebra and the representation of smooth curves via cubic splines.

Birkhoff published more than 200 papers and supervised more than 50 Ph.Ds. He was a member of the National Academy of Sciences and the American Academy of Arts and Sciences.

* 1967 (1940). Lattice Theory, 3rd ed. American Mathematical Society. * 1997 (1941) (with Saunders Mac Lane). A Survey of Modern Algebra. A K Peters. ISBN 1-56881-068-7 * 1978 (1950). Hydrodynamics: A study in logic, fact, and similitude . Greenwood Press. * 1957 (with E. Zarantello). Jets, Wakes, and Cavities. Academic Press. * 1989 (1962) (with Gian-Carlo Rota). Ordinary Differential Equations. John Wiley. * 1999 (1967) (with Saunders Mac Lane). Algebra. Chelsea. ISBN 0-8218-1646-2 * 1970 (with Thomas Bartee). Modern Applied Algebra. McGraw-Hill. * 1973. Source Book in Classical Analysis. Harvard Uni. Press.

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