Garrett Birkhoff
Garrett Birkhoff  

Born 
Princeton, New Jersey, United States

January 19, 1911
Died  November 22, 1996
Water Mill, New York, USA

(aged 85)
Alma mater 
Cambridge University Harvard University 
Known for  
Awards  
Scientific career  
Fields  Mathematics 
Institutions  Harvard University 
Academic advisors  
Doctoral students  
Other notable students  Richard S. Varga 
Influences  Constantin Carathéodory 
Influenced  GianCarlo Rota 
Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American mathematician. He is best known for his work in lattice theory.
The mathematician George Birkhoff (1884–1944) was his father.
Life[edit]
The son of the mathematician George David Birkhoff, Garrett was born in Princeton, New Jersey.^{[1]} 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 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 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. 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 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. Further, in 1935, Birkoff showed that any equivalence between expressions that holds for all possible forms of operator must have a finite proof using certain underlying rules about equality. However, as soon as one introduces actual axioms that constrain the operators this is no longer true—and in general it can be undecidable whether or not a particular equivalence holds.^{[2]}
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 overrelaxation (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 BirkhoffVarga collaboration led to many publications on positive operators and iterative methods for pcyclic 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 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.^{[3]}
Selected books[edit]
 Birkhoff, Garrett (1979) [1940], Lattice theory, American Mathematical Society Colloquium Publications, 25 (4th ed.), Providence, R.I.: American Mathematical Society, ISBN 9780821810255, MR 0598630 ^{[4]}
 ——; Mac Lane, Saunders (1997) [1941], A Survey of Modern Algebra, A.K. Peters, ISBN 1568810687 ^{[5]}
 —— (1978) [1950], Hydrodynamics: A study in logic, fact, and similitude, Greenwood Press
 ——; Zarantonello, E.H. (1957), Jets, Wakes, and Cavities, Academic Press
 ——; Rota, GianCarlo (1989) [1962], Ordinary Differential Equations, John Wiley
 ——; Mac Lane, Saunders (1999) [1967], Algebra, Chelsea, ISBN 0821816462
 ——; Bartee, Thomas (1970), Modern Applied Algebra, McGrawHill ^{[6]}
 —— (1973), Source Book in Classical Analysis, Harvard University Press
See also[edit]
 Birkhoff polytope
 Birkhoff's representation theorem
 Birkhoff's HSP theorem
 Birkhoff's theorem (disambiguation)
 Pierce–Birkhoff conjecture
 Poincaré–Birkhoff–Witt theorem
References[edit]
 ^ Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 19301980, p. 90. Institute for Advanced Study, 1980. Accessed November 20, 2015. "Birkhoff, Garrett 40s M Born 1911 Princeton, NJ."
 ^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media, Inc. p. 1172. ISBN 1579550088.
 ^ Birkhoff, Garrett (1973). "Current trends in algebra". Amer. Math. Monthly. 80: 760–782. doi:10.2307/2318163.
 ^ Wilcox, L. R. (1941). "Review: Lattice Theory, by Garrett Birkhoff" (PDF). Bull. Amer. Math. Soc. 47 (3): 194–196. doi:10.1090/S000299041941074094.
 ^ Thrall, R. M. (1942). "Review: A Survey of Modern Algebra, by Garrett Birkhoff and Saunders Mac Lane" (PDF). Bull. Amer. Math. Soc. 48 (5): 342–345. doi:10.1090/S000299041942076701.
 ^ Rheinboldt, Werner C. (1972). "Review: Modern Applied Algebra, by Garrett Birkhoff and Thomas C. Bartee" (PDF). Bull. Amer. Math. Soc. 78 (3): 383–385. doi:10.1090/s000299041972129082.
External links[edit]
 1911 births
 Members of the United States National Academy of Sciences
 1996 deaths
 20thcentury American mathematicians
 Guggenheim Fellows
 Harvard University faculty
 Institute for Advanced Study visiting scholars
 Algebraists
 Lattice theorists
 Fluid dynamicists
 Harvard University alumni
 People from Princeton, New Jersey
 Presidents of the Society for Industrial and Applied Mathematics
 American expatriates in the United Kingdom
 Mathematicians from New Jersey