Garrett Birkhoff
Born	(1911-01-19)January 19, 1911 Princeton,New Jersey, US
Died	November 22, 1996(1996-11-22) (aged 85) Water Mill,New York, US
Alma mater	Cambridge University Harvard University
Known for	Lattice theory Quantum logic
Awards	National Academy of Sciences American Academy of Arts and Sciences George David Birkhoff Prize(1978)
Scientific career
Fields	Mathematics
Institutions	Harvard University
Academic advisors	Ralph H. Fowler Philip Hall
Doctoral students	Richard Arens Jerry L. Bona Thomas Caywood Chandler Davis George Fix Emmett Keeler Uta Merzbach George Mostow Peter J. Olver R. Tyrrell Rockafellar Henry C. Wente Philip M. Whitman David M. Young, Jr. Vern Poythress
Other notable students	Richard S. Varga

Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an Americanmathematician. He is best known for his work inlattice theory.

The mathematicianGeorge Birkhoff (1884–1944) was his father.

The son of the mathematicianGeorge David Birkhoff, Garrett was born inPrinceton, New Jersey.^[1] He began theHarvard University BA course in 1928 after less than seven years of prior formal education. Upon completing his Harvard BA in 1932, he went toCambridge University to studymathematical physics but switched to studyingabstract algebra underPhilip Hall. While visiting theUniversity of Munich, he metConstantin Carathéodory who pointed him towards two important texts,Van der Waerden onabstract algebra andSpeiser ongroup 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'sSociety of Fellows, 1933–36, he spent the rest of his career teaching at Harvard.

During the 1930s, Birkhoff, along with his Harvard colleaguesMarshall Stone andSaunders Mac Lane, substantially advanced American teaching and research inabstract algebra. In 1941 he and Mac Lane publishedA Survey of Modern Algebra, the second undergraduate textbook in English on the subject (Cyrus Colton MacDuffee'sAn Introduction to Abstract Algebra was published in 1940). Mac Lane and Birkhoff'sAlgebra (1967) is a more advanced text onabstract algebra. A number of papers he wrote in the 1930s, culminating in his monograph,Lattice Theory (1940; the third edition remains in print), turnedlattice theory into a major branch ofabstract 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 universalalgebra and lattice theory acknowledged prior ideas ofCharles Sanders Peirce,Ernst Schröder, andAlfred North Whitehead; in fact, Whitehead had written an 1898 monograph entitledUniversal Algebra.

During and afterWorld War II, Birkhoff's interests gravitated towards what he called "engineering" mathematics. During the war, he worked on radar aiming and ballistics, including thebazooka. In the development of weapons, mathematical questions arose, some of which had not yet been addressed by the literature onfluid dynamics. Birkhoff's research was presented in his texts on fluid dynamics,Hydrodynamics (1950) andJets, Wakes and Cavities (1957).

Birkhoff, a friend ofJohn von Neumann, took a close interest in the rise of the electronic computer. Birkhoff supervised the Ph.D. thesis ofDavid M. Young on the numerical solution ofthe partial differential equation of Poisson, in which Young proposed thesuccessive over-relaxation (SOR) method. Birkhoff then worked withRichard S. Varga, a former student, who was employed atBettis Atomic Power Laboratory of theWestinghouse Electronic Corporation in Pittsburgh and was helping to design nuclear reactors. Extending the results of Young, the Birkhoff–Varga collaboration led to many publications onpositive operators anditerative methods forp-cyclic matrices.

Birkhoff's research and consulting work (notably forGeneral Motors) developed computational methods besidesnumerical linear algebra, notably the representation of smooth curves viacubic splines.

Birkhoff published more than 200 papers and supervised more than 50 Ph.D.s. He was a member of theNational Academy of Sciences,^[2] theAmerican Philosophical Society,^[3] and theAmerican Academy of Arts and Sciences.^[4] He was aGuggenheim Fellow for the academic year 1948–1949 and the president of theSociety for Industrial and Applied Mathematics for 1966–1968. He won aLester R. Ford Award in 1974.^[5]

• Birkhoff, Garrett (1979) [1940],Lattice theory, American Mathematical Society Colloquium Publications, vol. 25 (4th ed.), Providence, R.I.:American Mathematical Society,ISBN 978-0-8218-1025-5,MR 0598630^[6]
• ——;Mac Lane, Saunders (1997) [1941],A Survey of Modern Algebra, A.K. Peters,ISBN 1-56881-068-7^[7]
• —— (1978) [1950],Hydrodynamics: A study in logic, fact, and similitude, Greenwood Press^[8]2015 pbk reprint of 1960 2nd edition
• ——;Zarantonello, E.H. (1957),Jets, Wakes, and Cavities, Academic Press^[9]
• ——;Rota, Gian-Carlo (1989) [1962],Ordinary Differential Equations, John Wiley^[10]
• ——;Mac Lane, Saunders (1999) [1967],Algebra, Chelsea,ISBN 0-8218-1646-2
• ——; Bartee, Thomas (1970),Modern Applied Algebra, McGraw-Hill^[11]
• ——; Lynch, Robert Edward (1984).Numerical solution of elliptic problems. SIAM Studies in Applied Mathematics, vol. 6. Philadelphia: Society of Industrial and Applied Mathematics (SIAM).doi:10.1137/1.9781611970869.fm.
• —— (1973),Source Book in Classical Analysis, Harvard University Press
• Oliveira, J. S.; Rota, G.-C., eds. (1 January 1987).Selected Papers on Algebra and Topology by Garrett Birkhoff. Springer Science & Business Media.ISBN 978-0-8176-3114-7.

• Birkhoff algorithm
• Birkhoff's condition
• Birkhoff polytope
• Birkhoff's representation theorem
• Birkhoff's HSP theorem
• Birkhoff's theorem
• Birkhoff–Kakutani theorem
• Pierce–Birkhoff conjecture
• Pierce–Birkhoff ring
• Poincaré–Birkhoff–Witt theorem
• Algebraic statistics
• Median algebra
• Universal algebra

• Garrett Birkhoff at theMathematics Genealogy Project
• O'Connor, John J.;Robertson, Edmund F.,"Garrett Birkhoff",MacTutor History of Mathematics Archive,University of St Andrews

• 1911 births
• Members of the United States National Academy of Sciences
• 1996 deaths
• 20th-century American mathematicians
• Harvard University Department of Mathematics faculty
• Institute for Advanced Study visiting scholars
• American algebraists
• Lattice theorists
• American fluid dynamicists
• Harvard University alumni
• Scientists from Princeton, New Jersey
• Presidents of the Society for Industrial and Applied Mathematics
• American expatriates in the United Kingdom
• Mathematicians from New Jersey
• Members of the American Philosophical Society

• Webarchive template wayback links
• Articles with short description
• Short description matches Wikidata
• Articles with hCards