Garrett Birkhoff

{{Infobox scientist

| name = Garrett Birkhoff

| image = Birkhoff Garrett 3.jpeg

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| birth_date = {{birth date|1911|1|19}}

| birth_place = Princeton, New Jersey, US

| death_date = {{death date and age|1996|11|22|1911|1|19}}

| death_place = Water Mill, New York, US

| fields = Mathematics

| workplaces = Harvard University

| alma_mater = Cambridge University
Harvard University

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| notable_students = Richard S. Varga

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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

The son of the mathematician George David Birkhoff, Garrett was born in Princeton, New Jersey.Staff. [https://library.ias.edu/files/pdfs/hs/cos.pdf#page=90 A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930-1980] {{Webarchive|url=https://web.archive.org/web/20111124234612/http://library.ias.edu/files/pdfs/hs/cos.pdf#page=90 |date=2011-11-24 }}, p. 90. Institute for Advanced Study, 1980. Accessed November 20, 2015. "Birkhoff, Garrett 40s M Born 1911 Princeton, NJ." 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,{{Cite web |title=Garrett Birkhoff |url=http://www.nasonline.org/member-directory/deceased-members/57681.html |access-date=2022-11-30 |website=www.nasonline.org}} the American Philosophical Society,{{Cite web |title=APS Member History |url=https://search.amphilsoc.org/memhist/search?creator=Garrett+Birkhoff&title=&subject=&subdiv=&mem=&year=&year-max=&dead=&keyword=&smode=advanced |access-date=2022-11-30 |website=search.amphilsoc.org}} and the American Academy of Arts and Sciences.{{Cite web |title=Garrett Birkhoff |url=https://www.amacad.org/person/garrett-birkhoff |access-date=2022-11-30 |website=American Academy of Arts & Sciences |language=en}} 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.{{cite journal|author=Birkhoff, Garrett|title=Current trends in algebra|journal=Amer. Math. Monthly|volume=80|year=1973|issue=7 |pages=760–782|url=http://www.maa.org/programs/maa-awards/writing-awards/current-trends-in-algebra|doi=10.2307/2318163|jstor=2318163 }}

Selected books

{{Citation | last1=Birkhoff | first1=Garrett | title=Lattice theory | orig-year=1940 | url=https://books.google.com/books?id=o4bu3ex9BdkC | publisher=American Mathematical Society | location=Providence, R.I. | edition=4th | series=American Mathematical Society Colloquium Publications | isbn=978-0-8218-1025-5 | mr=598630 | year=1979 | volume=25}}{{cite journal|author=Wilcox, L. R.|title=Review: Lattice Theory, by Garrett Birkhoff|journal=Bull. Amer. Math. Soc.|year=1941|volume=47|issue=3|pages=194–196|url=https://www.ams.org/journals/bull/1941-47-03/S0002-9904-1941-07409-4/S0002-9904-1941-07409-4.pdf|doi=10.1090/S0002-9904-1941-07409-4|doi-access=free}}
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 |year=1997 | first2=Saunders |last2=Mac Lane |author2-link=Saunders Mac Lane |title=A Survey of Modern Algebra |publisher=A.K. Peters |isbn=1-56881-068-7 |orig-year=1941}}{{cite journal|author=Thrall, R. M.|author-link=Robert M. Thrall|title=Review: A Survey of Modern Algebra, by Garrett Birkhoff and Saunders Mac Lane|journal=Bull. Amer. Math. Soc.|year=1942|volume=48|issue=5|pages=342–345|url=https://www.ams.org/journals/bull/1942-48-05/S0002-9904-1942-07670-1/S0002-9904-1942-07670-1.pdf|doi=10.1090/S0002-9904-1942-07670-1|doi-access=free}}
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 |year=1978 |orig-year=1950 |title=Hydrodynamics: A study in logic, fact, and similitude |publisher=Greenwood Press }}{{cite journal|last1=Stoker|first1=J. J.|author-link=James J. Stoker|title=Book Review: Hydrodynamics, a study in logic, fact, and similitude|journal=Bulletin of the American Mathematical Society|volume=57|issue=6|year=1951|pages=497–500|issn=0002-9904|doi=10.1090/S0002-9904-1951-09552-X|doi-access=free}} [https://books.google.com/books?id=Vj7WCgAAQBAJ 2015 pbk reprint of 1960 2nd edition]
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 | year=1957 | first2=E.H. |last2=Zarantonello |author2-link=Eduardo Héctor Zarantonello|title=Jets, Wakes, and Cavities| publisher=Academic Press}}{{cite journal|author=Lighthill, M. J.|author-link=James Lighthill|title=Review of Jets, Wakes and Cavities by Garrett Birkhoff and E. H. Zarantonello|journal=Journal of Fluid Mechanics|volume=3|issue=4|year=1958|pages=437–440|doi=10.1017/S0022112058210100 |s2cid=123461272 |url=https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/abs/jets-wakes-and-cavities-by-garrett-birkhoff-and-e-h-zarantonello-new-york-academic-press-1957-353-pp-1000-or-80s/95F0DFF393775181EF43EDF4959F81C2}}
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 | year=1989 |orig-year=1962 | first2=Gian-Carlo |last2=Rota |author2-link=Gian-Carlo Rota |title=Ordinary Differential Equations |publisher=John Wiley}}{{cite journal|title=Review of Ordinary Differential Equations by G. Blrkhoff and G. Rota|author=Burkill, J. C.|author-link=John Charles Burkill|journal=The Mathematical Gazette|volume=48|issue=364|date=May 1964|pages=240–241|doi=10.1017/S0025557200051068|s2cid=195544336 |url=https://www.cambridge.org/core/journals/mathematical-gazette/article/abs/ordinary-differential-equations-by-g-blrkhoff-and-g-rota-pp-vi-318-850-1962-ginn-and-company/B7A4F33B7EF76E7E48C1C001AE699F8A}}
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 | year=1999 |orig-year=1967 |first2=Saunders |last2=Mac Lane |author2-link=Saunders Mac Lane |title=Algebra |publisher=Chelsea |isbn=0-8218-1646-2 |url=https://books.google.com/books?id=L6FENd8GHIUC}}
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 | year=1970 |first2=Thomas |last2=Bartee |title=Modern Applied Algebra |publisher=McGraw-Hill}}{{cite journal|author=Rheinboldt, Werner C.|title=Review: Modern Applied Algebra, by Garrett Birkhoff and Thomas C. Bartee|journal=Bull. Amer. Math. Soc.|year=1972|volume=78|issue=3|pages=383–385|url=https://www.ams.org/journals/bull/1972-78-03/S0002-9904-1972-12908-2/S0002-9904-1972-12908-2.pdf|doi=10.1090/s0002-9904-1972-12908-2|doi-access=free}}
{{cite book|title=Numerical solution of elliptic problems|author=Birkhoff, Garrett Birkhoff|author-mask=2|author2=Lynch, Robert Edward|year=1984|publisher=Society of Industrial and Applied Mathematics (SIAM)|location=Philadelphia|series=SIAM Studies in Applied Mathematics, vol. 6|doi=10.1137/1.9781611970869.fm |url=https://epubs.siam.org/doi/pdf/10.1137/1.9781611970869.fm}}
{{Citation | last1=Birkhoff | first1=Garrett |author-mask=2 | year=1973 |title=Source Book in Classical Analysis |publisher=Harvard University Press}}
{{cite book|editor1=Oliveira, J. S.|editor2=Rota, G.-C.|title=Selected Papers on Algebra and Topology by Garrett Birkhoff|url=https://books.google.com/books?id=BBykgLvJ3yAC|date=1 January 1987|publisher=Springer Science & Business Media|isbn=978-0-8176-3114-7}}