Garrett Birkhoff (January 19, 1911 – November 22, 1996) was an American

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

. He is best known for his work in

lattice theory A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ...

. The mathematician George Birkhoff (1884–1944) was his father.

Life

The son of the mathematician

George David Birkhoff George David Birkhoff (March 21, 1884 – November 12, 1944) was an American mathematician best known for what is now called the ergodic theorem. Birkhoff was one of the most important leaders in American mathematics in his generation, and durin ...

, Garrett was born in

Princeton, New Jersey Princeton is a municipality with a borough form of government in Mercer County, in the U.S. state of New Jersey. It was established on January 1, 2013, through the consolidation of the Borough of Princeton and Princeton Township, both of whi ...

. He began the

Harvard University Harvard University is a private Ivy League research university in Cambridge, Massachusetts. Founded in 1636 as Harvard College and named for its first benefactor, the Puritan clergyman John Harvard, it is the oldest institution of higher le ...

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 , mottoeng = Literal: From here, light and sacred draughts. Non literal: From this place, we gain enlightenment and precious knowledge. , established = , other_name = The Chancellor, Masters and Schola ...

to study

mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

but switched to studying

abstract algebra In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...

under

Philip Hall Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups. Biography He was educated first at Christ's Hospital, where he won the Thomps ...

. While visiting the

University of Munich The Ludwig Maximilian University of Munich (simply University of Munich or LMU; german: Ludwig-Maximilians-Universität München) is a public research university in Munich, Germany. It is Germany's List of universities in Germany, sixth-oldest u ...

, he met Carathéodory who pointed him towards two important texts,

Van der Waerden Bartel Leendert van der Waerden (; 2 February 1903 – 12 January 1996) was a Dutch mathematician and historian of mathematics. Biography Education and early career Van der Waerden learned advanced mathematics at the University of Amsterd ...

and Speiser on

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

. 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 The Society of Fellows is a group of scholars selected at the beginnings of their careers by Harvard University for their potential to advance academic wisdom, upon whom are bestowed distinctive opportunities to foster their individual and intell ...

, 1933–36, he spent the rest of his career teaching at Harvard. During the 1930s, Birkhoff, along with his Harvard colleagues

Marshall Stone Marshall Harvey Stone (April 8, 1903 – January 9, 1989) was an American mathematician who contributed to real analysis, functional analysis, topology and the study of Boolean algebras. Biography Stone was the son of Harlan Fiske Stone, who wa ...

and

Saunders Mac Lane Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville ...

, substantially advanced American teaching and research in

. In 1941 he and Mac Lane published ''A Survey of Modern Algebra'', the second undergraduate textbook in English on the subject (

Cyrus Colton MacDuffee Cyrus Colton MacDuffee (June 29, 1895 – August 21, 1961) from Oneida, New York was a professor of mathematics at University of Wisconsin. He wrote a number of influential research papers in abstract algebra. MacDuffee served on the Council of th ...

's ''An Introduction to Abstract Algebra'' was published in 1940). Mac Lane and Birkhoff's ''Algebra'' (1967) is a more advanced text on

. A number of papers he wrote in the 1930s, culminating in his monograph, ''Lattice Theory'' (1940; the third edition remains in print), turned

into a major branch of

. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics,

universal algebra Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures. For instance, rather than take particular groups as the object of study, ...

. Birkhoff's approach to this development of universal

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary a ...

and lattice theory acknowledged prior ideas of

Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism". Educated as a chemist and employed as a scientist for t ...

, Ernst Schröder, and

Alfred North Whitehead Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He is best known as the defining figure of the philosophical school known as process philosophy, which today has found applicat ...

; in fact, Whitehead had written an 1898 monograph entitled ''Universal Algebra''. During and after

World War II World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the vast majority of the world's countries—including all of the great powers—forming two opposin ...

, Birkhoff's interests gravitated towards what he called "engineering" mathematics. During the war, he worked on radar aiming and ballistics, including the

bazooka Bazooka () is the common name for a Man-portable anti-tank systems, man-portable recoilless Anti-tank warfare, anti-tank rocket launcher weapon, widely deployed by the United States Army, especially during World War II. Also referred to as the ...

. In the development of weapons, mathematical questions arose, some of which had not yet been addressed by the literature on

fluid dynamics In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...

. 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 John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...

, 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 In numerical linear algebra, the method of successive over-relaxation (SOR) is a variant of the Gauss–Seidel method for solving a linear system of equations, resulting in faster convergence. A similar method can be used for any slowly converging ...

(SOR) method. Birkhoff then worked with

Richard S. Varga Richard Steven Varga (October 9, 1928 - February 25, 2022) was an American mathematician who specialized in numerical analysis and linear algebra. He was an Emeritus University Professor of Mathematical Sciences at Kent State University and an a ...

, a former student, who was employed at

Bettis Atomic Power Laboratory Bettis Atomic Power Laboratory is a U.S. Government-owned research and development facility in the Pittsburgh suburb of West Mifflin, Pennsylvania, that works exclusively on the design and development of nuclear power for the U.S. Navy. It was one ...

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 operator In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \l ...

s and

iterative method In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived fr ...

s for ''p''-cyclic matrices. Birkhoff's research and consulting work (notably for

General Motors The General Motors Company (GM) is an American Multinational corporation, multinational Automotive industry, automotive manufacturing company headquartered in Detroit, Michigan, United States. It is the largest automaker in the United States and ...

) developed computational methods besides

numerical linear algebra Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which efficiently and accurately provide approximate answers to questions in continuous mathematic ...

, notably the representation of smooth curves via

cubic spline In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form, that is, by its values and first derivatives at the end points of the correspondin ...

s. 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 National Academy of Sciences (NAS) is a United States nonprofit, non-governmental organization. NAS is part of the National Academies of Sciences, Engineering, and Medicine, along with the National Academy of Engineering (NAE) and the Nati ...

, the

American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...

, and the

American Academy of Arts and Sciences The American Academy of Arts and Sciences (abbreviation: AAA&S) is one of the oldest learned societies in the United States. It was founded in 1780 during the American Revolution by John Adams, John Hancock, James Bowdoin, Andrew Oliver, and ...

. He was a

Guggenheim Fellow Guggenheim Fellowships are grants that have been awarded annually since by the John Simon Guggenheim Memorial Foundation to those "who have demonstrated exceptional capacity for productive scholarship or exceptional creative ability in the ar ...

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 Lester is an ancient Anglo-Saxon surname and given name. Notable people and characters with the name include: People Given name * Lester Bangs (1948–1982), American music critic * Lester W. Bentley (1908–1972), American artist from Wisc ...

in 1974.

Selected books

* * *2015 pbk reprint of 1960 2nd edition
* * * * * * *

References

External links

* * {{DEFAULTSORT:Birkhoff, Garrett 1911 births Members of the United States National Academy of Sciences 1996 deaths 20th-century American mathematicians 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 Members of the American Philosophical Society

Life

Selected books

See also

References

External links