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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.
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''.
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.<ref>{{cite book|last=Wolfram|first=Stephen|title=A New Kind of Science|publisher=Wolfram Media, Inc.|year=2002|page=1172|isbn=1-57955-008-8}}</ref>


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).
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).

Revision as of 16:34, 10 July 2018

Garrett Birkhoff
Cambridge University
Harvard University
Known for
Awards
Scientific career
FieldsMathematics
InstitutionsHarvard University
Academic advisors
Doctoral students
Other notable studentsRichard S. Varga

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

lattice theory
.

The mathematician

George Birkhoff
(1884–1944) was his father.

Life

The son of the mathematician

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

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

positive operators and iterative methods
for p-cyclic matrices.

Birkhoff's research and consulting work (notably for

cubic splines
.

Birkhoff published more than 200 papers and supervised more than 50 Ph.D.s. He was a member of the

Lester R. Ford Award in 1974.[3]

Selected books

See also

References

  1. ^ Staff. A COMMUNITY OF SCHOLARS: The Institute for Advanced Study Faculty and Members 1930-1980, p. 90. Institute for Advanced Study, 1980. Accessed November 20, 2015. "Birkhoff, Garrett 40s M Born 1911 Princeton, NJ."
  2. ISBN 1-57955-008-8
    .
  3. doi:10.2307/2318163
    .
  4. doi:10.1090/S0002-9904-1941-07409-4
    .
  5. doi:10.1090/S0002-9904-1942-07670-1
    .
  6. doi:10.1090/s0002-9904-1972-12908-2
    .

External links
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