Garrett Birkhoff
Encyclopedia
Garrett Birkhoff was an American
mathematician
. He is best known for his work in lattice theory.
The mathematician George Birkhoff (1884–1944) was his father.
, Garrett 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 a 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 method
s for p-cylic 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
.
United States
The United States of America is a federal constitutional republic comprising fifty states and a federal district...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
. 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 BirkhoffGeorge David Birkhoff
-External links:* − from National Academies Press, by Oswald Veblen....
, Garrett began the Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
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
England
England is a country that is part of the United Kingdom. It shares land borders with Scotland to the north and Wales to the west; the Irish Sea is to the north west, the Celtic Sea to the south west, with the North Sea to the east and the English Channel to the south separating it from continental...
to study mathematical physics
Mathematical physics
Mathematical physics refers to development of mathematical methods for application to problems in physics. The Journal of Mathematical Physics defines this area as: "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and...
but switched to studying abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
under Philip Hall
Philip Hall
Philip Hall FRS , was an English mathematician.His major work was on group theory, notably on finite groups and solvable groups.-Biography:...
. While visiting the University of Munich, he met Carathéodory
Constantin Carathéodory
Constantin Carathéodory was a Greek mathematician. He made significant contributions to the theory of functions of a real variable, the calculus of variations, and measure theory...
who pointed him towards two important texts, Van der Waerden
Bartel Leendert van der Waerden
Bartel Leendert van der Waerden was a Dutch mathematician and historian of mathematics....
on abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
and Speiser on group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
.
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
Saunders Mac Lane
Saunders Mac Lane was an American mathematician who cofounded category theory with Samuel Eilenberg.-Career:...
, substantially advanced American teaching and research in abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. 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
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. 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
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...
. His 1935 paper, "On the Structure of Abstract Algebras" founded a new branch of mathematics, universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
. Birkhoff's approach to this development of universal algebra and lattice theory acknowledged prior ideas of Charles Sanders Peirce, Ernst Schröder
Ernst Schröder
Ernst Schröder was a German mathematician mainly known for his work on algebraic logic. He is a major figure in the history of mathematical logic , by virtue of summarizing and extending the work of George Boole, Augustus De Morgan, Hugh MacColl, and especially Charles Peirce...
, and Alfred North Whitehead
Alfred North Whitehead
Alfred North Whitehead, OM FRS was an English mathematician who became a philosopher. He wrote on algebra, logic, foundations of mathematics, philosophy of science, physics, metaphysics, and education...
; in fact, Whitehead had written a 1898 monograph entitled Universal Algebra.
During and after World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
, Birkhoff's interests gravitated towards what he called "engineering" mathematics. During the war, he worked on radar aiming and ballistics, including the bazooka
Bazooka
Bazooka is the common name for a man-portable recoilless rocket antitank weapon, widely fielded by the U.S. Army. Also referred to as the "Stovepipe", the innovative bazooka was amongst the first-generation of rocket propelled anti-tank weapons used in infantry combat...
. 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
John von Neumann
John von Neumann was a Hungarian-American mathematician and polymath who made major contributions to a vast number of fields, including set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics and game theory, computer science, numerical analysis,...
, took a close interest in the rise of the electronic computer. Birkhoff supervised the Ph.D. thesis of David M. Young
David M. Young, Jr.
David M. Young, Jr. was an American mathematician and computer scientist who was one of the pioneers in the field of modern numerical analysis/scientific computing....
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
Richard S. Varga
Richard Steven Varga is an American mathematician who specializes in numerical analysis and linear algebra. He is currently a University Professor of Mathematical Sciences at Kent State University, Kent, Ohio...
, a former student, who was employed at Bettis Atomic Power Laboratory
Bettis Atomic Power Laboratory
Bettis Atomic Power Laboratory is a U.S. Government-owned, contractor-operated research and development facility located in the Pittsburgh suburb of West Mifflin, Pennsylvania. It solely focuses on the design and development of nuclear power for the U.S. Navy....
of the Westinghouse Electronic Corporation
Westinghouse Electric (1886)
Westinghouse Electric was an American manufacturing company. It was founded in 1886 as Westinghouse Electric Company and later renamed Westinghouse Electric Corporation by George Westinghouse. The company purchased CBS in 1995 and became CBS Corporation in 1997...
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 method
Iterative method
In computational mathematics, an iterative method is a mathematical procedure that generates a sequence of improving approximate solutions for a class of problems. A specific implementation of an iterative method, including the termination criteria, is an algorithm of the iterative method...
s for p-cylic 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
United States National Academy of Sciences
The National Academy of Sciences is a corporation in the United States whose members serve pro bono as "advisers to the nation on science, engineering, and medicine." As a national academy, new members of the organization are elected annually by current members, based on their distinguished and...
and the American Academy of Arts and Sciences
American Academy of Arts and Sciences
The American Academy of Arts and Sciences is an independent policy research center that conducts multidisciplinary studies of complex and emerging problems. The Academy’s elected members are leaders in the academic disciplines, the arts, business, and public affairs.James Bowdoin, John Adams, and...
.
See also
- Birkhoff polytope
- Birkhoff's representation theoremBirkhoff's representation theoremIn mathematics, Birkhoff's representation theorem for distributive lattices states that the elements of any finite distributive lattice can be represented as finite sets, in such a way that the lattice operations correspond to unions and intersections of sets...
- Birkhoff's HSP theorem
- Birkhoff's theoremBirkhoff's theoremBirkhoff's theorem may refer to several theorems named for the American mathematician George David Birkhoff:* Birkhoff's theorem * Birkhoff's theorem * Birkhoff's ergodic theorem...
s - Pierce–Birkhoff conjecturePierce–Birkhoff conjectureIn abstract algebra, the Pierce–Birkhoff conjecture asserts that any piecewise-polynomial function can be expressed as a maximum of finite minima of finite collections of polynomials. It was first stated, albeit in non-rigorous and vague wording, in the 1956 paper of Garrett Birkhoff and Richard S....