George David Birkhoff's contributions as a theoretical mathematician, a teacher, and a member of the international scientific community rank him as one of the foremost mathematicians of the 20th century. He made extensive contributions to the area of differential equations and continued the work of the great French mathematician Jules Henri Poincaré on celestial mechanics. He is considered the founder of the modern theory of dynamical systems.
Born in Overisel, Michigan, on March 21, 1884, Birkhoff was the eldest of six children born to David Birkhoff, a physician, and Jane Gertrude Droppers. When Birkhoff was two years old, his family moved to Chicago, where he spent most of his childhood. From 1896 to 1902 Birkhoff studied at the Lewis Institute (now the Illinois Institute of Technology). Following a year at the University of Chicago as an undergraduate, Birkhoff transferred to Harvard University in 1903. In 1904, while still an undergraduate, he wrote his first mathematics paper, on number theory. He earned a bachelor's degree at Harvard in 1905 and a master's degree in 1906. Returning to the University of Chicago for his doctorate, Birkhoff wrote a dissertation on differential equations under the guidance of Eliakim Hastings Moore. He was awarded a doctorate summa cum laude in 1907.
Birkhoff taught mathematics at the University of Wisconsin from 1907 to 1909, when he took a position as assistant professor at Princeton University. He joined the faculty of Harvard University in 1912, teaching there until his death in 1944. Birkhoff, though not a great lecturer, was an inspiring teacher. Many of the influential American mathematicians of the mid-twentieth century, including Marston Morse and Marshall Stone, studied with Birkhoff at the doctoral or post-doctoral level. Six of his former students went on to become members of the National Academy of Sciences. From 1935 to 1939 Birkhoff also served as Dean of the Faculty of Arts and Science at Harvard.
The single most important influence on Birkhoff's mathematical research was that of Poincaré. The two never met, but Birkhoff studied Poincaré's work and adopted some of the problems in differential equations and celestial mechanics Poincaré left behind at his death in 1912. In 1913 Birkhoff first attracted international attention by proving a geometrical theorem that Poincaré had proposed but not proved in his last published paper. Birkhoff's accomplishment marked a special advance in solving the problem of three bodies. The three-body problem of celestial mechanics concerns trajectories and orbits of bodies moving in systems in such a way that each body affects the motion of the others.
From Poincaré's theorem, Birkhoff went on to consider the entire field of dynamical systems and made contributions that have become fundamental to this branch of mathematics. In his book Dynamical Systems (1927), Birkhoff wrote that "the final aim of the theory of motions of a dynamical system must be directed towards the qualitative determination of all possible types of motions and the interrelation of these motions." Using ideas developed by Poincaré, Birkhoff laid the foundations for the topological theory of dynamical systems by defining and classifying possible types of dynamic motions. Another signal achievement came in 1931, when Birkhoff offered further proof of the so-called ergodic theorem, which demonstrates the conditions needed for the behavior of a large dynamical system, such as a container of a gas, to reach equilibrium. This problem had baffled scientists for more than 50 years.
Birkhoff's journal articles and books reflect the breadth of his talent and the diversity of his interests. Among his published works is a basic geometry text that for many years formed the basis of high-school geometry curricula. Birkhoff also wrote extensively about the theory of relativity and quantum mechanics. Although his ideas in this field are not widely accepted, the mathematical tools that he developed for his approach play an important role in modern relativity theory. Birkhoff's other work encompasses number theory and point-set theory and the famous four-color problem, which is concerned with the possibility of coloring any map using only four colors. In 1933, his life-long passion for art, music, and poetry led him to write Aesthetic Measure, in which he attempted to create a general mathematical theory of the fine arts, starting out from the Pythagorean notion that beauty is mathematical in nature. He later extended the theory to ethics. Birkhoff was also the editor of several mathematical journals, including the Annals of Mathematics,Transactions of the American Mathematical Society, and the American Journal of Mathematics.
During his lifetime, Birkhoff received many honors and honorary degrees from universities worldwide. Among others, he was awarded the Querini-Stampalia prize of the Royal Institute of Science, Letters and Arts, Venice (1918); the Bôcher prize of the American Mathematical Society (1923) for his research in dynamics; the annual prize of the American Association for the Advancement of Science (1926); and the biennial prize of the Pontifical Academy of Sciences (1933) for his research on systems of differential equations. Birkhoff was elected to membership in the National Academy of Sciences (1918), the American Philosophical Society, and the American Academy of Arts and Sciences. He was made an officer of the French Legion of Honor in 1936, and was an honorary member of the Edinburgh Mathematical Society, the London Mathematical Society, the Peruvian Philosophic Society, and the Scientific Society of Argentina.
Birkhoff was fluent in French, the language of his famous mathematical predecessor, Poincaré, and presented several of his fundamental papers in that language. He traveled widely, promoting his belief in international fellowship among scientists. Because of his preeminence in research, he was able to represent mathematics in international scientific circles, and he played an important role in the creation of the mathematical institutes at Göttingen and Paris after World War I.
Birkhoff married Margaret Elizabeth Grafius of Chicago on September 2, 1908, and they had three children: Barbara, Garrett, and Rodney. Garrett went on to become a professor of mathematics at Harvard. Birkhoff died of a heart attack on November 12, 1944, in Cambridge, Massachusetts.
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