George David Birkhoff, well known mathematician, who made significant and substantial contributions both as a theorist and as a mentor (46 students in 30 years) during the first half of the XXth century. His contributions were so numerous it is hard to decide what he is best remembered for, like an American

David Hilbert, but perhaps it is his work in dynamic systems (1927) and then his

aha! of the

ergodic theorem in 1931-32 which provided the

giant's shoulders for others to stand on. He had, in 1913, at an age when many young people are still trying to find themselves (their talents, and their ambitions), taken

Poincaré's Last Geometric Problem and extended the solution beyond Poincaré's resolution of a few special cases. Analysis, the mathematics underpinning the physics of

relativity and

electricity and not just

Maxwell-

Boltzmann and their

gases, and work on the

four-color combinatoric problem dear to

topologists all interested him. Brilliant

theorist, he, like many others said to live in an

Ivory Tower, may have sought to generalize his "wisdom" beyond its domain of applicability, and not noticed cases where his intuition was biased.

Born: 21 March 1884 in Overisel, Michigan, USA

Died: 12 Nov 1944 in Cambridge, Massachusetts, USA

Awarded Ph.D. by the University of Chicago in 1907, for his dissertation: *Asymptotic Properties of Certain Ordinary Differential Equations with Applications to Boundary Value and Expansion Problems *; Advisor, E. H. Moore.

Incidentally, Moore also had as student Oswald Veblen (Ph.D. 1903, for *A System of Axioms for Geometry*), who went on to a brilliant career at Princeton, where he counted among his student Alonzo Church, a major figure in symbolic logic.)

After studies begun at the University of Chicago, continued at Harvard, and culminating in a Ph.D. from the University of Chicago, Birkhoff took a job teaching at the University of Wisconsin at Madison. While there, in 1908, he married Margaret Elizabeth Grafius. Then he spent a year as a preceptor in mathematics at Princeton University (New Jersey) and was promoted to professor in 1911. Off to Harvard beginning 1911-12 (he was then 27) as assistant professor, promoted to full professor at 35 (1919).

##### Meanwhile,

For perspective, other mathematico-scientific activity in these years and those just following included:

##### Back to Birkhoff

During this period of intense activity developing the foundations and links of probability and the likelihood understanding of it, statistics, functional analysis and measure theory, and physics, Birkhoff was in the midst of it all, synthesizing and seeking unification. For instance, Birkhoff supervised Ben Zion Linfield, whose 1923 dissertation treated : *On the Theory of Discrete Varieties*, as his advisor; then David Vernon Widder, whose 1924 thesis treated * Theorems of Mean Value and Trigonometric Interpolation*, and David Gordon Bourgin, whose 1926 thesis treated *Intensities of the Lines of the Hydrogen Chloride Fundamental Band *: all three clearly related to statistical and measurement problems, physical problems, or both.

He had a long association with the American Mathematical Society being
Vice-President in 1919, Colloquium lecturer in 1920 (when he lectured on Dynamical Systems), editor of the *Transactions of the American Mathematical Society* from 1921 to 1924. In 1923 Birkhoff received the first award of the Bôcher Memorial Prize (Maxime Bocher was his colleague at Harvard) from the American Mathematical Society for his memoir, *Dynamical systems with two degrees of freedom *which he had published in the Transactions of the American Mathematical Society in 1917. He was AMS President from 1925 to 1926.

Much like its cousin entropy, ergodic theory (1932) was first developed to help make sense of problems in thermodynamics and dynamic systems of a physical sort, particularly the statistical mechanics of J. Willard Gibbs which needed a coherent and reliable way to relate phase averages and time averages, but then was found applicable in the frontier territory of communications. Mapped out by the explorers of electrical engineering (under wraps during WWII), Norbert Wiener's time series analysis, cybernetics, and communications theory, in turn provided background and inspiration to Claude Shannon who, with Warren Weaver, developed *the mathematical theory of communication* with entropy as a significant feature.

Birkhoff and wife Margaret had three children, of whom one, Garrett, became a significant mathematician, too.

According to biographers *J J O'Connor and E F Robertson* of University of St Andrews, Scotland,

He developed a mathematical theory of aesthetics which he applied to art, music and poetry. Before writing Aesthetic measure he spent a year travelling round the world studying art, music and poetry and various countries. This is referred to in the citation
He has told us that the formal structure of western music, the riddle of melody, began to interest him in undergraduate days; somewhat intense consideration of the mathematical elements here involved led him to apply his theory also to aesthetic objects such as polygons, tilings, vases, and even poetry.

He was also perceived by

Einstein and others to be

*anti-semite*.

Jointly with O. D. Kellogg he published the monograph *Invariant points in function space* (1922), and with R. E. Langer, the monograph *Relativity and Modern Physics* in 1923. Other publications included:

- Dynamical Systems (1927),
- Aesthetic Measure (1933),
- On the combination, of Topologies (1936)
- Basic Geometry (1941).

Remarkable world-wide recognition for Birkhoff's outstanding contribution shown be his election to:

- the National Academy of Sciences,
- American Philosophical Society,
- American Academy of Arts and Sciences,
- Académie des Sciences in Paris,
- Accademia dei Lincei,
- Pontifical Academy,
- Circolo Matematico di Palermo,
- Royal Danish Academy of Sciences and Letters,
- Göttingen Academy of Sciences,
- Royal Institute of Bologna,
- Edinburgh Mathematical Society,
- London Mathematical Society, and
- National Academy of Sciences of Lima, Peru.

##### Acknowledgements

*Mathematics Genealogy Project* now centered at the Department of Mathematics, North Dakota State University, and
- the
* Mathematicians biographies* database developed by the School of Mathematics and Statistics, University of St Andrews, Scotland.