Andrew M. Gleason
Andrew Mattei Gleason was an American mathematician who made fundamental contributions to widely varied areas of mathematics, including the solution of Hilbert's fifth problem, and was a leader in reform and innovation in teaching at all levels. Gleason's theorem in quantum logic and the Greenwood–Gleason graph, an important example in Ramsey theory, are named for him.
As a young World War II naval officer, Gleason broke German and Japanese military codes. After the war he spent his entire academic career at Harvard University, from which he retired in 1992. His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and the Harvard Society of Fellows, and presidency of the American Mathematical Society. He continued to advise the United States government on cryptographic security, and the Commonwealth of Massachusetts on education for children, almost until the end of his life.
Gleason won the Newcomb Cleveland Prize in 1952 and the Gung–Hu Distinguished Service Award of the American Mathematical Society in 1996. He was a member of the National Academy of Sciences and of the American Philosophical Society, and held the Hollis Chair of Mathematics and Natural Philosophy at Harvard.
He was fond of saying that proofs "really aren't there to convince you that something is truethey're there to show you why it is true." The Notices of the American Mathematical Society called him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."
Biography
Gleason was born on November 4, 1921, in Fresno, California, the youngest of three children;his father Henry Gleason was a botanist and a member of the Mayflower Society, and his mother was the daughter of Swiss-American winemaker Andrew Mattei.
His older brother Henry Jr. became a linguist.
He grew up in Bronxville, New York, where his father was the curator of the New York Botanical Garden.
After briefly attending Berkeley High School
he graduated from Roosevelt High School in Yonkers, winning a scholarship to Yale University.
Though Gleason's mathematics education had gone only so far as some self-taught calculus, Yale mathematician William Raymond Longley urged him to try a course in mechanics normally intended for juniors.
One month later he enrolled in a differential equations course as well. When Einar Hille temporarily replaced the regular instructor, Gleason found Hille's style "unbelievably different ... He had a view of mathematics that was just vastly different ... That was a very important experience for me. So after that I took a lot of courses from Hille" including, in his sophomore year, graduate-level real analysis. "Starting with that course with Hille, I began to have some sense of what mathematics is about."
While at Yale he competed three times in the recently founded William Lowell Putnam Mathematical Competition, always placing among the top five entrants in the country.
After the Japanese attacked Pearl Harbor during his senior year, Gleason applied for a commission in the US Navy,
and on graduation
joined the team working to break Japanese naval codes.
He also collaborated with British researchers attacking the German Enigma cipher;
Alan Turing, who spent substantial time with Gleason while visiting Washington, called him "the brilliant young Yale graduate mathematician" in a report of his visit.
In 1946, at the recommendation of Navy colleague Donald Howard Menzel, Gleason was appointed a Junior Fellow at Harvard.
An early goal of the Junior Fellows program was to allow young scholars showing extraordinary promise to sidestep the lengthy PhD process; four years later Harvard appointed Gleason an assistant professor of mathematics,
though he was almost immediately recalled to Washington for cryptographic work related to the Korean War.
He returned to Harvard in the fall of 1952, and soon after published the most important of his results on Hilbert's fifth problem.
Harvard awarded him tenure the following year.
In January 1959 he married Jean Berko
whom he had met at a party featuring the music of Tom Lehrer.
Berko, a psycholinguist, was a professor at Boston University for many years.
They had three daughters.
In 1969 Gleason took the Hollis Chair of Mathematics and Natural Philosophy. Established in 1727, this is the oldest scientific endowed professorship in the US.
He retired from Harvard in 1992 but remained active in service to Harvard
and to mathematics: in particular, promoting the Harvard Calculus Reform Project and working with the Massachusetts Board of Education.
He died on October 17, 2008 from complications following surgery.
Teaching and education reform
Gleason said he "always enjoyed helping other people with math"a colleague said he "regarded teaching mathematicslike doing mathematicsas both important and also genuinely fun."At fourteen, during his brief attendance at Berkeley High School, he found himself not only bored with first-semester geometry, but also helping other students with their homeworkincluding those taking the second half of the course, which he soon began auditing.
At Harvard he "regularly taught at every level", including administratively burdensome multisection courses.
One class presented Gleason with a framed print of Picasso's Mother and Child in recognition of his care for them.
In 1964 he created "the first of the 'bridge' courses now ubiquitous for math majors, only twenty years before its time." Such a course is designed to teach new students, accustomed to rote learning of mathematics in secondary school, how to reason abstractly and construct mathematical proofs. That effort led to publication of his Fundamentals of Abstract Analysis, of which one reviewer wrote:
But Gleason's "talent for exposition" did not always imply that the reader would be enlightened without effort of his own. Even in a wartime memo on the urgently important decryption of the German Enigma cipher, Gleason and his colleagues wrote:
His notes and exercises on probability and statistics, drawn up for his lectures to code-breaking colleagues during the war remained in use in National Security Agency training for several decades; they were published openly in 1985.
In a 1964 Science article, Gleason wrote of an apparent paradox arising in attempts to explain mathematics to nonmathematicians:
Gleason was the first chairman of the advisory committee of the School Mathematics Study Group, which helped define the New Math of the 1960sambitious changes in American elementary and high school mathematics teaching emphasizing understanding of concepts over rote algorithms. Gleason was "always interested in how people learn"; as part of the New Math effort he spent most mornings over several months with second-graders. Some years later he gave a talk in which he described his goal as having been:
In 1986 he helped found the Calculus Consortium, which has published a successful and influential series of "calculus reform" textbooks for college and high school, on precalculus, calculus, and other areas. His "credo for this program as for all of his teaching was that the ideas should be based in equal parts of geometry for visualization of the concepts, computation for grounding in the real world, and algebraic manipulation for power." However, the program faced heavy criticism from the mathematics community for its omission of topics such as the mean value theorem, and for its perceived lack of mathematical rigor.
Cryptanalysis work
During World War II Gleason was part of OP-20-G, the U.S. Navy's signals intelligence and cryptanalysis group.One task of this group, in collaboration with British cryptographers at Bletchley Park such as Alan Turing, was to penetrate German Enigma machine communications networks. The British had great success with two of these networks, but the third, used for German-Japanese naval coordination, remained unbroken because of a faulty assumption that it employed a simplified version of Enigma. After OP-20-G's Marshall Hall observed that certain metadata in Berlin-to-Tokyo transmissions used letter sets disjoint from those used in Tokyo-to-Berlin metadata, Gleason hypothesized that the corresponding unencrypted letters sets were A-M and N-Z, then devised novel statistical tests by which he confirmed this hypothesis. The result was routine decryption of this third network by 1944.
OP-20-G then turned to the Japanese navy's "Coral" cipher. A key tool for the attack on Coral was the "Gleason crutch", a form of Chernoff bound on tail distributions of sums of independent random variables. Gleason's classified work on this bound predated Chernoff's work by a decade.
Toward the end of the war he concentrated on documenting the work of OP-20-G and developing systems for training new cryptographers.
In 1950 Gleason returned to active duty for the Korean War, serving as a Lieutenant Commander in the Nebraska Avenue Complex. His cryptographic work from this period remains classified, but it is known that he recruited mathematicians and taught them cryptanalysis.
He served on the advisory boards for the National Security Agency and the Institute for Defense Analyses, and he continued to recruit, and to advise the military on cryptanalysis, almost to the end of his life.
Mathematics research
Gleason made fundamental contributions to widely varied areas of mathematics, including the theory of Lie groups, quantum mechanics,and combinatorics.
According to Freeman Dyson's famous classification of mathematicians as being either birds or frogs,
Gleason was a frog: he worked as a problem solver rather than a visionary formulating grand theories.
Hilbert's fifth problem
In 1900 David Hilbert posed 23 problems he felt would be central to next century of mathematics research. Hilbert's fifth problem concerns the characterization of Lie groups by their actions on topological spaces: to what extent does their topology provide information sufficient to determine their geometry?The "restricted" version of Hilbert's fifth problem asks, more specifically, whether every locally Euclidean topological group is a Lie group. That is, if a group G has the structure of a topological manifold, can that structure be strengthened to a real analytic structure, so that within any neighborhood of an element of G, the group law is defined by a convergent power series, and so that overlapping neighborhoods have compatible power series definitions? Prior to Gleason's work, special cases of the problem had been solved by L. E. J. Brouwer, John von Neumann, Lev Pontryagin, and Garrett Birkhoff, among others.
Gleason's interest in the fifth problem began in the late 1940s, sparked by a course he took from George Mackey.
In 1949 he published a paper introducing the "no small subgroups" property of Lie groups that would eventually be crucial to its solution.
His 1952 paper on the subject, together with a paper published concurrently by Deane Montgomery and Leo Zippin, solves affirmatively the restricted version of Hilbert's fifth problem, showing that indeed every locally Euclidean group is a Lie group. Gleason's contribution was to prove that this is true when G has the no small subgroups property; Montgomery and Zippin showed every locally Euclidean group has this property. As Gleason told the story, the key insight of his proof was to apply the fact that monotonic functions are differentiable almost everywhere. On finding the solution, he took a week of leave to write it up, and it was printed in the Annals of Mathematics alongside the paper of Montgomery and Zippin; another paper a year later by Hidehiko Yamabe removed some technical side conditions from Gleason's proof.
The "unrestricted" version of Hilbert's fifth problem, closer to Hilbert's original formulation, considers both a locally Euclidean group G and another manifold M on which G has a continuous action. Hilbert asked whether, in this case, M and the action of G could be given a real analytic structure. It was quickly realized that the answer was negative, after which attention centered on the restricted problem. However, with some additional smoothness assumptions on G and M, it might yet be possible to prove the existence of a real analytic structure on the group action. The Hilbert–Smith conjecture, still unsolved, encapsulates the remaining difficulties of this case.