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Andrew Gleason
Berlin, 1959
Born	(1921-11-04)November 4, 1921 Fresno, California
Died	October 17, 2008(2008-10-17) (aged 86)
Nationality	American
Alma mater	Yale University
Known for	Hilbert's fifth problem Gleason's theorem Greenwood–Gleason graph
Spouse	Jean Berko Gleason
Awards	Newcomb Cleveland Prize (1952) Gung-Hu Distinguished Service to Mathematics Award (1996)
Scientific career
Fields	mathematics cryptography
Institutions	Harvard University
Doctoral advisor	none
Other academic advisors	George Mackey
Doctoral students	James Eells Jessie MacWilliams Richard Palais Joel Spencer

Andrew Mattei Gleason (November 4, 1921 – October 17, 2008) was an American mathematician. As a young World War II naval officer, he helped break Japanese and German military codes. Then over the succeeding sixty years he 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 mathematics teaching at all levels. He spent his entire academic career at Harvard, from which he retired in 1992 as the Hollis Professor of Mathematics and Natural Philosophy. Gleason's theorem and the Greenwood–Gleason graph are named for him.

Gleason's leadership posts in academic and scholarly societies included being Chairman of the Harvard Society of Fellows and President of the American Mathematical Society. He continued to advise the United States government on cryptographic security, and the Commonwealth of Massachusetts on mathematics education for children, almost until the end of his life. 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."

He was fond of saying that mathematical proofs "really aren't there to convince you that something is true—they're there to show you why it is true."

Biography

Gleason was born 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.

He briefly attended Berkeley High School in Berkeley, California where, bored with his geometry class, he simultaneously audited a more advanced course. He graduated from Roosevelt High School in Yonkers, then did his undergraduate studies at Yale University, where he took many mathematics classes from Einar Hille including graduate-level real analysis in his sophomore year. At Yale he competed in the recently-founded William Lowell Putnam Mathematical Competition (1940, 1941 and 1942), each time placing among the top five in the country and thereby becoming a three-time Putnam Fellow.

After the Japanese attacked Pearl Harbor during his final year at Yale, Gleason applied for a commission in the US Navy, and on graduation joined the team which broke Japanese naval codes. (Others on this team included his future collaborator Robert E. Greenwood and Yale professor Marshall Hall, Jr.) 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 in his series of results on Hilbert's fifth problem (see below). He received a master's degree, and tenure, the following year.

In January 1959 he married Jean Berko, a graduate student at Radcliffe College whom he had met at a party featuring the music of another Harvard mathematician, Tom Lehrer. (Jean Berko Gleason is a noted researcher in psycholinguistics at Boston University.) They had three daughters.

In 1969 Gleason took the Hollis Chair of Mathematics and Natural Philosophy, the oldest (est. 1727) scientific endowed professorship in the US. He retired from Harvard in 1992 but remained active in service to Harvard (as chair of the Society of Fellows, for example) and to mathematics: in particular, promoting the Harvard Calculus Reform Project and working with the Massachusetts Board of Education.

He died in 2008 from complications following surgery.

Awards and honors

In 1952 Gleason was awarded the Newcomb Cleveland Prize by the American Association for the Advancement of Science. He was elected to the National Academy of Sciences and the American Philosophical Society, and was a Fellow of the American Academy of Arts and Sciences.

In 1981 and 1982 he was president of the American Mathematical Society, and at various times held numerous other posts in professional and scholarly organizations, including chairmanship of the Harvard Department of Mathematics. In 1986 he chaired the organizing committee for the International Congress of Mathematicians in Berkeley, California, and was president of the Congress.

In 1996 a Harvard Society of Fellows symposium honored him on his retirement after seven years as its chairman; that same year, the Mathematics Association of America awarded him the Yueh-Gin Gung and Dr. Charles Y. Hu Distinguished Service to Mathematics Award. A past president of the Association wrote:

In thinking about, and admiring, Andy Gleason's career, your natural reference is the total profession of a mathematician: designing and teaching courses, advising on education at all levels, doing research, consulting for the users of mathematics, acting as a leader of the profession, cultivating mathematical talent, and serving one's institution. Andy Gleason is that rare individual who has done all of these superbly.

After his death a 32-page collection of essays in the Notices of the American Mathematical Society recalled "the life and work of eminent American mathematician", calling him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."

Cryptanalysis

During World War II, Gleason worked for OP-20-G, the signals intelligence and cryptanalysis group of the U.S. Navy.

One of the group's tasks in this period (in collaboration with Alan Turing and the other British cryptographers of Bletchley Park) was to decrypt transmissions in three separate German communication networks using Enigma machine ciphers. The British cryptographers had failed to break one of these networks, used by the German navy to communicate with their Japanese attaché, because of a faulty assumption that it was using a simplified version of the Enigma machine. Analyses made by the OP-20-G group refuted this assumption and led to the eventual decryption of this communication channel. After Marshall Hall observed that certain metadata associated with each transmission used disjoint sets of letters for Berlin-to-Tokyo and Tokyo-to-Berlin messages, with these sets changing each day, Gleason hypothesized that the corresponding unencrypted letters fell into the ranges A-M (in one direction) and N-Z (in the other), then confirmed this hypothesis with novel statistical tests. This work led to routine decryption of this network by 1944. The work of Gleason and the others in this group also involved deeper mathematics related to permutation groups and the graph isomorphism problem.

After this success, the interests of OP-20-G shifted to the "Coral" cipher used by the Japanese navy. One of the key tools used by the group in this effort was the "Gleason crutch", a form of the Chernoff bound on tail distributions of sums of independent random variables predating the work of Chernoff on the same problem.

During the course of his war work, Gleason wrote up many lecture notes and exercises on probability and statistics for the OP-20-G group. These documents continued to be used by the National Security Agency for years afterwards, and were eventually published openly in 1985. Towards the end of the war he became more directly involved in documenting the work of OP-20-G and in developing systems to train new cryptographers.

In 1950, Gleason returned to the Navy for the Korean War, serving as a Lieutenant Commander in their Nebraska Avenue Complex (much later to become the home of the DHS Cyber Security Division). Although his cryptographic work from this period remains classified, he also worked to recruit many mathematicians into the project, and taught the new recruits how to become cryptanalysts. He continued to advise the military on cryptanalysis after this time, serving on the NSA Scientific Advisory Board and the advisory committee for the Communications Research Division of the Institute for Defense Analyses, and working as a recruiter for both NSA and IDA CRD.

Mathematical 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 a famous speech at the International Congress of Mathematicians in 1900, David Hilbert posed 23 questions for the next century of research. Some of Hilbert's problems were solved quickly; others remain unsolved. Hilbert's fifth problem concerns the characterization of Lie groups by their actions on topological spaces: to what extent does their topology provide enough information to determine their geometry? The "restricted" version of Hilbert's fifth problem (solved by Gleason) 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 earlier, by Luitzen Egbertus Jan 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, and in 1949 he published a paper introducing the "no small subgroups" property of Lie groups (the existence of a neighborhood of the identity within which no nontrivial subgroup exists) that would eventually be crucial to its solution. His 1952 paper on the subject, together with a paper from the same time by Deane Montgomery and Leo Zippin, solves affirmatively the restricted version of Hilbert's problem, and proves that indeed every locally Euclidean group is a Lie group. Gleason's part in the solution 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.

Quantum mechanics

In 1956, Gleason and Harvard visitor Richard Kadison audited a class by George Mackey on quantum mechanics. The Born rule in quantum mechanics states that an observable property of a quantum system is defined by a Hermitian operator on a separable Hilbert space, that the only observable values of the property are the eigenvalues of the operator, and that the probability of being observed in a particular eigenvalue is the square of the absolute value of the complex number obtained by projecting the state vector (a point in the Hilbert space) onto the corresponding eigenvector. Mackey had asked earlier whether Born's rule is a necessary consequence of a particular set of axioms for quantum mechanics, and more specifically whether every measure on the lattice of projections of a Hilbert space can be defined by a positive operator with unit trace. This question caught the attention of both Gleason and Kadison. Kadison showed that the answer is false for two-dimensional Hilbert spaces; Gleason showed that it is true for higher dimensions, a result that he published in 1957 and that is now known as Gleason's theorem.

Gleason's theorem implies the nonexistence of certain types of hidden variable theories for quantum mechanics, strengthening a previous argument of John von Neumann. Von Neumann had claimed to show that hidden variable theories were impossible, but (as Grete Hermann pointed out) his demonstration made an assumption that quantum systems obeyed a form of additivity of expectation for noncommuting operators that might not hold a priori. In 1966, John Stewart Bell showed that Gleason's theorem could be used to remove this extra assumption from von Neumann's argument.

Ramsey theory

The Ramsey number R(k,l) is the smallest number r such that every graph with at least r vertices contains either a k-vertex clique or an l-vertex independent set. Ramsey numbers require enormous effort to compute; when max(k,l) ≥ 3 only finitely many of them are known precisely, and an exact computation of R(6,6) is believed to be out of reach. In 1953, the calculation of R(3,3) was given as a question in the Putnam Competition; in 1955, motivated by this problem, Gleason and his co-author Robert M. Greenwood made significant progress in the computation of Ramsey numbers with their proof that R(3,4) = 9, R(3,5) = 14, and R(4,4) = 18. Since then, only five more of these values have been found.

In the same 1955 paper, Greenwood and Gleason also computed the multicolor Ramsey number R(3,3,3): the smallest number r such that, if a complete graph on r vertices has its edges colored with three colors, then it necessarily contains a monochromatic triangle. As they showed, R(3,3,3) = 17; this remains the only nontrivial multicolor Ramsey number whose exact value is known. As part of their proof, they used an algebraic construction to show that a 16-vertex complete graph can be decomposed into three disjoint copies of a triangle-free 5-regular graph with 16 vertices and 40 edges; One of the names of this graph is the Greenwood–Gleason graph, after Greenwood and Gleason's use of it in Ramsey theory.

Ronald Graham writes that the paper by Greenwood and Gleason "is now recognized as a classic in the development of Ramsey theory". In the late 1960s, Gleason became the doctoral advisor of Joel Spencer, who also became known for his contributions to Ramsey theory.

Other areas

Gleason founded the theory of Dirichlet algebras, and made other mathematical contributions including work on coding theory and the enumerative combinatorics of permutations As well, he was not above publishing research in more elementary mathematics, such as the derivation of the set of polygons that can be constructed with compass, straightedge, and an angle trisector.

Selected publications

Research papers

• Gleason, A. M. (1952), "One-parameter subgroups and Hilbert's fifth problem", Proceedings of the [[International Congress of Mathematicians]], Cambridge, Mass., 1950, Vol. 2 (PDF), Providence, R. I.: American Mathematical Society, pp. 451–452, MR 0043788 {{citation}}: URL–wikilink conflict (help)
• Greenwood, R. E.; Gleason, A. M. (1955), "Combinatorial relations and chromatic graphs", Canadian Journal of Mathematics, 7: 1–7, doi:10.4153/CJM-1955-001-4, MR 0067467.
• Gleason, Andrew M. (1956), "Finite Fano planes", American Journal of Mathematics, 78: 797–807, doi:10.2307/2372469, MR 0082684.
• Gleason, Andrew M. (1957), "Measures on the closed subspaces of a Hilbert space", Journal of Mathematics and Mechanics, 6: 885–893, MR 0096113.
• Gleason, Andrew M. (1958), "Projective topological spaces", Illinois Journal of Mathematics, 2: 482–489, MR 0121775, Zbl 0083.17401.
• Gleason, Andrew M. (1967), "A characterization of maximal ideals", Journal d'Analyse Mathématique, 19: 171–172, MR 0213878.
• Gleason, Andrew M. (1971), "Weight polynomials of self-dual codes and the MacWilliams identities", Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3, Paris: Gauthier-Villars, pp. 211–215, MR 0424391.

Books

• Gleason, Andrew M. (1966), Fundamentals of Abstract Analysis, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., MR 0202509. Corrected reprint, Boston: Jones and Bartlett, 1991, MR1140189.
• Gleason, Andrew M.; Greenwood, Robert E.; Kelly, Leroy Milton (1980), The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938–1964, Mathematical Association of America, ISBN 9780883854624, MR 0588757.
• Gleason, Andrew M.; Penney, Walter F.; Wyllys, Ronald E. (1985), Elementary Course in Probability for the Cryptanalyst, Laguna Hills, CA: Aegean Park Press. Unclassified reprint of a book originally published in 1957 by the National Security Agency, Office of Research and Development, Mathematical Research Division.
• Gleason, Andrew M.; Hughes-Hallett, Deborah (1994), Calculus, Wiley. Since its original publications this book has been extended to many different editions and variations with additional co-authors.

Film

• Gleason, Andrew M. (1966), Nim and other oriented-graph games, Mathematical Association of America. 63 minutes, black & white. Produced by Richard G. Long and directed by Allan Hinderstein.

Notes

1. "Although Andy never earned a Ph.D., he thought of George as his mentor and advisor and lists himself as George's student on the Mathematics Genealogy Project website."

References

1. ^ Palais, Richard (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1243–1248 {{citation}}: |contribution= ignored (help).
2. ^ O'Connor, John J.; Robertson, Edmund F., "Andrew Mattei Gleason", MacTutor History of Mathematics Archive, University of St Andrews
3. ^ Castello, Caitlin (October 20, 2008), "Andrew Gleason; helped solve vexing geometry problem" (subscription required), Boston Globe.
4. ^ Bolker, Ethan D. (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1237–1239 {{citation}}: |contribution= ignored (help).
5. ^ Albers, Donald J.; Alexanderson, Gerald L.; Reid, Constance, eds. (1990), "Andrew M. Gleason", More Mathematical People, Harcourt Brace Jovanovich, p. 86.
6. ^ Albers, Alexanderson & Reid (1990), pp. 82–99.
7. ^ Gleason, Jean Berko (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1266–1267 {{citation}}: |contribution= ignored (help).
8. "Henry A. Gleason Papers". Mertz Library, New York Botanical Garden. Retrieved April 9, 2013.
9. Gallian, Joseph A., The Putnam Competition from 1938–2011 (PDF).
10. ^ Burroughs, John; Lieberman, David; Reeds, Jim (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1239–1243 {{citation}}: |contribution= ignored (help).
11. ^ Mazur, Barry; Gross, Benedict; Mumford, David (December 2010), "Andrew Gleason, 4 November 1921 – 17 October 2008" (PDF), Proceedings of the American Philosophical Society, 154 (4): 471–476.
12. Walsh, Colleen (May 3, 2012), "The oldest endowed professorship: 1721 gift led to long line of Hollis Chair occupants at Divinity School", Harvard Gazette.
13. ^ Ruder, Debra Bradley (May 9, 1996), "Symposium Will Celebrate Gleason and Society of Fellows", Harvard Gazette.
14. Hallett, Deborah Hughes; Stevens, T. Christine; Tecosky-Feldman, Jeff; Tucker, Thomas (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1260–1265 {{citation}}: |contribution= ignored (help).
15. ^ Pollak, H. O. (February 1996), "Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Andrew Gleason", American Mathematical Monthly, 103 (2): 105–106, JSTOR 2975102.
16. List of Newcomb Cleveland Prize winners
17. The Mathematical Association of America's Yueh-Gin Gung and Dr. Charles Y. Hu Distinguished Service to Mathematics Award.
18. "Features" (PDF), Notices of the American Mathematical Society, 56 (10): 1227, 2009 {{citation}}: Unknown parameter |month= ignored (help).
19. ^ Chernoff, Paul R. (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1253–1259 {{citation}}: |contribution= ignored (help).
20. ^ Spencer, Joel J. (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1251–1253 {{citation}}: |contribution= ignored (help).
21. Dyson, Freeman (February 2009), "Birds and frogs" (PDF), Notices of the American Mathematical Society, 56 (2): 212–223.
22. ^ Illman, Sören (2001), "Hilbert's fifth problem: review", Journal of Mathematical Sciences (New York), 105 (2): 1843–1847, doi:10.1023/A:1011323915468, MR 1871149.
23. Spencer, Joel J. (1994), Ten Lectures on the Probabilistic Method, SIAM, p. 4, ISBN 978-0-89871-325-1
24. ^ Graham, R. L. (1992), "Roots of Ramsey theory", in Bolker, E.; Cherno, P.; Costes, C.; Lieberman, D. (eds.), Andrew M. Gleason, Glimpses of a Life in Mathematics (PDF), pp. 39–47.
25. ^ Radziszowski, Stanisław (Updated August 22, 2011), "Small Ramsey Numbers", Electronic Journal of Combinatorics, DS1 {{citation}}: Check date values in: |date= (help).
26. Sun, Hugo S.; Cohen, M. E. (1984), "An easy proof of the Greenwood-Gleason evaluation of the Ramsey number R(3,3,3)" (PDF), The Fibonacci Quarterly, 22 (3): 235–238, MR 0765316.
27. Rigby, J. F. (1983), "Some geometrical aspects of a maximal three-coloured triangle-free graph", Journal of Combinatorial Theory, Series B, 34 (3): 313–322, doi:10.1016/0095-8956(83)90043-6, MR 0714453.
28. Andrew M. Gleason at the Mathematics Genealogy Project
29. Wermer, John (November 2009), Bolker, Ethan D. (ed.), "Andrew M. Gleason 1921–2008" (PDF), Notices of the American Mathematical Society, 56 (10): 1248–1251 {{citation}}: |contribution= ignored (help).

External links

• Ethan D. Bolker; et al. (2009). "Andrew M. Gleason, 1921–2008" (PDF). Notices of the AMS. 56 (10): 1236–1267. {{cite journal}}: Explicit use of et al. in: |author= (help); Unknown parameter |month= ignored (help)
• Debra Bradley Rudar (1996) Symposium celebrating Gleason and Society of Fellows, Harvard University Gazette.

Template:Persondata

• 1921 births
• 2008 deaths
• American mathematicians
• Mathematical analysts
• 20th-century mathematicians
• 21st-century mathematicians
• Harvard University faculty
• Putnam Fellows
• Yale University alumni
• Presidents of the American Mathematical Society
• Hollis Chair of Mathematics and Natural Philosophy
• Fellows of the American Academy of Arts and Sciences
• Members of the United States National Academy of Sciences
• People from Fresno, California