Andrew Mattei Gleason (1921–2008) 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 math­e­mat­ics teaching at all levels.

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.

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.

[6] 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.

[6] (Others on this team included his future collaborator Robert E. Greenwood and Yale professor Marshall Hall Jr.)[11] 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.

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,[6] though he was almost immediately recalled to Washington for cryptographic work related to the Korean War.

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 homework‍—‌including those taking the second half of the course, which he soon began auditing.

[19] That effort led to publication of his Fundamentals of Abstract Analysis, of which one reviewer wrote: This is a most unusual book ... Every working mathematician of course knows the difference between a lifeless chain of formalized propositions and the "feeling" one has (or tries to get) of a mathematical theory, and will probably agree that helping the student to reach that "inside" view is the ultimate goal of mathematical education; but he will usually give up any attempt at successfully doing this except through oral teaching.

[17]His notes and exercises on probability and statistics, drawn up for his lectures to code-breaking colleagues during the war (see below) remained in use in National Security Agency training for several decades; they were published openly in 1985.

At the end of his talk, someone asked Andy whether he had ever worried that teaching math to little kids wasn't how faculty at research institutions should be spending their time.

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.

[11] 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.

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.

[11] In 1950 Gleason returned to active duty for the Korean War, serving as a Lieutenant Commander in the Nebraska Avenue Complex (which much later became the home of the DHS Cyber Security Division).

[11] 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.

[11] Gleason made fundamental contributions to widely varied areas of mathematics, including the theory of Lie groups,[1] quantum mechanics,[18] and combinatorics.

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.

George Mackey had asked 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.

Though Richard Kadison proved this was false for two-dimensional Hilbert spaces, Gleason's theorem (published 1957) shows it to be true for higher dimensions.

[18] 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 1955, motivated by this problem,[30] Gleason and his co-author Robert E. 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.

[36] Pless, who had previously worked in abstract algebra but became one of the world's leading experts in coding theory during this time, writes that "these monthly meetings were what I lived for."

[12][39] Gleason founded the theory of Dirichlet algebras,[40] and made other math­e­mat­i­cal contributions including work on finite geometry[41] and on the enumerative combinatorics of permutations.

[7] In 1952 Gleason was awarded the American Association for the Advancement of Science's Newcomb Cleveland Prize[42] for his work on Hilbert's fifth problem.

[44] 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.