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{{short description|American mathematician and educator (1921–2008)}} 
{{good article}}
{{Infobox scientist
| name              = Andrew M. Gleason
| image             = [[File:GleasonAndrewMattei Berlin1959.jpg|frameless|upright=1.35]]
| caption           = Berlin, 1959
| birth_date        = {{birth date|1921|11|04|mf=y}}
| birth_place       = [[Fresno, California]]
| death_date        = {{death date and age|2008|10|17|1921|11|04|mf=y}}
| death_place       = [[Cambridge, Massachusetts]]
| fields            = [[Mathematics]], [[cryptography]]
| workplaces        = [[Harvard University]]
| alma_mater        = [[Yale University]]{{r|society}}
| doctoral_advisor  = None<!--this really needs to be "None", not just empty -- see article and note to academic_advisors, below -->
| academic_advisors = [[George Mackey]]{{#tag:ref|"Although Andy never earned a Ph.D., he thought of George [Mackey] as his mentor and advisor and lists himself as George's student on the Mathematics Genealogy Project website."<ref name="palais-h5p"/> It is customary at Harvard (as at many schools) to award a Harvard degree to tenured faculty who do not have such a degree already;<ref>{{citation|title=Honorary degrees at Harvard|last=Elkins|first=Kimball C.|journal=Harvard Library Bulletin|url=http://nrs.harvard.edu/urn-3:FHCL:419150|pages=326–353|volume=12|issue=3|year=1958}}. On pp. 327–328, Elkins writes "There is another type of degree, however, that must be classified as honorary, since it is so designated in the official records, although it differs somewhat from the sort usually understood by that term. This is the degree given by the University to persons on its own faculty who are not Harvard graduates, so as to make them, in the words of their diplomas, 'members of our flock' – ''ut in grege nostro numeretur''. The degree given for this purpose is Master of Arts (A.M.)."</ref> in conjunction with his tenure, therefore, Gleason received a Harvard master's degree in 1953.{{r|society}}|name=mentor|group=upper-alpha}}
| doctoral_students = {{Plainlist|
* [[Glen Bredon]]
* [[Daniel I. A. Cohen]]
* [[James Eells]]
* [[Jessie MacWilliams]]
* [[Richard Palais]]
* [[Joel Spencer]]
* [[T. Christine Stevens]]
}}
| known_for         = {{Plainlist|
* [[Hilbert's fifth problem]]
* [[Gleason's theorem]]
* [[Greenwood–Gleason graph]]
* [[Gleason–Prange theorem]]
* Gleason polynomials
}}
| awards            = {{Plainlist|
* [[Newcomb Cleveland Prize]] (1952)
* Gung{{ndash}}Hu Distinguished Service to Mathematics Award (1996)
}}
| spouse            = {{marriage|{{nowrap|[[Jean Berko Gleason]]}}|January 1959}}
}}
'''Andrew Mattei Gleason''' (1921{{ndash}}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 {{shy|math|e|mat|ics}} teaching at all levels.<ref
name="mactutor">{{MacTutor|id=Gleason|title=Andrew Mattei Gleason}}</ref><ref name=globe>{{citation|url=http://www.highbeam.com/doc/1P2-19351189.html|archive-url=https://web.archive.org/web/20130520085339/http://www.highbeam.com/doc/1P2-19351189.html|url-status=dead|archive-date=May 20, 2013|url-access=subscription |newspaper=[[Boston Globe]]|title=Andrew Gleason; helped solve vexing geometry problem|first=Caitlin|last=Castello|date=October 20, 2008}}.</ref> [[Gleason's theorem]] in [[quantum logic]] and the [[Greenwood–Gleason graph]], an important example in [[Ramsey theory]], are named for him.<!-- coupla other eponyms too -- see AMS-->

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 [[cryptography|cryptographic security]], and the Commonwealth of Massachusetts on {{shy|math|e|mat|ics}} 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 [[mathematical proof|{{shy|math|e|mat|ic|al}} proofs]] "really aren't there to convince you that something is true{{mdashb}}they're there to show you why it is true."<ref name=mmp/> 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."<ref name=50years/>

==Biography==
[[File:GleasonAndrewMattei ActiveDuty1940s.jpg|thumb|left|upright=0.8|<!--Lieutenant, ?-->US Navy, 1940s]]
Gleason was born on November 4, 1921, in [[Fresno, California]], the youngest of three children;
his father [[Henry Gleason]] was a [[botany|botanist]] and a member of the [[Mayflower Society]], and his mother was the daughter of Swiss-American winemaker [[Andrew Mattei]].<ref name="mmp"/><ref name="lww"/>
His older brother [[Henry Allan Gleason (linguist)|Henry Jr.]] became a linguist.<ref>
{{citation
 | title = Henry A. Gleason Papers
 | publisher = Mertz Library, New York Botanical Garden
 | url = http://library.nybg.org/finding_guide/archv/gleason_ppf.html
 | access-date = April 9, 2013
 | archive-url = https://web.archive.org/web/20100712222720/http://library.nybg.org/finding_guide/archv/gleason_ppf.html
 | archive-date = July 12, 2010
 | url-status = dead
 }}
</ref>
He grew up in [[Bronxville, New York]], where his father was the curator of the [[New York Botanical Garden]].<ref
name="mmp">
{{citation|contribution=Andrew M. Gleason|title=More Mathematical People|editor1-first=Donald J.|editor1-last=Albers|editor2-first=Gerald L.|editor2-last=Alexanderson|editor2-link=Gerald L. Alexanderson|editor3-first=Constance|editor3-last=Reid|editor3-link=Constance Reid|publisher=Harcourt Brace Jovanovich|year=1990|page=86}}.
</ref><ref name="lww"/>

After briefly attending [[Berkeley High School (Berkeley, California)|Berkeley High School]] (Berkeley, California)<ref name="mactutor"/>
he graduated from [[Roosevelt High School (Yonkers, New York)|Roosevelt High School]] in Yonkers, winning a scholarship<!--check cite--> to [[Yale University]].<ref name="mmp"/>
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.
{{blockquote|So I learned first year calculus and second year calculus and became the consultant to one end of the whole Old Campus&nbsp;... I used to do all the homework for all the sections of <!--what was the equivalent of Math 1--> [first-year calculus]. I got plenty of practice in doing elementary calculus problems. I don't think there exists a problem{{mdashb}}the classical kind of pseudo reality problem which first and second-year students are given{{mdashb}}that I haven't seen.<!-- At the same time I was working on this course with Longley--><ref name=mmp/>
}}

One month later he enrolled in a differential equations course ("mostly full of seniors") as well. When [[Einar Hille]] temporarily replaced the regular instructor, Gleason found Hille's style "unbelievably different&nbsp;... He had a view of mathematics that was just vastly different&nbsp;... 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."<ref name=mmp/>

While at Yale he competed three times (1940, 1941 and 1942) in the recently founded [[William Lowell Putnam Mathematical Competition]], always placing among the top five entrants in the country (making him the second three-time [[Putnam Fellow]]).<!--problem her is that Putnam Fellow redirects to P. Competition, so it seems dumb to have two links--><ref>{{citation|first=Joseph A.|last=Gallian|author-link=Joseph Gallian|url=http://www.d.umn.edu/~jgallian/putnam.pdf|title=The Putnam Competition from 1938&ndash;2013|access-date=2016-04-10}}.</ref>

After the [[Attack on Pearl Harbor|Japanese attacked Pearl Harbor]] during his senior year, Gleason applied for a commission in the US Navy,<ref
name="secret-life">{{citation|first1=John|last1=Burroughs|first2=David|last2=Lieberman|first3=Jim|last3=Reeds|title=The secret life of Andrew Gleason|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1239–1243|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref>
and on graduation<!-- tk year??-->
joined the team working to break [[Japanese naval codes]].<ref name="mmp"/> 
(Others on this team included his future collaborator [[Robert E. Greenwood]] and Yale professor [[Marshall Hall (mathematician)|Marshall Hall Jr.]])<ref name="secret-life"/>
He also collaborated with British researchers [[Cryptanalysis of the Enigma|attacking the German Enigma]] cipher;<!--there are several articles related to Enigma -- not sure this is the best choice here -->
[[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.<ref name="secret-life"/>

[[File:GleasonAndrewMattei WithJean1958.jpg|thumb|With [[Jean Berko Gleason|Jean Berko]], 1958]]
In 1946, at the recommendation of Navy colleague [[Donald Howard Menzel]], Gleason was appointed a [[Harvard Society of Fellows|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,<ref name="mmp"/>
though he was almost immediately recalled to Washington for cryptographic work related to the [[Korean War]].<ref name="mmp"/>
He returned to Harvard in the fall of 1952, and soon after published the most important of his results on Hilbert's fifth problem (see [[#Hilbert's fifth problem|below]]).
Harvard awarded him [[Tenure (academic)|tenure]] the following year.<ref name="mmp"/><ref name="aps"/><ref name=mentor group=upper-alpha/>

In January 1959 he married [[Jean Berko Gleason|Jean Berko]]<ref name="mmp"/> 
whom he had met at a party featuring the music of [[Tom Lehrer#NewMath|Tom Lehrer]].<ref
name="lww">{{citation|last1=Gleason|first1=Jean Berko|author1-link=Jean Berko Gleason|title=A life well lived|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1266–1267|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref>
Berko, a [[psycholinguistics|psycholinguist]], worked for many years at [[Boston University]].<ref name="aps"/>
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.<ref name="mactutor"/><ref>{{citation|title=The oldest endowed professorship: 1721 gift led to long line of Hollis Chair occupants at Divinity School|journal=Harvard Gazette|url=http://news.harvard.edu/gazette/story/2012/05/the-oldest-endowed-professorship/|first=Colleen|last=Walsh|date=May 3, 2012}}.<!-- This source doesn't mention Gleason but does give the founding date of the chair, not in MacTutor. The "oldest scientific endowed professorship" claim is in MacTutor, not here. EEng says: I'll likely find better sources re Hollis chair along the way --></ref>
<!-- there's a lot to say in between -->
He retired from Harvard in 1992 but remained active in service to Harvard (as chair of the [[Harvard Society of Fellows|Society of Fellows]], for example)<ref name="hsf96"/>
and to mathematics: in particular, promoting the Harvard Calculus Reform Project<ref name="hh">{{citation|last1=Hughes-Hallett|first1=Deborah|author1-link= Deborah Hughes Hallett |last2=Stevens|first2=T. Christine|author2-link=T. Christine Stevens|last3=Tecosky-Feldman|first3=Jeff|last4=Tucker|first4=Thomas|title=Andy Gleason: teacher|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1260–1265|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref> and working with the [[Massachusetts Board of Education]].<ref name="pollak"/><!--clarify-->

He died on October 17, 2008 from complications following surgery.<ref name="mactutor"/><ref name=globe/>

== Teaching and education reform==
[[File:GleasonAndrewMattei Koala1988.jpg|thumb|right|upright=0.8|Australia, 1988]]
Gleason said he "always enjoyed helping other people with math"{{mdashb}}a colleague said he "regarded teaching mathematics{{mdashb}}like doing mathematics{{mdashb}}as 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 homework{{mdashb}}including those taking the second half of the course, which he soon began auditing.<ref name=mmp/><ref
name="amsx">{{citation|title=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref>

At Harvard he "regularly taught at every level",<ref name="hh"/> 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.<ref name="ag-qm"/>

In 1964 he created "the first of the 'bridge' courses now ubiquitous for math majors, only twenty years before its time."<ref name="hh"/> 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.<ref>{{citation|url=http://www.wou.edu/~wardm/BridgeCourseSlidesWinterMeetings2007forWebPage.pdf|contribution=Everything you want to know about bridge courses–except whether they work: Preliminary findings from a national survey|first1=Jennifer|last1=Carmichael|first2=Michael B.|last2=Ward|title=Joint Mathematics Meeting|year=2007}}.</ref> That effort led to publication of his ''Fundamentals of Abstract Analysis'', of which one reviewer wrote:
{{blockquote|This is a most unusual book&nbsp;... 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. The originality of the author is that he has tried to attain that goal in a textbook, and in the reviewer's opinion, he has succeeded remarkably well in this all but impossible task. Most readers will probably be delighted (as the reviewer has been) to find, page after page, painstaking discussions and explanations of standard mathematical and logical procedures, always written in the most felicitous style, which spares no effort to achieve the utmost clarity without falling into the vulgarity which so often mars such attempts.<ref name=amsx/>}}

[[File:GleasonAndrewMattei Sphynx2001.jpg|thumb|The [[Great Sphinx of Giza|Sphinx]], 2001]]
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:
{{blockquote|The reader may wonder why so much is left to the reader. A book on swimming strokes may be nice to read, but one must practice the strokes while actually in the water before one can claim to be a swimmer. So if the reader desires to actually possess the knowledge for recovering wiring from a [[Cryptanalysis#Depth|depth]], let the reader get his paper and pencils, using perhaps four colors to avoid confusion in the connecting links, and go to work.<ref name=amsx/>}}

His notes and exercises on probability and statistics, drawn up for his lectures to code-breaking colleagues during the war (see [[#Cryptanalysis|below]]) remained in use in [[National Security Agency]] training for several decades; they were published openly in 1985.<ref name=amsx/>

In a 1964 ''Science'' article, Gleason wrote of an apparent paradox arising in attempts to explain mathematics to nonmathematicians:
{{blockquote|It is notoriously difficult to convey the proper impression of the frontiers of mathematics to nonspecialists. Ultimately the difficulty stems from the fact that mathematics is an easier subject than the other sciences. Consequently, many of the important primary problems of the subject{{mdashb}}that is, problems which can be understood by an intelligent outsider{{mdashb}}have either been solved or carried to a point where an indirect approach is clearly required. The great bulk of pure mathematical research is concerned with secondary, tertiary, or higher-order problem, the very statement of which can hardly be understood until one has mastered a great deal of technical mathematics.<ref>
Andrew M. Gleason. "Evolution of an active mathematical theory", Science ''31'' (July 1964), pp. 451–457.
</ref>
}}

[[File:GleasonAndrewMattei HorseAndClipboard.jpg|thumb|right|"With the inevitable clipboard under his arm",<ref name="hh"/> <!--November, -->1989]]
Gleason was the first chairman of the advisory committee of the [[School Mathematics Study Group]], which helped define the [[New Math]] of the 1960s{{mdashb}}ambitious 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:
{{blockquote|to find out how much they could figure out for themselves, given appropriate activities and the right guidance. 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] quick and decisive response: "No, I didn't think about that at all. I had a ball!"<ref name=amsx/>}}

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."<ref name=aps/> However, the program faced heavy criticism from the mathematics community for its omission of topics such as the [[mean value theorem]],<ref>{{citation|journal=PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies|volume=4|issue=3|year=1994|title=Reflections on the Harvard calculus approach|doi=10.1080/10511979408965753|first=Patti Frazer|last=Lock|author-link= Patti Frazer Lock |pages=229–234}}.</ref> and for its perceived lack of mathematical rigor.<ref>{{citation|title=The Mathematics Education Reform: Why You Should be Concerned and What You Can Do|first=H.|last=Wu|journal=[[American Mathematical Monthly]]|volume=104|issue=10|year=1997|pages=946–954|jstor=2974477|doi=10.2307/2974477|url=http://www.math.berkeley.edu/~wu/reform5.pdf}}.</ref><ref>{{citation|title=On the Harvard Consortium Calculus|first=Saunders|last=Mac Lane|author-link=Saunders Mac Lane|department=Letters to the Editor|journal=[[Notices of the American Mathematical Society]]|year=1997|volume=44|issue=8|url=https://www.ams.org/notices/199708/letters.pdf|page=893}}.</ref><ref>{{citation|title=Calculus Reform—For the $Millions|journal=[[Notices of the American Mathematical Society]]|year=1997|volume=44|issue=10|pages=1324–1325|url=https://www.ams.org/notices/199710/comm-klein.pdf|first1=David|last1=Klein|first2=Jerry|last2=Rosen}}.</ref>

==Cryptanalysis work==
[[File:Enigma Wiring Recovery from the Reading of a Depth-p24.jpg|upright=1.3|thumb|Report (1945) by Gleason and colleagues [[Cryptanalysis of the Enigma|{{shy|re|gard|ing}} the German Enigma]]. "The recovery of wiring from a depth can be a very {{shy|in|ter|est|ing prob|lem. Let the read|er sur|round him|self with pleas|ant work|ing con|dit|ions}} and try it."]]
During World War II Gleason was part of [[OP-20-G]], the U.S. Navy's signals intelligence and [[cryptanalysis]] group.<ref name="secret-life"/>
One task of this group, in collaboration with British cryptographers at [[Government Communications Headquarters|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 (mathematician)|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 (in one direction) and N-Z (in the other), then devised novel statistical tests by which he confirmed this hypothesis. The result was routine decryption of this third network by 1944. (This work also involved deeper {{shy|math|e|mat|ics}} related to [[permutation group]]s and the [[graph isomorphism problem]].)<ref name="secret-life"/>

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.<ref name="secret-life"/>

Toward the end of the war he concentrated on documenting the work of OP-20-G and developing systems for training new cryptographers.<ref name="secret-life"/>

In 1950 Gleason returned to active duty for the [[Korean War]], serving as a [[Lieutenant commander (United States)|Lieutenant Commander]] in the [[United States Department of Homeland Security|Nebraska Avenue Complex]] (which much later became the home of the [[DHS Cyber Security Division]]). His cryptographic work from this period remains classified, but it is known that he recruited mathematicians and taught them cryptanalysis.<ref name="secret-life"/>
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.<ref name="secret-life"/>

==Mathematics research==
Gleason made fundamental contributions to widely varied areas of mathematics, including the theory of [[Lie group]]s,<ref name="palais-h5p"/> [[quantum mechanics]],<ref
name="ag-qm">{{citation|last=Chernoff|first=Paul R.|title=Andy Gleason and quantum mechanics|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1253–1259|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref>
and [[combinatorics]].<ref name="spencer-dm"/>
According to [[Freeman Dyson]]'s famous classification of mathematicians as being either birds or frogs,<ref>{{citation
|url=https://www.ams.org/notices/200902/rtx090200212p.pdf|title=Birds and frogs|first=Freeman|last=Dyson|author-link=Freeman Dyson|journal=Notices of the American Mathematical Society|volume=56|issue=2|date=February 2009|pages=212–223}}.
</ref>
Gleason was a frog: he worked as a problem solver rather than a visionary formulating grand theories.<ref
name=50years>{{citation|last=Bolker|first=Ethan D.|title=50+ years&nbsp;... |department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1237–1239|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.</ref>

===Hilbert's fifth problem===
[[File:GleasonAndrewMattei HilbertFifthJournal.jpg|thumb|upright=1.3|Journal entry (1947): "July 10. We hung out the clothes to dry this {{shy|morn|ing and Char|les wash|ed the car. I did a lit|tle work on the Hil|bert}} fifth."]]
In 1900 [[David Hilbert]] posed [[Hilbert's problems|23 problems]] he felt would be central to next century of mathematics research. [[Hilbert's fifth problem]] concerns the [[characterization (mathematics)|characterization]] of [[Lie group]]s by their [[Group action (mathematics)|actions]] on [[topological space]]s: to what extent does their topology provide information sufficient to determine their geometry?

The "restricted" version of Hilbert's fifth problem (solved by Gleason) asks, more specifically, whether every [[Local property|locally]] [[Euclidean space|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 [[Analytic manifold|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.<ref name="palais-h5p">{{citation|first=Richard|last=Palais|author-link=Richard Palais|title=Gleason's contribution to the solution of Hilbert's Fifth Problem|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1243–1248|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref><ref
name="illman">{{citation
 | last = Illman | first = Sören | doi = 10.1023/A:1011323915468
 | issue = 2 | journal = Journal of Mathematical Sciences
 | mr = 1871149 | pages = 1843–1847 | title = Hilbert's fifth problem: review
 | volume = 105 | year = 2001| s2cid = 115527342 | doi-access = free }}.
</ref>

[[File:GleasonAndrewMattei withGeorgeMackey2000.jpg|thumb|left|With his mentor<ref name=mentor group=upper-alpha/> [[George Mackey]] at Alice Mackey's 80th birthday (2000).]]
Gleason's interest in the fifth problem began in the late 1940s, sparked by a course he took from [[George Mackey]].<ref name="mmp"/>
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.<ref name="palais-h5p"/>
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.<ref name="palais-h5p"/><ref name="illman"/> 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.<ref name="palais-h5p"/><ref name="illman"/> As Gleason told the story, the key insight of his proof was to apply the fact that [[monotonic function]]s are [[Differentiable function|differentiable]] [[almost everywhere]].<ref name="mmp"/> 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.<ref name="mmp"/>{{efn-ua|
In a 1959 description of his own research, Gleason simply said that he had written "a number of papers" which "contributed substantially" to the solution of Hilbert's Fifth.{{r|society}}
}}

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 function|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.<ref name="palais-h5p"/><ref name="illman"/> 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.<ref name="palais-h5p"/><ref name="illman"/> The [[Hilbert–Smith conjecture]], still unsolved, encapsulates the remaining difficulties of this case.<ref>See, e.g., {{Citation | first = John | last = Pardon | author-link = John Pardon | title = The Hilbert–Smith conjecture for three-manifolds | periodical = [[Journal of the American Mathematical Society]] | volume = 26 | issue = 3 |date= 2013 | pages = 879–899 | doi=10.1090/s0894-0347-2013-00766-3| arxiv = 1112.2324 | s2cid = 96422853 }}.</ref>

===Quantum mechanics===
[[File:GleasonAndrewMattei Fred.jpg|thumb|upright=0.8|With family cat Fred about 1966]]
{{main|Gleason's theorem}}
The [[Born rule]] 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 [[eigenvalue]]s of the operator, and that the probability of the system 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. 
[[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 (mathematics)|measure]] on the lattice of projections of a Hilbert space can be defined by a positive operator with unit [[Trace (linear algebra)|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.<ref name="ag-qm"/>

Gleason's theorem implies the nonexistence of certain types of [[Hidden variable theory|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 [[Expected value#Linearity|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.<ref name="ag-qm"/>

===Ramsey theory===
[[File:Clebsch Lombardi.svg|thumb|left|upright=0.7|The [[Greenwood–Gleason graph]]]]
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 (graph theory)|clique]] or an ''l''-vertex [[Independent set (graph theory)|independent set]]. Ramsey numbers require enormous effort to compute; when max(''k'',''l'')&nbsp;≥&nbsp;3 only finitely many of them are known precisely, and an exact computation of ''R''(6,6) is believed to be out of reach.<ref>{{citation|title=Ten Lectures on the Probabilistic Method|page=[https://archive.org/details/tenlecturesonpro0000spen/page/4 4]|last=Spencer|first=Joel J.|author-link=Joel H. Spencer|year=1994|publisher=[[Society for Industrial and Applied Mathematics|SIAM]]|isbn=978-0-89871-325-1|url=https://archive.org/details/tenlecturesonpro0000spen/page/4}}</ref> In 1953, the calculation of ''R''(3,3) was given as a question in the [[William Lowell Putnam Mathematical Competition|Putnam Competition]]. In 1955, motivated by this problem,<ref name="rlg-ramsey">
{{citation|contribution=Roots of Ramsey theory|url=http://www.math.ucsd.edu/~ronspubs/92_08_ramsey_roots.pdf|first=R. L.|last=Graham|author-link=Ronald Graham|title=Andrew M. Gleason, Glimpses of a Life in Mathematics|pages=39–47|year=1992|editor1-first=E.|editor1-last=Bolker|editor2-first=P.|editor2-last=Cherno|editor3-first=C.|editor3-last=Costes|editor4-first=D.|editor4-last=Lieberman}}.</ref> Gleason and his co-author Robert E. Greenwood made significant progress in the computation of Ramsey numbers with their proof that ''R''(3,4)&nbsp;=&nbsp;9, ''R''(3,5)&nbsp;=&nbsp;14, and ''R''(4,4)&nbsp;=&nbsp;18. Since then, only five more of these values have been found.<ref name="rad-ejc">{{citation|first=Stanisław|last=Radziszowski|journal=Electronic Journal of Combinatorics|title=Small Ramsey Numbers|volume=DS1|date=August 22, 2011|url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/DS1/html|access-date=April 5, 2013|archive-url=https://web.archive.org/web/20121018023743/http://www.combinatorics.org/ojs/index.php/eljc/article/view/ds1/html|archive-date=October 18, 2012|url-status=dead}}.</ref> 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)&nbsp;=&nbsp;17; this remains the only nontrivial multicolor Ramsey number whose exact value is known.<ref name="rad-ejc"/> 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 graph|triangle-free]] [[regular graph|5-regular graph]] with 16 vertices and 40 edges<ref
name="spencer-dm">{{citation|last=Spencer|first=Joel J.|author-link=Joel H. Spencer|title=Andrew Gleason's discrete mathematics|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1251–1253|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref><ref>
{{citation
 | last1 = Sun | first1 = Hugo S. | last2 = Cohen | first2 = M. E.
 | mr = 765316 | issue = 3 | journal = The Fibonacci Quarterly | pages = 235–238
 | title = An easy proof of the Greenwood-Gleason evaluation of the Ramsey number ''R''(3,3,3)
 | url = http://www.fq.math.ca/Scanned/22-3/sun.pdf | volume = 22 | year = 1984| doi = 10.1080/00150517.1984.12429887 }}.
</ref>
(sometimes called the [[Greenwood–Gleason graph]]).<ref>{{citation
 | last = Rigby | first = J. F. | doi = 10.1016/0095-8956(83)90043-6
 | issue = 3 | journal = [[Journal of Combinatorial Theory]] | mr = 714453 | pages = 313–322 | series = Series B 
 | title = Some geometrical aspects of a maximal three-coloured triangle-free graph
 | volume = 34 | year = 1983| doi-access = free }}.</ref>

[[Ronald Graham]] writes that the paper by Greenwood and Gleason "is now recognized as a classic in the development of Ramsey theory".<ref name="rlg-ramsey"/> In the late 1960s, Gleason became the [[doctoral advisor]] of [[Joel Spencer]], who also became known for his contributions to Ramsey theory.<ref name="spencer-dm"/><ref
name="mathgenealogy">{{MathGenealogy|id=13307}}
</ref>

===Coding theory===
[[File:GleasonAndrewMattei withHenryAllanGleason.jpg|thumb|upright=0.8|With his brother, linguist [[Henry Allan Gleason (linguist)|Henry Allan Gleason Jr.]], in Toronto, 1969]]
Gleason published few contributions to [[coding theory]], but they were influential ones,<ref name="spencer-dm"/> and included "many of the seminal ideas and early results" in algebraic coding theory.<ref>{{citation | title = Review of ''The Mathematical Theory of Coding'', E. F. Assmus, Jr. (1977) | journal = SIAM Review | volume = 19 | issue = 1| pages = 175–176 | doi = 10.1137/1019032 }}</ref> During the 1950s and 1960s, he attended monthly meetings on coding theory with [[Vera Pless]] and others at the Air Force Cambridge Research Laboratory.<ref name="own">{{citation|first=Vera|last=Pless|author-link=Vera Pless|title=In Her Own Words|journal=Notices of the AMS|volume=38|issue=7|date=September 1991|pages=702–706|url=http://www.awm-math.org/articles/notices/199107/six/node5.html|access-date=2013-05-06|archive-url=https://web.archive.org/web/20160304220209/http://www.awm-math.org/articles/notices/199107/six/node5.html|archive-date=2016-03-04|url-status=dead}}.</ref> 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." She frequently posed her mathematical problems to Gleason and was often rewarded with a quick and insightful response.<ref name="spencer-dm"/>

The [[Gleason–Prange theorem]] is named after Gleason's work with AFCRL researcher [[Eugene Prange]]; it was originally published in a 1964 AFCRL research report by H. F. Mattson Jr. and E. F. Assmus Jr.
It concerns the [[quadratic residue code]] of order ''n'', extended by adding a single parity check bit. This "remarkable theorem"<ref name="blahut">{{citation
 | last = Blahut | first = R. E. | author-link = R. E. Blahut
 | date = September 2006
 | doi = 10.1109/18.133245
 | issue = 5
 | journal = IEEE Trans. Inf. Theory
 | location = Piscataway, NJ, USA
 | pages = 1269–1273
 | publisher = IEEE Press
 | title = The Gleason-Prange theorem
 | volume = 37}}.</ref> shows that this code is highly symmetric, having the [[projective linear group]] ''PSL''<sub>2</sub>(''n'') as a subgroup of its symmetries.<ref name="spencer-dm"/><ref name="blahut"/>

Gleason is the namesake of the Gleason polynomials, a system of polynomials that generate the [[Enumerator polynomial|weight enumerators]] of [[linear code]]s.<ref name="spencer-dm"/><ref>{{citation|contribution=8.4 Gleason polynomials|pages=134–138|contribution-url=https://books.google.com/books?id=4v6_Hcqbm-8C&pg=PA134|title=Introduction to the Theory of Error-Correcting Codes|title-link= Introduction to the Theory of Error-Correcting Codes |volume=48|series=Wiley Series in Discrete Mathematics and Optimization|first=Vera|last=Pless|author-link=Vera Pless|edition=3rd|publisher=John Wiley & Sons|year=2011|isbn=978-1-118-03099-8}}.</ref> These polynomials take a particularly simple form for [[Dual code|self-dual code]]s: in this case there are just two of them, the two bivariate polynomials ''x''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>2</sup> and ''x''<sup>8</sup>&nbsp;+&nbsp;14''x''<sup>2</sup>''y''<sup>2</sup>&nbsp;+&nbsp;''y''<sup>8</sup>.<ref name="spencer-dm"/> Gleason's student [[Jessie MacWilliams]] continued Gleason's work in this area, proving a relationship between the weight enumerators of codes and their duals that has become known as the [[MacWilliams identity]].<ref name="spencer-dm"/>

In this area, he also did pioneering work in [[experimental mathematics]], performing computer experiments in 1960. This work studied the average distance to a codeword, for a code related to the [[Berlekamp switching game]].<ref name="aps"/><ref>{{citation | last1 = Brown | first1 = Thomas A. | last2 = Spencer | first2 = Joel H. | doi = 10.4064/cm-23-1-165-171 | journal = Colloquium Mathematicum | mr = 307944 | pages = 165–171, 177 | title = Minimization of <math>\pm 1</math> matrices under line shifts | volume = 23 | year = 1971| doi-access = free }}</ref>

===Other areas===
Gleason founded the theory of [[Dirichlet algebra]]s,<ref>{{citation|last=Wermer|first=John|title=Gleason's work on Banach algebras|department=Andrew M. Gleason 1921–2008|editor-first=Ethan D.|editor-last=Bolker|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009|pages=1248–1251|url=https://www.ams.org/notices/200910/rtx091001236p.pdf}}.
</ref>
and made other {{shy|math|e|mat|i|cal}} contributions including work on
[[finite geometry]]<ref>See his 1956 paper "Finite Fano planes".</ref> and on
the [[enumerative combinatorics]] of [[permutation]]s.<ref name="50years"/>
(In 1959 he wrote that his research "sidelines" included "an intense interest in combinatorial problems."){{r|society}}
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 and straightedge constructions|compass, straightedge]], and an [[angle trisection|angle trisector]].<ref name="50years"/>

==Awards and honors==
[[File:GleasonAndrewMattei NavalReserves1960s.jpg|thumb|upright=0.8|In Naval Reserve uniform, 1960s]]
In 1952 Gleason was awarded the [[American Association for the Advancement of Science]]'s [[Newcomb Cleveland Prize]]<ref>{{citation|url=http://archives.aaas.org/awards.php?a_id=7|title=AAAS Newcomb Cleveland Prize|publisher=[[American Association for the Advancement of Science]]|access-date=2016-04-10|archive-date=2011-06-07|archive-url=https://web.archive.org/web/20110607023735/http://archives.aaas.org/awards.php?a_id=7|url-status=dead}}.</ref>
for his work on [[Hilbert's fifth problem]].{{r|society}}
He was elected to the [[National Academy of Sciences]] and the [[American Philosophical Society]], was a Fellow of the [[American Academy of Arts and Sciences]],<ref name="mmp"/><ref name="aps">{{citation|title=Andrew Gleason, 4 November 1921&nbsp;– 17 October 2008|journal=[[Proceedings of the American Philosophical Society]]|volume=154|issue=4|date=December 2010|url=https://www.amphilsoc.org/sites/default/files/proceedings/1540408.pdf|first1=Barry|last1=Mazur|author1-link=Barry Mazur|first2=Benedict|last2=Gross|author2-link=Benedict Gross|first3=David|last3=Mumford|author3-link=David Mumford|pages=471–476|access-date=10 April 2016|archive-url=https://web.archive.org/web/20161220070304/https://www.amphilsoc.org/sites/default/files/proceedings/1540408.pdf|archive-date=20 December 2016|url-status=dead}}.</ref>
and belonged to the [[Société Mathématique de France]].{{refn|name=society|{{citation
|editor-last=Brinton|editor-first=Crane|work= Society of Fellows
|location=Cambridge|publisher=Society of Fellows of Harvard University|year=1959
|title=Andrew Mattei Gleason|pages=135{{ndash}}136
}} }}
 
In 1981 and 1982 he was president of the [[American Mathematical Society]],<ref name="mmp"/><!--apparently a two-year term, need check-->
and at various times held numerous other posts in professional and scholarly organizations,
including chairmanship of the Harvard Department of Mathematics.<ref>
{{citation
| title=Hironaka to Teach Math|date=October 23, 1967|journal=Harvard Crimson
|url=http://www.thecrimson.com/article/1967/10/23/hironaka-to-teach-math-pheisuke-hironaka
}}
</ref>
In 1986 he chaired the organizing committee for the [[International Congress of Mathematicians]] in [[Berkeley, California]], and was president of the Congress.<ref name="pollak"/>

In 1996 the [[Harvard Society of Fellows]] held a special symposium honoring Gleason on his retirement after seven years as its chairman;<ref name="hsf96">{{citation|title=Symposium Will Celebrate Gleason and Society of Fellows|journal=Harvard Gazette|first=Debra Bradley|last=Ruder|date=May 9, 1996|url=http://news.harvard.edu/gazette/1996/05.09/SymposiumWillCe.html}}.</ref>
that same year, the [[Mathematics Association of America]] awarded him the [[Yueh-Gin Gung and Dr. Charles Y. Hu Distinguished Service to Mathematics Award]].<ref>{{citation|url=http://www.maa.org/awards/gunghu.html |title=Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service|publisher=[[Mathematics Association of America]]|access-date=2016-08-05}}.</ref>
A past president of the Association wrote:
{{blockquote|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 {{shy|math|e|mat|i|cal}} talent, and serving one's institution. Andy Gleason is that rare individual who has done all of these superbly.<ref name="pollak">{{citation|title=Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Andrew Gleason|first=H. O.|last=Pollak|journal=[[American Mathematical Monthly]]|volume=103|issue=2|date=February 1996|pages=105–106|doi=10.1080/00029890.1996.12004708|jstor=2975102}}.</ref>
}}

After his death a 32-page collection of essays in the ''Notices of the American Mathematical Society'' recalled "the life and work of [this] eminent American mathematician",<ref>
{{citation|title=Features|page=1227|journal=Notices of the American Mathematical Society|volume=56|issue=10|date=November 2009
|url=https://www.ams.org/notices/200910/200910-full-issue.pdf<!-- this is a different url than used in the other Notices cites -- this one is to entire issue, to show front matter, which contains cited phrase-->
}}.</ref>
calling him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."<ref name=50years/>

== Selected publications ==
;Research papers
*{{citation | last = Gleason | first = A. M. | contribution = One-parameter subgroups and Hilbert's fifth problem | location = Providence, R. I. | mr = 0043788 | pages = 451–452 | publisher = American Mathematical Society | title = Proceedings of the [[International Congress of Mathematicians]], Cambridge, Mass., 1950, Vol. 2 | contribution-url = http://www.mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0451.0452.ocr.pdf | year = 1952 | access-date = 2013-04-05 | archive-date = 2014-12-14 | archive-url = https://web.archive.org/web/20141214200430/http://www.mathunion.org/ICM/ICM1950.2/Main/icm1950.2.0451.0452.ocr.pdf | url-status = dead }}
*{{citation
 | last = Gleason | first = A. M. | author-mask = 2 | doi = 10.2307/2372469 
 | journal = [[American Journal of Mathematics]] | mr = 0082684 | pages = 797–807 
 | title = Finite Fano planes | volume = 78 | year = 1956| issue = 4 | jstor = 2372469 }}.
*{{citation
  | last = Gleason | first = A. M. | author-mask = 2 | journal = [[Journal of Mathematics and Mechanics]]
 | mr = 0096113 | pages = 885–893 | title = Measures on the closed subspaces of a Hilbert space
 | url = http://www.iumj.indiana.edu/IUMJ/FULLTEXT/1957/6/56050 | volume = 6 | year = 1957 | issue = 4 | doi=10.1512/iumj.1957.6.56050| doi-access = free | url-access = subscription }}.
*{{citation
 | last = Gleason | first = A. M. | author-mask = 2 | journal = Illinois Journal of Mathematics
 | mr = 0121775 | pages = 482–489 | title = Projective topological spaces
 | url = http://projecteuclid.org/euclid.ijm/1255454110 | volume = 2 | year = 1958 | issue = 4A | doi = 10.1215/ijm/1255454110 | zbl = 0083.17401| doi-access = free | url-access = subscription }}.
*{{citation
 | last = Gleason | first = A. M. | author-mask = 2 | journal = [[Journal d'Analyse Mathématique]]
 | mr = 0213878 | pages = 171–172 | title = A characterization of maximal ideals
 | volume = 19 | year = 1967 | doi=10.1007/bf02788714| doi-access=free | s2cid = 121062823 }}.
*{{citation
 | last = Gleason | first = A. M. | author-mask = 2 
 | contribution = Weight polynomials of self-dual codes and the MacWilliams identities
 | location = Paris | mr = 0424391 | pages = 211–215 | publisher = Gauthier-Villars
 | title = Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3 | year = 1971}}.
*{{citation
 | last1 = Greenwood | first1 = R. E. | last2 = Gleason | first2 = A. M.
 | doi = 10.4153/CJM-1955-001-4 | mr = 0067467
 | journal = [[Canadian Journal of Mathematics]] | pages = 1–7
 | title = Combinatorial relations and chromatic graphs | volume = 7 | year = 1955| s2cid = 124255697 | doi-access = free }}.

;Books
*{{citation
 | last = Gleason | first = Andrew M. | mr = 0202509
 | publisher = Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.
 | title = Fundamentals of Abstract Analysis
 | year = 1966}}. Corrected reprint, Boston: Jones and Bartlett, 1991, {{MR|1140189}}.
*{{citation|title=The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938–1964 | last1 = Gleason | first1 = A. M. | author-mask = 2 |first2=Robert E.|last2=Greenwood|first3=Leroy Milton|last3=Kelly|author3-link=Leroy Milton Kelly|publisher=[[Mathematical Association of America]]|year=1980|isbn=978-0-88385-462-4|mr=0588757}}.
*{{citation | last1 = Gleason | first1 = A. M. | author-mask = 2 |first2=Walter F.|last2=Penney|first3=Ronald E.|last3=Wyllys|title=Elementary Course in Probability for the Cryptanalyst|publisher=Aegean Park Press|location=Laguna Hills, CA|year=1985}}. Unclassified reprint of a book originally published in 1957 by the National Security Agency, Office of Research and Development, Mathematical Research Division.
*{{citation | last1 = Gleason | first1 = A. M. | author-mask = 2 |first2=Deborah|last2=Hughes-Hallett|author2-link= Deborah Hughes Hallett |title=Calculus|publisher=Wiley|year=1994}}. Since its original publications this book has been extended to many different editions and variations with additional co-authors.

;Film
*{{citation|first=Andrew M.|last=Gleason|title=Nim and other oriented-graph games|publisher=[[Mathematical Association of America]]|year=1966| url=https://archive.org/details/nimandotherorientedgraphgames}}. 63 minutes, black & white. Produced by Richard G. Long and directed by Allan Hinderstein.

==See also==
*[[John Stewart Bell#Critique of von Neumann's proof|Bell's critique of von Neumann's proof]]
*[[Pierpont prime]], a class of [[prime number]]s conjectured by Gleason to be infinite

==Notes==
{{Reflist|group=upper-alpha}}

==References==
{{Reflist|2}}

==External links==
{{Commons category|Andrew Gleason}}
*{{citation|title=Faculty of Arts and Sciences{{snd}}Memorial Minute. Andrew Mattei Gleason|date=April 1, 2010|work=Harvard Gazette|url=http://news.harvard.edu/gazette/story/2010/04/andrew-mattei-gleason/}}

{{Hollisian Professors of Mathematics}}
{{AMS Presidents|state=collapsed}}

{{Authority control}}

{{DEFAULTSORT:Gleason, Andrew}}
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