Wikipedia

Andrew M. Gleason

Andrew Mattei Gleason(1921–2008) was an Americanmathematicianwho made fundamental contributions to widely varied areas of mathematics, including the solution ofHilbert's fifth problem,and was a leader in reform and innovation in math­e­mat­ics teaching at all levels.[4][5]Gleason's theoreminquantum logicand theGreenwood–Gleason graph,an important example inRamsey theory,are named for him.

Andrew M. Gleason
Berlin, 1959
Born(1921-11-04)November 4, 1921
Fresno, California
DiedOctober 17, 2008(2008-10-17)(aged 86)
Cambridge, Massachusetts
Alma materYale University[3]
Known for
Spouse
​
​
(m.1959)​
Awards
Scientific career
FieldsMathematics,cryptography
InstitutionsHarvard University
Doctoral advisorNone
Other academic advisorsGeorge Mackey[A]
Doctoral students

As a young World War II naval officer, Gleason broke German and Japanese military codes. After the war he spent his entire academic career atHarvard University,from which he retired in 1992. His numerous academic and scholarly leadership posts included chairmanship of the Harvard Mathematics Department and theHarvard Society of Fellows,and presidency of theAmerican Mathematical Society.He continued to advise the United States government oncryptographic security,and the Commonwealth of Massachusetts on math­e­mat­ics education for children, almost until the end of his life.

Gleason won theNewcomb Cleveland Prizein 1952 and the Gung–Hu Distinguished Service Award of the American Mathematical Society in 1996. He was a member of theNational Academy of Sciencesand of theAmerican Philosophical Society,and held theHollis Chair of Mathematics and Natural Philosophyat Harvard.

He was fond of saying thatmath­e­mat­ic­al proofs"really aren't there to convince you that something is true‍—they're there to show you why it is true."[6]TheNotices of the American Mathematical Societycalled him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."[7]

Biography

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US Navy, 1940s

Gleason was born on November 4, 1921, inFresno, California,the youngest of three children; his fatherHenry Gleasonwas abotanistand a member of theMayflower Society,and his mother was the daughter of Swiss-American winemakerAndrew Mattei.[6][8] His older brotherHenry Jr.became a linguist.[9] He grew up inBronxville, New York,where his father was the curator of theNew York Botanical Garden.[6][8]

After briefly attendingBerkeley High School(Berkeley, California)[4] he graduated fromRoosevelt High Schoolin Yonkers, winning a scholarship toYale University.[6] Though Gleason's mathematics education had gone only so far as some self-taught calculus, Yale mathematicianWilliam Raymond Longleyurged him to try a course inmechanicsnormally intended for juniors.

So I learned first year calculus and second year calculus and became the consultant to one end of the whole Old Campus... I used to do all the homework for all the sections of [first-year calculus]. I got plenty of practice in doing elementary calculus problems. I don't think there exists a problem‍—the classical kind of pseudo reality problem which first and second-year students are given‍—that I haven't seen.[6]

One month later he enrolled in a differential equations course ( "mostly full of seniors" ) as well. WhenEinar Hilletemporarily 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."[6]

While at Yale he competed three times (1940, 1941 and 1942) in the recently foundedWilliam Lowell Putnam Mathematical Competition,always placing among the top five entrants in the country (making him the second three-timePutnam Fellow).[10]

After theJapanese attacked Pearl Harborduring his senior year, Gleason applied for a commission in the US Navy,[11] and on graduation joined the team working to breakJapanese naval codes.[6] (Others on this team included his future collaboratorRobert E. Greenwoodand Yale professorMarshall Hall Jr.)[11] He also collaborated with British researchersattacking the German Enigmacipher; 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.[11]

WithJean Berko,1958

In 1946, at the recommendation of Navy colleagueDonald Howard Menzel,Gleason was appointed aJunior Fellowat 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,[6] though he was almost immediately recalled to Washington for cryptographic work related to theKorean War.[6] He returned to Harvard in the fall of 1952, and soon after published the most important of his results on Hilbert's fifth problem (seebelow). Harvard awarded himtenurethe following year.[6][12][A]

In January 1959 he marriedJean Berko[6] whom he had met at a party featuring the music ofTom Lehrer.[8] Berko, apsycholinguist,worked for many years atBoston University.[12] They had three daughters.

In 1969 Gleason took theHollis Chair of Mathematics and Natural Philosophy.Established in 1727, this is the oldest scientificendowed professorshipin the US.[4][13] He retired from Harvard in 1992 but remained active in service to Harvard (as chair of theSociety of Fellows,for example)[14] and to mathematics: in particular, promoting the Harvard Calculus Reform Project[15]and working with theMassachusetts Board of Education.[16]

He died on October 17, 2008 from complications following surgery.[4][5]

Teaching and education reform

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Australia, 1988

Gleason said he "always enjoyed helping other people with math" ‍—a colleague said he "regarded teaching mathematics‍—like doing mathematics‍—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—including those taking the second half of the course, which he soon began auditing.[6][17]

At Harvard he "regularly taught at every level",[15]including administratively burdensome multisection courses. One class presented Gleason with a framed print of Picasso'sMother and Childin recognition of his care for them.[18]

In 1964 he created "the first of the 'bridge' courses now ubiquitous for math majors, only twenty years before its time."[15]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.[19]That effort led to publication of hisFundamentals 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. 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.[17]

TheSphinx,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:

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 adepth,let the reader get his paper and pencils, using perhaps four colors to avoid confusion in the connecting links, and go to work.[17]

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

In a 1964Sciencearticle, Gleason wrote of an apparent paradox arising in attempts to explain mathematics to nonmathematicians:

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—that is, problems which can be understood by an intelligent outsider‍—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.[20]

"With the inevitable clipboard under his arm",[15]1989

Gleason was the first chairman of the advisory committee of theSchool Mathematics Study Group,which helped define theNew Mathof the 1960s‍—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:

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!"[17]

In 1986 he helped found theCalculus 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."[12]However, the program faced heavy criticism from the mathematics community for its omission of topics such as themean value theorem,[21]and for its perceived lack of mathematical rigor.[22][23][24]

Cryptanalysis work

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Report (1945) by Gleason and colleaguesre­gard­ing the German Enigma."The recovery of wiring from a depth can be a very in­ter­est­ing prob­lem. Let the read­er sur­round him­self with pleas­ant work­ing con­dit­ions and try it."

During World War II Gleason was part ofOP-20-G,the U.S. Navy's signals intelligence andcryptanalysisgroup.[11] One task of this group, in collaboration with British cryptographers atBletchley Parksuch asAlan Turing,was to penetrate GermanEnigma machinecommunications 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'sMarshall Hallobserved that certainmetadatain 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 math­e­mat­ics related topermutation groupsand thegraph isomorphism problem.)[11]

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 ofChernoff boundon tail distributions of sums of independent random variables. Gleason's classified work on this bound predated Chernoff's work by a decade.[11]

Toward the end of the war he concentrated on documenting the work of OP-20-G and developing systems for training new cryptographers.[11]

In 1950 Gleason returned to active duty for theKorean War,serving as aLieutenant Commanderin theNebraska Avenue Complex(which much later became the home of theDHS Cyber Security Division). His cryptographic work from this period remains classified, but it is known that he recruited mathematicians and taught them cryptanalysis.[11] He served on the advisory boards for theNational Security Agencyand theInstitute for Defense Analyses,and he continued to recruit, and to advise the military on cryptanalysis, almost to the end of his life.[11]

Mathematics research

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Gleason made fundamental contributions to widely varied areas of mathematics, including the theory ofLie groups,[1]quantum mechanics,[18] andcombinatorics.[25] According toFreeman Dyson's famous classification of mathematicians as being either birds or frogs,[26] Gleason was a frog: he worked as a problem solver rather than a visionary formulating grand theories.[7]

Hilbert's fifth problem

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Journal entry (1947): "July 10. We hung out the clothes to dry this morn­ing and Char­les wash­ed the car. I did a lit­tle work on the Hil­bert fifth."

In 1900David Hilbertposed23 problemshe felt would be central to next century of mathematics research.Hilbert's fifth problemconcerns thecharacterizationofLie groupsby theiractionsontopological spaces: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 everylocallyEuclideantopological groupis a Lie group. That is, if a groupGhas the structure of atopological manifold,can that structure be strengthened to areal analytic structure,so that within anyneighborhoodof an element ofG,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 byL. E. J. Brouwer,John von Neumann,Lev Pontryagin,andGarrett Birkhoff,among others.[1][27]

With his mentor[A]George Mackeyat Alice Mackey's 80th birthday (2000).

Gleason's interest in the fifth problem began in the late 1940s, sparked by a course he took fromGeorge Mackey.[6] 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.[1] His 1952 paper on the subject, together with a paper published concurrently byDeane MontgomeryandLeo Zippin,solves affirmatively the restricted version of Hilbert's fifth problem, showing that indeed every locally Euclidean group is a Lie group.[1][27]Gleason's contribution was to prove that this is true whenGhas the no small subgroups property; Montgomery and Zippin showed every locally Euclidean group has this property.[1][27]As Gleason told the story, the key insight of his proof was to apply the fact thatmonotonic functionsaredifferentiablealmost everywhere.[6]On finding the solution, he took a week of leave to write it up, and it was printed in theAnnals of Mathematicsalongside the paper of Montgomery and Zippin; another paper a year later byHidehiko Yamaberemoved some technical side conditions from Gleason's proof.[6][B]

The "unrestricted" version of Hilbert's fifth problem, closer to Hilbert's original formulation, considers both a locally Euclidean groupGand another manifoldMon whichGhas acontinuousaction. Hilbert asked whether, in this case,Mand the action ofGcould be given a real analytic structure. It was quickly realized that the answer was negative, after which attention centered on the restricted problem.[1][27]However, with some additional smoothness assumptions onGandM,it might yet be possible to prove the existence of a real analytic structure on the group action.[1][27]TheHilbert–Smith conjecture,still unsolved, encapsulates the remaining difficulties of this case.[28]

Quantum mechanics

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With family cat Fred about 1966
Main article:Gleason's theorem

TheBorn rulestates that an observable property of a quantum system is defined by aHermitian operatoron aseparable Hilbert space,that the only observable values of the property are theeigenvaluesof 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 Mackeyhad asked whether Born's rule is a necessary consequence of a particular set of axioms for quantum mechanics, and more specifically whether everymeasureon the lattice of projections of a Hilbert space can be defined by a positive operator with unittrace.ThoughRichard Kadisonproved 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 ofhidden variable theoriesfor quantum mechanics, strengthening a previous argument ofJohn von Neumann.Von Neumann had claimed to show that hidden variable theories were impossible, but (asGrete Hermannpointed out) his demonstration made an assumption that quantum systems obeyed a form ofadditivity of expectationfor noncommuting operators that might not hold a priori. In 1966,John Stewart Bellshowed that Gleason's theorem could be used to remove this extra assumption from von Neumann's argument.[18]

Ramsey theory

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TheGreenwood–Gleason graph

TheRamsey numberR(k,l) is the smallest numberrsuch that every graph with at leastrvertices contains either ak-vertexcliqueor anl-vertexindependent 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 ofR(6,6) is believed to be out of reach.[29]In 1953, the calculation ofR(3,3) was given as a question in thePutnam Competition.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 thatR(3,4) = 9,R(3,5) = 14, andR(4,4) = 18. Since then, only five more of these values have been found.[31]In the same 1955 paper, Greenwood and Gleason also computed the multicolor Ramsey numberR(3,3,3): the smallest numberrsuch that, if acomplete graphonrvertices 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.[31]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 atriangle-free5-regular graphwith 16 vertices and 40 edges[25][32] (sometimes called theGreenwood–Gleason graph).[33]

Ronald Grahamwrites that the paper by Greenwood and Gleason "is now recognized as a classic in the development of Ramsey theory".[30]In the late 1960s, Gleason became thedoctoral advisorofJoel Spencer,who also became known for his contributions to Ramsey theory.[25][34]

Coding theory

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With his brother, linguistHenry Allan Gleason Jr.,in Toronto, 1969

Gleason published few contributions tocoding theory,but they were influential ones,[25]and included "many of the seminal ideas and early results" in algebraic coding theory.[35]During the 1950s and 1960s, he attended monthly meetings on coding theory withVera Plessand others at the Air Force Cambridge Research Laboratory.[36]Pless, who had previously worked inabstract algebrabut 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.[25]

TheGleason–Prange theoremis named after Gleason's work with AFCRL researcherEugene Prange;it was originally published in a 1964 AFCRL research report by H. F. Mattson Jr. and E. F. Assmus Jr. It concerns thequadratic residue codeof ordern,extended by adding a single parity check bit. This "remarkable theorem"[37]shows that this code is highly symmetric, having theprojective linear groupPSL2(n) as a subgroup of its symmetries.[25][37]

Gleason is the namesake of the Gleason polynomials, a system of polynomials that generate theweight enumeratorsoflinear codes.[25][38]These polynomials take a particularly simple form forself-dual codes:in this case there are just two of them, the two bivariate polynomialsx2+y2andx8+ 14x2y2+y8.[25]Gleason's studentJessie MacWilliamscontinued Gleason's work in this area, proving a relationship between the weight enumerators of codes and their duals that has become known as theMacWilliams identity.[25]

In this area, he also did pioneering work inexperimental mathematics,performing computer experiments in 1960. This work studied the average distance to a codeword, for a code related to theBerlekamp switching game.[12][39]

Other areas

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Gleason founded the theory ofDirichlet algebras,[40] and made other math­e­mat­i­cal contributions including work on finite geometry[41]and on theenumerative combinatoricsofpermutations.[7] (In 1959 he wrote that his research "sidelines" included "an intense interest in combinatorial problems." )[3] 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 withcompass, straightedge,and anangle trisector.[7]

Awards and honors

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In Naval Reserve uniform, 1960s

In 1952 Gleason was awarded theAmerican Association for the Advancement of Science'sNewcomb Cleveland Prize[42] for his work onHilbert's fifth problem.[3] He was elected to theNational Academy of Sciencesand theAmerican Philosophical Society,was a Fellow of theAmerican Academy of Arts and Sciences,[6][12] and belonged to theSociété Mathématique de France.[3]

In 1981 and 1982 he was president of theAmerican Mathematical Society,[6] and at various times held numerous other posts in professional and scholarly organizations, including chairmanship of the Harvard Department of Mathematics.[43] In 1986 he chaired the organizing committee for theInternational Congress of MathematiciansinBerkeley, California,and was president of the Congress.[16]

In 1996 theHarvard Society of Fellowsheld a special symposium honoring Gleason on his retirement after seven years as its chairman;[14] that same year, theMathematics Association of Americaawarded him theYueh-Gin Gung and Dr. Charles Y. Hu Distinguished Service to Mathematics Award.[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. Andy Gleason is that rare individual who has done all of these superbly.[16]

After his death a 32-page collection of essays in theNotices of the American Mathematical Societyrecalled "the life and work of [this] eminent American mathematician",[45] calling him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."[7]

Selected publications

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Research papers
  • Gleason, A. M. (1952),"One-parameter subgroups and Hilbert's fifth problem"(PDF),Proceedings of theInternational Congress of Mathematicians,Cambridge, Mass., 1950, Vol. 2,Providence, R. I.: American Mathematical Society, pp.451–452,MR0043788,archived fromthe original(PDF)on 2014-12-14,retrieved2013-04-05
  • —— (1956), "Finite Fano planes",American Journal of Mathematics,78(4):797–807,doi:10.2307/2372469,JSTOR2372469,MR0082684.
  • —— (1957),"Measures on the closed subspaces of a Hilbert space",Journal of Mathematics and Mechanics,6(4):885–893,doi:10.1512/iumj.1957.6.56050,MR0096113.
  • —— (1958),"Projective topological spaces",Illinois Journal of Mathematics,2(4A):482–489,doi:10.1215/ijm/1255454110,MR0121775,Zbl0083.17401.
  • —— (1967), "A characterization of maximal ideals",Journal d'Analyse Mathématique,19:171–172,doi:10.1007/bf02788714,MR0213878,S2CID121062823.
  • —— (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,MR0424391.
  • 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,MR0067467,S2CID124255697.
Books
  • Gleason, Andrew M. (1966),Fundamentals of Abstract Analysis,Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont.,MR0202509.Corrected reprint, Boston: Jones and Bartlett, 1991,MR1140189.
  • ——; Greenwood, Robert E.;Kelly, Leroy Milton(1980),The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938–1964,Mathematical Association of America,ISBN978-0-88385-462-4,MR0588757.
  • ——; 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.
  • ——;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

See also

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Notes

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  1. ^abc"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."[1]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;[2]in conjunction with his tenure, therefore, Gleason received a Harvard master's degree in 1953.[3]
  2. ^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.[3]

References

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  1. ^abcdefghPalais, Richard(November 2009), Bolker, Ethan D. (ed.),"Gleason's contribution to the solution of Hilbert's Fifth Problem"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1243–1248.
  2. ^Elkins, Kimball C. (1958),"Honorary degrees at Harvard",Harvard Library Bulletin,12(3):326–353.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.). "
  3. ^abcdefBrinton, Crane, ed. (1959), "Andrew Mattei Gleason",Society of Fellows,Cambridge: Society of Fellows of Harvard University, pp.135–136
  4. ^abcdO'Connor, John J.;Robertson, Edmund F.,"Andrew Mattei Gleason",MacTutor History of Mathematics Archive,University of St Andrews
  5. ^abCastello, Caitlin (October 20, 2008),"Andrew Gleason; helped solve ve xing geometry problem",Boston Globe,archived fromthe originalon May 20, 2013.
  6. ^abcdefghijklmnopq Albers, Donald J.;Alexanderson, Gerald L.;Reid, Constance,eds. (1990), "Andrew M. Gleason",More Mathematical People,Harcourt Brace Jovanovich, p. 86.
  7. ^abcdeBolker, Ethan D. (November 2009), Bolker, Ethan D. (ed.),"50+ years..."(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1237–1239.
  8. ^abcGleason, Jean Berko(November 2009), Bolker, Ethan D. (ed.),"A life well lived"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1266–1267.
  9. ^ Henry A. Gleason Papers,Mertz Library, New York Botanical Garden, archived fromthe originalon July 12, 2010,retrievedApril 9,2013
  10. ^Gallian, Joseph A.,The Putnam Competition from 1938–2013(PDF),retrieved2016-04-10.
  11. ^abcdefghiBurroughs, John; Lieberman, David; Reeds, Jim (November 2009), Bolker, Ethan D. (ed.),"The secret life of Andrew Gleason"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1239–1243.
  12. ^abcdeMazur, 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, archived fromthe original(PDF)on 20 December 2016,retrieved10 April2016.
  13. ^Walsh, Colleen (May 3, 2012),"The oldest endowed professorship: 1721 gift led to long line of Hollis Chair occupants at Divinity School",Harvard Gazette.
  14. ^abRuder, Debra Bradley (May 9, 1996),"Symposium Will Celebrate Gleason and Society of Fellows",Harvard Gazette.
  15. ^abcdHughes-Hallett, Deborah;Stevens, T. Christine;Tecosky-Feldman, Jeff; Tucker, Thomas (November 2009), Bolker, Ethan D. (ed.),"Andy Gleason: teacher"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1260–1265.
  16. ^abcPollak, 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,doi:10.1080/00029890.1996.12004708,JSTOR2975102.
  17. ^abcdeBolker, Ethan D., ed. (November 2009),"Andrew M. Gleason 1921–2008"(PDF),Notices of the American Mathematical Society,56(10).
  18. ^abcdChernoff, Paul R. (November 2009), Bolker, Ethan D. (ed.),"Andy Gleason and quantum mechanics"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1253–1259.
  19. ^Carmichael, Jennifer; Ward, Michael B. (2007), "Everything you want to know about bridge courses–except whether they work: Preliminary findings from a national survey",Joint Mathematics Meeting(PDF).
  20. ^ Andrew M. Gleason. "Evolution of an active mathematical theory", Science31(July 1964), pp. 451–457.
  21. ^Lock, Patti Frazer(1994), "Reflections on the Harvard calculus approach",PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies,4(3):229–234,doi:10.1080/10511979408965753.
  22. ^Wu, H. (1997),"The Mathematics Education Reform: Why You Should be Concerned and What You Can Do"(PDF),American Mathematical Monthly,104(10):946–954,doi:10.2307/2974477,JSTOR2974477.
  23. ^Mac Lane, Saunders(1997),"On the Harvard Consortium Calculus"(PDF),Letters to the Editor,Notices of the American Mathematical Society,44(8): 893.
  24. ^Klein, David; Rosen, Jerry (1997),"Calculus Reform—For the $Millions"(PDF),Notices of the American Mathematical Society,44(10):1324–1325.
  25. ^abcdefghiSpencer, Joel J.(November 2009), Bolker, Ethan D. (ed.),"Andrew Gleason's discrete mathematics"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1251–1253.
  26. ^Dyson, Freeman(February 2009),"Birds and frogs"(PDF),Notices of the American Mathematical Society,56(2):212–223.
  27. ^abcdeIllman, Sören (2001), "Hilbert's fifth problem: review",Journal of Mathematical Sciences,105(2):1843–1847,doi:10.1023/A:1011323915468,MR1871149,S2CID115527342.
  28. ^See, e.g.,Pardon, John(2013), "The Hilbert–Smith conjecture for three-manifolds",Journal of the American Mathematical Society,vol. 26, no. 3, pp.879–899,arXiv:1112.2324,doi:10.1090/s0894-0347-2013-00766-3,S2CID96422853.
  29. ^Spencer, Joel J.(1994),Ten Lectures on the Probabilistic Method,SIAM,p.4,ISBN978-0-89871-325-1
  30. ^ab 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.
  31. ^abRadziszowski, Stanisław (August 22, 2011),"Small Ramsey Numbers",Electronic Journal of Combinatorics,DS1,archived fromthe originalon October 18, 2012,retrievedApril 5,2013.
  32. ^ Sun, Hugo S.; Cohen, M. E. (1984),"An easy proof of the Greenwood-Gleason evaluation of the Ramsey numberR(3,3,3) "(PDF),The Fibonacci Quarterly,22(3):235–238,doi:10.1080/00150517.1984.12429887,MR0765316.
  33. ^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,MR0714453.
  34. ^Andrew M. Gleasonat theMathematics Genealogy Project
  35. ^"Review ofThe Mathematical Theory of Coding,E. F. Assmus, Jr. (1977) ",SIAM Review,19(1):175–176,doi:10.1137/1019032
  36. ^Pless, Vera(September 1991),"In Her Own Words",Notices of the AMS,38(7):702–706, archived fromthe originalon 2016-03-04,retrieved2013-05-06.
  37. ^abBlahut, R. E.(September 2006), "The Gleason-Prange theorem",IEEE Trans. Inf. Theory,37(5), Piscataway, NJ, USA: IEEE Press:1269–1273,doi:10.1109/18.133245.
  38. ^Pless, Vera(2011),"8.4 Gleason polynomials",Introduction to the Theory of Error-Correcting Codes,Wiley Series in Discrete Mathematics and Optimization, vol. 48 (3rd ed.), John Wiley & Sons, pp.134–138,ISBN978-1-118-03099-8.
  39. ^Brown, Thomas A.; Spencer, Joel H. (1971), "Minimization ofmatrices under line shifts ",Colloquium Mathematicum,23:165–171, 177,doi:10.4064/cm-23-1-165-171,MR0307944
  40. ^Wermer, John (November 2009), Bolker, Ethan D. (ed.),"Gleason's work on Banach algebras"(PDF),Andrew M. Gleason 1921–2008,Notices of the American Mathematical Society,56(10):1248–1251.
  41. ^See his 1956 paper "Finite Fano planes".
  42. ^AAAS Newcomb Cleveland Prize,American Association for the Advancement of Science,archived fromthe originalon 2011-06-07,retrieved2016-04-10.
  43. ^ "Hironaka to Teach Math",Harvard Crimson,October 23, 1967
  44. ^Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service,Mathematics Association of America,retrieved2016-08-05.
  45. ^ "Features"(PDF),Notices of the American Mathematical Society,56(10): 1227, November 2009.
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