Andrew M. Gleason - Knowledge

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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.

1226: 912: 824: 692:, which has published a successful and influential series of "calculus reform" textbooks for college and high school, on precalculus, calculus, and other areas. His "credo for this program as for all of his teaching was that the ideas should be based in equal parts of geometry for visualization of the concepts, computation for grounding in the real world, and algebraic manipulation for power." However, the program faced heavy criticism from the mathematics community for its omission of topics such as the 663: 1091: 1974:. 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' – 705: 1886:"Although Andy never earned a Ph.D., he thought of George as his mentor and advisor and lists himself as George's student on the Mathematics Genealogy Project website." 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; in conjunction with his tenure, therefore, Gleason received a Harvard master's degree in 1953. 658:

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.

1381: 380: 498: 613: 44: 580: 1009: 927:. In 1949 he published a paper introducing the "no small subgroups" property of Lie groups (the existence of a neighborhood of the identity within which no nontrivial subgroup exists) that would eventually be crucial to its solution. His 1952 paper on the subject, together with a paper published concurrently by 607:

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

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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.

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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

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In 1964 he created "the first of the 'bridge' courses now ubiquitous for math majors, only twenty years before its time." Such a course is designed to teach new students, accustomed to rote learning of mathematics in secondary school, how to reason abstractly and construct mathematical proofs. That

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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

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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. quick

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temporarily replaced the regular instructor, Gleason found Hille's style "unbelievably different ... He had a view of mathematics that was just vastly different ... That was a very important experience for me. So after that I took a lot of courses from Hille" including, in his sophomore

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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 . I got plenty of practice in doing elementary calculus problems. I don't think there exists a problem‍—‌the

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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

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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

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at Harvard. An early goal of the Junior Fellows program was to allow young scholars showing extraordinary promise to sidestep the lengthy PhD process; four years later Harvard appointed Gleason an assistant professor of mathematics, though he was almost immediately recalled to Washington for

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of 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

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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.

1364:. (In 1959 he wrote that his research "sidelines" included "an intense interest in combinatorial problems.") 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 1249:

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.

1192:(3,3,3) = 17; this remains the only nontrivial multicolor Ramsey number whose exact value is known. As part of their proof, they used an algebraic construction to show that a 16-vertex complete graph can be decomposed into three disjoint copies of a 736:

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

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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:

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recalled "the life and work of eminent American mathematician", calling him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."

712:. "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." 1423:, and at various times held numerous other posts in professional and scholarly organizations, including chairmanship of the Harvard Department of Mathematics. In 1986 he chaired the organizing committee for the 987:

could be given a real analytic structure. It was quickly realized that the answer was negative, after which attention centered on the restricted problem. However, with some additional smoothness assumptions on

892:, 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 1241:, but they were influential ones, and included "many of the seminal ideas and early results" in algebraic coding theory. During the 1950s and 1960s, he attended monthly meetings on coding theory with 935:, solves affirmatively the restricted version of Hilbert's fifth problem, showing that indeed every locally Euclidean group is a Lie group. Gleason's contribution was to prove that this is true when 939:

has the no small subgroups property; Montgomery and Zippin showed every locally Euclidean group has this property. As Gleason told the story, the key insight of his proof was to apply the fact that

783:). His cryptographic work from this period remains classified, but it is known that he recruited mathematicians and taught them cryptanalysis. He served on the advisory boards for the 4343: 588:

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.

815:'s famous classification of mathematicians as being either birds or frogs, Gleason was a frog: he worked as a problem solver rather than a visionary formulating grand theories. 827:

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."

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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.

1172:(4,4) = 18. Since then, only five more of these values have been found. In the same 1955 paper, Greenwood and Gleason also computed the multicolor Ramsey number 591:

At Harvard he "regularly taught at every level", including administratively burdensome multisection courses. One class presented Gleason with a framed print of Picasso's

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Carmichael, Jennifer; Ward, Michael B. (2007), "Everything you want to know about bridge courses–except whether they work: Preliminary findings from a national survey",

1160:. In 1955, motivated by this problem, Gleason and his co-author Robert E. Greenwood made significant progress in the computation of Ramsey numbers with their proof that 435: 3260: 4328: 1806:. Unclassified reprint of a book originally published in 1957 by the National Security Agency, Office of Research and Development, Mathematical Research Division. 371:

called him "one of the quiet giants of twentieth-century mathematics, the consummate professor dedicated to scholarship, teaching, and service in equal measure."

4363: 689: 479: 494:, who spent substantial time with Gleason while visiting Washington, called him "the brilliant young Yale graduate mathematician" in a report of his visit. 3213: 1209:

writes that the paper by Greenwood and Gleason "is now recognized as a classic in the development of Ramsey theory". In the late 1960s, Gleason became the

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mathematical talent, and serving one's institution. Andy Gleason is that rare individual who has done all of these superbly.

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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

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had asked whether Born's rule is a necessary consequence of a particular set of axioms for quantum mechanics, and more specifically whether every

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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

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

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The "unrestricted" version of Hilbert's fifth problem, closer to Hilbert's original formulation, considers both a locally Euclidean group

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The result was routine decryption of this third network by 1944. (This work also involved deeper mathematics related to

522:. 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 423: 764:

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

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year, graduate-level real analysis. "Starting with that course with Hille, I began to have some sense of what mathematics is about."

427: 3637: 1820:. Since its original publications this book has been extended to many different editions and variations with additional co-authors. 1485: 1424: 339:, and the Commonwealth of Massachusetts on mathematics education for children, almost until the end of his life. 757:

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

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His notes and exercises on probability and statistics, drawn up for his lectures to code-breaking colleagues during the war (see

634:, let the reader get his paper and pencils, using perhaps four colors to avoid confusion in the connecting links, and go to work. 3218:, Wiley Series in Discrete Mathematics and Optimization, vol. 48 (3rd ed.), John Wiley & Sons, pp. 134–138, 1365: 1269:, extended by adding a single parity check bit. This "remarkable theorem" shows that this code is highly symmetric, having the 725: 1413: 3223: 2933: 1782: 1409: 474:

during his senior year, Gleason applied for a commission in the US Navy, and on graduation joined the team working to break

4308: 2053: 1561: 1336:, performing computer experiments in 1960. This work studied the average distance to a codeword, for a code related to the 772: 1637: 1852: 1838: 1774: 1188:

vertices has its edges colored with three colors, then it necessarily contains a monochromatic triangle. As they showed,

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in 1952 and the Gung–Hu Distinguished Service Award of the American Mathematical Society in 1996. He was a member of the

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Pollak, H. O. (February 1996), "Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Andrew Gleason",

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classical kind of pseudo reality problem which first and second-year students are given‍—‌that I haven't seen.

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held a special symposium honoring Gleason on his retirement after seven years as its chairman; that same year, the

565: 365:"really aren't there to convince you that something is true‍—‌they're there to show you why it is true." The 2372: 4323: 4098: 3053: 1695: 1518: 4189: 2566: 1405: 1230: 1137: 862:

The "restricted" version of Hilbert's fifth problem (solved by Gleason) asks, more specifically, whether every

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article, Gleason wrote of an apparent paradox arising in attempts to explain mathematics to nonmathematicians:

434:. Though Gleason's mathematics education had gone only so far as some self-taught calculus, Yale mathematician 408: 351: 17: 4353: 4213: 3647: 3093: 2637: 2460: 1420: 738: 483: 332: 4173: 3630: 2373:"The oldest endowed professorship: 1721 gift led to long line of Hollis Chair occupants at Divinity School" 1401: 997: 791:, and he continued to recruit, and to advise the military on cryptanalysis, almost to the end of his life. 347: 1499: 1200: 1095: 1082:

showed that Gleason's theorem could be used to remove this extra assumption from von Neumann's argument.

996:, it might yet be possible to prove the existence of a real analytic structure on the group action. The 780: 709: 704: 487: 313: 119: 2991: 2346: 451:

One month later he enrolled in a differential equations course ("mostly full of seniors") as well. When

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Gleason made fundamental contributions to widely varied areas of mathematics, including the theory of

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in the US. He retired from Harvard in 1992 but remained active in service to Harvard (as chair of the

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who made fundamental contributions to widely varied areas of mathematics, including the solution of

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Gleason's interest in the fifth problem began in the late 1940s, sparked by a course he took from

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Rigby, J. F. (1983), "Some geometrical aspects of a maximal three-coloured triangle-free graph",

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on the lattice of projections of a Hilbert space can be defined by a positive operator with unit

1031: 944: 859:: to what extent does their topology provide information sufficient to determine their geometry? 343: 177: 3777: 3547: 3539: 2954:(1992), "Roots of Ramsey theory", in Bolker, E.; Cherno, P.; Costes, C.; Lieberman, D. (eds.), 2419: 2214: 2114: 1842:. 63 minutes, black & white. Produced by Richard G. Long and directed by Allan Hinderstein. 1811: 1676:—— (1971), "Weight polynomials of self-dual codes and the MacWilliams identities", 1262: 1245:

and others at the Air Force Cambridge Research Laboratory. Pless, who had previously worked in

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second-graders. Some years later he gave a talk in which he described his goal as having been:

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Gleason is the namesake of the Gleason polynomials, a system of polynomials that generate the

463:, always placing among the top five entrants in the country (making him the second three-time 3999: 3991: 3943: 3836: 3820: 3761: 1287: 1129: 1044: 952: 950:. On finding the solution, he took a week of leave to write it up, and it was printed in the 836: 557: 2955: 1832: 1225: 1148:) ≥ 3 only finitely many of them are known precisely, and an exact computation of 4273: 4268: 3951: 3769: 3409: 3285: 3242: 3076: 3037: 2842: 2423: 1792: 1758: 1750: 1718: 1685: 1660: 1619: 1584: 1547: 1495: 1428: 1326: 1056: 1017: 911: 877: 617: 510: 475: 412: 305: 278: 114: 1627: 823: 8: 3844: 3753: 3729: 3678: 2022: 1693:

Greenwood, R. E.; Gleason, A. M. (1955), "Combinatorial relations and chromatic graphs",

1556: 1380: 1193: 1075: 1070:. Von Neumann had claimed to show that hidden variable theories were impossible, but (as 976: 1074:

pointed out) his demonstration made an assumption that quantum systems obeyed a form of

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Burroughs, John; Lieberman, David; Reeds, Jim (November 2009), Bolker, Ethan D. (ed.),

2182: 2178: 2147: 1934: 1766: 1722: 1664: 1535: 1322: 1027: 940: 693: 534: 502: 388: 362: 324: 263: 253: 215: 143: 67: 3439:"Faculty of Arts and Sciences – Memorial Minute. Andrew Mattei Gleason" 1352:, and made other mathematical contributions including work on 379: 4023: 3967: 3892: 3876: 3860: 3737: 3593: 3561: 3219: 3067: 2929: 2850: 2672: 2598: 1778: 1726: 1668: 1349: 1079: 947: 881: 869: 856: 804: 747: 546: 542: 527: 400: 2897: 2889: 2705: 1771:

The William Lowell Putnam Mathematical Competition: Problems and Solutions 1938–1964

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While at Yale he competed three times (1940, 1941 and 1942) in the recently founded

4197: 4082: 3975: 3935: 3868: 3670: 3271: 3187: 3118: 3062: 2885: 2828: 2646: 2630:"The Mathematics Education Reform: Why You Should be Concerned and What You Can Do" 2610: 2473: 2469: 1704: 1646: 1623: 1605: 1570: 1527: 1369: 1246: 1210: 1067: 928: 905: 897: 893: 631: 497: 222: 3745: 4015: 3916: 3531: 3491: 3281: 3072: 3033: 2911: 2838: 2750: 2676: 2426:; Tecosky-Feldman, Jeff; Tucker, Thomas (November 2009), Bolker, Ethan D. (ed.), 1788: 1755: 1746: 1714: 1681: 1656: 1615: 1593: 1580: 1543: 1491: 1353: 1052: 957: 866: 612: 431: 96: 43: 2064: 579: 4165: 4074: 3983: 3959: 2629: 2321: 2235: 2118: 1930: 1857: 1181: 901: 863: 758: 733: 724:

group. One task of this group, in collaboration with British cryptographers at

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Yueh-Gin Gung and Dr. Charles Y. Hu Distinguished Service to Mathematics Award

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PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies

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alongside the paper of Montgomery and Zippin; another paper a year later by

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and decisive response: "No, I didn't think about that at all. I had a ball!"

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Andrew M. Gleason. "Evolution of an active mathematical theory", Science

2317: 1361: 1298:: in this case there are just two of them, the two bivariate polynomials 1291: 729: 491: 452: 258: 248: 201: 3383: 3015:"An easy proof of the Greenwood-Gleason evaluation of the Ramsey number 1745:, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1000:, still unsolved, encapsulates the remaining difficulties of this case. 4157: 4050: 3479: 3205: 3135: 2658: 2481: 2003:, Cambridge: Society of Fellows of Harvard University, pp. 135–136 1651: 1539: 1242: 1035: 1026:

states that an observable property of a quantum system is defined by a

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Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 3

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Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service

1935:"Gleason's contribution to the solution of Hilbert's Fifth Problem" 1090: 742: 675: 2880: 1152:(6,6) is believed to be out of reach. In 1953, the calculation of 1078:

for noncommuting operators that might not hold a priori. In 1966,

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he felt would be central to next century of mathematics research.

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training for several decades; they were published openly in 1985.

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He died on October 17, 2008 from complications following surgery.

717: 1217:, who also became known for his contributions to Ramsey theory. 670:

Gleason was the first chairman of the advisory committee of the

1635:—— (1967), "A characterization of maximal ideals", 1062:

Gleason's theorem implies the nonexistence of certain types of

396: 1978:. The degree given for this purpose is Master of Arts (A.M.)." 1140:. Ramsey numbers require enormous effort to compute; when max( 403:, and his mother was the daughter of Swiss-American winemaker 1800:——; Penney, Walter F.; Wyllys, Ronald E. (1985), 960:

removed some technical side conditions from Gleason's proof.

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Brown, Thomas A.; Spencer, Joel H. (1971), "Minimization of

2868:(2013), "The Hilbert–Smith conjecture for three-manifolds", 1066:

for quantum mechanics, strengthening a previous argument of

1059:(published 1957) shows it to be true for higher dimensions. 2538:

Chernoff, Paul R. (November 2009), Bolker, Ethan D. (ed.),

2213:, Mertz Library, New York Botanical Garden, archived from 2146:

Bolker, Ethan D. (November 2009), Bolker, Ethan D. (ed.),

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proved this was false for two-dimensional Hilbert spaces,

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Members of the United States National Academy of Sciences

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Illman, Sören (2001), "Hilbert's fifth problem: review",

2395:"Symposium Will Celebrate Gleason and Society of Fellows" 478:. (Others on this team included his future collaborator 335:. He continued to advise the United States government on 183:

Gung–Hu Distinguished Service to Mathematics Award (1996)

2601:(1994), "Reflections on the Harvard calculus approach", 2330:"Andrew Gleason, 4 November 1921 – 17 October 2008" 1294:. These polynomials take a particularly simple form for 2280: 1754:. Corrected reprint, Boston: Jones and Bartlett, 1991, 3298:

Wermer, John (November 2009), Bolker, Ethan D. (ed.),

2112: 2054:"Andrew Gleason; helped solve vexing geometry problem" 1455:

After his death a 32-page collection of essays in the

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Brinton, Crane, ed. (1959), "Andrew Mattei Gleason",

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Elementary Course in Probability for the Cryptanalyst

1764: 1557:"Measures on the closed subspaces of a Hilbert space" 1477:"One-parameter subgroups and Hilbert's fifth problem" 556:. Established in 1727, this is the oldest scientific 4339:

Fellows of the American Academy of Arts and Sciences

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Harvard University Department of Mathematics faculty

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Introduction to the Theory of Error-Correcting Codes

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Andrew M. Gleason, Glimpses of a Life in Mathematics

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is named after Gleason's work with AFCRL researcher

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with 16 vertices and 40 edges (sometimes called the

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Hollis Chair of Mathematicks and Natural Philosophy

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American Association for the Advancement of Science

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American Association for the Advancement of Science

666:"With the inevitable clipboard under his arm", 1989 4334:Hollis Chair of Mathematics and Natural Philosophy 3254: 2063: 554:Hollis Chair of Mathematics and Natural Philosophy 537:whom he had met at a party featuring the music of 356:Hollis Chair of Mathematics and Natural Philosophy 3186:(5), Piscataway, NJ, USA: IEEE Press: 1269–1273, 2338:Proceedings of the American Philosophical Society 2316: 2020: 509:In 1946, at the recommendation of Navy colleague 4260: 3178:(September 2006), "The Gleason-Prange theorem", 1692: 767:In 1950 Gleason returned to active duty for the 574: 523: 486:) He also collaborated with British researchers 4329:Presidents of the American Mathematical Society 2874:, vol. 26, no. 3, pp. 879–899, 2564: 2276: 2274: 2272: 2270: 2268: 2266: 2264: 2262: 2260: 461:William Lowell Putnam Mathematical Competition 3631: 3465: 3083: 2946: 2944: 2745: 2743: 2741: 2739: 2737: 2735: 2733: 2731: 2729: 1680:, Paris: Gauthier-Villars, pp. 211–215, 1516:—— (1956), "Finite Fano planes", 1446:. A past president of the Association wrote: 1332:In this area, he also did pioneering work in 979:action. Hilbert asked whether, in this case, 391:, the youngest of three children; his father 4364:United States Navy personnel of World War II 3417:Notices of the American Mathematical Society 3307:Notices of the American Mathematical Society 3238: 2977: 2871:Journal of the American Mathematical Society 2792:Notices of the American Mathematical Society 2762:Notices of the American Mathematical Society 2714:Notices of the American Mathematical Society 2685:Notices of the American Mathematical Society 2547:Notices of the American Mathematical Society 2513:Notices of the American Mathematical Society 2435:Notices of the American Mathematical Society 2290:Notices of the American Mathematical Society 2190:Notices of the American Mathematical Society 2155:Notices of the American Mathematical Society 1942:Notices of the American Mathematical Society 1925: 1923: 1921: 1919: 1917: 1915: 1913: 1911: 1457:Notices of the American Mathematical Society 368:Notices of the American Mathematical Society 2978:Radziszowski, Stanisław (August 22, 2011), 2257: 2241:The Putnam Competition from 1938–2013 720:, the U.S. Navy's signals intelligence and 363:mathematical proofs 3638: 3624: 3472: 3458: 2973: 2971: 2941: 2726: 2703: 2016: 2014: 2012: 2010: 880:, can that structure be strengthened to a 818: 415:, where his father was the curator of the 42: 3275: 3170: 3168: 3066: 2879: 2832: 2814: 2812: 2810: 2808: 2806: 2753:(November 2009), Bolker, Ethan D. (ed.), 2453: 2451: 2449: 2181:(November 2009), Bolker, Ethan D. (ed.), 2141: 2139: 2137: 2135: 2133: 1994: 1992: 1990: 1988: 1986: 1984: 1933:(November 2009), Bolker, Ethan D. (ed.), 1908: 1708: 1650: 1609: 1574: 1419:In 1981 and 1982 he was president of the 779:(which much later became the home of the 426:(Berkeley, California) he graduated from 387:Gleason was born on November 4, 1921, in 3324:See his 1956 paper "Finite Fano planes". 3012: 2917:Ten Lectures on the Probabilistic Method 2671: 2537: 2533: 2531: 2529: 2527: 2388: 2386: 2051: 1882: 1880: 1878: 1486:International Congress of Mathematicians 1463: 1425:International Congress of Mathematicians 1379: 1224: 1089: 1007: 910: 822: 716:During World War II Gleason was part of 708:Report (1945) by Gleason and colleagues 703: 661: 611: 578: 496: 378: 2968: 2910: 2755:"Andrew Gleason's discrete mathematics" 2749: 2504:Bolker, Ethan D., ed. (November 2009), 2499: 2497: 2495: 2493: 2491: 2177: 2125:, Harcourt Brace Jovanovich, p. 86 2047: 2045: 2032:MacTutor History of Mathematics Archive 2007: 1998: 1830: 1740: 1675: 1634: 1591: 1554: 1515: 1474: 1237:Gleason published few contributions to 919:at Alice Mackey's 80th birthday (2000). 794: 710:regarding the German Enigma 14: 4261: 3297: 3174: 3165: 2950: 2864: 2818: 2803: 2503: 2457: 2446: 2312: 2310: 2308: 2306: 2304: 2145: 2130: 2108: 2106: 2104: 2102: 2100: 2098: 2096: 2052:Castello, Caitlin (October 20, 2008), 1981: 1959: 1929: 1853:Bell's critique of von Neumann's proof 1765:——; Greenwood, Robert E.; 3619: 3453: 3204: 3134: 3050: 2779: 2524: 2414: 2412: 2410: 2408: 2392: 2383: 2370: 2173: 2171: 2169: 2094: 2092: 2090: 2088: 2086: 2084: 2082: 2080: 2078: 2076: 1875: 1864:conjectured by Gleason to be infinite 1804:, Laguna Hills, CA: Aegean Park Press 1431:, and was president of the Congress. 1410:American Academy of Arts and Sciences 1156:(3,3) was given as a question in the 699: 595:in recognition of his care for them. 430:in Yonkers, winning a scholarship to 4284:21st-century American mathematicians 4279:20th-century American mathematicians 2706:"Calculus Reform—For the $ Millions" 2677:"On the Harvard Consortium Calculus" 2597: 2540:"Andy Gleason and quantum mechanics" 2488: 2393:Ruder, Debra Bradley (May 9, 1996), 2121:, eds. (1990), "Andrew M. Gleason", 2042: 1562:Journal of Mathematics and Mechanics 1375: 1120:such that every graph with at least 1003: 872:is a Lie group. That is, if a group 160: 3300:"Gleason's work on Banach algebras" 3128: 3013:Sun, Hugo S.; Cohen, M. E. (1984), 2984:Electronic Journal of Combinatorics 2704:Klein, David; Rosen, Jerry (1997), 2301: 2283:"The secret life of Andrew Gleason" 2234: 1839:Mathematical Association of America 1775:Mathematical Association of America 1283:) as a subgroup of its symmetries. 27:American mathematician and educator 24: 3390:Mathematics Association of America 2627: 2405: 2166: 2073: 1834:Nim and other oriented-graph games 1440:Mathematics Association of America 639: 518:cryptographic work related to the 25: 4375: 3431: 3107:The Mathematical Theory of Coding 1743:Fundamentals of Abstract Analysis 601:Fundamentals of Abstract Analysis 599:effort led to publication of his 411:became a linguist. He grew up in 2821:Journal of Mathematical Sciences 1488:, Cambridge, Mass., 1950, Vol. 2 1388:In 1952 Gleason was awarded the 1220: 1085: 566:Massachusetts Board of Education 3402: 3376: 3358: 3327: 3318: 3305:, Andrew M. Gleason 1921–2008, 3291: 3232: 3198: 3098: 3054:Journal of Combinatorial Theory 3044: 3006: 2904: 2890:10.1090/s0894-0347-2013-00766-3 2857: 2773: 2760:, Andrew M. Gleason 1921–2008, 2697: 2665: 2621: 2591: 2578: 2558: 2545:, Andrew M. Gleason 1921–2008, 2433:, Andrew M. Gleason 1921–2008, 2364: 2288:, Andrew M. Gleason 1921–2008, 2228: 2201: 2188:, Andrew M. Gleason 1921–2008, 2153:, Andrew M. Gleason 1921–2008, 1940:, Andrew M. Gleason 1921–2008, 1889: 1696:Canadian Journal of Mathematics 1598:Illinois Journal of Mathematics 1594:"Projective topological spaces" 1519:American Journal of Mathematics 1384:In Naval Reserve uniform, 1960s 1012:With family cat Fred about 1966 603:, of which one reviewer wrote: 442:normally intended for juniors. 156: 4359:Mathematicians from California 4349:People from Fresno, California 4294:American mathematical analysts 2474:10.1080/00029890.1996.12004708 2371:Walsh, Colleen (May 3, 2012), 1953: 1638:Journal d'Analyse Mathématique 1414:Société Mathématique de France 1406:American Philosophical Society 1348:Gleason founded the theory of 1343: 789:Institute for Defense Analyses 672:School Mathematics Study Group 472:Japanese attacked Pearl Harbor 352:American Philosophical Society 13: 1: 3648:American Mathematical Society 3262:matrices under line shifts", 3109:, E. F. Assmus, Jr. (1977)", 3094:Mathematics Genealogy Project 2638:American Mathematical Monthly 2506:"Andrew M. Gleason 1921–2008" 2461:American Mathematical Monthly 1962:"Honorary degrees at Harvard" 1902: 1421:American Mathematical Society 1176:(3,3,3): the smallest number 575:Teaching and education reform 438:urged him to try a course in 333:American Mathematical Society 3336:AAAS Newcomb Cleveland Prize 3150:(7): 702–706, archived from 3068:10.1016/0095-8956(83)90043-6 2345:(4): 471–476, archived from 1976:ut in grege nostro numeretur 1402:National Academy of Sciences 688:In 1986 he helped found the 549:. They had three daughters. 374: 348:National Academy of Sciences 296:(1921–2008) was an American 232:Other academic advisors 7: 4309:American quantum physicists 1960:Elkins, Kimball C. (1958), 1846: 1831:Gleason, Andrew M. (1966), 1741:Gleason, Andrew M. (1966), 1229:With his brother, linguist 1124:vertices contains either a 781:DHS Cyber Security Division 545:, worked for many years at 533:In January 1959 he married 488:attacking the German Enigma 361:He was fond of saying that 10: 4380: 3901:Theophil Henry Hildebrandt 1436:Harvard Society of Fellows 1168:(3,5) = 14, and 1015: 732:, was to penetrate German 674:, which helped define the 513:, Gleason was appointed a 329:Harvard Society of Fellows 316:, an important example in 4149: 4034: 3927: 3812: 3705: 3654: 3487: 3423:(10): 1227, November 2009 3210:"8.4 Gleason polynomials" 2682:, Letters to the Editor, 2615:10.1080/10511979408965753 2588:(July 1964), pp. 451–457. 2568:Joint Mathematics Meeting 1576:10.1512/iumj.1957.6.56050 1358:enumerative combinatorics 1116:) is the smallest number 1076:additivity of expectation 752:graph isomorphism problem 552:In 1969 Gleason took the 417:New York Botanical Garden 287: 241: 231: 221: 211: 197: 190: 170: 136: 102: 92: 74: 53: 41: 34: 3578:John Hasbrouck Van Vleck 3366:"Hironaka to Teach Math" 2123:More Mathematical People 2037:University of St Andrews 1966:Harvard Library Bulletin 1868: 1400:. He was elected to the 1338:Berlekamp switching game 1334:experimental mathematics 1064:hidden variable theories 998:Hilbert–Smith conjecture 785:National Security Agency 644:National Security Agency 422:After briefly attending 331:, and presidency of the 86:Cambridge, Massachusetts 4289:American cryptographers 4123:Cathleen Synge Morawetz 3695:Robert Simpson Woodward 3663:John Howard Van Amringe 3570:Percy Williams Bridgman 3277:10.4064/cm-23-1-165-171 3264:Colloquium Mathematicum 3180:IEEE Trans. Inf. Theory 3026:The Fibonacci Quarterly 2834:10.1023/A:1011323915468 2428:"Andy Gleason: teacher" 2420:Hughes-Hallett, Deborah 2210:Henry A. Gleason Papers 2115:Alexanderson, Gerald L. 2027:"Andrew Mattei Gleason" 1812:Hughes-Hallett, Deborah 1592:—— (1958), 1555:—— (1957), 1475:Gleason, A. M. (1952), 1398:Hilbert's fifth problem 1394:Newcomb Cleveland Prize 1271:projective linear group 1231:Henry Allan Gleason Jr. 1201:Greenwood–Gleason graph 1096:Greenwood–Gleason graph 1032:separable Hilbert space 882:real analytic structure 876:has the structure of a 841:Hilbert's fifth problem 819:Hilbert's fifth problem 777:Nebraska Avenue Complex 526:). Harvard awarded him 436:William Raymond Longley 344:Newcomb Cleveland Prize 314:Greenwood–Gleason graph 302:Hilbert's fifth problem 178:Newcomb Cleveland Prize 120:Greenwood–Gleason graph 110:Hilbert's fifth problem 4324:Yale University alumni 3778:Leonard Eugene Dickson 3548:Wallace Clement Sabine 3540:Benjamin Osgood Peirce 3256: 2980:"Small Ramsey Numbers" 1710:10.4153/CJM-1955-001-4 1611:10.1215/ijm/1255454110 1453: 1412:, and belonged to the 1408:, was a Fellow of the 1385: 1263:quadratic residue code 1255:Gleason–Prange theorem 1234: 1098: 1013: 920: 828: 741:observed that certain 713: 686: 667: 660: 636: 621: 610: 584: 506: 449: 384: 337:cryptographic security 125:Gleason–Prange theorem 4000:Charles B. Morrey Jr. 3992:Abraham Adrian Albert 3944:Gordon Thomas Whyburn 3837:Earle Raymond Hedrick 3821:George David Birkhoff 3762:Edward Burr Van Vleck 3257: 3255:{\displaystyle \pm 1} 2424:Stevens, T. Christine 1464:Selected publications 1448: 1383: 1366:compass, straightedge 1228: 1164:(3,4) = 9, 1093: 1011: 967:and another manifold 953:Annals of Mathematics 914: 884:, so that within any 826: 707: 681: 665: 655: 642:) remained in use in 627: 615: 605: 582: 558:endowed professorship 500: 444: 428:Roosevelt High School 382: 320:, are named for him. 294:Andrew Mattei Gleason 4354:American topologists 3952:Raymond Louis Wilder 3770:Ernest William Brown 3243: 2148:"50+ years ..." 2023:Robertson, Edmund F. 1429:Berkeley, California 1327:MacWilliams identity 1321:. Gleason's student 878:topological manifold 795:Mathematics research 773:Lieutenant Commander 530:the following year. 511:Donald Howard Menzel 476:Japanese naval codes 424:Berkeley High School 413:Bronxville, New York 407:. His older brother 399:and a member of the 279:T. Christine Stevens 3845:Luther P. Eisenhart 3754:Henry Burchard Fine 3730:William Fogg Osgood 3679:George William Hill 2994:on October 18, 2012 2352:on 20 December 2016 2183:"A life well lived" 2179:Gleason, Jean Berko 2113:Albers, Donald J.; 2021:O'Connor, John J.; 1767:Kelly, Leroy Milton 1484:Proceedings of the 941:monotonic functions 690:Calculus Consortium 482:and Yale professor 480:Robert E. Greenwood 129:Gleason polynomials 3853:Arthur Byron Coble 3794:Gilbert Ames Bliss 3646:Presidents of the 3372:, October 23, 1967 3252: 3144:Notices of the AMS 3140:"In Her Own Words" 3138:(September 1991), 2673:Mac Lane, Saunders 2599:Lock, Patti Frazer 2236:Gallian, Joseph A. 2001:Society of Fellows 1652:10.1007/bf02788714 1386: 1350:Dirichlet algebras 1323:Jessie MacWilliams 1288:weight enumerators 1235: 1233:, in Toronto, 1969 1158:Putnam Competition 1099: 1028:Hermitian operator 1014: 983:and the action of 921: 857:topological spaces 829: 748:permutation groups 714: 700:Cryptanalysis work 694:mean value theorem 668: 622: 585: 562:Society of Fellows 507: 389:Fresno, California 385: 325:Harvard University 264:Jessie MacWilliams 254:Daniel I. A. Cohen 216:Harvard University 144:Jean Berko Gleason 68:Fresno, California 4256: 4255: 4067:Andrew M. Gleason 4024:Saunders Mac Lane 3968:Edward J. McShane 3893:Marshall H. Stone 3877:Griffith C. Evans 3861:Solomon Lefschetz 3738:Henry Seely White 3613: 3612: 3606: 3598: 3594:Bertrand Halperin 3590: 3582: 3574: 3566: 3558: 3552: 3544: 3536: 3528: 3520: 3512: 3504: 3496: 3225:978-1-118-03099-8 3192:10.1109/18.133245 3090:Andrew M. Gleason 2935:978-0-89871-325-1 2785:"Birds and frogs" 2783:(February 2009), 2328:(December 2010), 1784:978-0-88385-462-4 1376:Awards and honors 1080:John Stewart Bell 1057:Gleason's theorem 1018:Gleason's theorem 1004:Quantum mechanics 948:almost everywhere 888:of an element of 870:topological group 805:quantum mechanics 547:Boston University 484:Marshall Hall Jr. 401:Mayflower Society 306:Gleason's theorem 291: 290: 242:Doctoral students 192:Scientific career 115:Gleason's theorem 36:Andrew M. Gleason 16:(Redirected from 4371: 4299:Coding theorists 4249: 4241: 4233: 4225: 4217: 4209: 4201: 4198:Eric Friedlander 4193: 4185: 4177: 4169: 4161: 4142: 4134: 4126: 4118: 4110: 4102: 4094: 4086: 4083:Irving Kaplansky 4078: 4070: 4062: 4054: 4046: 4027: 4019: 4011: 4003: 3995: 3987: 3979: 3976:Deane Montgomery 3971: 3963: 3955: 3947: 3939: 3936:John von Neumann 3920: 3912: 3904: 3896: 3888: 3880: 3872: 3869:Robert Lee Moore 3864: 3856: 3848: 3840: 3832: 3824: 3805: 3797: 3789: 3781: 3773: 3765: 3757: 3749: 3741: 3733: 3725: 3717: 3698: 3690: 3682: 3674: 3671:Emory McClintock 3666: 3640: 3633: 3626: 3617: 3616: 3604: 3596: 3588: 3580: 3572: 3564: 3556: 3550: 3542: 3534: 3526: 3518: 3510: 3502: 3494: 3474: 3467: 3460: 3451: 3450: 3446: 3426: 3424: 3414: 3406: 3400: 3398: 3397: 3396: 3380: 3374: 3373: 3362: 3356: 3354: 3353: 3352: 3343:, archived from 3331: 3325: 3322: 3316: 3314: 3304: 3295: 3289: 3288: 3279: 3270:: 165–171, 177, 3261: 3259: 3258: 3253: 3236: 3230: 3228: 3202: 3196: 3194: 3172: 3163: 3161: 3160: 3159: 3132: 3126: 3125: 3102: 3096: 3087: 3081: 3079: 3070: 3048: 3042: 3040: 3023: 3010: 3004: 3002: 3001: 2999: 2990:, archived from 2975: 2966: 2964: 2963:, pp. 39–47 2962: 2948: 2939: 2938: 2912:Spencer, Joel J. 2908: 2902: 2900: 2883: 2861: 2855: 2853: 2836: 2827:(2): 1843–1847, 2816: 2801: 2799: 2789: 2777: 2771: 2769: 2759: 2751:Spencer, Joel J. 2747: 2724: 2722: 2710: 2701: 2695: 2693: 2681: 2669: 2663: 2661: 2634: 2625: 2619: 2617: 2595: 2589: 2582: 2576: 2574: 2573: 2562: 2556: 2554: 2544: 2535: 2522: 2520: 2510: 2501: 2486: 2484: 2455: 2444: 2442: 2432: 2416: 2403: 2401: 2390: 2381: 2379: 2368: 2362: 2360: 2359: 2357: 2351: 2334: 2314: 2299: 2297: 2287: 2278: 2255: 2253: 2252: 2251: 2246: 2232: 2226: 2225: 2224: 2222: 2217:on July 12, 2010 2205: 2199: 2197: 2187: 2175: 2164: 2162: 2152: 2143: 2128: 2126: 2110: 2071: 2069: 2067: 2062:, archived from 2049: 2040: 2039: 2018: 2005: 2004: 1996: 1979: 1973: 1957: 1951: 1949: 1939: 1927: 1896: 1893: 1887: 1884: 1841: 1819: 1810:——; 1805: 1795: 1753: 1729: 1712: 1688: 1671: 1654: 1630: 1613: 1587: 1578: 1550: 1512: 1511: 1510: 1504: 1498:, archived from 1481: 1442:awarded him the 1396:for his work on 1247:abstract algebra 1211:doctoral advisor 1180:such that, if a 1068:John von Neumann 929:Deane Montgomery 915:With his mentor 908:, among others. 906:Garrett Birkhoff 898:John von Neumann 894:L. E. J. Brouwer 845:characterization 593:Mother and Child 342:Gleason won the 223:Doctoral advisor 164: 162: 158: 146: 81: 78:October 17, 2008 64:November 4, 1921 63: 61: 46: 32: 31: 21: 4379: 4378: 4374: 4373: 4372: 4370: 4369: 4368: 4304:Graph theorists 4259: 4258: 4257: 4252: 4244: 4236: 4228: 4220: 4212: 4204: 4196: 4188: 4180: 4172: 4164: 4156: 4145: 4137: 4129: 4121: 4113: 4105: 4099:William Browder 4097: 4089: 4081: 4073: 4065: 4057: 4049: 4041: 4030: 4022: 4016:Nathan Jacobson 4014: 4006: 3998: 3990: 3982: 3974: 3966: 3958: 3950: 3942: 3934: 3923: 3917:Joseph L. Walsh 3915: 3907: 3899: 3891: 3883: 3875: 3867: 3859: 3851: 3843: 3835: 3827: 3819: 3808: 3800: 3792: 3784: 3776: 3768: 3760: 3752: 3744: 3736: 3728: 3720: 3712: 3701: 3693: 3685: 3677: 3669: 3661: 3650: 3644: 3614: 3609: 3532:Joseph Lovering 3508:Samuel Williams 3492:Isaac Greenwood 3483: 3478: 3445:, April 1, 2010 3443:Harvard Gazette 3437: 3434: 3429: 3412: 3408: 3407: 3403: 3394: 3392: 3382: 3381: 3377: 3370:Harvard Crimson 3364: 3363: 3359: 3350: 3348: 3333: 3332: 3328: 3323: 3319: 3313:(10): 1248–1251 3302: 3296: 3292: 3244: 3241: 3240: 3237: 3233: 3226: 3203: 3199: 3173: 3166: 3157: 3155: 3133: 3129: 3123:10.1137/1019032 3104: 3103: 3099: 3088: 3084: 3049: 3045: 3021: 3011: 3007: 2997: 2995: 2976: 2969: 2960: 2949: 2942: 2936: 2909: 2905: 2862: 2858: 2817: 2804: 2787: 2778: 2774: 2768:(10): 1251–1253 2757: 2748: 2727: 2721:(10): 1324–1325 2708: 2702: 2698: 2679: 2670: 2666: 2651:10.2307/2974477 2645:(10): 946–954, 2632: 2628:Wu, H. (1997), 2626: 2622: 2596: 2592: 2583: 2579: 2571: 2563: 2559: 2553:(10): 1253–1259 2542: 2536: 2525: 2508: 2502: 2489: 2456: 2447: 2441:(10): 1260–1265 2430: 2417: 2406: 2399:Harvard Gazette 2391: 2384: 2377:Harvard Gazette 2369: 2365: 2355: 2353: 2349: 2332: 2322:Gross, Benedict 2315: 2302: 2296:(10): 1239–1243 2285: 2279: 2258: 2249: 2247: 2244: 2233: 2229: 2220: 2218: 2207: 2206: 2202: 2196:(10): 1266–1267 2185: 2176: 2167: 2161:(10): 1237–1239 2150: 2144: 2131: 2119:Reid, Constance 2111: 2074: 2068:on May 20, 2013 2050: 2043: 2019: 2008: 1997: 1982: 1958: 1954: 1948:(10): 1243–1248 1937: 1931:Palais, Richard 1928: 1909: 1905: 1900: 1899: 1894: 1890: 1885: 1876: 1871: 1849: 1785: 1604:(4A): 482–489, 1532:10.2307/2372469 1508: 1506: 1502: 1479: 1469:Research papers 1466: 1378: 1370:angle trisector 1354:finite geometry 1346: 1310: + 14 1296:self-dual codes 1278: 1223: 1197:5-regular graph 1138:independent set 1088: 1053:Richard Kadison 1020: 1006: 958:Hidehiko Yamabe 821: 811:. According to 797: 771:, serving as a 702: 583:Australia, 1988 577: 432:Yale University 377: 354:, and held the 283: 186: 166: 154: 150: 147: 142: 132: 97:Yale University 93:Alma mater 88: 83: 79: 70: 65: 59: 57: 49: 37: 28: 23: 22: 15: 12: 11: 5: 4377: 4367: 4366: 4361: 4356: 4351: 4346: 4341: 4336: 4331: 4326: 4321: 4319:Putnam Fellows 4316: 4311: 4306: 4301: 4296: 4291: 4286: 4281: 4276: 4271: 4254: 4253: 4251: 4250: 4242: 4234: 4226: 4218: 4210: 4202: 4194: 4190:George Andrews 4186: 4178: 4170: 4166:David Eisenbud 4162: 4153: 4151: 4147: 4146: 4144: 4143: 4135: 4127: 4119: 4111: 4103: 4095: 4087: 4079: 4075:Julia Robinson 4071: 4063: 4055: 4047: 4038: 4036: 4032: 4031: 4029: 4028: 4020: 4012: 4004: 3996: 3988: 3984:Joseph L. Doob 3980: 3972: 3964: 3960:Richard Brauer 3956: 3948: 3940: 3931: 3929: 3925: 3924: 3922: 3921: 3913: 3905: 3897: 3889: 3881: 3873: 3865: 3857: 3849: 3841: 3833: 3825: 3816: 3814: 3810: 3809: 3807: 3806: 3798: 3790: 3782: 3774: 3766: 3758: 3750: 3742: 3734: 3726: 3718: 3709: 3707: 3703: 3702: 3700: 3699: 3691: 3683: 3675: 3667: 3658: 3656: 3652: 3651: 3643: 3642: 3635: 3628: 3620: 3611: 3610: 3608: 3607: 3599: 3591: 3586:Andrew Gleason 3583: 3575: 3567: 3562:Theodore Lyman 3559: 3553: 3545: 3537: 3529: 3521: 3513: 3505: 3497: 3488: 3485: 3484: 3477: 3476: 3469: 3462: 3454: 3448: 3447: 3433: 3432:External links 3430: 3428: 3427: 3401: 3375: 3357: 3326: 3317: 3290: 3251: 3248: 3231: 3224: 3197: 3164: 3127: 3117:(1): 175–176, 3097: 3082: 3061:(3): 313–322, 3043: 3032:(3): 235–238, 3005: 2967: 2940: 2934: 2903: 2856: 2802: 2781:Dyson, Freeman 2772: 2725: 2696: 2664: 2620: 2609:(3): 229–234, 2590: 2577: 2557: 2523: 2487: 2468:(2): 105–106, 2445: 2404: 2382: 2363: 2326:Mumford, David 2300: 2256: 2227: 2200: 2165: 2129: 2072: 2041: 2006: 1980: 1952: 1906: 1904: 1901: 1898: 1897: 1888: 1873: 1872: 1870: 1867: 1866: 1865: 1858:Pierpont prime 1855: 1848: 1845: 1844: 1843: 1827: 1826: 1822: 1821: 1807: 1797: 1783: 1762: 1737: 1736: 1732: 1731: 1690: 1673: 1632: 1589: 1569:(4): 885–893, 1552: 1526:(4): 797–807, 1513: 1471: 1470: 1465: 1462: 1377: 1374: 1345: 1342: 1276: 1222: 1219: 1182:complete graph 1087: 1084: 1016:Main article: 1005: 1002: 945:differentiable 902:Lev Pontryagin 820: 817: 796: 793: 759:Chernoff bound 734:Enigma machine 726:Bletchley Park 701: 698: 576: 573: 543:psycholinguist 383:US Navy, 1940s 376: 373: 289: 288: 285: 284: 282: 281: 276: 271: 269:Richard Palais 266: 261: 256: 251: 245: 243: 239: 238: 233: 229: 228: 225: 219: 218: 213: 209: 208: 199: 195: 194: 188: 187: 185: 184: 181: 174: 172: 168: 167: 152: 148: 141: 140: 138: 134: 133: 131: 130: 127: 122: 117: 112: 106: 104: 103:Known for 100: 99: 94: 90: 89: 84: 82:(aged 86) 76: 72: 71: 66: 55: 51: 50: 47: 39: 38: 35: 26: 18:Andrew Gleason 9: 6: 4: 3: 2: 4376: 4365: 4362: 4360: 4357: 4355: 4352: 4350: 4347: 4345: 4342: 4340: 4337: 4335: 4332: 4330: 4327: 4325: 4322: 4320: 4317: 4315: 4312: 4310: 4307: 4305: 4302: 4300: 4297: 4295: 4292: 4290: 4287: 4285: 4282: 4280: 4277: 4275: 4272: 4270: 4267: 4266: 4264: 4247: 4243: 4239: 4235: 4231: 4227: 4223: 4219: 4215: 4214:Robert Bryant 4211: 4207: 4203: 4199: 4195: 4191: 4187: 4183: 4179: 4175: 4171: 4167: 4163: 4159: 4155: 4154: 4152: 4148: 4140: 4139:Felix Browder 4136: 4132: 4128: 4124: 4120: 4116: 4115:Ronald Graham 4112: 4108: 4107:Michael Artin 4104: 4100: 4096: 4092: 4091:George Mostow 4088: 4084: 4080: 4076: 4072: 4068: 4064: 4060: 4056: 4052: 4048: 4044: 4040: 4039: 4037: 4033: 4025: 4021: 4017: 4013: 4009: 4008:Oscar Zariski 4005: 4001: 3997: 3993: 3989: 3985: 3981: 3977: 3973: 3969: 3965: 3961: 3957: 3953: 3949: 3945: 3941: 3937: 3933: 3932: 3930: 3926: 3918: 3914: 3910: 3906: 3902: 3898: 3894: 3890: 3886: 3885:Marston Morse 3882: 3878: 3874: 3870: 3866: 3862: 3858: 3854: 3850: 3846: 3842: 3838: 3834: 3830: 3829:Virgil Snyder 3826: 3822: 3818: 3817: 3815: 3811: 3803: 3802:Oswald Veblen 3799: 3795: 3791: 3787: 3783: 3779: 3775: 3771: 3767: 3763: 3759: 3755: 3751: 3747: 3746:Maxime Bôcher 3743: 3739: 3735: 3731: 3727: 3723: 3719: 3715: 3711: 3710: 3708: 3704: 3696: 3692: 3688: 3687:Simon Newcomb 3684: 3680: 3676: 3672: 3668: 3664: 3660: 3659: 3657: 3653: 3649: 3641: 3636: 3634: 3629: 3627: 3622: 3621: 3618: 3603: 3600: 3595: 3592: 3587: 3584: 3579: 3576: 3571: 3568: 3563: 3560: 3554: 3549: 3546: 3541: 3538: 3533: 3530: 3525: 3522: 3517: 3516:Samuel Webber 3514: 3509: 3506: 3501: 3500:John Winthrop 3498: 3493: 3490: 3489: 3486: 3482: 3475: 3470: 3468: 3463: 3461: 3456: 3455: 3452: 3444: 3440: 3436: 3435: 3422: 3418: 3411: 3405: 3391: 3387: 3386: 3379: 3371: 3367: 3361: 3347:on 2011-06-07 3346: 3342: 3338: 3337: 3330: 3321: 3312: 3308: 3301: 3294: 3287: 3283: 3278: 3273: 3269: 3265: 3249: 3246: 3235: 3227: 3221: 3217: 3216: 3211: 3207: 3201: 3193: 3189: 3185: 3181: 3177: 3176:Blahut, R. E. 3171: 3169: 3154:on 2016-03-04 3153: 3149: 3145: 3141: 3137: 3131: 3124: 3120: 3116: 3112: 3108: 3101: 3095: 3091: 3086: 3078: 3074: 3069: 3064: 3060: 3056: 3055: 3047: 3039: 3035: 3031: 3027: 3020: 3018: 3009: 2993: 2989: 2985: 2981: 2974: 2972: 2959: 2958: 2953: 2952:Graham, R. L. 2947: 2945: 2937: 2931: 2927: 2923: 2919: 2918: 2913: 2907: 2899: 2895: 2891: 2887: 2882: 2877: 2873: 2872: 2867: 2860: 2852: 2848: 2844: 2840: 2835: 2830: 2826: 2822: 2815: 2813: 2811: 2809: 2807: 2797: 2793: 2786: 2782: 2776: 2767: 2763: 2756: 2752: 2746: 2744: 2742: 2740: 2738: 2736: 2734: 2732: 2730: 2720: 2716: 2715: 2707: 2700: 2691: 2687: 2686: 2678: 2674: 2668: 2660: 2656: 2652: 2648: 2644: 2640: 2639: 2631: 2624: 2616: 2612: 2608: 2604: 2600: 2594: 2587: 2581: 2570: 2569: 2561: 2552: 2548: 2541: 2534: 2532: 2530: 2528: 2518: 2514: 2507: 2500: 2498: 2496: 2494: 2492: 2483: 2479: 2475: 2471: 2467: 2463: 2462: 2454: 2452: 2450: 2440: 2436: 2429: 2425: 2421: 2415: 2413: 2411: 2409: 2400: 2396: 2389: 2387: 2378: 2374: 2367: 2348: 2344: 2340: 2339: 2331: 2327: 2323: 2319: 2313: 2311: 2309: 2307: 2305: 2295: 2291: 2284: 2277: 2275: 2273: 2271: 2269: 2267: 2265: 2263: 2261: 2243: 2242: 2237: 2231: 2216: 2212: 2211: 2204: 2195: 2191: 2184: 2180: 2174: 2172: 2170: 2160: 2156: 2149: 2142: 2140: 2138: 2136: 2134: 2124: 2120: 2116: 2109: 2107: 2105: 2103: 2101: 2099: 2097: 2095: 2093: 2091: 2089: 2087: 2085: 2083: 2081: 2079: 2077: 2066: 2061: 2060: 2055: 2048: 2046: 2038: 2034: 2033: 2028: 2024: 2017: 2015: 2013: 2011: 2002: 1995: 1993: 1991: 1989: 1987: 1985: 1977: 1971: 1967: 1963: 1956: 1947: 1943: 1936: 1932: 1926: 1924: 1922: 1920: 1918: 1916: 1914: 1912: 1907: 1892: 1883: 1881: 1879: 1874: 1863: 1862:prime numbers 1860:, a class of 1859: 1856: 1854: 1851: 1850: 1840: 1836: 1835: 1829: 1828: 1824: 1823: 1817: 1813: 1808: 1803: 1798: 1794: 1790: 1786: 1780: 1776: 1772: 1768: 1763: 1760: 1757: 1752: 1748: 1744: 1739: 1738: 1734: 1733: 1728: 1724: 1720: 1716: 1711: 1706: 1702: 1698: 1697: 1691: 1687: 1683: 1679: 1674: 1670: 1666: 1662: 1658: 1653: 1648: 1644: 1640: 1639: 1633: 1629: 1625: 1621: 1617: 1612: 1607: 1603: 1599: 1595: 1590: 1586: 1582: 1577: 1572: 1568: 1564: 1563: 1558: 1553: 1549: 1545: 1541: 1537: 1533: 1529: 1525: 1521: 1520: 1514: 1505:on 2014-12-14 1501: 1497: 1493: 1489: 1487: 1478: 1473: 1472: 1468: 1467: 1461: 1458: 1452: 1447: 1445: 1441: 1437: 1432: 1430: 1426: 1422: 1417: 1415: 1411: 1407: 1403: 1399: 1395: 1391: 1382: 1373: 1371: 1367: 1363: 1359: 1355: 1351: 1341: 1339: 1335: 1330: 1328: 1324: 1320: 1317: + 1316: 1313: 1309: 1305: 1302: + 1301: 1297: 1293: 1289: 1284: 1282: 1275: 1272: 1268: 1264: 1260: 1259:Eugene Prange 1256: 1251: 1248: 1244: 1240: 1239:coding theory 1232: 1227: 1221:Coding theory 1218: 1216: 1212: 1208: 1207:Ronald Graham 1204: 1202: 1198: 1195: 1194:triangle-free 1191: 1187: 1183: 1179: 1175: 1171: 1167: 1163: 1159: 1155: 1151: 1147: 1143: 1139: 1135: 1131: 1127: 1123: 1119: 1115: 1111: 1107: 1104: 1103:Ramsey number 1097: 1092: 1086:Ramsey theory 1083: 1081: 1077: 1073: 1072:Grete Hermann 1069: 1065: 1060: 1058: 1054: 1050: 1046: 1042: 1041:George Mackey 1037: 1033: 1029: 1025: 1019: 1010: 1001: 999: 995: 991: 986: 982: 978: 974: 970: 966: 961: 959: 955: 954: 949: 946: 942: 938: 934: 930: 926: 925:George Mackey 918: 917:George Mackey 913: 909: 907: 903: 899: 895: 891: 887: 883: 879: 875: 871: 868: 865: 860: 858: 854: 850: 846: 843:concerns the 842: 838: 834: 833:David Hilbert 825: 816: 814: 813:Freeman Dyson 810: 809:combinatorics 806: 802: 792: 790: 786: 782: 778: 774: 770: 765: 762: 760: 755: 753: 749: 744: 740: 739:Marshall Hall 735: 731: 727: 723: 722:cryptanalysis 719: 711: 706: 697: 695: 691: 685: 680: 677: 673: 664: 659: 654: 652: 647: 645: 641: 635: 633: 626: 619: 614: 609: 604: 602: 596: 594: 589: 581: 572: 569: 567: 563: 559: 555: 550: 548: 544: 540: 536: 531: 529: 525: 521: 516: 515:Junior Fellow 512: 504: 499: 495: 493: 489: 485: 481: 477: 473: 468: 466: 465:Putnam Fellow 462: 457: 454: 448: 443: 441: 437: 433: 429: 425: 420: 418: 414: 410: 406: 405:Andrew Mattei 402: 398: 394: 393:Henry Gleason 390: 381: 372: 370: 369: 364: 359: 357: 353: 349: 345: 340: 338: 334: 330: 326: 321: 319: 318:Ramsey theory 315: 311: 310:quantum logic 307: 303: 299: 298:mathematician 295: 286: 280: 277: 275: 272: 270: 267: 265: 262: 260: 257: 255: 252: 250: 247: 246: 244: 240: 237: 236:George Mackey 234: 230: 226: 224: 220: 217: 214: 210: 207: 203: 200: 196: 193: 189: 182: 179: 176: 175: 173: 169: 145: 139: 135: 128: 126: 123: 121: 118: 116: 113: 111: 108: 107: 105: 101: 98: 95: 91: 87: 77: 73: 69: 56: 52: 45: 40: 33: 30: 19: 4238:Ruth Charney 4174:James Arthur 4131:Arthur Jaffe 4066: 3786:Frank Morley 3722:Thomas Fiske 3585: 3442: 3420: 3416: 3404: 3393:, retrieved 3384: 3378: 3369: 3360: 3349:, retrieved 3345:the original 3335: 3329: 3320: 3310: 3306: 3293: 3267: 3263: 3234: 3214: 3200: 3183: 3179: 3156:, retrieved 3152:the original 3147: 3143: 3130: 3114: 3110: 3106: 3100: 3085: 3058: 3057:, Series B, 3052: 3046: 3029: 3025: 3016: 3008: 2996:, retrieved 2992:the original 2987: 2983: 2956: 2916: 2906: 2869: 2866:Pardon, John 2859: 2824: 2820: 2798:(2): 212–223 2795: 2791: 2775: 2765: 2761: 2718: 2712: 2699: 2689: 2683: 2667: 2642: 2636: 2623: 2606: 2602: 2593: 2585: 2580: 2567: 2560: 2550: 2546: 2516: 2512: 2465: 2459: 2438: 2434: 2398: 2376: 2366: 2354:, retrieved 2347:the original 2342: 2336: 2318:Mazur, Barry 2293: 2289: 2248:, retrieved 2240: 2230: 2219:, retrieved 2215:the original 2209: 2203: 2193: 2189: 2158: 2154: 2122: 2065:the original 2059:Boston Globe 2057: 2030: 2000: 1975: 1972:(3): 326–353 1969: 1965: 1955: 1945: 1941: 1891: 1833: 1815: 1801: 1770: 1742: 1700: 1694: 1677: 1642: 1636: 1601: 1597: 1566: 1560: 1523: 1517: 1507:, retrieved 1500:the original 1483: 1456: 1454: 1449: 1434:In 1996 the 1433: 1418: 1387: 1362:permutations 1347: 1331: 1318: 1314: 1311: 1307: 1303: 1299: 1292:linear codes 1285: 1280: 1273: 1266: 1252: 1236: 1215:Joel Spencer 1205: 1189: 1185: 1177: 1173: 1169: 1165: 1161: 1153: 1149: 1145: 1141: 1133: 1125: 1121: 1117: 1113: 1109: 1105: 1100: 1061: 1021: 993: 989: 984: 980: 972: 968: 964: 962: 951: 936: 922: 889: 886:neighborhood 873: 861: 830: 798: 766: 763: 756: 715: 687: 682: 669: 656: 650: 648: 637: 628: 623: 606: 600: 597: 592: 590: 586: 570: 551: 532: 508: 469: 458: 450: 445: 421: 386: 366: 360: 358:at Harvard. 341: 322: 293: 292: 274:Joel Spencer 212:Institutions 206:cryptography 191: 80:(2008-10-17) 48:Berlin, 1959 29: 4274:2008 deaths 4269:1921 births 4248:(2023–2024) 4240:(2021–2022) 4232:(2019–2020) 4230:Jill Pipher 4224:(2017–2018) 4216:(2015–2016) 4208:(2013–2014) 4206:David Vogan 4200:(2011–2012) 4192:(2009–2010) 4184:(2007–2008) 4182:James Glimm 4176:(2005–2006) 4168:(2003–2004) 4160:(2001–2002) 4141:(1999–2000) 4133:(1997–1998) 4125:(1995–1996) 4117:(1993–1994) 4109:(1991–1992) 4101:(1989–1990) 4093:(1987–1988) 4085:(1985–1986) 4077:(1983–1984) 4069:(1981–1982) 4061:(1979–1980) 4053:(1977–1978) 4045:(1975–1976) 4043:Lipman Bers 4026:(1973–1974) 4018:(1971–1972) 4010:(1969–1970) 4002:(1967–1968) 3994:(1965–1966) 3986:(1963–1964) 3978:(1961–1962) 3970:(1959–1960) 3962:(1957–1958) 3954:(1955–1956) 3946:(1953–1954) 3938:(1951–1952) 3919:(1949–1950) 3911:(1947–1948) 3909:Einar Hille 3903:(1945–1946) 3895:(1943–1944) 3887:(1941–1942) 3879:(1939–1940) 3871:(1937–1938) 3863:(1935–1936) 3855:(1933–1934) 3847:(1931–1932) 3839:(1929–1930) 3831:(1927–1928) 3823:(1925–1926) 3804:(1923–1924) 3796:(1921–1922) 3788:(1919–1920) 3780:(1917–1918) 3772:(1915–1916) 3764:(1913–1914) 3756:(1911–1912) 3748:(1909–1910) 3740:(1907–1908) 3732:(1905–1906) 3724:(1903–1904) 3716:(1901–1902) 3714:E. H. Moore 3697:(1899–1900) 3689:(1897–1898) 3681:(1895–1896) 3673:(1891–1894) 3665:(1888–1890) 3602:Cumrun Vafa 3557:(1919-1921) 3524:John Farrar 3206:Pless, Vera 3136:Pless, Vera 3111:SIAM Review 3105:"Review of 2863:See, e.g., 1645:: 171–172, 1356:and on the 1344:Other areas 1036:eigenvalues 837:23 problems 730:Alan Turing 541:. Berko, a 492:Alan Turing 453:Einar Hille 350:and of the 259:James Eells 249:Glen Bredon 202:Mathematics 4263:Categories 4158:Hyman Bass 4051:R. H. Bing 3410:"Features" 3395:2016-08-05 3351:2016-04-10 3158:2013-05-06 2924:, p. 2250:2016-04-10 1903:References 1628:0083.17401 1509:2013-04-05 1243:Vera Pless 977:continuous 933:Leo Zippin 849:Lie groups 801:Lie groups 769:Korean War 649:In a 1964 539:Tom Lehrer 535:Jean Berko 520:Korean War 503:Jean Berko 470:After the 60:1921-11-04 4246:Bryna Kra 4222:Ken Ribet 4150:2001–2024 4059:Peter Lax 4035:1975–2000 3928:1951–1974 3813:1925–1950 3706:1901–1924 3655:1888–1900 3247:± 2881:1112.2324 2851:115527342 1727:124255697 1669:121062823 1368:, and an 1265:of order 1051:. 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