GREAT MATHEMATICIANS MATHEMATICIANS th
OF THE 20 CENTURY
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Contents
Pavel Sergeevih Ale!san"rov $%it&en Eg'ert%s (an )ro%*er ,lie (ose-h Cartan Israil Moiseevi Gel/an" %rt G1"el Feli3 Ha%s"or// 5avi" Hil'ert Hein& Ho-/ An"re7 Ni!olaevih ol8ogorov Ni!olai Ni!olaevih $%&in (ohn von Ne%8ann E887 A8alie Noether Roger Penrose $ev Se8enovih Pontr7agin Sergei $vovih So'olev An"rei Ni!olaevih Ti!honov Alan Mathison T%ring Pavel Sa8%ilovih Ur7sohn Her8ann la%s H%go ;e7l Nor'ert ;iener Re/erenes
# + .0 .2 . .4 2. 26 #9 #. # #: 6. 6 64 0 2 + : +# +9
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Pavel Sergeevih Ale!san"rov )orn< 9 Ma7 .4:+ in )ogoro"s! =also alle" Nogins!>? R%ssia 5ie"< .+ Nov .:42 in Moso*? R%ssia =USSR> Like most Russian mathematician mathematicianss there are different different ways to translitera transliterate te Ale!san"rov's Ale!san"rov's name into the Roman alphabet. The most common way, other than Aleksandrov, is to write it as Ale3an"ro//@ avel !er"eevich Aleksandrov's father !er"e# Aleksandrovich Aleksandrov was a medical "raduate from $oscow %niversity who had decided not to follow an academic career but instead had chosen to use his skills in helpin" people and so he worked as a "eneral practitioner in &aroslavskii. Later he worked in more senior positions in a hospital in o"orodskii, which is where he was when avel !er"eevich was born. (hen avel !er"eevich was one year old his father moved to !molensk !tate hospital, where he was to earn the reputation of bein" a very fine sur"eon, and the family lived from this time in !molensk. The city of !molensk is on the )nieper River *2+ km west of $oscow. avel !er"eevich's early education was from his mother, Tseariya Akimovna Aleksandrova, who applied all her considerable talents to brin"in" up and educatin" her children. -t was from her that Aleksandrov learnt rench and also /erman. 0is home was one that was always filled with music as his brothers and sisters all had "reat talent i n that area. The fine start which his mother "ave him meant that he always ecelled at the "rammar school in !molensk which he attended. 0is mathematics teacher Alesander Romanovich i"es soon realised that his pupil had a remarkable talent for the sub#ect and 3456 and 4*6789 ... at grammar school he studied celestial mechanics and mathematical analysis. But his interest was mainly mainly direct directed ed toward towardss fundam fundament ental al proble problems ms of mathem mathemati atics: cs: the founda foundatio tions ns of geometry and non-euclidean geometry. Eiges had a proper appreciation of his pupil and exerted a decisive influence on his choice of a career in mathematics. -n 1:15 Aleksandrov "raduated from the "rammar school bein" du of the school and winnin" the "old medal. ;ertainly at this time he had already decided on a career in mathematics, but he had not set his si"hts as hi"h as a university teacher, rather he was aimin" to become a secondary school teacher of mathematics. i"es was the role model who he was aspirin" to match at this sta"e, for i"es had done more than teach Aleksandrov mathematics, he had also influenced his tastes in literature and the arts. Aleksandrov entered $oscow %niversity in 1:15 and immediately he was helped by !tepanov. !tepanov, who was workin" at $oscow %niversity, was seven years older than Aleksandrov but his home was also in !molensk and he often visited the Aleksandrov home there. !tepanov was an important influence on Aleksandro Aleksandrovv at this time and su""ested su""ested that Aleksandrov Aleksandrov #oin "orov's seminar seminar even in the first year of his studies in $oscow. -n Aleksandrov's second year of study he came in contact with Luin who had #ust returned to $oscow. Aleksandrov wrote 3see for eample 456 or 4*6789 After Luzins lecture ! turned to him for advice on how best to continue my mathematical studies and was struc" most of all by Luzins "indness to the man addressing him - an #$-year old student student ... ! then became became a student student of Luzin% during his most creative creative period ... &o see Luzin in those years was to see a display of what is called an inspired relationship to science. ! learnt not only mathematics from him% ! received also a lesson in what ma"es a true scholar and what a university professor can and should be. &hen% too% ! saw that the pursuit of science and the raining of young people in it are two facets of one and the same activity - that of a scholar. Aleksandrov proved his first important result in 1:1<, namely that every non9denumerable orel set contains a perfect subset. -t was not only the result which was important for set theory, but also the methods which Aleksandrov used which turned out to be one of the most useful methods in descriptive set theory. After Aleksandrov's "reat successes Luin did what many a supervisor mi"ht do, he realised that he had one of the 5
"reatest mathematical talents in Aleksandrov so he thou"ht that it was worth askin" him to try to solve the bi""est open problem in set theory, namely the continuum hypothesis. After Aleksandrov failed to solve the continuum hypothesis 3which is not surprisin" since it can neither be proved or disproved as was shown by ;ohen in the 1:=+s7 he thou"ht he was not capable of a mathematical career. Aleksandrov went to >ov"orod9!everskii and became a theatre producer. 0e then went to ;hernikov where, in addition to theatrical work, he lectured on Russian and forei"n lan"ua"es, becomin" friends with poets, artists and musicians. After a short term in #ail in 1:1: at the time of the Russian revolution, Aleksandrov returned to $oscow in 1:2+. Luin and "orov had built up an impressive research "roup at the %niversity of $oscow which the students called 'Luitania' and they, to"ether with rivalov and !tepanov, were very welcomin" to Aleksandrov on his return. -t was not an immediate return to $oscow for Aleksandrov, however, for he spent 1:2+921 back home in !molensk where he tau"ht at the %niversity. )urin" this time he worked on his research, "oin" to $oscow about once every month to keep in touch with the mathematicians there and to prepare himself for his eaminations. At around this time Aleksandrov became friendly with %rysohn, who was a member of 'Luitania', and the friendship would soon develop into a ma#or mathematical collaboration. After takin" his eaminations in 1:21, Aleksandrov was appointed as a lecturer at $oscow university and lectured on a variety of topics includin" functions of a real variable, topolo"y and /alois theory. -n ?uly 1:22 Aleksandrov and %rysohn went to spend the summer at olshev, near to $oscow, where they be"an to study concepts in topolo"y. 0ausdorff, buildin" on work by r@chet and others, had created a theory of topolo"ical and metric spaces in his famous book 'rundz(ge der )engenlehre published in 1:1*. Aleksandrov and %rysohn now be"an to push the theory forward with work on countably compact spaces producin" results of fundamental importance. The notion of a compact space and a locally compact space is due to them. -n the summers of 1:25 and 1:2* Aleksandrov and %rysohn visited /ttin"en and impressed mmy >oether, ;ourant and 0ilbert with their results. The mathematicians in /ttin"en were particularly impressed with their results on when a topolo"ical space is metrisable. -n the summer of 1:2* they also visited 0ausdorff in onn and he was fascinated to hear the ma#or new directions that the two were takin" in topolo"y. 0owever while visitin" 0ausdorff in onn 3456 and 4*6789 Every day Ale"sandrov and *rysohn swam across the +hine - a feat that was far from being safe and provo"ed ,ausdorffs displeasure. Aleksandrov and %rysohn then visited rouwer in 0olland and aris in Au"ust 1:2* before havin" a holiday in the fishin" villa"e of our" de at in rittany. Bf course mathematicians continue to do mathematics while on holiday and they were both workin" hard. Bn the mornin" of 1C Au"ust %rysohn be"an to write a new paper but tra"ically he drowned while swimmin" in the Atlantic later that day. Aleksandrov determined that no ideas of his "reat friend and collaborator should be lost and he spent part of 1:2< and 1:2= in 0olland workin" with rouwer on preparin" %rysohn's paper for publication. The atmosphere in /ttin"en had proved very helpful to Aleksandrov, particularly after the death of %rysohn, and he went there every summer from 1:2< until 1:52. 0e became close friends with 0opf and the two held a topolo"ical seminar in /ttin"en. Bf course Aleksandrov also tau"ht in $oscow %niversity and from 1:2* he or"anised a topolo"y seminar there. At /ttin"en, Aleksandrov also lectured and participated in mmy >oether's seminar. -n fact Aleksandrov always included mmy >oether and 0ilbert amon" his teachers, as well as rouwer in Amsterdam and Luin and "orov in $oscow. rom 1:2= Aleksandrov and 0opf were close friends workin" to"ether. They spent some time in 1:2= in the south of rance with >eu"ebauer. Then Aleksandrov and 0opf spent the academic year 1:2C92D at rinceton in the %nited !tates. This was an important year in the development of topolo"y with Aleksandrov and 0opf in rinceton and able to collaborate with Lefschet, Eeblen and Aleander. )urin" their year in rinceton, Aleksandrov and 0opf planned a #oint multi9volume work on &opology the first volume of which did not appear until 1:5<. This was the only one of the three intended volumes to appear since (orld (ar -prevented further collaboration on the remainin" two volumes. -n fact before the #oint work with 0opf appeared in print, Aleksandrov had be"un yet another important friendship and collaboration. -n 1:2: Aleksandrov's friendship with Folmo"orov be"an and the y 3456 and 4*6789 ... ourneyed a lot along the olga% the /nieper% and other rivers% and in the 0aucuses% the *
0rimea% and the south of 1rance. The year 1:2: marks not only the be"innin" of the friendship with Folmo"orov but also the appointment of Aleksandrov as rofessor of $athematics at $oscow %niversity. -n 1:5< Aleksandrov went to &alta with Folmo"orov, then finished the work on his &opology book in the nearby ;rimea and the book was published in that year. The 'Fomarovski' period also be"an in that year 3456 and 4*6789 2ver the last forty years% many of the events in the history of mathematics in the *niversity of )oscow have been lin"ed with 3omarov"a% a small village outside )oscow. ,ere is the house owned since 1:5< by Ale"sandrov and 3olmogorov. )any famous foreign mathematicians also visited 3omarov"a - ,adamard% 1r4chet% Banach% ,opf% 3uratows"i% and others. -n 1:5D91:5: a number of leadin" mathematicians from the $oscow %niversity, amon" them Aleksandrov, #oined the !teklov $athematical -nstitute of the %!!R Academy of !ciences but at the same time they kept their positions at the %niversity. Aleksandrov wrote about 5++ scientific works in his lon" career. As early as 1:2* he introduced the concept of a locally finite coverin" which he used as a basis for his criteria for the metrisability of topolo"ical spaces. 0e laid the foundations of homolo"y theory in a series of fundamental papers between 1:2< and 1:2:. 0is methods allowed ar"uments of combinatorial and al"ebraic topolo"y to be applied to point set topolo"y and brou"ht to"ether these areas. Aleksandrov's work on homolo"y moved forward with his homolo"ical theory of dimension around 1:2D95+ Aleksandrov was the first to use the phrase 'kernel of a homomorphism' and around 1:*+9*1 he discovered the in"redients of an eact seGuence. 0e worked on the theory of continuous mappin"s of topolo"ical spaces. -n 1:<* he or"anised a seminar on this last topic aimed at first year students at $oscow %niversity and in this he showed one of the aspects of his career which was of ma#or importance to him, namely the education of students. This is described in 3456 and 4*6789 &o the training of these students and those who came after them% Ale"sandrov literally devoted all his strength. ,is influence on the class of young men studying topology under him was never purely mathematical% however real and significant that was. &here were physical days exercise on topological wal"s% in long outings lasting several days by boat% ... in swimming across the olga or other broad stretches of water% in s"iing excursions lasting for hours on the slopes outside )oscow% slopes to which Ale"sandrov gave stri"ing% fantastic names... $any honours were "iven to Aleksandrov for his outstandin" contribution to mathematics. 0e was president of the $oscow $athematical !ociety from 1:52 to =*, vice president of the -nternational ;on"ress of $athematicians from 1:ational Academy of !ciences of the %nited !tates, the London $athematical !ociety, the American hilosophical !ociety, and the )utch $athematical !ociety. 0e edited several mathematical #ournals, in particular the famous !oviet ?ournal *spe"hi )atematiches"i"h 5au"% and he received many !oviet awards, includin" the !talin rie in 1:*5 and five Brders of Lenin. Today the )epartment of /eneral Topolo"y and /eometry of $oscow !tate %niversity is Russia's leadin" centre of research in set9theoretic topolo"y. After Aleksandrov's death in >ovember 1:D2, his collea"ues from the )epartment of 0i"her /eometry and Topolo"y, in which he had held the chair, sent a letter to $oscow %niversity's rector A A Lo"unov proposin" that one of Aleksandrov's former students should become 0ead of the )epartment, to preserve Aleksandrov's scientific school. Bn 2D )ecember 1:D2 the rector issued a circular creatin" the )epartment of "eneral topolo"y and /eometry. Eitaly Eitalievich edorchuk was elected 0ead of the )epartment. Also in memory of Aleksandrov's contributions to topolo"y at $oscow %niversity and his work with the $oscow $athematical !ociety, there is an annual topolo"ical symposium Ale"sandrov 6roceedings held every $ay.
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$%it&en Eg'ert%s (an )ro%*er )orn< 29 Fe' .44. in Overshie =no* a s%'%r' o/ Rotter"a8>? Netherlan"s 5ie"< 2 5e .:++ in )lari%8? Netherlan"s $ E ( )ro%*er is usually known by this form of his name with full initials, but he was known to his friends as ertus, an abbreviation of the second of his three forenames. 0e attended hi"h school in 0oorn, a town on the Huideree north of Amsterdam. 0is performance there was outstandin" and he completed his studies by the a"e of fourteen. 0e had not studied /reek or Latin at hi"h school but both were reGuired for entry into university, so rouwer spent the net two years studyin" these topics. )urin" this time his family moved to 0aarlem, #ust west of Amsterdam, and it was in the /ymnasium there in 1D:C that he sat the entrance eaminations for the %niversity of Amsterdam. Fortewe" was the professor of mathematics at the %niversity of Amsterdam when rouwer be"an his studies, and he Guickly realised that in rouwer he had an outstandin" student. (hile still an under"raduate rouwer proved ori"inal results on continuous motions in four dimensional space and Fortewe" encoura"ed him to present them for publication. This he did, and it became his first paper published by the Royal Academy of !cience in Amsterdam in 1:+*. Bther topics which interested rouwer were topolo"y and the foundations of mathematics. 0e learnt somethin" of these topics from lectures at the university but he also read many works on the topics on his own. 0e obtained his master's de"ree in 1:+* and in the same year married Lie de 0oll who was eleven years older that rouwer and had a dau"hter from a previous marria"e. After the marria"e, which would produce no children, the couple moved to laricum, near Amsterdam. Three years later Lie Gualified as a pharmacist and rouwer helped her in many ways from doin" bookkeepin" to servin" in the chemists shop. 0owever, rouwer did not "ain the affection of his step9dau"hter and relations between them was strained. rom an early sta"e rouwer was interested in the philosophy of mathematics, but he was also fascinated by mysticism and other philosophical Guestions relatin" to human society. 0e published his own ideas on this topic in 1:+< in his treatise Leven% 3unst% en )ystie" 3Life, art, and mysticism7. -n this work he 41689 ... considers as one of the important moving principles in human activity the transition from goal to means% which after some repetitions may result in activities opposed to the original goal. rouwer's doctoral dissertation, published in 1:+C, made a ma#or contribution to the on"oin" debate between Russell and oincar@ on the lo"ical foundations of mathematics. 0is doctoral thesis 415689 ... revealed the twin interests in mathematics that dominated his entire career7 his fundamental concern with critically assessing the foundations of mathematics% which led to his creation of intuitionism% and his deep interest in geometry% which led to his seminal wor" in topology ... 0e Guickly discovered that his ideas on the foundations of mathematics would not be readily accepted 415689 Brouwer 8uic"ly found that his philosophical ideas spar"ed controversy. 3orteweg% his thesis advisor% had not been pleased with the more philosophical aspects of the thesis% and had even demanded that several parts of the original draft be cut from the final presentation. 3orteweg urged Brouwer to concentrate on more 9respectable9 mathematics% so that the young man might enhance his mathematical reputation and thus secure an academic career. Brouwer was fiercely independent and did not follow in anybodys footsteps% but he apparently too" his teachers advice ... rouwer continued to develop the ideas of his thesis in &he *nreliability of the Logical 6rinciples published in 1:+D. The research which rouwer now undertook was in two areas. 0e continued his study of the lo"ical =
foundations of mathematics and he also put a very lar"e effort into studyin" various problems which he attacked because they appeared on 0ilbert's list of problems proposed at the aris -nternational ;on"ress of $athematicians in 1:++. -n particular rouwer attacked 0ilbert's fifth problem concernin" the theory of continuous "roups. 0e addressed the -nternational ;on"ress of $athematicians in Rome in 1:+D on the topolo"ical foundations of Lie "roups. 0owever, after studyin" !chnflies' report on set theory, he wrote to 0ilbert89 ! discovered all of a sudden that the choenfliesian investigations concerning topology of the plane% on which ! had relied in the fullest way% could not be ta"en as correct in all parts% so that my group-theoretic results also became doubt. -n 1:+: he was appointed as an privatdocent at the %niversity of Amsterdam. 0e "ave his inau"ural lecture on 12 Bctober 1:+: on 'The nature of "eometry' in which he outlined his research pro"ramme. A couple of months later he made an important visit to aris, around ;hristmas 1:+:, and there met oincar@, 0adamard and orel. rompted by discussions in aris, he be"an workin" on the problem of the invariance of dimension. rouwer was elected to the Royal Academy of !ciences in 1:12 and, in the same year, was appointed etraordinary professor of set theory, function theory and aiomatics at the %niversity of AmsterdamI he would hold the post until he retired in 1:<1. 0ilbert wrote a warm letter of recommendation which helped rouwer to "ain his chair in 1:12. )espite the substantial contributions he had made to topolo"y by this time, rouwer chose to "ive his inau"ural professorial lecture on intuitionism and formalism. -n the followin" year Fortewe" resi"ned his chair so that t hat rouwer could be appointed as ordinary professor. Althou"h he had helped rouwer to obtain his chair in Amsterdam, in 1:1: 0ilbert tried to tempt him away with an offer of a chair in /ttin"en. 0e was also offered the chair at erlin in the same year. These must have been temptin" offers, but despite their attractions rouwer turned them down. erhaps the eceptional way he was treated by Amsterdam, mentioned in the followin" Guote by Ean der (aerden, helped him make these decisions. Ean der (aerden, who studied at Amsterdam from 1:1: to 1:25, wrote about rouwer as a lecturer 3see for eample 41*6789 Brouwer came 4to the university6 university6 to give his courses but lived in Laren. ,e came only once a wee". !n general that would have not been permitted - he should have lived in Amsterdam - but for him an exception was made. ... ! once interrupted him during a lecture to as" a 8uestion. Before the next wee"s lesson% his assistant came to me to say that Brouwer did not want 8uestions put to him in class. ,e ust did not want them% he was always loo"ing at the blac" blac"bo boar ard% d% never never towa toward rdss the the stud studen ents ts.. ... ... Even Even thou though gh his his most most impo import rtan antt rese resear arch ch contributions were in topology% Brouwer never gave courses on topology% but always on -- and only on -- the foundations of intuitionism. !t seemed that he was no longer convinced of his results in topology because they were not not correct from the point of view of intuitionism% and he udged everything he had done before% his greatest output% false according to his philosophy. ,e was a very strange person% crazy in love with his philosophy. As is mentioned in this Guotation, rouwer was a ma#or contributor to the theory of topolo"y and he is considered by many to be its founder. The status of the sub#ect when he be"an his research is well described in 415689 ;hen Brouwer was beginning his career as a mathematician% set-theoretic topology was in a primitive state. 0ontroversy surrounded 0antors general set theory because of the set-theoretic paradoxes or contradictions. 6oint set theory was widely applied in analysis and somewhat less widely applied in geometry% but it did not have the character of a unified theory. &here were some perceived benchmar"s. 1or example7 the generally held view that dimension was invariant under one-to-one continuous mappings ... 0e did almost all his work in topolo"y early in his career between 1:+: and 1:15. 0e discovered characterisations of topolo"ical mappin"s of the ;artesian plane and a number of fied point theorems. 0is first fied point theorem, which showed that an orientation preservin" continuous one9one mappin" of the C
sphere to itself always fies at least one point, came out of his researches on 0ilbert's fifth problem. Bri"inally proved for a 29dimensional sphere, rouwer later "eneralised the result to spheres in n dimensions. Another result of eceptional importance was provin" the invariance of dimension. As well as provin" theorems of ma#or importance in topolo"y, rouwer also developed methods which have become standard tools in the sub#ect. -n particular he used simplicial approimation, which approimated continuous mappin"s by piecewise linear ones. 0e also introduced the idea of the de"ree of a mappin", "eneralised the ?ordan curve theorem to n9dimensional space, and defined topolo"ical spaces in 1:15. Ean der (aerden, in the above Guote, said that rouwer would not lecture on his own topolo"ical results since they did not fit with mathematical intuitionism. -n fact rouwer is best known to many mathematicians as the founder of the doctrine of mathematical intuitionism, which views mathematics as the formulation of mental constructions that are "overned by self9evident laws. 0is doctrine differed substantially from the formalism of 0ilbert and the lo"icism of Russell. 0is doctoral thesis in 1:+C attacked the lo"ical foundations of mathematics and marks the be"innin" of the -ntuitionist !chool. 0is views had more in common with those of oincar@ and if one asks which side of the debate between Russell and oincar@ he came down on then it would have with the latter. -n his 1:+D paper &he *nreliability *nreliability of the Logical Logical 6rinciples 6rinciples rouwer rouwer re#ected re#ected in mathematical mathematical proofs the rinciple of the cluded $iddle, which states that any mathematical statement is either true or false. -n 1:1D he published a set theory developed without usin" the rinciple of the cluded $iddle 1ounding $iddle 1ounding et &heory !ndependently of the 6rinciple of the Excluded Excl uded )iddle. 6art 2ne% 'eneral et &heory. 0is &heory. 0is 1:2+ lecture /oes lecture /oes Every +eal 5umber ,ave a /ecimal Expansion< was Expansion< was published in the followin" year. The answer to the Guestion of the title which rouwer "ives is JnoJ. Also in 1:2+ he published !ntuitionistic et &heory% then &heory% then in 1:2C he developed a theory of functions 2n the /omains of /efinition of 1unctions without 1unctions without the use of the rinciple of the cluded $iddle. 0is constructive theories were not easy to set up since the notion of a set could not be taken as a basic concept but had to be built up usin" more basic notions which, in rouwer's case, were choice seGuences. Loosely speakin", that the elements of a set had property p, meant to rouwer that he had a construction which allowed him to decide after a finite number of steps whether each element of the set had property p. !uch ideas are fundamental to theoretical computer science today. The later part of rouwer's career contains some controversial episodes. 0e had been appointed to the editorial editorial board of )athematische of )athematische Annalen in Annalen in 1:1* but in 1:2D 0ilbert decided that rouwer was becomin" too powerful, particularly since 0ilbert felt that he himself did not have lon" to live 3in fact he lived until 1:*57. 0e tried to remove rouwer from the board in a way which was not compatible with the way the board was set up. rouwer vi"orously opposed the move and he was stron"ly supported by other board members such as instein and ;arath@odory. -n the end 0ilbert mana"ed to "et his own way but it was a devastatin" episode for rouwer who was left mentally brokenI see 42=6 for details. -n 1:5< rouwer entered local politics when he was elected as >eutral arty candidate for the municipal council of laricum. 0e continued to serve on the council until 1:*1. 0e was also active settin" up a new #ournal and he became a foundin" editor editor of 0ompositio )athematica which )athematica which be"an publication in 1:5*. urther controversy arose due to his actions in (orld (ar --. rouwer was active in helpin" the )utch resistance, and in particular he supported ?ewish students durin" this difficult period. 0owever, in 1:*5 the /ermans insisted that the students si"n a declaration of loyalty to /ermany and rouwer encoura"ed his students to do so. 0e afterwards said that he did so in order that his students mi"ht have a chance to complete their studies and to work for the )utch resistance a"ainst the /ermans. 0owever, after Amsterdam was liberated, rouwer was suspended from his post for a few months because of his actions. A"ain he was deeply hurt and considered emi"ration. After retirin" in 1:<1, rouwer lectured in !outh Africa in 1:<2, and the %nited !tates and ;anada in 1:<5. 0is wife died in 1:<: at the a"e of D: and rouwer, who himself was CD, was offered a one year post in the %niversity of ritish ;olumbia in EancouverI he declined. -n 1:=2, despite bein" well into his D+s, he was offered a post in $ontana. 0e died in 1:== in laricum as the result of a traffic accident. Fneebone writes in 456 about rouwer's contributions to the philosophy of mathematics89 Brouwer is most famous ... for his contribution to the philosophy of mathematics and his attempt to build up mathematics anew on an !ntuitionist foundation% in order to meet his own searching D
criticism of hitherto un8uestioned assumptions. Brouwer was somewhat li"e 5ietzsche in his ability to step outside the established cultural tradition in order to subect its most hallowed presuppositions to cool and obective scrutiny7 and his 8uestioning of principles of thought led him to a 5ietzschean revolution in the domain of logic. ,e in fact reected the universally accepted logic of deductive reasoning which had been codified initially by Aristotle% handed down with very little change into modern times% and very recently extended and generalised out of all recognition with the aid of mathematical symbolism. Fneebone also writes in 456 about the influence that rouwer's views on the foundations of mathematics had on his fellow mathematicians89 Brouwers proected reconstruction of of the whole edifice of mathematics remained a dream% but his ideal of constructiv constructivism ism is now woven into our whole fabric fabric of mathematical mathematical thought% and it has inspired% as it still continues to inspire% a wide variety of in8uiries in the constructivist spirit which have led to maor advances in mathematical "nowledge. )espite failin" to convert mathematicians to his way of thinkin", rouwer received many honours for his outstandin" contributions. (e mentioned his election to the Royal )utch Academy of !ciences above. Bther honours included election to the Royal !ociety of London, the erlin Academy of !ciences, and the /ttin"en /ttin"en Academy of !ciences. !ciences. 0e was awarded honorary honorary doctorates the %niversity of Bslo in 1:2:, and the %niversity of ;ambrid"e in 1:<*. 0e was made Fni"ht in the Brder of the )utch Lion in 1:52.
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,lie (ose-h Cartan )orn< : A-ril .4+: in 5olo8ie% =near Cha8'r7>? Savoie? RhBneAl-es? Frane 5ie"< + Ma7 .:. in Paris? Frane ,lie Cartan's mother was Anne ;otta and his father was ?oseph ;artan who was a blacksmith. The family were very poor and in late 1: th century rance it was not possible for children from poor families to obtain a university education. -t was Klie's eceptional abilities, to"ether with a lot of luck, which made a hi"h Guality education possible for him. (hen he was in primary school he showed his remarkable talents which impressed the youn" school inspector, later important politician, Antonin )ubost. )ubost was at this time employed as an inspector of primary schools and it was on a visit to the primary school in )olomieu, in the rench Alps, that he discovered the remarkable youn" Klie. )ubost was able to obtain state funds which paid for Klie to attend the Lyc@e in Lyons, where he completed his school education with distinction in mathematics. The state stipend was etended to allow him to study at the Kcole >ormale !up@rieure in aris. ;artan became a student at the Kcole >ormale !up@rieure in 1DDD and obtained his doctorate in 1D:*. 0e was then appointed to the %niversity at $ontpellier where he lectured from 1D:* to 1D:=. ollowin" this he was appointed as a lecturer at the %niversity of Lyon, where he tau"ht from 1D:= to 1:+5. -n 1:+5 ;artan was appointed as a professor at the %niversity of >ancy and he remained there until 1:+: when he moved to aris. 0is appointment in 1:+: was as a lecturer at the !orbonne but three years later he was appointed to the ;hair of )ifferential and -nte"ral ;alculus in aris. 0e was appointed as rofessor of Rational $echanics in 1:2+, and then rofessor of 0i"her /eometry from 1:2* to 1:*+. 0e retired in 1:*+. 0e married $arie9Louise ianconi in 1:+5 and they had four children, one of them 0enri ;artan was to produce brilliant work in mathematics. Two other sons died tra"ically. ?ean, a composer, died of tuberculosis at the a"e of 2< while their son Louis was a member of the Resistance fi"htin" in rance a"ainst the occupyin" /erman forces. After his arrest in ebruary 1:*5 the family received no further news but they feared the worst. Bnly in $ay 1:*< did they learn that he had been beheaded by the >ais in )ecember 1:*5. y the time they received the news of Louis' murder by the /ermans, ;artan was C< years old and it was a devastatin" blow for him. Their fourth child was a dau"hter. ;artan worked on continuous "roups, Lie al"ebras, differential eGuations and "eometry. 0is work achieved a synthesis between these areas. 0e added "reatly to the theory of continuous "roups which had been initiated by Lie. 0is doctoral thesis of 1D:* contains a ma#or contribution to Lie al"ebras where he completed the classification of the semisimple al"ebras over the comple field which Fillin" had essentially found. 0owever, althou"h Fillin" had shown that only certain eceptional simple al"ebras were possible, he had not proved that in fact these al"ebras eist. This was shown by ;artan in his thesis when he constructed each of the eceptional simple Lie al"ebras over the comple field. 0e later classified the semisimple Lie al"ebras over the real field and found all the irreducible linear representations of the simple Lie al"ebras. 0e turned to the theory of associative al"ebras and investi"ated the structure for these al"ebras over the real and comple field. (edderburn would complete ;artan's work in this area. 0e then turned to representations of semisimple Lie "roups. 0is work is a strikin" synthesis of Lie theory, classical "eometry, differential "eometry and topolo"y which was to be found in all ;artan's work. 0e applied /rassmann al"ebra to the theory of eterior differential forms. 0e developed this theory between 1D:* and 1:+* and applied his theory of eterior differential forms to a wide variety of problems in differential "eometry, dynamics and relativity. )ieudonn@ writes in 41689 ,e discussed a large number of examples% treating them in an extremely elliptic style that was made possible only by his uncanny algebraic and geometric insight and that has baffled two generations of mathematicians. -n 1:*< he published the book Les syst=mes diff4rentiels ext4rieurs et leurs applications g4om4tri8ues.
1+
y 1:+* ;artan was writin" papers on differential eGuations and in many ways this work is his most impressive. A"ain his approach was totally innovative and he formulated problems so that they were invariant and did not depend on the particular variables or unknown functions. This enabled ;artan to define what the "eneral solution of an arbitrary differential system really is but he was not only interested in the "eneral solution for he also studied sin"ular solutions. 0e did this by movin" from a "iven system to a new associated system whose "eneral solution "ave the sin"ular solutions to the ori"inal system. 0e failed to show that all sin"ular solutions were "iven by his techniGue, however, and this was not achieved until four years after his death. rom 1:1= on he published mainly on differential "eometry. Flein's Erlanger 6rogramm was seen to be inadeGuate as a "eneral description of "eometry by (eyl and Eeblen and ;artan was to play a ma#or role. 0e eamined a space acted on by an arbitrary Lie "roup of transformations, developin" a theory of movin" frames which "eneralises the kinematical theory of )arbou. -n fact this work led ;artan to the notion of a fibre bundle althou"h he does not "ive an eplicit definition of the concept in his work. ;artan further contributed to "eometry with his theory of s ymmetric spaces which have their ori"ins in papers he wrote in 1:2=. -t developed ideas first studied by ;lifford and ;ayley and used topolo"ical methods developed by (eyl in 1:2<. This work was completed by 1:52 and so provides 41689 ... one of the few instances in which the initiator of a mathematical theory was also the one who brought it to completion. ;artan then went on to eamine problems on a topic first studied by oincar@. y this sta"e his son, 0enri ;artan, was makin" ma#or contributions to mathematics and Klie ;artan was able to build on theorems proved by his son. 0enri ;artan said 4:689 4 )y father6 "new more than ! did about Lie groups% and it was necessary to use this "nowledge for the determination of all bounded circled domains which admit a transitive group. o we wrote an article on the subect together 4 Les transformations des domaines cercl4s born4s% 0. +. Acad. ci. 6aris 1:2 31:517, C+:9C126. But in general my father wor"ed in his corner% and ! wor"ed in mine. ;artan discovered the theory of spinors in 1:15. These are comple vectors that are used to transform three9 dimensional rotations into two9dimensional representations and they later played a fundamental role in Guantum mechanics. ;artan published the two volume work Le>ons sur la th4orie des spineurs in 1:5D. 0e is certainly one of the most important mathematicians of the first half of the 2+ th century. )ieudonn@ writes in 41689 0artans recognition as a first rate mathematician came to him only in his old age7 before 1:5+ 6oincar4 and ;eyl were probably the only prominent mathematicians who correctly assessed his uncommon powers and depth. &his was due partly to his extreme modesty and partly to the fact that in 1rance the main trend of mathematical research after 1:++ was in the field of function theory% but chiefly to his extraordinary originality. !t was only after 1:5+ that a younger generation started to explore the rich treasure of ideas and results that lay buried in his papers. ince then his influence has been steadily increasing% and with the exception of 6oincar4 and ,ilbert% probably no one else has done so much to give the mathematics of our day its present shape and viewpoints. or his outstandin" contributions ;artan received many honours, but as )ieudonn@ eplained in the above Guote, these did not come until late in career. 0e received honorary de"rees from the %niversity of Lie"e in 1:5*, and from 0arvard %niversity in 1:5=. -n 1:*C he was awarded three honorary de"rees from the ree %niversity of erlin, the %niversity of ucharest and the ;atholic %niversity of Louvain. -n the followin" year he was awarded an honorary doctorate by the %niversity of isa. 0e was elected a ellow of the Royal !ociety of London on 1 $ay 1:*C, the Accademia dei Lincei and the >orwe"ian Academy. lected to the rench Academy of !ciences on : $arch 1:51 he was vice9president of the Academy in 1:*< and resident in 1:*=.
11
Israil Moiseevi Gel/an" )orn< 2 Se-t .:.# in rasn7e O!n7? O"essa? U!raine? R%ssia 5ie"< Ot 200: in Ne* )r%ns*i!? N@(@? USA Israil Gel/an" went to $oscow at the a"e of 1=, in 1:5+, before completin" his secondary education. There he took on a variety of different #obs such as door keeper at the Lenin library, but he also be"an to teach mathematics. There were many different institutes in $oscow where mathematics was tau"ht in evenin" classes and /elfand tau"ht elementary mathematics in various of these institutes, then a little later pro"ressin" to teach more advanced mathematics. (hile he did this evenin" teachin" he also attended lectures at $oscow %niversity, the first course he attended bein" the theory of functions of a comple variable by Lavrentev. -n 1:52 /elfand was admitted as a research student under Folmo"orov's supervision. 0is work was in functional analysis and he was fortunate to be in a stron" school of functional analysis so he received much support from other mathematicians such as A lessner and L A Lyusternik. /elfand presented his thesis Abstract functions and linear operators in 1:5< which contains important results, but is perhaps even more important for the methods that he used, studyin" functions on normed spaces by applyin" linear functionals to them and usin" classical analysis to study the resultin" functions. /elfand's net ma#or achievement was the theory of commutative normed rin"s which he created and studied in his ).!c. thesis submitted in 1:5D. The importance of this work is brou"ht out in 42+6 and 421689 ... it was 4'elfand6 who brought to light the fundamental concept of a maximal ideal which made it possible to unite previously uncoordinated facts and to create an interesting new theory. 'elfands theory of normed rings revealed close connections between Banachs general functional analysis and classical analysis. )urin" the time that he was carryin" out this work for his ).!c., /elfand tau"ht at the %!!R Academy of !ciences. 0e held a post at the Academy from 1:5< until 1:*1 when he was appointed as professor at $oscow !tate %niversity. -n #oint work with >aimark in the early 1:*+s, /elfand worked on noncommutative normed rin"s with an involution. They showed that these rin"s could always be represented as a rin" of linear operators on a 0ilbert space. -t is impossible to do any #ustice to the ran"e of work covered by /elfand in this short article. 0owever, we should mention some of the main strands of his work. Bne important area which he started work on in the early 1:*+s was the theory of representations of non9compact "roups. This work followed on from the representation theory of finite "roups by robenius and !chur and the representation theory of compact "roups by (eyl. Another important area of his work is that on differential eGuations where he worked on the inverse !turm9 Liouville problem. 0e saw the importance of the work of !obolev and !chwart on the theory of "eneralised functions and distributions, and he developed this theory in a series of mono"raphs. 0e worked on computational mathematics, developin" "eneral methods for solvin" the eGuations of mathematical physics by numerical means. -n this area he also worked on difference operators. /elfand's work on "roup representations led him to study inte"ral "eometry 3a term due to laschke7 which in turn he was led to by a study of the Radon transform. etween 1:=D and 1:C2 /elfand produced a series of important papers on the cohomolo"y of infinite dimensional Lie al"ebras. rom 1:
theoretical work which was his first interest. 0is work in biolo"y is described in 4:6 and 41+6 as follows89 2n the basis of actual biological results% he developed important general principles of the organisation of control in complex multi-cell systems. &hese ideas% apart from their biological significance% served as a starting point for the creation of new methods of finding an extremum% which were succesfully applied to problems of ?-ray structural analysis% problems of recognition% ..... /elfand's interests were certainly not confined to research despite his incredible record of havin" published over <++ papers in mathematics, applied mathematics and biolo"y. 0e established a correspondence school in mathematics which 4*689 ... helped to bring rich mathematical experiences to students all over the oviet *nion. 0is style of teachin" is described in 42+6 and 421689 2ne of the characteristic features of !srail )oiseevics activities has been the extremely close bond between his research wor" and his teaching. &he formulation of new problems and unexpected 8uestions% a tendency to loo" at even well "nown things from a new point of view characterises 'elfand as a teacher% regardless of whether at a given moment he is holding a conversation with schoolchildren or with his own colleagues. -n 1:C5 /elfand was awarded the Brder of Lenin for the third time 34:6 and 41+6789 1or services in the development of mathematics% the training of scientific specialists and in celebration of his sixtieth birthday ... This is only one of a very lar"e number of honours which have been "iven to /elfand over many years of outstandin" contributions. 0e was president of the $oscow $athematical !ociety durin" 1:=D9C+. 0e was elected an honorary member of the American >ational Academy of !cience, the American Academy of Arts and !ciences, the Royal -rish Academy, the American $athematical !ociety, the London $athematical !ociety. 0e has been awarded many honorary doctorates includin" one from the %niversity of Bford. -n 1:D:9:+ /elfand tau"ht at 0arvard %niversity and in 1::+ he also tau"ht at $assachusetts -nstitute of Technolo"y. -n that same year, 1::+, he emi"rated to the %nited !tates where he became )istin"uished Eisitin" rofessor at Rut"ers. 0e also holds a chair in the departments of mathematics and biolo"y at the ;enter for $athematics, !cience, and ;omputer ducation in the -nstitute for )iscrete $athematics and ;omputer !cience at Rut"ers %niversity. -n 1::2 /elfand set up a pro"ramme in the %nited !tates similar to the correspondence school in mathematics which he had run in Russia. The /elfand Butreach ro"ram 4*689 ... fosters mathematical excellence in high school students. -n 1::* /elfand was awarded a $acArthur ellowship from the ?ohn T and ;atherine ) $acArthur oundation. The $acArthur ellowships are 4*689 ... no-strings-attached awards that are intended to foster creativity in a wide range of human endeavours. &he award to 'elfand is 5C<,+++% to be paid over five years. Let us mention two other honours "iven to /elfand. -n 1:D: he received the Fyoto rie from the -namori oundation, an international award to honor those who have contributed si"nificantly to the scientific, cultural, and spiritual betterment of mankind. The announcement states89 !zrail )oiseevich 'elfand is one of the highest authorities in modern mathematical sciences. &hrough his pioneering and monumental wor" in mathematical sciences% especially in functional analysis - which has experienced tremendous development this century and not only affected other areas of mathematics but has also provided indispensable mathematical tools for the physics of elementary particles and 8uantum mechanics - he has brought up and inspired many prominent mathematicians in the course of his creative career. ,e has provided "ey ideas and deep insights to the whole of mathematical sciences and made outstanding contributions to the 15
advancement of the field. -n 2++< /elfand received the Leroy !teele rie of the American $athematical !ociety for Lifetime Achievement89 ... for profoundly influencing many fields of research through his own wor" and through his interactions with other mathematicians and students.
1*
%rt G1"el )orn< 24 A-ril .:0+ in )rDnn? A%striaH%ngar7 =no* )rno? C&eh Re-%'li> 5ie"< .6 (an .:94 in Prineton? Ne* (erse7? USA %rt G1"el's father was Rudolf /del whose family were from Eienna. Rudolf did not take his academic studies far as a youn" man, but had done well for himself becomin" mana"in" director and part owner of a ma#or tetile firm in rMnn. Furt's mother, $arianne 0andschuh, was from the Rhineland and the dau"hter of /ustav 0andschuh who was also involved with tetiles in rMnn. Rudolf was 1* years older than $arianne who, unlike Rudolf, had a literary education and had undertaken part of her school studies in rance. Rudolf and $arianne /del had two children, both boys. The elder they named Rudolf after his father, and the youn"er was Furt. Furt had Guite a happy childhood. 0e was very devoted to his mother but seemed rather timid and troubled when his mother was not in the home. 0e had rheumatic fever when he was si years old, but after he recovered life went on much as before. 0owever, when he was ei"ht years old be be"an to read medical books about the illness he had suffered from, and learnt that a weak heart was a possible complication. Althou"h there is no evidence that he did have a weak heart, Furt became convinced that he did, and concern for his health became an everyday worry for him. Furt attended school in rMnn, completin" his school studies in 1:25. 0is brother Rudolf said89 Even in ,igh chool my brother was somewhat more one-sided than me and to the astonishment of his teachers and fellow pupils had mastered university mathematics by his final 'ymnasium years. ... )athematics and languages ran"ed well above literature and history. At the time it was rumoured that in the whole of his time at ,igh chool not only was his wor" in Latin always given the top mar"s but that he had made not a single grammatical error. /del entered the %niversity of Eienna in 1:25 still without havin" made a definite decision whether he wanted to specialise in mathematics or theoretical physics. 0e was tau"ht by urtwNn"ler, 0ahn, (irtin"er, $en"er, 0elly and others. The lectures by urtwNn"ler made the most impact on /del and because of them he decided to take mathematics as his main sub#ect. There were two reasons8 urtwNn"ler was an outstandin" mathematician and teacher, but in addition he was paralysed from the neck down so lectured from a wheel chair with an assistant who wrote on the board. This would make a bi" impact on any student, but on /del who was very conscious of his own health, it had a ma#or influence. As an under"raduate /del took part in a seminar run by !chlick which studied Russell's book !ntroduction to mathematical philosophy. Bl"a Taussky9Todd, a fellow student of /del's, wrote89 !t became slowly obvious that he would stic" with logic% that he was to be ,ahns student and not chlic"s% that he was incredibly talented. ,is help was much in demand. 0e completed his doctoral dissertation under 0ahn's supervision in 1:2: submittin" a thesis provin" the completeness of the first order functional calculus. 0e became a member of the faculty of the %niversity of Eienna in 1:5+, where he belon"ed to the school of lo"ical positivism until 1:5D. /del's father died in 1:2: and, havin" had a successful business, the family were left financially secure. After the death of her husband, /del's mother purchased a lar"e flat in Eienna and both her sons lived in it with her. y this time /del's older brother was a successful radiolo"ist. (e mentioned above that /del's mother had a literary education and she was now able to en#oy the culture of Eienna, particularly the theatre accompanied by Rudolf and Furt. /del is best known for his proof of J/del's -ncompleteness TheoremsJ. -n 1:51 he published these results in @ber formal unentscheidbare tze der 6rincipia )athematica und verwandter ysteme. 0e proved fundamental results about aiomatic systems, showin" in any aiomatic mathematical system there are propositions that cannot be proved or disproved within t he aioms of the system. -n particular the consistency 1<
of the aioms cannot be proved. This ended a hundred years of attempts to establish aioms which would put the whole of mathematics on an aiomatic basis. Bne ma#or attempt had been by ertrand Russell with 6rincipia )athematica 31:1+9157. Another was 0ilbert's formalism which was dealt a severe blow by /del's results. The theorem did not destroy the fundamental idea of formalism, but it did demonstrate that any system would have to be more comprehensive than that envisa"ed by 0ilbert. /del's results were a landmark in 2+th9century mathematics, showin" that mathematics is not a finished ob#ect, as had been believed. -t also implies that a computer can never be pro"rammed to answer all mathematical Guestions. /del met Hermelo in ad lster in 1:51. Bl"a Taussky9Todd, who was at the same meetin", wrote89 &he trouble with ermelo was that he felt he had already achieved 'Cdels most admired result himself. cholz seemed to thin" that this was in fact the case% but he had not announced it and perhaps would never have done so. ... &he peaceful meeting between ermelo and 'Cdel at Bad Elster was not the start of a scientific friendship between two logicians. !ubmittin" his paper on incompleteness to the %niversity of Eienna for his habilitation, this was accepted by 0ahn on 1 )ecember 1:52. /del became a rivatdoent at the %niversity of Eienna in $ arch 1:55. >ow 1:55 was the year that 0itler came to power. At first this had no effect on /del's life in EiennaI he had little interest in politics. -n 1:5* /del "ave a series of lectures at rinceton entitled 2n undecidable propositions of formal mathematical systems. At Eeblen's su""estion Fleene, who had #ust completed his h.). thesis at rinceton, took notes of these lectures which have been subseGuently published. 0owever, /del suffered a nervous breakdown as he arrived back in urope and telephoned his brother Rudolf from aris to say he was ill. 0e was treated by a psychiatrist and spent several months in a sanatorium recoverin" from depression. )espite the health problems, /del's research was pro"ressin" well and he proved important results on the consistency of the axiom of choice with the other aioms of set theory in 1:5<. 0owever after !chlick, whose seminar had aroused /del's interest in lo"ic, was murdered by a >ational !ocialist student in 1:5=, /del was much affected and had another breakdown. 0is brother Rudolf wrote89 &his event was surely the reason why my brother went through a severe nervous crisis for some time% which was of course of great concern% above all for my mother. oon after his recovery he received the first call to a 'uest 6rofessorship in the *A. 0e visited /ttin"en in the summer of 1:5D, lecturin" there on his set theory research. 0e returned to Eienna and married Adele orkert in the autumn of 1:5D. -n fact he had met her in 1:2C in )er >achtfalter ni"ht club in Eienna. !he was si years older than /del and had been married before and both his parents, but particularly his father, ob#ected to the idea that they marry. !he was not the first "irl that /del's parents had ob#ected to, the first he had met around the time he went to university was ten years older than him. -n $arch 1:5D Austria had became part of /ermany but /del was not much interested and carried on his life much as normal. 0e visited rinceton for the second time, spendin" the first term of session 1:5D95: at the -nstitute for Advanced !tudy. The second term of that academic year he "ave a beautiful lecture course at >otre )ame. $ost who held the title of privatdoent in Austria became paid lecturers after the country became part of /ermany but /del did not and his application made on 2< !eptember 1:5: was "iven an unenthusiastic response. -t seems that he was thou"ht to be ?ewish, but in fact this was entirely wron", althou"h he did have many ?ewish friends. Bthers also mistook him for a ?ew, and he was once attacked by a "an" of youths, believin" him to be a ?ew, while out walkin" with his wife in Eienna. (hen the war started /del feared that he mi"ht be conscripted into the /erman army. Bf course he was also convinced that he was in far too poor health to serve in the army, but if he could be mistaken for a ?ew he mi"ht be mistaken for a healthy man. 0e was not prepared to risk this, and after len"thy ne"otiation to obtain a %.!. visa he was fortunate to be able to return to the %nited !tates, althou"h he had to travel via Russia and ?apan to do so. 0is wife accompanied him. -n 1:*+ /del arrived in the %nited !tates, becomin" a %.!. citien in 1:*D 3in fact he believed he had found an inconsistency in the %nited !tates ;onstitution, but the #ud"e had more sense than to listen durin" his interviewO7. 0e was an ordinary member of the -nstitute for Advanced !tudy from 1:*+ to 1:*= 3holdin" year lon" appointments which were renewed every year7, then he was a permanent member until 1:<5. 0e held a 1=
chair at rinceton from 1:<5 until his death, holdin" a contract which eplicitly stated that he had no lecturin" duties. Bne of /del's closest friends at rinceton was instein. They each had a hi"h re"ard for the other and they spoke freGuently. -t is unclear how much instein influenced /del to work on relativity, but he did indeed contribute to that sub#ect. 0e received the instein Award in 1:<1, and >ational $edal of !cience in 1:C*. 0e was a member of the >ational Academy of !ciences of the %nited !tates, a fellow of the Royal !ociety, a member of the -nstitute of rance, a fellow of the Royal Academy and an 0onorary $ember of the London $athematical !ociety. 0owever, it says much about his feelin"s towards Austria that he refused membership of the Academy of !ciences in Eienna, then later when he was elected to honorary membership he a"ain refused the honour. 0e also refused to accept the hi"hest >ational $edal for scientific and artistic achievement that Austria offered him. 0e certainly felt bitter at his own treatment but eGually so about that of his family. /del's mother had left Eienna before he did, for in 1:5C she returned to her villa in rno where she was openly critical of the >ational !ocialist re"ime. /del's brother Rudolf had remained in Eienna but by 1:** both epected /erman defeat, and Rudolf's mother #oined him in Eienna. -n terms of the treaty ne"otiated after the war between the Austrians and the ;echs, she received one tenth of the value for her villa in rno. -t was an in#ustice which infuriated /delI in fact he always took such in#ustices as personal even althou"h lar"e numbers suffered in the same way. After settlin" in the %nited !tates, /del a"ain produced work of the "reatest importance. 0is masterpiece 0onsistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory 31:*+7 is a classic of modern mathematics. -n this he proved that if an aiomatic system of set theory of the type proposed by Russell and (hitehead in 6rincipia )athematica is consistent, then it will remain so when the aiom of choice and the "eneralied continuum9hypothesis are added to the system. This did not prove that these aioms were independent of the other aioms of set theory, but when this was finall y established by ;ohen in 1:=5 he built on these ideas of /del. ;oncerns with his health became increasin"ly worryin" to /del as the years went by. Rudolf, /del's brother, was a medical doctor so the medical details "iven by him in the followin" will be accurate. 0e wrote89 )y brother had a very individual and fixed opinion about everything and could hardly be convinced otherwise. *nfortunately he believed all his life that he was always right not only in mathematics but also in medicine% so he was a very difficult patient for doctors. After severe bleeding from a duodenal ulcer ... for the rest of his life he "ept to an extremely strict 3over strict<7 diet which caused him slowly to lose weight. Adele, /del's wife, was a "reat support to him and she did much to ease the tensions which troubled him. 0owever she herself be"an to suffer health problems, havin" two strokes and a ma#or operation. Towards the end of his life /del became convinced that he was bein" poisoned and, refusin" to eat to avoid bein" poisoned, essentially starved himself to death 45689 A slight person and very fastidious% 'Cdel was generally worried about his health and did not travel or lecture widely in later years. ,e had no doctoral students% but through correspondence and personal contact with the constant succession of visitors to 6rinceton% many people benefited from his extremely 8uic" and incisive mind. 1riend to Einstein% von 5eumann and )orgenstern% he particularly enoyed philosophical discussion. 0e died 41D689 ... sitting in a chair in his hospital room at 6rinceton% in the afternoon of 1* Danuary 1:CD. -t would be fair to say that /del's ideas have chan"ed the course of mathematics 45689 ... it seems clear that the fruitfulness of his ideas will continue to stimulate new wor". 1ew mathematicians are granted this "ind of immortality.
1C
Feli3 Ha%s"or// )orn< 4 Nov .4+4 in )resla%? Ger8an7 =no* ;roa*? Polan"> 5ie"< 2+ (an .:62 in )onn? Ger8an7 Feli3 Ha%s"or// 's father was Louis 0ausdorff, who was a merchant dealin" in tetiles, and his mother was 0edwi" TietI both were ?ewish. eli was born into a wealthy family and this had Guite an influence on his life and career since he never had the problem of havin" to work to support himself financially. eli was stll a youn" boy when the family moved from reslau to Leipi", and it was in Leipi" that he "rew up. At school he had wide interests and, in addition to mathematics, he was attracted to literature and music. -n fact he wanted to pursue a career in music as a composer but his parents put pressure on him to "ive up the idea of becomin" a composer. They achieved this, but only after Guite a n effort for eli had his heart set on the idea, and after this he turned towards mathematics as the sub#ect to study at university. 0ausdorff studied at Leipi" %niversity under 0einrich runs and Adolph $ayer, "raduatin" in 1D:1 with a doctorate in applications of mathematics to astronomy. 0is thesis was titled ur &heorie der astronomischen trahlenbrechung and studied refraction and etinction of li"ht in the atmosphere. 0e published four papers on astronomy and optics over the net few years and he submitted his habilitation thesis to Leipi" in 1D:<, also based on his research into astronomy and optics. 0is methods were based on those of runs who had developed his own method of determinin" refraction and etinction, based on an idea of essel. 0owever 0ausdorff's main interests were in literature and philosophy and his circle of friends consisted almost entirely of writers and artists, such as the composer $a Re"er, rather than scientists. 0e also seemed keen to make a name for himself in the world of literature, more so than in the world of mathematics, and he published his literary work under the pseudonym of aul $on"r@. -n 1D:C he published his first literary work ant !lario: &houghts from arathustras 0ountry which was a work of 5CD pa"es. 0e published a philosophy book /as 0haos in "osmischer Auslese 31D:D7 which is a critiGue of metaphysics contrastin" the empirical with the transcendental world that he re#ected. 0is net ma#or literary work was a book of poem E"stases 31:++7 which deals with nature, life, death and erotic passion, and in addition he wrote many articles on philosophy and literature. As !e"al writes in 4=689 As the child of a wealthy family% he did not have to worry about ma"ing a career as a mathematician7 for him% mathematics% both as research and as a subect to teach% was more an avocation than anything else. 0ausdorff married ;harlotte !ara /oldschmidt in Leipi" in 1D::. ;harlotte and her sister dith were from ?ewish parents but had converted to Lutheranism. Althou"h still a rivatdoent, 0ausdorff was well off, so marria"e at this sta"e in his career presented no financial difficulties. -n 1:+2 he was promoted to an etraordinary professorship of mathematics at Leipi" and turned down the offer of a similar appointment at /ttin"en. This clearly indicates that at this time 0ausdorff was keener to remain in his literary and artistic circle in Leipi" than he was to pro"ress his career in mathematics. 0e continued his literary interests and in 1:+* published a farce /er Arzt seiner Ehre. -n many ways this marked the end of his literary interests but this farce was performed in 1:12 and was very successful. After 1:+* 0ausdorff be"an workin" in the area for which he is famous, namely topolo"y and set theory. 0e introduced the concept of a partially ordered set and from 1:+1 to 1:+: he proved a series of results on ordered sets. -n 1:+C he introduced special types of ordinals in an attempt to prove ;antor's continuum hypothesis. 0e also posed a "eneralisation of the continuum hypothesis by askin" if 2 to the power ℵa was eGual to ℵaP1. 0ausdorff proved further results on the cardinality of orel sets in 1:1=. 0ausdorff tau"ht at Leipi" until 1:1+ when he went to onn. -t was !tudy who in many ways motivated 0ausdorff to become more involved in both mathematical research and also in developin" his career in mathematics. artly the lack of mathematical drive in his early career had been due to his etreme modesty, 1D
so his friendship with !tudy was an important factor in turnin" him towards important problems and his subseGuent rise to fame. 0avin" encoura"ed 0ausdorff to move to onn, !tudy encoura"ed him to move a"ain in 1:15, this time to become an ordinary professorship at /reifswalf. A year later, in 1:1*, 0ausdorff published his famous tet 'rundz(ge der )engenlehre which builds on work by r@chet and others to created a theory of topolo"ical and metric spaces. arlier results on topolo"y fitted naturally into the framework set up by 0ausdorff as Fatetov eplains in 41689 4 ,ausdorffs6 broad approach% his aesthetic feeling% and his sense of balance may have played a substantial part. ,e succeeded in creating a theory of topological and metric spaces into which the previous results fitted well% and he enriched it with many new notions and theorems. 1rom the modern point of view% the 'rundz(ge contained% in addition to other special topics% the beginnings of the theories of topological and metric spaces% which are now included in all textboo"s on the subect. The 'rundz(ge was republished in revised form in 1:2C and 1:5C. The 1:1* edition was reprinted in 1:*: and 1:=< by ;helsea, the 1:2C edition was published in 1:5C in Russian, and the 1:5C edition was translated into n"lish and also published by ;helsea in 1:ai re"ime. Althou"h as early as 1:52 he sensed the oncomin" calamity of >aism he made no attempt to emi"rate while it was still possible. 0e swore the necessary oath to 0itler in >ovember 1:5* but by the followin" ?anuary a new law forced him to "ive up his position. 0e continued to undertake research in topolo"y and set theory but the results could not be published in /ermany. ;ertainly he wanted to continue research and wished to emi"rate for in 1:5: he wrote to ;ourant askin" if he could find a research fellowship for him. !adly ;ourant could not do so. As a ?ew his position became more and more difficult. -n 1:*1 he was scheduled to "o to an internment camp but mana"ed to avoid bein" sent. rich essel90a"en, the only collea"ue from onn who kept in touch with 0ausdorff after his forced retirement, wrote in a letter to a friend in the summer of 1:*1 3see 42*6 and 4=6789 ! often had great anxiety about the ,ausdorffs. )rs ,ausdorff was for a long time seriously ill from an old ailment - ! dont "now what it is. carcely was she over the worst than there came the agitation about the intended internment of the Dews. ,ere the procedure was mad. !n the early part of the year% old nuns were forcibly driven out of a cloister on the 3reuzberg7 these poor old women who never harmed anyone and only carried on a retiring life devoted to their pious usages ... 5ow all Dews still living in Bonn will be compulsorily interned in this stolen building7 they must either auction their things% or place them for preservation in 9faithful9 hands. onn %niversity reGuested that the 0ausdorffs be allowed to remain in their home and this was "ranted. y Bctober 1:*1 they were forced to wear the Jyellow starJ and around the end of the year they were informed that they would be sent to ;olo"ne. essel90a"en wrote that he knew this was 3see 42*6 and 4=6789 ... a preliminary to deportation to 6oland. And what one hears concerning the accommodation and treatment of Dews there is completely unimaginable. They were not sent to ;olo"ne but in ?anuary 1:*2 they were informed that they were to be interned in ndenich. To"ether with his wife and his wife's sister, he committed suicide on 2= ?anuary. 0e wrote to a friend on !unday 2< ?anuary 3see 42*6 and 4=6789 /ear 1riend ;ollstein By the time you receive these lines% we three will have solved the problem in another way - in the way which you have continually attempted to dissuade us. ... ;hat has been done against the Dews in recent months arouses well-founded anxiety that we will no longer be allowed to experience a bearable situation. ... 1orgive us% that we still cause you trouble beyond death7 ! am convinced that you will do what you are able to do and which perhaps is not very muchF. 1orgive us also our desertionG ;e wish you and all our friends will experience better times Hours faithfully 1:
1elix ,ausdorff Bn the ni"ht of !unday 2< ?anuary all three took barbiturates. oth 0ausdorff and his wife ;harlotte were dead by the mornin" of the 2= ?anuary. dith, ;harlotte's sister, survived for a few days in a coma before dyin". (e have mentioned above 0ausdorff's early work on astronomy, his work on philosophy, and his literature. (e also mentioned his work on ordered sets and his masterpiece on set theory and topolo"y 'rundz(ge der )engenlehre 31:1*7. Let us add that one famous paradoical result, namely that half a sphere and one third of a sphere can be con"ruent to each other, is contained in this work 3see 42D6 for details7. Let us now eamine other important contributions made by 0ausdorff. -n 1:1: he introduced the notion of 0ausdorff dimension in the seminal paper /imension und usseres )ass. The idea was a "eneralisation of one which had been introduced five years earlier by ;arath@odory but 0ausdorff realised that ;arath@odory's construction made sense, and was useful, for definin" fractional dimensions. 0ausdorff's paper includes a proof that the dimension of the middle9third ;antor set is lo" 2Qlo" 5. ;hatter#i writes 41+689 ;ithin the mathematical wor" of ,ausdorff the two publications devoted explicitly to measure theory occupy a significant place: they are not only important for measure theory but have also contributed fundamentally to its development. !t is not well "nown that throughout his life ,ausdorff had been interested in various fundamental problems of measure and integration theory and had made important contributions at different times. &his becomes 8uite evident if one studies his lecture notes and other 5achlass papers. Bne such lecture course was "iven on probability theory by 0ausdorff in onn in the summer of 1:25. 0e studied the /aussian law of errors, limit theorems and problems of moments, and set theory and the stron" law of lar"e numbers, which he based on measure theory.
2+
5avi" Hil'ert )orn< 2# (an .4+2 in 1nigs'erg? Pr%ssia =no* aliningra"? R%ssia> 5ie"< .6 Fe' .:6# in G1ttingen? Ger8an7 5avi" Hil'ert attended the "ymnasium in his home town of Fni"sber". After "raduatin" from the "ymnasium, he entered the %niversity of Fni"sber". There he went on to study under Lindemann for his doctorate which he received in 1DD< for a thesis entitled @ber invariante Eigenschaften specieller binrer 1ormen% insbesondere der 3ugelfunctionen. Bne of 0ilbert's friends there was $inkowski, who was also a doctoral student at Fni"sber", and they were to stron"ly influence each others mathematical pro"ress. -n 1DD* 0urwit was appointed to the %niversity of Fni"sber" and Guickly became friends with 0ilbert, a friendship which was another important factor in 0ilbert's mathematical development. 0ilbert was a member of staff at Fni"sber" from 1DD= to 1D:<, bein" a rivatdoent until 1D:2, then as traordinary rofessor for one year before bein" appointed a full professor in 1D:5. -n 1D:2 !chwar moved from /ttin"en to erlin to occupy (eierstrass's chair and Flein wanted to offer 0ilbert the vacant /ttin"en chair. 0owever Flein failed to persuade his collea"ues and 0einrich (eber was appointed to the chair. Flein was probably not too unhappy when (eber moved to a chair at !trasbour" three years later since on this occasion he was successful in his aim of appointin" 0ilbert. !o, in 1D:<, 0ilbert was appointed to the chair of mathematics at the %niversity of /ttin"en, where he continued to teach for the rest of his career. 0ilbert's eminent position in the world of mathematics after 1:++ meant that other institutions would have liked to tempt him to leave /ttin"en and, in 1:+2, the %niversity of erlin offered 0ilbert uchs' chair. 0ilbert turned down the erlin chair, but only after he had used the offer to bar"ain with /ttin"en and persuade them to set up a new chair to brin" his friend $inkowski to /ttin"en. 0ilbert's first work was on invariant theory and, in 1DDD, he proved his famous asis Theorem. Twenty years earlier /ordan had proved the finite basis theorem for binary forms usin" a hi"hly computational approach. Attempts to "eneralise /ordan's work to systems with more than two variables failed since the computational difficulties were too "reat. 0ilbert himself tried at first to follow /ordan's approach but soon realised that a new line of attack was necessary. 0e discovered a completely new approach which proved the finite basis theorem for any number of variables but in an entirely abstract way. Althou"h he proved that a finite basis eisted his methods did not construct such a basis. 0ilbert submitted a paper provin" the finite basis theorem to )athematische Annalen. 0owever /ordan was the epert on invariant theory for )athematische Annalen and he found 0ilbert's revolutionary approach difficult to appreciate. 0e refereed the paper and sent his comments to Flein89 &he problem lies not with the form ... but rather much deeper. ,ilbert has scorned to present his thoughts following formal rules% he thin"s it suffices that no one contradict his proof ... he is content to thin" that the importance and correctness of his propositions suffice. ... for a comprehensive wor" for the Annalen this is insufficient. 0owever, 0ilbert had learnt throu"h his friend 0urwit about /ordan's letter to Flein and 0ilbert wrote himself to Flein in forceful terms89 ... ! am not prepared to alter or delete anything% and regarding this paper% ! say with all modesty% that this is my last word so long as no definite and irrefutable obection against my reasoning is raised. At the time Flein received these two letters from 0ilbert and /ordan, 0ilbert was an assistant lecturer while /ordan was the reco"nised leadin" world epert on invariant theory and also a close friend of Flein's. 0owever Flein reco"nised the importance of 0ilbert's work and assured him that it would appear in the 21
Annalen without any chan"es whatsoever, as indeed it did. 0ilbert epanded on his methods in a later paper, a"ain submitted to the )athematische Annalen and Flein, after readin" the manuscript, wrote to 0ilbert sayin"89 ! do not doubt that this is the most important wor" on general algebra that the Annalen has ever published. -n 1D:5 while still at Fni"sber" 0ilbert be"an a work ahlbericht on al"ebraic number theory. The /erman $athematical !ociety reGuested this ma#or report three years after the !ociety was created in 1D:+. The ahlbericht 31D:C7 is a brilliant synthesis of the work of Fummer, Fronecker and )edekind but contains a wealth of 0ilbert's own ideas. The ideas of the present day sub#ect of ';lass field theory' are all contained in this work. Rowe, in 41D6, describes this work as89 ... not really a Bericht in the conventional sense of the word% but rather a piece of original research revealing that ,ilbert was no mere specialist% however gifted. ... he not only synthesized the results of prior investigations ... but also fashioned new concepts that shaped the course of research on algebraic number theory for many years to come. 0ilbert's work in "eometry had the "reatest influence in that area after uclid. A systematic study of the aioms of uclidean "eometry led 0ilbert to propose 21 such aioms and he analysed their si"nificance. 0e published 'rundlagen der 'eometrie in 1D:: puttin" "eometry in a formal aiomatic settin". The book continued to appear in new editions and was a ma#or influence in promotin" the aiomatic approach to mathematics which has been one of the ma#or characteristics of the sub#ect throu"hout the 2+ th century. 0ilbert's famous 25 aris problems challen"ed 3and still today challen"e7 mathematicians to solve fundamental Guestions. 0ilbert's famous speech &he 6roblems of )athematics was delivered to the !econd -nternational ;on"ress of $athematicians in aris. -t was a speech full of optimism for mathematics in the comin" century and he felt that open problems were the si"n of vitality in the sub#ect89 &he great importance of definite problems for the progress of mathematical science in general ... is undeniable. ... 4for6 as long as a branch of "nowledge supplies a surplus of such problems% it maintains its vitality. ... every mathematician certainly shares ..the conviction that every mathematical problem is necessarily capable of strict resolution ... we hear within ourselves the constant cry: &here is the problem% see" the solution. Hou can find it through pure thought... 0ilbert's problems included the continuum hypothesis, the well orderin" of the reals, /oldbach's con#ecture, the transcendence of powers of al"ebraic numbers, the Riemann hypothesis, the etension of )irichlet's principle and many more. $any of the problems were solved durin" this century, and each time one of the problems was solved it was a ma#or event for mathematics. Today 0ilbert's name is often best remembered throu"h the concept of 0ilbert space. -rvin" Faplansky, writin" in 426, eplains 0ilbert's work which led to this concept89 ,ilberts wor" in integral e8uations in about 1:+: led directly to IJth-century research in functional analysis 3the branch of mathematics in which functions are studied collectively7. &his wor" also established the basis for his wor" on infinite-dimensional space% later called ,ilbert space% a concept that is useful in mathematical analysis and 8uantum mechanics. )a"ing use of his results on integral e8uations% ,ilbert contributed to the development of mathematical physics by his important memoirs on "inetic gas theory and the theory of radiations. $any have claimed that in 1:1< 0ilbert discovered the correct field eGuations for "eneral relativity before instein but never claimed priority. The article 4116 however, shows that this view is in error. -n this paper the authors show convincin"ly that 0ilbert submitted his article on 2+ >ovember 1:1<, five days before instein submitted his article containin" the correct field eGuations. instein's article appeared on 2 )ecember 1:1< but the proofs of 0ilbert's paper 3dated = )ecember 1:1<7 do not contain the field eGuations. As the authors of 4116 write89 !n the printed version of his paper% ,ilbert added a reference to Einsteins conclusive paper and a concession to the latters priority: 9&he differential e8uations of gravitation that result are% as 22
it seems to me% in agreement with the magnificent theory of general relativity established by Einstein in his later papers9. !f ,ilbert had only altered the dateline to read 9submitted on IJ 5ovember #K#% revised on 4any date after I /ecember #K#% the date of Einsteins conclusive paper6%9 no later priority 8uestion would have arisen. -n 1:5* and 1:5: two volumes of 'rundlagen der )athemati" were published which were intended to lead to a 'proof theory', a direct check for the consistency of mathematics. /del's paper of 1:51 showed that this aim is impossible. 0ilbert contributed to many branches of mathematics, includin" invariants, al"ebraic number fields, functional analysis, inte"ral eGuations, mathematical physics, and the calculus of variations. 0ilbert's mathematical abilities were nicely summed up by Btto lumenthal, his first student89 !n the analysis of mathematical talent one has to differentiate between the ability to create new concepts that generate new types of thought structures and the gift for sensing deeper connections and underlying unity. !n ,ilberts case% his greatness lies in an immensely powerful insight that penetrates into the depths of a 8uestion. All of his wor"s contain examples from farflung fields in which only he was able to discern an interrelatedness and connection with the problem at hand. 1rom these% the synthesis% his wor" of art% was ultimately created. !nsofar as the creation of new ideas is concerned% ! would place )in"ows"i higher% and of the classical great ones% 'auss% 'alois% and +iemann. But when it comes to penetrating insight% only a few of the very greatest were the e8ual of ,ilbert. Amon" 0ilbert's students were 0ermann (eyl, the famous world chess champion Lasker, and Hermelo. 0ilbert received many honours. -n 1:+< the 0un"arian Academy of !ciences "ave a special citation for 0ilbert. -n 1:5+ 0ilbert retired and the city of Fni"sber" made him an honorary citien of the city. 0e "ave an address which ended with si famous words showin" his enthusiasm for mathematics and his life devoted to solvin" mathematical problems89 ;ir m(ssen wissen% wir werden wissen - ;e must "now% we shall "now.
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Hein& Ho-/ )orn< .: Nov .4:6 in Gr'shen =near )resla%>? Ger8an7 =no* ;roa*? Polan"> 5ie"< # (%ne .:9. in olli!on? S*it&erlan" Hein& Ho-/ 's father was (ilhelm 0opf and his mother was liabeth Firchner. (ilhelm 0opf was from a ?ewish family. 0e #oined 0einrich Firchner at his brewery in reslau in 1DDC. (ilhelm married liabeth, 0einrich Firchner's eldest dau"hter, in 1D:2 and by that time he owned the brewery firm. They had two children, the eldest 0edwi" was born in 1D:5 while 0ein was born in the followin" year. liabeth 0opf was a rotestant and, in 1D:<, (ilhelm converted to his wife's reli"ion. 0ein attended )r Farl $ittelhaus' school from 1:+1 until 1:+* and followin" this he be"an his studies at the Fni"9(ilhelm /ymnasium in reslau. 0e attended the /ymnasium until 1:15 and it was at this school that his talent for mathematics first became clear to his teachers. -n his other sub#ects, however, his results were less "ood and it is probable that he devoted too much time to sport, he was particularl y fond of swimmin" and tennis, and not enou"h to his academic sub#ects. 0e left the /ymnasium with the mathematics report statin"89 ,e has shown an extraordinary gift in this topic% especially in the algebraic direction. -n April 1:15 0opf entered the !ilesian riedrich (ilhelms %niversity in reslau to read for a de"ree in mathematics. There he was tau"ht by Fneser, !chmidt, and Rudolf !turm. 0e also attended lectures by )ehn and !teinit who tau"ht at the polytechnic in reslau. 0owever, his studies were interrupted by the outbreak of (orld (ar - in 1:1*. 0e immediately enlisted and for the duration of the war he fou"ht on the (estern front as a lieutenant. )urin" a fortni"ht's leave from military service in 1:1C 0opf went to a class by !chmidt on set theory at the %niversity of reslau. rom that time on he knew that he wanted to undertake research in mathematics. 0e wrote in 4156 about the influence !chmidt's lectures had on him 89 ! was fascinated7 this fascination - of the power of the method of the mapping degree - has never left me since% but has influenced maor parts of my wor". And when ! loo" for the cause of this effect% ! see particularly two things: firstly% chmidts vividness and enthusiasm in his lecture% and secondly my own increased receptiveness during a fortnight off after many years of military service. After the war 0opf returned to his studies in reslau but after about a year he left and went to the %niversity of 0eidelber". y this time !chmidt had left reslau and it appears that 0opf wanted to "o to 0eidelber" to be with his sister who had be"un her studies there in the previous year. At 0eidelber" 0opf took courses in philosophy and psycholo"y as well as attendin" courses by erron and !tNckel. -n 1:2+ 0opf went to study for his doctorate at the %niversity of erlin where !chmidt was now teachin". 0e attended several courses by !chur in erlin and he received his doctorate in 1:2< with a thesis, supervised by !chmidt, studyin" the topolo"y of manifolds. Amon" other results, he classified simply connected Riemannian 59manifolds of constant curvature in this thesis. -t was an impressive piece of work which received the followin" praise from !chmidt in his report 3see for eample 4116789 &he boldness of the 8uestions deserves as much admiration as the surprising results of the solutions. But the most beautiful thing in the thesis is the method of proving% which is% particularly rarely found in wor"s in that area% abstract and comprehensible in every step% and which% due to the abstractness% shows e8ually clearly the richness of the concrete geometric imagination. ieberbach and !chmidt eamined him in mathematics, while lanck eamined him in physics. 0opf went to /ttin"en in 1:2< where he met mmy >oether. 0er contributions would play an important part in 0opf's developin" ideas. erhaps even more si"nificant was the fact that Aleksandrov was also 2*
spendin" time in /ttin"en and 0opf wrote in 415689 )y most important experience in 'Cttingen was to meet 6avel Ale"sandrov. &he meeting soon became friendship7 not only topology% not only mathematics was discussed7 it was a fortunate and also a very happy time% not restricted to 'Cttingen but continued on many oint ourneys. )urin" this year in /ttin"en 0opf worked on his habilitation thesis which was completed by the autumn of 1:2=. The thesis contains a different proof of the fact #ust shown by Lefschet that for any closed manifold the sum of the indices of a "eneric vector field is a topolo"ical invariant, namely the uler characteristic. Aleksandrov and 0opf spent some time in 1:2= in the south of rance with >eu"ebauer. Then the two spent the academic year 1:2C92D at rinceton in the %nited !tates. This was an important year in the development of topolo"y with Aleksandrov and 0opf in rinceton and able to collaborate with Lefschet, Eeblen and Aleander. )urin" their year in rinceton, Aleksandrov and 0opf planned a #oint multi9volume work on &opology the first volume of which did not appear until 1:5<. This was the only one of the three intended volumes to appear since (orld (ar -- prevented further collaboration on the remainin" two volumes. 0opf married An#a von $ickwit in Bctober 1:2D. 0e was offered an assistant professorship by rinceton in )ecember 1:2: but he re#ected the offer. -n 1:5+ (eyl left his chair in the T0 in Hurich to take up a chair at /ttin"en and in 1:51 0opf was approached to see if he was interested in acceptin" this chair. -n part the offer had been prompted by a very positive recommendation which !chur had sent to Hurich89 ,opf is an excellent lecturer% a mathematician of strong temperament and strong influence% a leading example in his discipline ... ! cannot wish you a better colleague in respect to his manners% his education and his sympathetic nature. 0opf replied to the approach of the T0 in Hurich indicatin" that he would accept a formal offer89 A call to witzerland% to the beautiful city of urich% could indeed tempt and honour me% particularly to such a famous chair. ! therefore declare that ! am in principle willing to accept such an offer. 0owever, before receivin" the formal offer from Hurich, 0opf received the offer of a chair at reibur" but he waited for the Hurich offer and accepted it. 0e took up his duties in Hurich in April 1:51. The net few years were not easy ones for 0opf. After the >ais came to power in /ermany in 1:55, 0opf's father, bein" ?ewish, came under increasin" pressure. 0opf continued to visit his parents in reslau up until 1:5:. !eein" the difficulties that his father faced 0opf arran"ed for his parents to receive immi"ration papers for !witerland. 0owever, his father fell ill and could not travel. 0opf was able to provide refu"e in !witerland for friends who had to flee /ermany under the >ais. -n particular !chur came for a while before finally "oin" to alestine in 1:5:. 0opf's own position became more difficult, however, for he was still a /erman citien. Lefschet, realisin" 0opf's difficulties, invited him to rinceton but 0opf refused. Then in 1:*5 he was told to move back to /ermany or he would lose his /erman citienship. aced with this he had little choice but to Guickly apply for !wiss citienship, which was soon "ranted. (ith the end of (orld (ar -- 0opf was able to help his /erman friends a"ain. 0e did much more than this, however, for he put much ener"y into tryin" to re9establish a mathematical community in /ermany. 0is visit to the research centre in Bberwolfach in Au"ust 1:*= was part of his efforts. !oon after the Bberwolfach visit, 0opf went to the %nited !tates where he spent si months and there he renewed many old friendships. 0e was offered professorships by many of the most presti"ious of the American universities but, after careful consideration, he decided to remain loyal to Hurich. Bver the net few years he en#oyed invitations to lecture at leadin" international conferences, and he visited many places includin" aris, russels, Rome and Bford. 0e spent the academic year 1:<<9<= with his wife in the %nited !tates. $ost of 0opf's work was in al"ebraic topolo"y where he can be thou"ht of as continuin" rouwer's work. 0e studied homotopy classes and vector fields producin" a formula about the inte"ral curvature. 0opf etended Lefschet's fied point formula in work which he undertook in 1:2D. -t is in this 1:2D paper that he first eplicitly used homolo"y "roups. 0is work on the homolo"y of manifolds, undertaken in 2<
rinceton in 1:2C92D, led to his definition of the intersection rin" by definin" a product on cycles by their intersection. This idea was later seen to be connected to cohomolo"y. 0e defined what is now known as the '0opf invariant' in 1:51. This was done in his work on maps between spheres of different dimensions which cannot be distin"uished homolo"ically so reGuired the introduction of a new invariant. -n 1:5: he eamined the homolo"y of a compact Lie "roup. This was to attack Guestions posed to him by Klie ;artan. The ideas which he introduced in this investi"ation led to him definin" what is today called a 0opf al"ebra. -n the early 1:*+s 0opf published 411689 &he paper 1undamentalgruppe und zweite Bettische 'ruppe 4which6 is legitimately regarded to be the beginning of homological algebra. !t opened the way for the definition for the homology and cohomology of a group. &his step was made independently at different places shortly after the paper became "nown ... The honours which 0opf received are almost too numerous to list. 0e was resident of the -nternational $athematical %nion from 1:<< until 1:
2=
An"re7 Ni!olaevih ol8ogorov )orn< 2 A-ril .:0# in Ta8'ov? Ta8'ov -rovine? R%ssia 5ie"< 20 Ot .:49 in Moso*? R%ssia =USSR> An"rei Ni!olaevih ol8ogorov's parents were not married and his father took no part in his upbrin"in". 0is father >ikolai Fataev, the son of a priest, was an a"riculturist who was eiled. 0e returned after the Revolution to head a )epartment in the A"ricultural $inistry but died in fi"htin" in 1:1:. Folmo"orov's mother also, tra"ically, took no part in his upbrin"in" since she died in childbirth at Folmo"orov's birth. 0is mother's sister, Eera &akovlena, brou"ht Folmo"orov up and he always had the deepest affection for her. -n fact it was chance that had Folmo"orov born in Tambov since the family had no connections with that place. Folmo"orov's mother had been on a #ourney from the ;rimea back to her home in Tunoshna near &aroslavl and it was in the home of his maternal "randfather in Tunoshna that Folmo"orov spent his youth. Folmo"orov's name came from his "randfather, &akov !tepanovich Folmo"orov, and not from his own father. &akov !tepanovich was from the nobility, a difficult status to have in Russia at this time, and there is certainly stories told that an ille"al printin" press was operated from his house. After Folmo"orov left school he worked for a while as a conductor on the railway. -n his spare time he wrote a treatise on >ewton's laws of mechanics. Then, in 1:2+, Folmo"orov entered $oscow !tate %niversity but at this sta"e he was far from committed to mathematics. 0e studied a number of sub#ects, for eample in addition to mathematics he studied metallur"y and Russian history. >or should it be thou"ht that Russian history was merely a topic to fill out his course, indeed he wrote a serious scientific thesis on the ownin" of property in >ov"orod in the 1< th and 1=th centuries. There is an anecdote told by ) / Fendall in 41+6 re"ardin" this thesis, his teacher sayin"89 Hou have supplied one proof of your thesis% and in the mathematics that you study this would perhaps suffice% but we historians prefer to have at least ten proofs. Folmo"orov may have told this story as a #oke but nevertheless #okes are only funny if there is some truth in them and undoubtedly this is the case here. -n mathematics Folmo"orov was influenced at an early sta"e by a number of outstandin" mathematicians. ! Aleksandrov was be"innin" his research 3for the second time7 at $oscow around the time Folmo"orov be"an his under"raduate career. Luin and "orov were runnin" their impressive research "roup at this time which the students called 'Luitania'. -t included $ &a !uslin and ! %rysohn, in addition to Aleksandrov. 0owever the person who made the deepest impression on Folmo"orov at this time was !tepanov who lectured to him on tri"onometric series. -t is remarkable that Folmo"orov, althou"h only an under"raduate, be"an research and produced results of international importance at this sta"e. 0e had finished writin" a paper on operations on sets by the sprin" of 1:22 which was a ma#or "eneralisation of results obtained by !uslin. y ?une of 1:22 he had constructed a summable function which diver"ed almost everywhere. This was wholly unepected by the eperts and Folmo"orov's name be"an to be known around the world. The authors of 4C6 and 4D6 note that89 Almost simultaneously 4 3olmogorov6 exhibited his interest in a number of other areas of classical analysis: in problems of differentiation and integration% in measures of sets etc. !n every one of his papers% dealing with such a variety of topics% he introduced an element of originality% a breadth of approach% and a depth of thought. Folmo"orov "raduated from $oscow !tate %niversity in 1:2< and be"an research under Luin's supervision in that year. -t is remarkable that Folmo"orov published ei"ht papers in 1:2<, all written while he was still an under"raduate. Another milestone occurred in 1:2<, namely Folmo"orov's first paper on probability appeared. This was published #ointly with Fhinchin and contains the 'three series' theorem as well as results 2C
on ineGualities of partial sums of random variables which would become the basis for martin"ale ineGualities and the stochastic calculus. -n 1:2: Folmo"orov completed his doctorate. y this time he had 1D publications and Fendall writes in 41+689 &hese included his versions of the strong law of large numbers and the law of the iterated logarithm% some generalisations of the operations of differentiation and integration% and a contribution to intuitional logic. ,is papers ... on this last topic are regarded with awe by specialists in the field. &he +ussian language edition of 3olmogorovs collected wor"s contains a retrospective commentary on these papers which 4 3olmogorov6 evidently regarded as mar"ing an important development in his philosophical outloo". An important event for Folmo"orov was his friendship with Aleksandrov which be"an in the summer of 1:2: when they spent three weeks to"ether. Bn a trip startin" from &aroslavl, they went by boat down the Eol"a then across the ;aucasus mountains to Lake !evan in Armenia. There Aleksandrov worked on the topolo"y book which he co9authored with 0opf, while Folmo"orov worked on $arkov processes with continuous states and continuous time. Folmo"orov's results from his work by the Lake were published in 1:51 and mark the be"innin" of diffusion theory. -n the summer of 1:51 Folmo"orov and Aleksandrov made another lon" trip. They visited erlin, /ttin"en, $unich, and aris where Folmo"orov spent many hours in deep discussions with aul L@vy. After this they spent a month at the seaside with r@chet Folmo"orov was appointed a professor at $oscow %niversity in 1:51. 0is mono"raph on probability theory 'rundbegriffe der ;ahrscheinlich"eitsrechnung published in 1:55 built up probability theory in a ri"orous way from fundamental aioms in a way comparable with uclid's treatment of "eometry. Bne success of this approach is that it provides a ri"orous definition of conditional epectation. As noted in 41+689 &he year 1:51 can be regarded as the beginning of the second creative stage in 3olmogorovs life. Broad general concepts advanced by him in various branched of mathematics are characteristic of this stage. After mentionin" the hi"hly si"nificant paper Analytic methods in probability theory which Folmo"orov published in 1:5D layin" the foundations of the theory of $arkov random processes, they continue to describe89 ... his ideas in set-theoretic topology% approximation theory% the theory of turbulent flow% functional analysis% the foundations of geometry% and the history and methodology of mathematics. 4 ,is contributions to6 each of these branches ... 4is6 a single whole% where a serious advance in one field leads to a substantial enrichment of the others. Aleksandrov and Folmo"orov bou"ht a house in Fomarovka, a small villa"e outside $oscow, in 1:5<. $any famous mathematicians visited Fomarovka8 0adamard, r@chet, anach, 0opf, Furatowski, and others. /nedenko and other "raduate students went on 34C6 and 4D6789 ... mathematical outings 4which6 ended in 3omarov"a% where 3olmogorov and Ale"sandrov treated the whole company to dinner. &ired and full of mathematical ideas% happy from the consciousness that we had found out something which one cannot find in boo"s% we would return in the evening to )oscow. Around this time $alcev and /elfand and others were "raduate students of Folmo"orov alon" with /nedenko who describes what it was like bein" supervised by Folmo"orov 34C6 and 4D6789 &he time of their graduate studies remains for all of 3olmogorovs students an unforgettable period in their lives% full of high scientific and cultural strivings% outbursts of scientific progress and a dedication of all ones powers to the solutions of the problems of science. !t is impossible to forget the wonderful wal"s on undays to which 4 3olmogorov6 invited all his own students 3graduates and undergraduates7% as well as the students of other supervisors. &hese outings in the environs of Bolshevo% 3lyazma% and other places about MJ-M "ilometres away% were full of discussions about the current problems of mathematics 3and its applications7% as well as discussions about the 8uestions of the progress of culture% especially painting% architecture and 2D
literature. -n 1:5D91:5: a number of leadin" mathematicians from the $oscow %niversity #oined the !teklov $athematical -nstitute of the %!!R Academy of !ciences while retainin" their positions at the %niversity. Amon" them were Aleksandrov, /elfand, Folmo"orov, etrovsky, and Fhinchin. The )epartment of robability and !tatistics was set up at the -nstitute and Folmo"orov was appointed as 0ead of )epartment. Folmo"orov later etended his work to study the motion of the planets and the turbulent flow of air from a #et en"ine. -n 1:*1 he published two papers on turbulence which are of fundamental importance. -n 1:<* he developed his work on dynamical systems in relation to planetary motion. 0e thus demonstrated the vital role of probability theory in physics. (e must mention #ust a few of the numerous other ma#or contributions which Folmo"orov made in a whole ran"e of different areas of mathematics. -n topolo"y Folmo"orov introduced the notion of cohomolo"y "roups at much the same time, and independently of, Aleander. -n 1:5* Folmo"orov investi"ated chains, cochains, homolo"y and cohomolo"y of a finite cell comple. -n further papers, published in 1:5=, Folmo"orov defined cohomolo"y "roups for an arbitrary locally compact topolo"ical space. Another contribution of the hi"hest si"nificance in this area was his definition of the cohomolo"y rin" which he announced at the -nternational Topolo"y ;onference in $oscow in 1:5<. At this conference both Folmo"orov and Aleander lectured on their independent work on cohomolo"y. -n 1:<5 and 1:<* two papers by Folmo"orov, each of four pa"es in len"th, appeared. These are on the theory of dynamical systems with applications to 0amiltonian dynamics. These papers mark the be"innin" of FA$9 theory, which is named after Folmo"orov, Arnold and $oser. Folmo"orov addressed the -nternational ;on"ress of $athematicians in Amsterdam in 1:<* on this topic with his important talk 'eneral theory of dynamical systems and classical mechanics. > 0 in"ham 41+6 notes Folmo"orov's ma#or part in settin" up the theory to answer the probability part of 0ilbert's !ith roblem Jto treat ... by means of aioms those physical sciences in which mathematics plays an important partI in the first rank are the theory of probability and mechanicsJ in his 1:55 mono"raph 'rundbegriffe der ;ahrscheinlich"eitsrechnung. in"ham also notes89 ... 6aul L4vy writes poignantly of his realisation% immediately on seeing the 9'rundbegriffe9% of the opportunity which he himself had neglected to ta"e. A rather different perspective is supplied by the elo8uent writings of )ar" 3ac on the struggles that 6olish mathematicians of the calibre teinhaus and himself had in the #KMJs% even armed with the 9'rundbegriffe9% to understand the 3apparently perspicuous7 notion of stochastic independence. -f Folmo"orov made a ma#or contribution to 0ilbert's sith problem, he completely solved 0ilbert's Thirteenth roblem in 1:
!ociety 31:<:7, the American hilosophical !ociety 31:=17, The -ndian !tatistical -nstitute 31:=27, the >etherlands Academy of !ciences 31:=57, the Royal !ociety of London 31:=*7, the >ational Academy of the %nited !tates 31:=C7, the rench Academy of !ciences 31:=D7. -n addition to the pries mentioned above, Folmo"orov was awarded the alan -nternational rie in 1:=2. $any universities awarded him an honorary de"ree includin" aris, !tockholm, and (arsaw. Folmo"orov had many interests outside mathematics, in particular he was interested i n the form and structure of the poetry of the Russian author ushkin.
5+
Ni!olai Ni!olaevih $%&in )orn< : 5e .44# in Ir!%ts!? R%ssia 5ie"< 2 Fe' .:0 in Moso*? R%ssia =USSR> Ni!olai Ni!olaevih $%&in was born in -rkutsk, and his birthplace was not, as is incorrectly stated in a number of sources, Tomsk. >ikolai's father was a businessman, half Russian and half uryat. >ikolai was the only son of his parents and the family moved to Tomsk when he was about eleven years old so that he could attend the /ymnasium there. Bne mi"ht epect that >ikolai would have shown a special talent for mathematics at the /ymnasium, but this was far from the case 341<6 and 41=6789 &his was because the system of instruction ... was based on mechanical memory: it was re8uired to learn the theorems by heart and to reproduce their proofs exactly. 1or Luzin this was torture. ,is progress in mathematics at the 'ymnasium became worse and worse% so that his father was obliged to engage a tutor ... ortunately the tutor was a talented youn" man who Guickly discovered that, despite Luin's poor performance in mathematics, he could solve hard problems but often usin" a novel method that the tutor had never seen before. !oon the tutor had shown Luin that mathematics was not a sub#ect where one had to learn lon" lists of facts, but a topic where creativity and ima"ination pla yed a ma#or role. -n 1:+1 Luin left the /ymnasium and at this time his father sold his business and the family moved to $oscow. There Luin entered the aculty of hysics and $athematics at $oscow %niversity intendin" to train to become an en"ineer. At first Luin lived in the new family home in $oscow, but Luin's father be"an to "amble on the stock echan"e with the money he had made from the sale of his business. The family soon hit hard times as Luin's father lost all their savin"s and the family had to leave their home. Luin, to"ether with a friend, moved into a room owned by the widow of a doctor. 0is friend soon became involved with the Revolution and was forced into hidin". Luin stayed on by himself in the room but he clearly "ot on well with the owners since he later, in 1:+D, married the widow's dau"hter. At $oscow %niversity Luin studied under u"aev, learnin" from him the theory of functions which was to influence "reatly the direction his research would eventually take. 0owever he was only an avera"e student who seemed to show little flair for mathematics. 0owever, althou"h Luin appeared to lack talent in mathematics, one of his teachers "orov spotted his "reat talent, invited him to his home, and be"an to set him hard problems. There was a mathematics student at the university, avel lorensky, who eperienced a crisis after "raduatin" and turned to reli"ion and the study of theolo"y. This had a ma#or effect on Luin, who was a close friend of lorensky, as we shall describe below. After "raduatin" in the autumn of 1:+< Luin seemed unsure whether to devote himself to mathematics. -n fact Luin's crisis had hit him in the sprin" of 1:+< and, on 1 $ay 1:+=, Luin wrote to lorensky from aris where "orov had sent him five months earlier in an attempt to "et him t hrou"h the crisis 3see 4:6789 Hou found me a mere child at the *niversity% "nowing nothing. ! dont "now how it happened% but ! cannot be satisfied any more with analytic functions and &aylor series ... it happened about a year ago. ... &o see the misery of people% to see the torment of life% to wend my way home from a mathematical meeting ... where% shivering in the cold% some women stand waiting in vain for dinner purchased with horror - this is an unbearable sight. !t is unbearable% having seen this% to calmly study 3in fact to enoy7 science. After that ! could not study only mathematics% and ! wanted to transfer to the medical school. ... ! have been here about five months% but have only recently begun to study.
51
Luin was not only upset by seein" the prostitutes, he also says in the letter how he had been affected by the 'terrible days' of the 1:+< Revolution. There are letters from "orov at this time pleadin" with Luin not to "ive up mathematics. After returnin" to Russia, Luin studied medicine and theolo"y as well as mathematics. 0owever in April 1:+D he wrote of the #oy he was findin" in nu mber theory 3see 4:6789 !t is a mysterious area that envelops me deeper and deeper. -n the same letter he says that he has #ust married and89 ... my wife is also very interested and shares my commitment to the search for the profound truths of life. Lar"ely Luin's crisis seems to have been solved by lorensky to whom Luin wrote in ?uly 1:+D89 &wo times ! was very close to suicide - then ! came ... loo"ing to tal" with you% and both times ! felt as if ! had leaned on a pillar and with this feeling of support ! returned home ... ! owe my interest in life to you... 0is interest in mathematics slowly returned but it was not until 1:+: that Luin seems to have finally committed himself completely to mathematics. %nder "orov's supervision he worked on his master's thesis. -n 1:1+ he was appointed as assistant lecturer in ure $athematics at $oscow %niversity. 0e worked for a year with "orov and they went on to publish #oint papers on function theory which mark the be"innin"s of the $oscow school of function theory. -n 1:1+ Luin travelled abroad visitin" /ttin"en where he was influenced by dmund Landau. 0e returned to $oscow in 1:1* and he completed his thesis &he integral and trigonometric series which he submitted in 1:1<. After his oral eamination he was awarded a doctorate, despite havin" submitted his thesis for the $aster's )e"ree. "orov was etraordinarily impressed by the work and had pressed for the award of the doctorate, but it was written in a style Guite different from the accepted Russian style of the time. !ome of the results were not ri"orously proved but were #ustified usin" phrases such as 'it seems to me' and '- am convinced'. Bther mathematicians were not so impressed at the time, for eample !teklov wrote comments in the mar"in such as 'it seems to him, but it doesn't seem to me' a nd '/ttin"en chatter'. 0owever, the work was of fundamental importance as is stated in 41<6 and 41=689 &he influence of Luzins dissertation on the future development of the theory of functions cannot be overestimated. !ts fundamental results% deep methods of investigation and fundamental statements of problems put it into the ran"s of wor"s with which it is difficult to compare any dissertation or monograph of the time. -n 1:1* Luin and his wife separated for a short time and a"ain lorensky seems to have helped them throu"h the difficult time. 0e wrote to Luin's wife 3see 4:6789 5i"olai 5i"olaevich is a very sweet and fine person7 but in personal relationships he is not at all mature% especially in intuitively perceiving the hidden currents of life. ... Hou will have to ta"e the relationship in hand and create a family tone% simplicity. !nstead% as ! perceive it ... you have established the tone of an ac8uaintanceship rather than a family. lorensky seems to have "iven "ood advice since Luin and his wife returned to a successful marria"e. -n 1:1C Luin was appointed as rofessor of ure $athematics at $oscow %niversity #ust before the Revolution. The Revolution caused Luin to rethink some of the same thou"hts as he had done at the time of his crisis and a"ain he echan"ed letters with lorensky. y this sta"e, however, his mathematical career was etremely successful and the second crisis did not materialise. Bver the net ten years Luin and "orov built up an impressive research "roup at the %niversity of $oscow which the students called 'Luitania'. The first students included ! Aleksandrov, $ &a !uslin, ) $enshov and A &a Fhinchin. The net students included ! %rysohn, A > Folmo"orov, > F ari, rnik and > / !hnirelman. -n 1:25 ! >ovikov and L E Feldysh #oined the "roup. Another of the members of the Luitania research "roup at this time was Lavrentev. -n fact Lavrentev draws 52
the followin" picture of the "roup89 ;hereas Egorov was reserved and formal% Luzin was extroverted and theatrical% inspiring real devotion among these students and young colleagues. ... &here was intense camaraderie ... inspired by Luzin. Luin's main contributions are in the area of foundations of mathematics and measure theory. 0e also made si"nificant contributions to descriptive set topolo"y. -n the theory of boundary properties of analytic functions he proved an important result in 1:1: on the invariance of sets of boundary points under conformal mappin"s. 0e also studied, to"ether with rivalov, boundary uniGueness properties of analytic functions. rom 1:1C onwards, Luin studied descriptive set theory. 0e stated the fundamental problem 341<6 and 41=6789 &he aim of set theory is a 8uestion of great importance: can we regard a line atomistically as a set of points: incidentally this 8uestion is not new% but goes bac" to the 'ree"s. $uch of Luin's work on set theory involved the study of effective sets, that is sets which can be constructed without the aiom of choice. Feldysh describes this work in 4126 and 415689 ... Luzin proceeded from the point of view of the 1rench school 3Borel% Lebesgue7% which greatly influenced him. But whereas the 1rench had analysed set-theoretical constructions carried out with the help of the Axiom of 0hoice% Luzin went considerably further and considered difficulties arising within the theory of effective sets. &he study of effective sets that he embar"ed upon was pursued intensively for more than two decades and led to the solution of many important problems of set theory ... Luin's school was at its peak durin" the years 1:22 to 1:2=, but then Luin concentrated on writin" his second mono"raph on the theory of functions and spent less time with the youn" mathematicians in the school. $any of these mathematicians turned to other topics such as topolo"y, differential eGuations, and functions of a comple variable. -n 1:2C Luin was elected as a member of the %!!R Academy of !ciences. Two years later he became a full member of first the )epartment of hilosophy, then to the )epartment of ure $athematics. 0e worked from this time until his death in the %!!R Academy of !ciences. rom 1:5< he headed the )epartment of the Theory of unctions of Real Eariables at the !teklov -nstitute. -n 1:51 Luin himself turned to a new area when he be"an to study differential eGuations and their application to "eometry and to control theory. 0is work in this area led him to study the bendin" of surfaces which is described in 41<6 and 41=689 &he bending of a surface on a principal base is a continuous bending of a surface under which the conugacy of the net of certain curves on the surface is preserved. ... 1ini"ov had derived differential e8uations that determine all principal on a given surface% and Byushgens had obtained differential e8uations that determine surfaces which have a given linear element and admit a bending on a principal base. ,owever% the 8uestion of solubility of these e8uations% in general% remained unclear. ... no example was found in which the e8uations ... were insoluble ... up to 1:5D% when Luzin% by means of a subtle analysis of these e8uations% established that the existence of a principal base is rather rare. -t has been drawn to our attention by 41:6, that in 1:5=, Luin was the victim of a violent political campai"n or"anied by the !oviet authorities throu"h the newspaper 6ravda. 0e was accused of anti9!oviet propa"anda and sabota"e by publishin" all his important results abroad and only minor papers in !oviet #ournals. T he aim was obviously to "et rid of Luin as a representative of the old pre9!oviet mathematical school of $oscow8 his master, "orov, had been himself the victim of such a campai"n in 1:5+ 3based on his reli"ious sympathies7 and died shortly after in 1:51 in despair and misery. A contemporary record of the JLuin affairJ has been miraculously preserved and recently edited in $oscow by )emidov and Levchin 456, 4256. -t shows that Luin had had a narrow escape from a tra"ic fate as the !oviet authorities may have feared the international conseGuences of a too stron" attack on a scientist so famous abroad. The main visible conseGuence of the Luin affair was that, from this precise moment, !oviet mathematicians be"an to publish 55
almost eclusively in !oviet #ournals and in Russian. Luin always had an interest in the history of mathematics and late in his career he wrote important articles on >ewton and on uler. As a teacher his remarkable talents are described by Funetsov 341<6 or 41=6789 ,is presentation was always very elegant and at first sight apparently unnecessarily simple - the result of his great pedagogic talent. &he solution of the large problems that he undertoo" is distinguished by their subtlety% elegance% and simplicity of presentation. Feldysh and >ovikov wrote in 41*689 &han"s to his exceptional intuition and his ability to see deeply into the heart of a 8uestion% Luzin fre8uently predicted mathematical facts whose proof turned out to be possible only after many years and re8uired the creation of completely new mathematical methods. ,e was one of the outstanding mathematicians and thin"ers of our time ...
5*
(ohn von Ne%8ann )orn< 24 5e .:0# in )%"a-est? H%ngar7 5ie"< 4 Fe' .:9 in ;ashington 5@C@? USA (ohn von Ne%8ann was born ?nos von >eumann. 0e was called ?ancsi as a child, a diminutive form of ?nos, then later he was called ?ohnny in the %nited !tates. 0is father, $a >eumann, was a top banker and he was brou"ht up in a etended family, livin" in udapest where as a child he learnt lan"ua"es from the /erman and rench "overnesses that were employed. Althou"h the family were ?ewish, $a >eumann did not observe the strict practices of that reli"ion and the household seemed to mi ?ewish and ;hristian traditions. -t is also worth eplainin" how $a >eumann's son acGuired the JvonJ to become ?nos von >eumann. -n 1:15 $a >eumann purchased a title but did not chan"e his name. 0is son, however, used the /erman form von >eumann where the JvonJ indicated the title. As a child von >eumann showed he had an incredible memory. oundstone, in 4D6, writes89 At the age of six% he was able to exchange o"es with his father in classical 'ree". &he 5eumann family sometimes entertained guests with demonstrations of Dohnnys ability to memorise phone boo"s. A guest would select a page and column of the phone boo" at random. Houng Dohnny read the column over a few times% then handed the boo" bac" to the guest. ,e could answer any 8uestion put to him 3who has number such and such<7 or recite names% addresses% and numbers in order. -n 1:11 von >eumann entered the Lutheran /ymnasium. The school had a stron" academic tradition which seemed to count for more than the reli"ious affiliation both in the >eumann's eyes and in those of the school. 0is mathematics teacher Guickly reco"nised von >eumann's "enius and special tuition was put on for him. The school had another outstandin" mathematician one year ahead of von >eumann, namely u"ene (i"ner. (orld (ar - had relatively little effect on von >eumann's education but, after the war ended, @la Fun controlled 0un"ary for five months in 1:1: with a ;ommunist "overnment. The >eumann family fled to Austria as the affluent came under attack. 0owever, after a month, they returned to face the problems of udapest. (hen Fun's "overnment failed, the fact that it had been lar"ely composed of ?ews meant that ?ewish people were blamed. !uch situations are devoid of lo"ic and the fact that the >eumann's were opposed to Fun's "overnment did not save them from persecution. -n 1:21 von >eumann completed his education at the Lutheran /ymnasium. 0is first mathematics paper, written #ointly with ekete the assistant at the %niversity of udapest who had been tutorin" him, was published in 1:22. 0owever $a >eumann did not want his son to take up a sub#ect that would not brin" him wealth. $a >eumann asked Theodore von Frmn to speak to his son and persuade him to follow a career in business. erhaps von Frmn was the wron" person to ask to undertake such a task but in the end all a"reed on the compromise sub#ect of chemistry for von >eumann's university studies. 0un"ary was not an easy country for those of ?ewish descent for many reasons and there was a strict limit on the number of ?ewish students who could enter the %niversity of udapest. Bf course, even with a strict Guota, von >eumann's record easily won him a place to study mathematics in 1:21 but he did not attend lectures. -nstead he also entered the %niversity of erlin in 1:21 to study chemistry. Eon >eumann studied chemistry at the %niversity of erlin until 1:25 when he went to Hurich. 0e achieved outstandin" results in the mathematics eaminations at the %niversity of udapest despite not attendin" any courses. Eon >eumann received his diploma in chemical en"ineerin" from the Technische 0ochschule in HMrich in 1:2=. (hile in Hurich he continued his interest in mathematics, despite studyin" chemistry, and interacted with (eyl and Slya who were both at Hurich. 0e even took over one of (eyl's courses when he was absent from Hurich for a time. Slya said 41D689 5<
Dohnny was the only student ! was ever afraid of. !f in the course of a lecture ! stated an unsolved problem% the chances were hed come to me as soon as the lecture was over% with the complete solution in a few scribbles on a slip of paper. Eon >eumann received his doctorate in mathematics from the %niversity of udapest, also in 1:2=, with a thesis on set theory. 0e published a definition of ordinal numbers when he was 2+, the definition is the one used today. Eon >eumann lectured at erlin from 1:2= to 1:2: and at 0ambur" from 1:2: to 1:5+. 0owever he also held a Rockefeller ellowship to enable him to undertake postdoctoral studies at the %niversity of /ttin"en. 0e studied under 0ilbert at /ttin"en durin" 1:2=92C. y this time von >eumann had achieved celebrity status 4D689 By his mid-twenties% von 5eumanns fame had spread worldwide in the mathematical community. At academic conferences% he would find himself pointed out as a young genius. Eeblen invited von >eumann to rinceton to lecture on Guantum theory in 1:2:. Replyin" to Eeblen that he would come after attendin" to some personal matters, von >eumann went to udapest where he married his fianc@e $arietta Fovesi before settin" out for the %nited !tates. -n 1:5+ von >eumann became a visitin" lecturer at rinceton %niversity, bein" appointed professor there in 1:51. etween 1:5+ and 1:55 von >eumann tau"ht at rinceton but this was not one of his stron" points 4D689 ,is fluid line of thought was difficult for those less gifted to follow. ,e was notorious for dashing out e8uations on a small portion of the available blac"board and erasing expressions before students could copy them. -n contrast, however, he had an ability to eplain complicated ideas in physics 45689 1or a man to whom complicated mathematics presented no difficulty% he could explain his conclusions to the uninitiated with amazing lucidity. After a tal" with him one always came away with a feeling that the problem was really simple and transparent. 0e became one of the ori"inal si mathematics professors 3? ( Aleander, A instein, $ $orse, B Eeblen, ? von >eumann and 0 (eyl7 in 1:55 at the newly founded -nstitute for Advanced !tudy in rinceton, a position he kept for the remainder of his life. )urin" the first years that he was in the %nited !tates, von >eumann continued to return to urope durin" the summers. %ntil 1:55 he still held academic posts in /ermany but resi"ned these when the >ais came to power. %nlike many others, von >eumann was not a political refu"ee but rather he went to the %nited !tates mainly because he thou"ht that the prospect of academic positions there was better than in /ermany. -n 1:55 von >eumann became co9editor of the Annals of )athematics and, two years later, he became co9 editor of 0ompositio )athematica. 0e held both these editorships until his death. Eon >eumann and $arietta had a dau"hter $arina in 1:5= but their marria"e ended in divorce in 1:5C. The followin" year he married Flra )n, also from udapest, whom he met on one of his uropean visits. After marryin", they sailed to the %nited !tates and made their home in rinceton. There von >eumann lived a rather unusual lifestyle for a top mathematician. 0e had always en#oyed parties 4D689 6arties and nightlife held a special appeal for von 5eumann. ;hile teaching in 'ermany% von 5eumann had been a denizen of the 0abaret-era Berlin nightlife circuit. >ow married to Flra the parties continued 41D689 &he parties at the von 5eumanns house were fre8uent% and famous% and long. %lam summarises von >eumann's work in 45<6. 0e writes89 !n his youthful wor"% he was concerned not only with mathematical logic and the axiomatics of set theory% but% simultaneously% with the substance of set theory itself% obtaining interesting results in measure theory and the theory of real variables. !t was in this period also that he 5=
began his classical wor" on 8uantum theory% the mathematical foundation of the theory of measurement in 8uantum theory and the new statistical mechanics. 0is tet )athematische 'rundlagen der Nuantenmechani" 31:527 built a solid framework for the new Guantum mechanics. Ean 0ove writes in 45=689 Nuantum mechanics was very fortunate indeed to attract% in the very first years after its discovery in 1:2<% the interest of a mathematical genius of von 5eumanns stature. As a result% the mathematical framewor" of the theory was developed and the formal aspects of its entirely novel rules of interpretation were analysed by one single man in two years 31:2C91:2:7. !elf9ad#oint al"ebras of bounded linear operators on a 0ilbert space, closed in the weak operator topolo"y, were introduced in 1:2: by von >eumann in a paper in )athematische Annalen . Fadison eplains in 422689 ,is interest in ergodic theory% group representations and 8uantum mechanics contributed significantly to von 5eumanns realisation that a theory of operator algebras was the next important stage in the development of this area of mathematics. !uch operator al"ebras were called Jrin"s of operatorsJ by von >eumann and later they were called (9 al"ebras by some other mathematicians. ? )imier, in 1:eumann al"ebrasJ in his mono"raph Algebras of operators in ,ilbert space von 5eumann algebrasF. -n the second half of the 1:5+'s and the early 1:*+s von >eumann, workin" with his collaborator ? $urray, laid the foundations for the study of von >eumann al"ebras in a fundamental series of papers. 0owever von >eumann is know for the wide variety of different scientific studies. %lam eplains 45<6 how he was led towards "ame theory89 on 5eumanns awareness of results obtained by other mathematicians and the inherent possibilities which they offer is astonishing. Early in his wor"% a paper by Borel on the minimax property led him to develop ... ideas which culminated later in one of his most original creations% the theory of games. -n "ame theory von >eumann proved the minima theorem. 0e "radually epanded his work in "ame theory, and with co9author Bskar $or"enstern, he wrote the classic tet &heory of 'ames and Economic Behaviour 31:**7. %lam continues in 45<689 An idea of 3oopman on the possibilities of treating problems of classical mechanics by means of operators on a function space stimulated him to give the first mathematically rigorous proof of an ergodic theorem. ,aars construction of measure in groups provided the inspiration for his wonderful partial solution of ,ilberts fifth problem% in which he proved the possibility of introducing analytical parameters in compact groups. -n 1:5D the American $athematical !ociety awarded the Ucher rie to ?ohn von >eumann for his memoir Almost periodic functions and groups. This was published in two parts in the &ransactions of the American )athematical ociety% the first part in 1:5* and the second part in the followin" year. Around this time von >eumann turned to applied mathematics 45<689 !n the middle 5+s% Dohnny was fascinated by the problem of hydrodynamical turbulence. !t was then that he became aware of the mysteries underlying the subect of non-linear partial differential e8uations. ,is wor"% from the beginnings of the econd ;orld ;ar% concerns a study of the e8uations of hydrodynamics and the theory of shoc"s. &he phenomena described by these non-linear e8uations are baffling analytically and defy even 8ualitative insight by present methods. 5umerical wor" seemed to him the most promising way to obtain a feeling for the behaviour of such systems. &his impelled him to study new possibilities of computation on electronic machines ... Eon >eumann was one of the pioneers of computer science makin" si"nificant contributions to the development of lo"ical desi"n. !hannon writes in 42:689 5C
on 5eumann spent a considerable part of the last few years of his life wor"ing in 4automata theory6. !t represented for him a synthesis of his early interest in logic and proof theory and his later wor"% during ;orld ;ar !! and after% on large scale electronic computers. !nvolving a mixture of pure and applied mathematics as well as other sciences% automata theory was an ideal field for von 5eumanns wide-ranging intellect. ,e brought to it many new insights and opened up at least two new directions of research. 0e advanced the theory of cellular automata, advocated the adoption of the bit as a measurement of computer memory, and solved problems in obtainin" reliable answers from unreliable computer components. )urin" and after (orld (ar --, von >eumann served as a consultant to the armed forces. 0is valuable contributions included a proposal of the implosion method for brin"in" nuclear fuel to eplosion and his participation in the development of the hydro"en bomb. rom 1:*+ he was a member of the !cientific Advisory ;ommittee at the allistic Research Laboratories at the Aberdeen rovin" /round in $aryland. 0e was a member of the >avy ureau of Brdnance from 1:*1 to 1:<<, and a consultant to the Los Alamos !cientific Laboratory from 1:*5 to 1:<<. rom 1:<+ to 1:<< he was a member of the Armed orces !pecial (eapons ro#ect in (ashin"ton, ).;. -n 1:<< resident isenhower appointed him to the Atomic ner"y ;ommission, and in 1:<= he received its nrico ermi Award, knowin" that he was incurably ill with cancer. u"ene (i"ner wrote of von >eumann's death 41D689 ;hen von 5eumann realised he was incurably ill% his logic forced him to realise that he would cease to exist% and hence cease to have thoughts ... !t was heartbrea"ing to watch the frustration of his mind% when all hope was gone% in its struggle with the fate which appeared to him unavoidable but unacceptable. -n 4<6 von >eumann's death is described in these terms89 ... his mind% the amulet on which he had always been able to rely% was becoming less dependable. &hen came complete psychological brea"down7 panic% screams of uncontrollable terror every night. ,is friend Edward &eller said% 9! thin" that von 5eumann suffered more when his mind would no longer function% than ! have ever seen any human being suffer.9 on 5eumanns sense of invulnerability% or simply the desire to live% was struggling with unalterable facts. ,e seemed to have a great fear of death until the last... 5o achievements and no amount of influence could save him now% as they always had in the past. Dohnny von 5eumann% who "new how to live so fully% did not "now how to die. -t would be almost impossible to "ive even an idea of the ran"e of honours which were "iven to von >eumann. 0e was ;olloGuium Lecturer of the American $athematical !ociety in 1:5C and received the its Ucher rie as mentioned above. 0e held the /ibbs Lectureship of the American $athematical !ociety in 1:*C and was resident of the !ociety in 1:<19<5. 0e was elected to many academies includin" the Academia >acional de ;iencias actas 3Lima, eru7, Academia >aionale dei Lincei 3Rome, -taly7, American Academy of Arts and !ciences 3%!A7, American hilosophical !ociety 3%!A7, -nstituto Lombardo di !ciene e Lettere 3$ilan, -taly7, >ational Academy of !ciences 3%!A7 and Royal >etherlands Academy of !ciences and Letters 3Amsterdam, The >etherlands7. Eon >eumann received two residential Awards, the $edal for $erit in 1:*C and the $edal for reedom in 1:<=. Also in 1:<= he received the Albert instein ;ommemorative Award and the nrico ermi Award mentioned above. eierls writes 45689 ,e was the antithesis of the 9long-haired9 mathematics don. Always well groomed% he had as lively views on international politics and practical affairs as on mathematical problems.
5D
E887 A8alie Noether )orn< 2# Marh .442 in Erlangen? )avaria? Ger8an7 5ie"< .6 A-ril .:# in )r7n Ma*r? Penns7lvania? USA E887 Noether's father $a >oether was a distin"uished mathematician and a professor at rlan"en. 0er mother was -da Faufmann, from a wealthy ;olo"ne family. oth mmy's parents were of ?ewish ori"in and mmy was the eldest of their four children, the three youn"er children bein" boys. mmy >oether attended the 0here Tchter !chule in rlan"en from 1DD: until 1D:C. !he studied /erman, n"lish, rench, arithmetic and was "iven piano lessons. !he loved dancin" and looked forward to parties with children of her father's university collea"ues. At this sta"e her aim was to become a lan"ua"e teacher and after further study of n"lish and rench she took the eaminations of the !tate of avaria and, in 1:++, became a certificated teacher of n"lish and rench in avarian "irls schools. 0owever >oether never became a lan"ua"e teacher. -nstead she decided to take the difficult route for a woman of that time and study mathematics at university. (omen were allowed to study at /erman universities unofficially and each professor had to "ive permission for his course. >oether obtained permission to sit in on courses at the %niversity of rlan"en durin" 1:++ to 1:+2. Then, havin" taken and passed the matriculation eamination in >Mrnber" in 1:+5, she went to the %niversity of /ttin"en. )urin" 1:+59+* she attended lectures by lumenthal, 0ilbert, Flein and $inkowski. -n 1:+* >oether was permitted to matriculate at rlan"en and in 1:+C was "ranted a doctorate after workin" under aul /ordan. 0ilbert's basis theorem of 1DDD had "iven an eistence result for finiteness of invariants in n variables. /ordan, however, took a constructive approach and looked at constructive methods to arrive at the same results. >oether's doctoral thesis followed this constructive approach of /ordan and listed systems of 551 covariant forms. 0avin" completed her doctorate the normal pro"ression to an academic post would have been the habilitation. 0owever this route was not open to women so >oether remained at rlan"en, helpin" her father who, particularly because of his own disabilities, was "rateful for his dau"hter's help. >oether also worked on her own research, in particular she was influenced by ischer who had succeeded /ordan in 1:11. This influence took >oether towards 0ilbert's abstract approach to the sub#ect and away from the constructive approach of /ordan. >oether's reputation "rew Guickly as her publications appeared. -n 1:+D she was elected to the ;ircolo $atematico di alermo, then in 1:+: she was invited to become a member of the )eutsche $athematiker9 Eereini"un" and in the same year she was invited to address the annual meetin" of the !ociety in !albur". -n 1:15 she lectured in Eienna. -n 1:1< 0ilbert and Flein invited >oether to return to /ttin"en. They persuaded her to remain at /ttin"en while they fou"ht a battle to have her officially on the aculty. -n a lon" battle with the university authorities to allow >oether to obtain her habilitation there were many setbacks and it was not until 1:1: that permission was "ranted. )urin" this time 0ilbert had allowed >oether to lecture by advertisin" her courses under his own name. or eample a course "iven in the winter semester of 1:1=91C appears in the catalo"ue as89 )athematical 6hysics eminar: 6rofessor ,ilbert% with the assistance of /r E 5oether% )ondays from *9=% no tuition. mmy >oether's first piece of work when she arrived in /ttin"en in 1:1< is a result in theoretical physics sometimes referred to as >oether's Theorem, which proves a relationship between symmetries in physics and conservation principles. This basic result in the "eneral theory of relativity was praised by instein in a letter to 0ilbert when he referred to >oether's penetrating mathematical thin"ing. 5:
-t was her work in the theory of invariants which led to formulations for several concepts of instein's "eneral theory of relativity. At /ttin"en, after 1:1:, >oether moved away from invariant theory to work on ideal theory, producin" an abstract theory which helped develop rin" theory into a ma#or mathematical topic. !dealtheorie in +ingbereichen 31:217 was of fundamental importance in the development of modern al"ebra. -n this paper she "ave the decomposition of ideals into intersections of primary ideals in any commutative rin" with ascendin" chain condition. Lasker 3the world chess champion7 had already proved this result for polynomial rin"s. -n 1:2* L van der (aerden came to /ttin"en and spent a year studyin" with >oether. After returnin" to Amsterdam van der (aerden wrote his book )oderne Algebra in two volumes. The ma#or part of the second volume consists of >oether's work. rom 1:2C on >oether collaborated with 0elmut 0asse and Richard rauer in work on non9 commutative al"ebras. -n addition to teachin" and research, >oether helped edit )athematische Annalen. $uch of her work appears in papers written by collea"ues and students, rather than under her own name. urther reco"nition of her outstandin" mathematical contributions came with invitations to address the -nternational $athematical ;on"ress at olo"na in 1:2D and a"ain at Hurich in 1:52. -n 1:52 she also received, #ointly with Artin, the Alfred Ackermann9Teubner $emorial rie for the Advancement of $athematical Fnowled"e. -n 1:55 her mathematical achievements counted for nothin" when the >ais caused her dismissal from the %niversity of /ttin"en because she was ?ewish. !he accepted a visitin" professorship at ryn $awr ;olle"e in the %!A and also lectured at the -nstitute for Advanced !tudy, rinceton in the %!A. (eyl in his $emorial Address 42D6 said89 ,er significance for algebra cannot be read entirely from her own papers% she had great stimulating power and many of her suggestions too" shape only in the wor"s of her pupils and co-wor"ers. -n 42=6 van der (aerden writes89 1or Emmy 5oether% relationships among numbers% functions% and operations became transparent% amenable to generalisation% and productive only after they have been dissociated from any particular obects and have been reduced to general conceptual relationships.
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Roger Penrose )orn< 4 A%g .:#. in Colhester? Esse3? Englan" Roger Penrose's parents, Lionel !harples enrose and $ar"aret Leathes, were both medically trained. $ar"aret was a doctor while Lionel was a medical "eneticist who was elected a ellow of the Royal !ociety. 0e was involved with a pro#ect called the ;olchester survey which aimed to discover whether inherited factors or environmental factors were the most si"nificant in determinin" if someone would be likely to suffer from mental heath problems. 0e was in ;olchester carryin" out this work at the time Ro"er was born. Ro"er's brother, Bliver enrose, had been born two years earlier. Bliver went on to become professor of mathematics first at the Bpen %niversity, then at 0eriot9(att %niversity in dinbur"h, !cotland. Ro"er also had a youn"er brother ?onathan who went on to become a lecturer in psycholo"y. ?onathan was ritish ;hess ;hampion ten times between 1:
interested in physics. 0e described how three courses which he attended durin" his first year at ;ambrid"e influenced him 3426 or 456789 ! remember going to three courses% none of which had anything to do with the research ! was supposed to be doing. 2ne was a course by ,ermann Bondi on general relativity which was fascinating ... Another was a course by 6aul /irac on 8uantum mechanics which was beautiful in a completely different way ... And the third course ... was a course on mathematical logic by teen. ! learnt about &uring machines and 'Cdels theorem ... The first ma#or influence promptin" his interest in physics had been )ennis !ciama, a physicist friend of his brother. enrose said 3426 or 456789 4ciama6 was very influential on me. ,e taught me a great deal of physics% and the excitement of doing physics came through7 he was that "ind of person% who conveyed the excitement of what was currently going on in physics ... (hile at ;ambrid"e workin" towards his doctorate he be"an to publish articles on semi"roups, and on rin"s of matrices. -n 1:<< he published A generalized inverse for matrices in the 6roceedings of the 0ambridge 6hilosophical ociety. -n this paper enrose defined a "eneralied inverse ? of a comple rectan"ular 3or possibly sGuare and sin"ular7 matri A to be the uniGue solution to the eGuations A?A V A, ?A? V ? , 3 A? 7& V A? , 3 ?A7& V ?A. 0e used this "eneralied inverse for problems such as solvin" systems of matri eGuations, and findin" a new type of spectral decomposition. 0is second publication of 1:<< was A note on inverse semigroups published in the same #ournal and co9authored with )ou"las $unn. An inverse semi"roup is a "eneralisation of a "roup and continues to be the sub#ect of many research papers. This early paper "ave several alternative definitions. -n the followin" year enrose published 2n best approximation solutions of linear matrix e8uations which used the "eneralied inverse of a matri to find the best approimate solution ? to A? V B where A is rectan"ular and non9sGuare or sGuare and sin"ular. enrose spent the academic year 1:<=9ATB Research ellowship which enabled him to spend the years 1:<:9=1 in the %nited !tates, first at rinceton and then at !yracuse %niversity. ack in n"land, enrose spent the followin" two years 1:=19=5 as a Research associate at Fin"'s ;olle"e, London before returnin" to the %nited !tates to spend the year 1:=59=* as a Eisitin" Associate rofessor at the %niversity of Teas at Austin. -n 1:=* enrose was appointed as a Reader at irkbeck ;olle"e, London and two years later he was promoted to rofessor of Applied $athematics there. -n 1:C5 he was appointed Rouse all rofessor of $athematics at the %niversity of Bford and he continued to hold this until he became meritus Rouse all rofessor of $athematics in 1::D. -n that year he was appointed /resham rofessor of /eometry at /resham ;olle"e, London. e"innin" in 1:<:, enrose published a series of important papers on cosmolo"y. The first was &he apparent shape of a relativistically moving sphere while in 1:=+ he published A spinor approach to general relativity. This latter paper was described as follows89 An elegant and detailed exposition ... of the mathematical apparatus of gravitation theory% with emphasis on the geometrical theory of the +iemann tensor. As well as important papers on cosmolo"y, enrose continues to publish papers on pure mathematics. To"ether with 0enry (hitehead and ;hristopher Heeman he published !mbedding of manifolds in euclidean space in 1:=1. Amon" other results, the authors prove in this paper that, if + W 2 m X n, then every closed 3m9179connected n9manifold can be imbedded in R 2n9mP1. This time with ra >ewman, enrose published An approach to gravitational radiation by a method of spin coefficients in the followin" year in which they show that89 ... the two-component spinor formalism leads to the consideration of a tetrad in space-time consisting of two real null-vectors and two complex conugate ones.
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-n 1:=<, usin" topolo"ical methods, enrose proved an important theorem which, under conditions which he called the eistence of a trapped surface, proved that a sin"ularity must occur in a "ravitational collapse. asically under these conditions space9time cannot be continued and classical "eneral relativity breaks down. enrose looked for a unified theory combinin" relativity and Guantum theory since Guantum effects become dominant at the sin"ularity. Bne of enrose's ma#or breakthrou"hs was his introduction of twistor theory in an attempt to unite relativity and Guantum theory. This is a remarkable mathematical theory combinin" powerful al"ebraic and "eometric methods. To"ether with (olf"an" Rindler, enrose published this first volume of pinors and space-time in 1:D*. This volume covered two9spinor calculus and relativistic fields while the second volume coverin" spinor and twistor methods in space9time "eometry appeared two years later. -t is for a number of outstandin" popular books that enrose is perhaps best known. 0e published &he Emperors 5ew )ind : 0oncerning computers% minds% and the laws of physics in 1:D:. -n the followin" year the book was awarded the Rhone9oulenc !cience ook rie. !klar, reviewin" the book, writes that its aim is89 ... to expound and critically attac" one recent view of the nature of mind ... ta"en as reducing mental activity to the carrying out of an algorithmic process% and to propose that a more ade8uate theory of mind will have to be founded on an as yet not existing physical theory ade8uate to the "nown nature of the material world. !n the process of the argument elegant expositions% at a level suitable for the unlearned but reasonably sophisticated reader% are given of a wide variety of topics ranging from the nature of algorithms and abstract computability% through results on undecidability and incompleteness% the basic structures of classical physics% the basic structures and philosophical puzzles in 8uantum mechanics% the basic features of entropic asymmetry and its relation to cosmological structure% the search for an ade8uate 8uantum theory of gravity% to some of the results of neuro-anatomy and research into the functioning of the brain. -n 1::* enrose published hadows of the mind : A search for the missing science of consciousness which continues to develop the topic of &he emperors new mind. -n 1::= enrose and 0awkin" published &he nature of space and time. This book is a record of a debate between the two at the -saac >ewton -nstitute of $athematical !ciences at the %niversity of ;ambrid"e in 1::*. ach of the two "ave three lectures "iven alternately so that each could respond to the other's ar"uments, and then, in a final session, there is a debate between the two. (e Guote from enrose's contribution since he states clearly his own position, and that of 0awkin"89 At the beginning of this debate tephen said that he thin"s that he is a positivist% whereas ! am a 6latonist. ! am happy with him being a positivist% but ! thin" that the crucial point here is% rather% that ! am a realist. Also% if one compares this debate with the famous debate of Bohr and Einstein% some seventy years ago% ! should thin" that tephen plays the role of Bohr% whereas ! play Einsteins roleG 1or Einstein argued that there should exist something li"e a real world% not necessarily represented by a wave function% whereas Bohr stressed that the wave function doesnt describe a 9real9 microworld but only 9"nowledge9 that is useful for ma"ing predictions. There is one further aspect of enrose's work which we must mention. This is his work on non9periodic tilin"s, an interest which he took up while a "raduate student at ;ambrid"e. 0is first attempts led to success but with a lar"e number of tiles. urther work over many years led to enrose discoverin" that he could find non9periodic tilin"s with only si tiles, then finally he achieved the seemin"ly impossible with findin" non9 periodic tilin"s with only two tiles. y non9periodic we mean that the tilin"s are not invariant under any translation. 0ere are some properties of the tilin"8 in any finite tiled re"ion, only one tilin" is possibleI in an infinite tilin" of the plane, any tilin" of a re"ion that occurs is repeated infinitely often elsewhere in the plane and must reoccur within twice the diameter of the re"ion from where you first found it. -n fact the tilin" of any finite re"ion will eventually appear in every enrose tilin". -n addition to enrose's main appointments which we have mentioned above, he also held a number of visitin" and part9time posts. 0e held visitin" positions at &eshiva, rinceton and ;ornell durin" 1:==9=C and 1:=:. rom 1:D5 until 1:DC he was Lovett rofessor at Rice %niversity in 0ouston. 0e then became )istin"uished rofessor of hysics and $athematics at !yracuse %niversity in >ew &ork until 1::5 when he *5
became rancis and 0elen ent )istin"uished rofessor of hysics and $athematics at ennsylvania !tate %niversity. enrose has received many honours for his contributions. 0e was elected a ellow of the Royal !ociety of London 31:C27 and a orei"n Associate of the %nited !tates >ational Academy of !ciences 31::D7. (e mentioned the !cience ook rie 31::+7 which he received for The mperor's >ew $ind but this is only one of many pries. Bthers include the Adams rie from ;ambrid"e %niversityI the (olf oundation rie for hysics 3#ointly with !tephen 0awkin" for their understandin" of the universe78 the )annie 0einemann rie from the American hysical !ociety and the American -nstitute of hysicsI the Royal !ociety Royal $edalI the )irac $edal and $edal of the ritish -nstitute of hysicsI the ddin"ton $edal of the Royal Astronomical !ocietyI the >aylor rie of the London $athematical !ocietyI and the Albert instein rie and $edal of the Albert instein !ociety. -n 1::* he was kni"hted for services to science. -n 2+++ he received the Brder of $erit. 0e was awarded the )e $or"an $edal by the London $athematical !ociety in 2++*. art of the citation reads89 ,is deep wor" on 'eneral +elativity has been a maor factor in our understanding of blac" holes. ,is development of &wistor &heory has produced a beautiful and productive approach to the classical e8uations of mathematical physics. ,is tilings of the plane underlie the newly discovered 8uasi-crystals. The Royal !ociety awarded enrose their ;opley $edal in 2++<. The announcement reads89 ir +oger 6enrose% 2)% 1+ has been awarded the +oyal ocietys 0opley medal the worlds oldest prize for scientific achievement for his exceptional contributions to geometry and mathematical physics. ir +oger% Emeritus +ouse Ball 6rofessor of )athematics at the *niversity of 2xford% has made outstanding contributions to general relativity theory and cosmology% most notably for his wor" on blac" holes and the Big Bang. $artin Rees, resident of the Royal !ociety, eplained enrose's eceptional contributions which led to the award89 +oger has been producing original and important scientific ideas for half a century. ,is wor" is characterised by exceptional geometrical and physical insight. ,e applied new mathematical techni8ues to Einsteins theory% and led the renaissance in gravitation theory in the 1:=+s. ,is novel ideas on space and time and his concept of twistors are increasingly influential. Even his recreations have had intellectual impact: for instance the impossible figures popularised in Eschers artwor"% and the never-repeating patterns of 6enrose tiling. ,e has influenced and stimulated a wide public through his lectures% and his best-selling and wide-ranging boo"s. Bn receivin" the award, enrose said89 &he award of the +oyal ocietys 0opley )edal came as a complete surprise to me. !t is an extraordinary honour% this being the +oyal ocietys oldest and most distinguished award% first given ust 2++ years before ! was born. ! feel most humbled for my name to be added to that enormously distinguished list of previous recipients. !everal universities have awarded enrose an honorary de"ree includin" >ew runswick %niversity 31::27, the %niversity of !urrey 31::57, the %niversity of ath 31::*7, the %niversity of London 31::<7, the %niversity of /las"ow 31::=7, sse %niversity 31::=7, the %niversity of !t Andrews 31::C7, !antiniketon %niversity 31::D7, (arsaw %niversity 32++<7, Fatholieke %niversiteit Leuven 32++<7 and the %niversity of &ork 32++=7.
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$ev Se8enovih Pontr7agin )orn< # Se-t .:04 in Moso*? R%ssia 5ie"< # Ma7 .:44 in Moso*? R%ssia =USSR> $ev Se8enovih Pontr7agin's father, !emen Akimovich ontrya"in was a civil servant. ontrya"in's mother, Tat'yana Andreevna ontrya"ina, was 2: years old when he was born and she was a remarkable woman who played a crucial role in his path to becomin" a mathematician. erhaps the description of 'civil servant', althou"h accurate, "ives the wron" impression that the family were reasonably well off. -n fact !emen Akimovich's #ob left the family without enou"h money to allow them to "ive their son a "ood education and Tat'yana Andreevna worked usin" her sewin" skills to help out the family finances. ontrya"in attended the town school where the standard of education was well below that of the better schools but the family's poor circumstances put these well out of reach financially. At the a"e of 1* years ontrya"in suffered an accident and an eplosion left him blind. This mi"ht have meant an end to his education and career but his mother had other ideas and devoted herself to help him succeed despite the almost impossible difficulties of bein" blind. The help that she "ave ontrya"in is described in 416 and 42689 1rom this moment &atyana Andreevna assumed complete responsibility for ministering to the needs of her son in all aspects of his life. !n spite of the great difficulties with which she had to contend% she was so successful in her self-appointed tas" that she truly deserves the gratitude ... of science throughout the world. 1or many years she wor"ed% in effect% as 6ontryagins secretary% reading scientific wor"s aloud to him% writing in the formulas in his manuscripts% correcting his wor" and so on. !n order to do this she had% in particular% to learn to read foreign languages. &atyana Andreevna helped 6ontryagin in all other respects% seeing to his needs and ta"ing very great care of him. -t is not unreasonable to pause for a moment and think about how Tat'yana Andreevna, with no mathematical trainin" or knowled"e, made by her determination and etreme efforts a ma#or contribution to mathematics by allowin" ontrya"in to become a mathematician a"ainst all the odds. There must be many other non9 mathematicians, perhaps many of whom are unrecorded by history, who have also by their unselfish acts allowed mathematics to flourish. As we try to show in this archive, the development of mathematics depends on a wide number of influences other than the talents of the mathematicians themselves8 political influences, economic influences, social influences, and the acts of non9mathematicians like Tat'yana Andreevna. ut how does one read a mathematics paper without knowin" any mathematicsY Bf course it is full of mysterious symbols and Tat'yana Andreevna, not knowin" their mathematical meanin" or name, could only describe them by their appearance. or eample an intersection si"n became a 'tails down' while a union symbol became a 'tails up'. -f she read ' A tails ri"ht B' then ontrya"in knew that A was a subset of BO ontrya"in entered the %niversity of $oscow in 1:2< and it Guickly became apparent to his lecturers that he was an eceptional student. Bf course that a blind student who could not make notes yet was able to remember the most complicated manipulations with symbols was in itself truly remarkable. ven more remarkable was the fact that ontrya"in could 'see' 3if you will ecuse the bad pun7 far more clearly than any of his fellow students the depth of meanin" in the topics presented to him. Bf the advanced courses he took, ontrya"in felt less happy with Fhinchin's analysis course but he took a special likin" to Aleksandrov's courses. ontrya"in was stron"ly influenced by Aleksandrov and the direction of Aleksandrov's research was to determine the area of ontrya"in's work for many years. 0owever this was as much to do with Aleksandrov himself as with his mathematics 3416 and 426789 Ale"sandrovs personal charm% his attention and helpfulness influenced the formation of 6ontryagins scientific interests to a remar"able extent% as much in fact as the personal abilities and inclinations of the young scholar himself.
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The year 1:2C was the year of the death of ontrya"in's father. y 1:2C, althou"h he was still only 1: years old, ontrya"in had be"un to produce important results on the Aleander duality theorem. 0is main tool was to use link numbers which had been introduced by rouwer and, by 1:52, he had produced the most si"nificant of these duality results when he proved the duality between the homolo"y "roups of bounded closed sets in uclidean space and the homolo"y "roups in the complement of the space. ontrya"in "raduated from the %niversity of $oscow in 1:2: and was appointed to the $echanics and $athematics aculty. -n 1:5* he became a member of the !teklov -nstitute and in 1:5< he became head of the )epartment of Topolo"y and unctional Analysis at the -nstitute. ontrya"in worked on problems in topolo"y and al"ebra. -n fact his own description of this area that he worked on was89 ... problems where these two domains of mathematics come together. The si"nificance of this work of ontrya"in on duality 3416 and 426789 ... lies not merely in its effect on the further development of topology7 of e8ual significance is the fact that his theorem enabled him to construct a general theory of characters for commutative topological groups. &his theory% historically the first really exceptional achievement in a new branch of mathematics% that of topological algebra% was one of the most fundamental advances in the whole of mathematics during the present century... Bne of the 25 problems posed by 0ilbert in 1:++ was to prove his con#ecture that any locally uclidean topolo"ical "roup can be "iven the structure of an analytic manifold so as to become a Lie "roup. This became known as 0ilbert's ifth roblem. -n 1:2: von >eumann, usin" inte"ration on "eneral compact "roups which he had introduced, was able to solve 0ilbert's ifth roblem for compact "roups. -n 1:5* ontrya"in was able to prove 0ilbert's ifth roblem for abelian "roups usin" the theory of characters on locally compact abelian "roups which he had introduced. Amon" ontrya"in's most important books on the above topics is topological groups 31:5D7. The authors of 416 and 426 ri"htly assert89 &his boo" belongs to that rare category of mathematical wor"s that can truly be called classical - boo" which retain their significance for decades and exert a formative influence on the scientific outloo" of whole generations of mathematicians. -n 1:5* ;artan visited $oscow and lectured in the $echanics and $athematics aculty. ontrya"in attended ;artan's lecture which was in rench but ontrya"in did not understand rench so he listened to a whispered translation by >ina ari who sat beside him. ;artan's lecture was based around the problem of calculatin" the homolo"y "roups of the classical compact Lie "roups. ;artan had some ideas how this mi"ht be achieved and he eplained these in the lecture but, the followin" year, ontrya"in was able to solve the problem completely usin" a totally different approach to the one su""ested by ;artan. -n fact ontrya"in used ideas introduced by $orse on eGuipotential surfaces. ontrya"in's name is attached to many mathematical concepts. The essential tool of cobordism theory is the ontrya"in9Thom construction. A fundamental theorem concernin" characteristic classes of a manifold deals with special classes called the ontrya"in characteristic class of the manifold. Bne of the main problems of characteristic classes was not solved until !er"ei >ovikov proved their topolo"ical invariance. -n 1:<2 ontrya"in chan"ed the direction of his research completely. 0e be"an to study applied mathematics problems, in particular studyin" differential eGuations and control theory. -n fact this chan"e of direction was not Guite as sudden as it appeared. rom the 1:5+s ontrya"in had been friendly with the physicist A A Andronov and had re"ularly discussed with him problems in the theory of oscillations and the theory of automatic control on which Andronov was workin". 0e published a paper with Andronov on dynamical systems in 1:52 but the bi" shift in ontrya"in's work in 1:<2 occurred around the time of Andronov's death. -n 1:=1 he published &he )athematical &heory of 2ptimal 6rocesses with his students E / oltyanskii, R E /amrelide and $ishchenko. The followin" year an n"lish translation appeared and, also in 1:=2, ontrya"in received the Lenin prie for his book. 0e then produced a series of papers on differential "ames which etends his work on control theory. ontrya"in's work in control theory is discussed in the historical *=
survey 456. Another book by ontrya"in 2rdinary differential e8uations appeared in n"lish translation, also in 1:=2. ontrya"in received many honours for his work. 0e was elected to the Academy of !ciences in 1:5:, becomin" a full member in 1:<:. -n 1:*1 he was of one the first recipients of the !talin pries 3later called the !tate ries7. 0e was honoured in 1:C+ by bein" elected Eice9resident of the -nternational $athematical %nion.
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Sergei $vovih So'olev )orn< + Ot .:04 in St Peters'%rg? R%ssia 5ie"< # (an .:4: in $eningra" =no* St Peters'%rg>? R%ssia =USSR> Sergei $vovih So'olev's father, Lev Aleksandrovich !obolev, was an important layer and barrister. 0is mother, >ataliya /eor"ievna, played an important role in !obolev's upbrin"in", particularly after the death of !obolev's father when !obolev was 1* years old. 0e studied at the Fhar'kov (orkers' Technical !chool preparin" to enter the hi"h school which he did in 1:22 around the time of his father's death. The hi"h school which he entered was called the 1:+ th !chool of Lenin"rad at the time althou"h previously it had been called the Lentovskii 0i"h !chool. -n 4<6 3see also 4*67 it is eplained that this school 89 ... was founded during the 1irst +ussian +evolution by the foremost t 6etersburg teachers for pupils who had been excluded from the tate chools and &echnical 0olleges because of their participation in the revolutionary movement. After "raduatin" from hi"h school in 1:2<, !obolev entered the hysics and $athematics aculty of Lenin"rad !tate %niversity where his talents were Guickly spotted by !mirnov who had returned to Lenin"rad three years earlier. !obolev became interested in differential eGuations, a topic which would dominate his research throu"hout his life, and even at this sta"e in his career he produced new results which he published. y 1:2: !obolev had completed his university education and he be"an to teach in a number of different educational establishments. or eample his first appointment was in 1:2: at the Theoretical )epartment of the !eismolo"ical -nstitute of the %!!R Academy of !ciences. 0owever, in addition, he tau"ht at the Lenin"rad lectrotechnic -nstitute in 1:5+951. -n 1:52 the !teklov -nstitute of hysics and $athematics was divided into separate )epartments of $athematics and of hysics. Eino"radov headed the $athematics )epartment and invited !obolev to #oin the )epartment. y this time, however, !obolev had already 34*6 and 4<6789 ... published a number of profound papers in which he put forward a new method for the solution of an important class of partial differential e8uations. (orkin" with !mirnov, !obolev studied functionally invariant solutions of the wave eGuation. These methods allowed them to find closed form solutions to the wave eGuation describin" the oscillations of an elastic medium. The methods also led them to a complete theory of Ra ylei"h surface waves and !obolev went on to solve problems on diffraction. !obolev was honoured for this outstandin" work by election as ;orrespondin" $ember of the %!!R Academy of !ciences in 1:55. Bn 2D April 1:5*, at the "eneral meetin" of the )ivision of $athematical and >atural !ciences of the %!!R Academy of !ciences, a decision was taken to split the )epartments of the !teklov -nstitute of hysics and $athematics, which had been created two years earlier, into two independent -nstitutes, the !teklov $athematical -nstitute and the Lebedev hysical -nstitute. -n the same year the !teklov $athematical -nstitute was moved from Lenin"rad to $oscow and !obolev went with the new -nstitute to $oscow. y 1:5< !obolev was head of the )epartment of the Theory of )ifferential Guations at the -nstitute. )urin" the 1:5+s !obolev introduced notions which were fundamental in the development of several different areas of mathematics 422689 &he study of obolev function spaces% which he introduced in the #KMJs% immediately became a whole area of functional analysis. obolevs notion of generalised function 3distribution7 turned out to be especially important7 with further developments by chwartz and 'elfand% it became one of the central notions of mathematics.
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(hile workin" in $oscow, !obolev built on the standard variational method for solvin" elliptic boundary value problems by introducin" these !obolev function spaces. 0e "ave ineGualities on the norms on these spaces which were important in the theory of embeddin" function spaces. 0e applied his methods to solve difficult problems in mathematical physics. -n 1:5: !obolev was elected a full member of the %!!R Academy of !ciences. 0e was only 51 years of a"e at the time of his election which was a remarkable achievement. -t made him the youn"est full member of the Academy of !ciences and in fact he remained the youn"est member for Guite a few years. At the be"innin" of (orld (ar --, the !teklov $athematical -nstitute was moved from $oscow to Faan. -n the Bctober of that year !obolev was appointed as )irector of the -nstitute and in the sprin" of 1:*5 he supervised the move of the -nstitute back to $oscow. !obolev became one of the first recipients of a !talin prie 3later called a !tate prie7 in the first presentation of these pries in 1:*1. 0is period as )irector of the -nstitute ended in ebruary 1:**. A new area of his research involved the study of the motion of a fluid in a rotatin" vessel. 0e was led to study a number of new problems which 34*6 and 4<6789 ... led him to lay the foundations of the theory of operators in a space with an indefinite metric% and to introduce new ideas in the spectral theory of operators. &hese ideas in the main concern generalised solutions of non-classical boundary value problems. -n 1:<+ he published his famous tet Applications of functional analysis in mathematical physics 3in Russian7. An n"lish translation was published by the American $athematical !ociety in 1:=5. !chwart's book on the the theory of distributions appeared in the same year. -n the early 1:<+s !obolev's work turned towards computational mathematics and in 1:<2 he became head of the first department of computational mathematics in the !oviet %nion when he or"anised the first such department at $oscow !tate %niversity. 0owever in 1:<= he #oined with a number of collea"ues in proposin" ways in which the lar"e areas of Russia in the east could be opened up with educational initiatives. The scheme was to set up a number of -nstitutes for !cientific research to balance the lar"e number of hi"h Guality educational establishments in the east of the !oviet %nion. After the plan was approved, !obolev spent some time in $oscow recruitin" staff and or"anisin" the establishment of an -nstitute in >ovosibirsk. -n 4*6 and 4<6 his contribution to the -nstitute in >ovosibirsk is stressed89 /uring the difficult formative years of the !nstitute obolev% by his excellent example% infused his young colleagues with the best habits for scientific wor". !n ten years% under the leadership of obolev% the !nstitute of )athematics of the iberian Branch of the *+ Academy of ciences has become one of the greatest centres for the mathematical sciences of international status. -n 1:ovosibirsk %niversity. )urin" the 1:=+s much of !obolev's research was directed towards numerical methods, in particular to interpolation. Althou"h interpolation for functions of a sin"le variable was well worked out, the problem of interpolation in many dimensions was lar"ely unsolved. !obolev applied his theories of "eneralised functions and of embeddin"s of function spaces to cubature formulae, the multi9dimensional analo"ues of Guadrature formulae for functions of one variable. A"ain a ma#or tet by !obolev !ntroduction to the theory of cubature formulae has been etremely influential in this area. !obolev received many honours for his fundamental contributions to mathematics. 0e was elected to many scientific societies, includin" the %!!R Academy of !ciences, the Acad@mie des !ciences de rance, and the Accademia >aionale dei Lincei. 0e was awarded many pries, includin" three !tate ries and the 1:DD $ E Lomonosov /old $edal from the %!!R Academy of !ciences.
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An"rei Ni!olaevih Ti!honov )orn< #0 Ot .:0+ in G&hats!a? S8olens!? R%ssia 5ie"< 4 Nov .::# in Moso*? R%ssia Like most Russian mathematicians there are different ways to transliterate An"rei Ni!olaevih Ti!honov's name into the Roman alphabet. The most common way, other than Andrei >ikolaevich Tikhonov, is to write it as An"re7 Ni!ola7evih T7hono//@ Andrei >ikolaevich Tikhonov attended secondary school as a day pupil and entered the $oscow %niversity in 1:22, the year in which he completed his school education. 0is studied in the $athematics )epartment of the aculty of $athematics and hysics at $oscow %niversity and made remarkable pro"ress, havin" his first paper published in 1:2< while he was still in the middle of his under"raduate course. This first work was related to results of Aleksandrov and %rysohn on conditions for a topolo"ical space to be metrisable. 0owever he did not stop there and continued his investi"ations in topolo"y. y 1:2= he had discovered the topolo"ical construction which is today named after him, the Tikhonov topolo"y defined on the product of topolo"ical spaces. Aleksandrov, recallin" in 4*6 how he failed to appreciate the si"nificance of Tikhonov's ideas at the time he proposed them, remembered89 ... very well with what mistrust he met &i"honovs proposed definition. ,ow was it possible that a topology introduced by means of such enormous neighbourhoods% which are only distinguished from the whole space by a finite number of the coordinates% could catch any of the essntial characteristics of a topological product< Tikhonov certainly had "iven the ri"ht definition and this idea, which was counterintuitive to even as "reat a topolo"ist as Aleksandrov, allowed Tikhonov to "o on and prove such important topolo"ical results as the product of any set of compact topolo"ical spaces is compact. ew mathematicians have "ained a worldwide reputation before they even start their research careers but this was essentially how it was for Tikhonov. 0is results on the Tikhonov topolo"y of products were achieved before he "raduated in 1:2C. (ith this impressive record he became a research student at $oscow %niversity in 1:2C. -t mi"ht be thou"ht that someone who had clearly such an intuitive "rasp of topolo"ical ideas would be only too pleased to use his talents in that area. Tikhonov, however, had eGual talents for other areas of mathematics. The ran"e of his work is summarised in 45689 ;e owe to &i"honov deep and fundamental results in a wide range of topics in modern mathematics. ,is first-class achievements in topology and functional analysis% in the theory of ordinary and partial differential e8uations% in the mathematical problems of geophysics and electrodynamics% in computational mathematics and in mathematical physics are all widely "nown. &i"honovs scientific wor" is characterised by magnificent achievements in very abstract fields of so-called pure mathematics% combined with deep investigations into the mathematical disciplines directly connected with practical re8uirements. -n fact Tikhonov's work led from topolo"y to functional analysis with his famous fied point theorem for continuous maps from conve compact subsets of locally conve topolo"ical spaces in 1:5<. These results are of importance in both topolo"y and functional analysis and were applied by Tikhonov to solve problems in mathematical physics. 0e defended his habilitation thesis in 1:5= on 1unctional e8uations of olterra type and their applications to mathematical physics. The thesis applied an etension of Kmile icard's method of approimatin" the solution of a differential eGuation and "ave applications to heat conduction, in particular coolin" which obeys the law "iven by ?osef !tefan and oltmann. After successfully defendin" his thesis, Tikhonov was appointed as a professor at $oscow %niversity in 1:5= and then, three years later, he was elected as a <+
;orrespondin" $ember of the %!!R Academy of !ciences. Tikhonov's approach to problems in mathematical physics is described in 41*689 A characteristic of &i"honovs research is to combine a concrete theme in natural science with investigations into a fundamental mathematical problem. !n discussing some general problem in nature he always "nows how to pic" out a typical concrete physical problem and to give it a clear mathematical formulation. ,owever% his mathematical investigations are never confined to the solution of a given concrete problem% but serve as the starting point for stating a general mathematical problem that is a broad generalisation of the first problem. &he extremely deep investigations of &i"honov into a number of general problems in mathematical physics grew out of his interest in geophysics and electrodynamics. &hus% his research on the Earths crust lead to investigations on well-posed 0auchy problems for parabolic e8uations and to the construction of a method for solving general functional e8uations of olterra type. ... Tikhonov's work on mathematical physics continued throu"hout the 1:*+s and he was awarded the !tate rie for this work in 1:<5. 0owever, in 1:*D he be"an to study a new type of problem when he considered the behaviour of the solutions of systems of eGuations with a small parameter in the term with the hi"hest derivative. After a series of fundamental papers introducin" the topic, the work was carried on by his students. Another area in which Tikhonov made fundamental contributions was that of computational mathematics 34116 and 4126789 *nder his guidance many algorithms for the solution of various problems of electrodynamics% geophysics% plasma physics% gas dynamics% ... and other branches of the natural sciences were evolved and put into practice. ... 2ne of the most outstanding achievemnets in computational mathematics is the theory of homogeneous difference schemes% which &i"honov developed in collaboration with amars"ii.... -n the 1:=+s Tikhonov be"an to produce an important series of papers on ill9posed problems. 0e defined a class of re"ularisable ill9posed problems and introduced the concept of a re"ularisin" operator which was used in the solution of these problems. ;ombinin" his computin" skills with solvin" problems of this type, Tikhonov "ave computer implementations of al"orithms to compute the operators which he used in the solution of these problems. Tikhonov was awarded the Lenin rie for his work on ill9posed problems in 1:==. -n the same year he was elected to full membership of the %!!R Academy of !ciences. Tikhonov's wide interests throu"hout mathematics led him to hold a number of different chairs at $oscow %niversity, in particular a chair in the $athematical hysics aculty and a chair of ;omputational $athematics in the n"ineerin" $athematics aculty. 0e also became dean of the aculty of ;omputin" and ;ybernetics at $oscow %niversity. Tikhonov was appointed as )eputy )irector of the -nstitute of Applied $athematics of the %!!R Academy of !ciences, a position he held for many years.
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Alan Mathison T%ring )orn< 2# (%ne .:.2 in $on"on? Englan" 5ie"< 9 (%ne .:6 in ;il8slo*? Cheshire? Englan" Alan T%ring was born at addin"ton, London. 0is father, ?ulius $athison Turin", was a ritish member of the -ndian ;ivil !ervice and he was often abroad. Alan's mother, thel !ara !toney, was the dau"hter of the chief en"ineer of the $adras railways and Alan's parents had met and married in -ndia. (hen Alan was about one year old his mother re#oined her husband in -ndia, leavin" Alan in n"land with friends of the family. Alan was sent to school but did not seem to be obtainin" any benefit so he was removed from the school after a few months. >et he was sent to 0alehurst reparatory !chool where he seemed to be an 'avera"e to "ood' pupil in most sub#ects but was "reatly taken up with followin" his own ideas. 0e became interested in chess while at this school and he also #oined the debatin" society. 0e completed his ;ommon ntrance amination in 1:2= and then went to !herborne !chool. >ow 1:2= was the year of the "eneral strike and when the strike was in pro"ress Turin" cycled =+ miles to the school from his home, not too demandin" a task for Turin" who later was to become a fine athlete of almost Blympic standard. 0e found it very difficult to fit into what was epected at this public school, yet his mother had been so determined that he should have a public school education. $any of the most ori"inal thinkers have found conventional schoolin" an almost incomprehensible process and this seems to have been the case for Turin". 0is "enius drove him in his own directions rather than those reGuired by his teachers. 0e was criticised for his handwritin", stru""led at n"lish, and even in mathematics he was too interested with his own ideas to produce solutions to problems usin" the methods tau"ht by his teachers. )espite producin" unconventional answers, Turin" did win almost every possible mathematics prie while at !herborne. -n chemistry, a sub#ect which had interested him from a ver y early a"e, he carried out eperiments followin" his own a"enda which did not please his teacher. Turin"'s headmaster wrote 3see for eample 4=6789 !f he is to stay at 6ublic chool% he must aim at becoming educated. !f he is to be solely a cientific pecialist% he is wasting his time at a 6ublic chool. This says far more about the school system that Turin" was bein" sub#ected to than it does about Turin" himself. 0owever, Turin" learnt deep mathematics while at school, althou"h his teachers were probably not aware of the studies he was makin" on his own. 0e read instein's papers on relativity and he also read about Guantum mechanics in ddin"ton's &he nature of the physical world. An event which was to "reatly affect Turin" throu"hout his life took place in 1:2D. 0e formed a close friendship with ;hristopher $orcom, a pupil in the year above hi m at school, and the two worked to"ether on scientific ideas. erhaps for the first time Turin" was able to find someone with whom he could share his thou"hts and ideas. 0owever $orcom died in ebruary 1:5+ and the eperience was a shatterin" one to Turin". 0e had a premonition of $orcom's death at the very instant that he was taken ill and felt that this was somethin" beyond what science could eplain. 0e wrote later 3see for eample 4=6789 !t is not difficult to explain these things away - but% ! wonderG )espite the difficult school years, Turin" entered Fin"'s ;olle"e, ;ambrid"e, in 1:51 to study mathematics. This was not achieved without difficulty. Turin" sat the scholarship eaminations in 1:2: and won an ehibition, but not a scholarship. >ot satisfied with this performance, he took the eaminations a"ain in the followin" year, this time winnin" a scholarship. -n many ways ;ambrid"e was a much easier place for unconventional people like Turin" than school had been. 0e was now much more able to eplore his own ideas and he read Russell's !ntroduction to mathematical philosophy in 1:55. At about the same time he read von >eumann's 1:52 tet on Guantum mechanics, a sub#ect he returned to a number of times throu"hout his life. <2
The year 1:55 saw the be"innin"s of Turin"'s interest in mathematical lo"ic. 0e read a paper to the $oral !cience ;lub at ;ambrid"e in )ecember of that year of which the followin" minute was recorded 3see for eample 4=6789 A ) &uring read a paper on 9)athematics and logic9. ,e suggested that a purely logistic view of mathematics was inade8uate7 and that mathematical propositions possessed a variety of interpretations of which the logistic was merely one. Bf course 1:55 was also the year of 0itler's rise in /ermany and of an anti9war movement in ritain. Turin" #oined the anti9war movement but he did not drift towards $arism, nor pacifism, as happened to many. Turin" "raduated in 1:5* then, in the sprin" of 1:5<, he attended $a >ewman's advanced course on the foundations of mathematics. This course studied /del's incompleteness results and 0ilbert's Guestion on decidability. -n one sense 'decidability' was a simple Guestion, namely "iven a mathematical proposition could one find an al"orithm which would decide if the proposition was true of false. or many propositions it was easy to find such an al"orithm. The real difficulty arose in provin" that for certain propositions no such al"orithm eisted. (hen "iven an al"orithm to solve a problem it was clear that it was indeed an al"orithm, yet there was no definition of an al"orithm which was ri"orous enou"h to allow one to prove that none eisted. Turin" be"an to work on these ideas. Turin" was elected a fellow of Fin"'s ;olle"e, ;ambrid"e, in 1:5< for a dissertation 2n the 'aussian error function which proved fundamental results on probability theory, namely the central limit theorem. Althou"h the central limit theorem had recently been discovered, Turin" was not aware of this and discovered it independently. -n 1:5= Turin" was a !mith's rieman. Turin"'s achievements at ;ambrid"e had been on account of his work in probability theory. 0owever, he had been workin" on the decidability Guestions since attendin" >ewman's course. -n 1:5= he published 2n 0omputable 5umbers% with an application to the Entscheidungsproblem. -t is in this paper that Turin" introduced an abstract machine, now called a JTurin" machineJ, which moved from one state to another usin" a precise finite set of rules 3"iven by a finite table7 and dependin" on a sin"le s ymbol it read from a tape. The Turin" machine could write a symbol on the tape, or delete a s ymbol from the tape. Turin" wrote 415689 ome of the symbols written down will form the se8uences of figures which is the decimal of the real number which is being computed. &he others are ust rough notes to 9assist the memory9. !t will only be these rough notes which will be liable to erasure. 0e defined a computable number as real number whose decimal epansion could be produced by a Turin" machine startin" with a blank tape. 0e showed that Z was computable, but since only countably many real numbers are computable, most real numbers are not computable. 0e then described a number which is not computable and remarks that this seems to be a parado since he appears to have described in finite terms, a number which cannot be described in finite terms. 0owever, Turin" understood the source of the apparent parado. -t is impossible to decide 3usin" another Turin" machine7 whether a Turin" machine with a "iven table of instructions will output an infinite seGuence of numbers. Althou"h this paper contains ideas which have proved of fundamental importance to mathematics and to computer science ever since it appeared, publishin" it in the 6roceedings of the London )athematical ociety did not prove easy. The reason was that Alono ;hurch published An unsolvable problem in elementary number theory in the American Dournal of )athematics in 1:5= which also proves that there is no decision procedure for arithmetic. Turin"'s approach is very different from that of ;hurch but >ewman had to ar"ue the case for publication of Turin"'s paper before the London $athematical !ociety would publish it. Turin"'s revised paper contains a reference to ;hurch's results and the paper, first completed in April 1:5=, was revised in this way in Au"ust 1:5= and it appeared in print in 1:5C. A "ood feature of the resultin" discussions with ;hurch was that Turin" became a "raduate student at rinceton %niversity in 1:5=. At rinceton, Turin" undertook research under ;hurch's supervision and he returned to n"land in 1:5D, havin" been back in n"land for the summer vacation in 1:5C when he first met (itt"enstein. The ma#or publication which came out of his work at rinceton was ystems of Logic Based on 2rdinals which was published in 1:5:. >ewman writes in 415689 &his paper is full of interesting suggestions and ideas. ... 4 !t 6 throws much light on &urings <5
views on the place of intuition in mathematical proof. efore this paper appeared, Turin" published two other papers on rather more conventional mathematical topics. Bne of these papers discussed methods of approimatin" Lie "roups by finite "roups. The other paper proves results on etensions of "roups, which were first proved by Reinhold aer, "ivin" a simpler and more unified approach. erhaps the most remarkable feature of Turin"'s work on Turin" machines was that he was describin" a modern computer before technolo"y had reached the point where construction was a realistic proposition. 0e had proved in his 1:5= paper that a universal Turin" machine eisted 415689 ... which can be made to do the wor" of any special-purpose machine% that is to say to carry out any piece of computing% if a tape bearing suitable 9instructions9 is inserted into it. Althou"h to Turin" a JcomputerJ was a person who carried out a computation, we must see in his description of a universal Turin" machine what we today think of as a computer with the tape as the pro"ram. (hile at rinceton Turin" had played with the idea of constructin" a computer. Bnce back at ;ambrid"e in 1:5D he startin" to build an analo"ue mechanical device to investi"ate the Riemann hypothesis, which many consider today the bi""est unsolved problem in mathematics. 0owever, his work would soon take on a new aspect for he was contacted, soon after his return, by the /overnment ;ode and ;ypher !chool who asked him to help them in their work on breakin" the /erman ni"ma codes. (hen war was declared in 1:5: Turin" immediately moved to work full9time at the /overnment ;ode and ;ypher !chool at letchley ark. Althou"h the work carried out at letchley ark was covered by the Bfficial !ecrets Act, much has recently become public knowled"e. Turin"'s brilliant ideas in solvin" codes, and developin" computers to assist break them, may have saved more lives of military personnel in the course of the war than any other. -t was also a happy time for him 415689 ... perhaps the happiest of his life% with full scope for his inventiveness% a mild routine to shape the day% and a congenial set of fellow-wor"ers. To"ether with another mathematician ( / (elchman, Turin" developed the Bombe, a machine based on earlier work by olish mathematicians, which from late 1:*+ was decodin" all messa"es sent by the ni"ma machines of the Luftwaffe. The ni"ma machines of the /erman navy were much harder to break but this was the type of challen"e which Turin" en#oyed. y the middle of 1:*1 Turin"'s statistical approach, to"ether with captured information, had led to the /erman navy si"nals bein" decoded at letchley. rom >ovember 1:*2 until $arch 1:*5 Turin" was in the %nited !tates liaisin" over decodin" issues and also on a speech secrecy system. ;han"es in the way the /ermans encoded their messa"es had meant that letchley lost the ability to decode the messa"es. Turin" was not directly involved with the successful breakin" of these more comple codes, but his ideas proved of the "reatest importance in this work. Turin" was awarded the B... in 1:*< for his vital contribution to the war effort. At the end of the war Turin" was invited by the >ational hysical Laboratory in London to desi"n a computer. 0is report proposin" the Automatic ;omputin" n"ine 3A;7 was submitted in $arch 1:*=. Turin"'s desi"n was at that point an ori"inal detailed desi"n and prospectus for a computer in the modern sense. The sie of stora"e he planned for the A; was re"arded by most who considered the report as hopelessly over9ambitious and there were delays in the pro#ect bein" approved. Turin" returned to ;ambrid"e for the academic year 1:*C9*D where his interests ran"ed over many topics far removed from computers or mathematicsI in particular he studied neurolo"y and physiolo"y. 0e did not for"et about computers durin" this period, however, and he wrote code for pro"rammin" computers. 0e had interests outside the academic world too, havin" taken up athletics seriously after the end of the war. 0e was a member of (alton Athletic ;lub winnin" their 5 mile and 1+ mile championship in record time. 0e ran in the A.A.A. $arathon in 1:*C and was placed fifth. y 1:*D >ewman was the professor of mathematics at the %niversity of $anchester and he offered Turin" a readership there. Turin" resi"ned from the >ational hysical Laboratory to take up the post in $anchester. >ewman writes in 4156 that in $anchester89 ... wor" was beginning on the construction of a computing machine by 1 0 ;illiams and & <*
3ilburn. &he expectation was that &uring would lead the mathematical side of the wor"% and for a few years he continued to wor"% first on the design of the subroutines out of which the larger programs for such a machine are built% and then% as this "ind of wor" became standardised% on more general problems of numerical analysis. -n 1:<+ Turin" published 0omputing machinery and intelligence in )ind. -t is another remarkable work from his brilliantly inventive mind which seemed to foresee the Guestions which would arise as computers developed. 0e studied problems which today lie at the heart of artificial intelli"ence. -t was in this 1:<+ paper that he proposed the Turin" Test which is still today the test people apply in attemptin" to answer whether a computer can be intelli"ent 41689 ... he became involved in discussions on the contrasts and similarities between machines and brains. &urings view% expressed with great force and wit% was that it was for those who saw an unbridgeable gap between the two to say ust where the difference lay. Turin" did not for"et about Guestions of decidability which had been the startin" point for his brilliant mathematical publications. Bne of the main problems in the theory of "roup presentations was the Guestion8 "iven any word in a finitely presented "roups is there an al"orithm to decide if the word is eGual to the identity. ost had proved that for semi"roups no such al"orithm eist. Turin" thou"ht at first that he had proved the same result for "roups but, #ust before "ivin" a seminar on his proof, he discovered an error. 0e was able to rescue from his faulty proof the fact that there was a cancellative semi"roup with insoluble word problem and he published this result in 1:<+. oone used the ideas from this paper by Turin" to prove the eistence of a "roup with insoluble word problem in 1:ot only did he press forward with further study of morpho"enesis, but he also worked on new ideas in Guantum theory, on the representation of elementary particles by spinors, and on relativity theory. Althou"h he was completely open about his seuality, he had a further unhappiness which he was forbidden to talk about due to the Bfficial !ecrets Act. The decodin" operation at letchley ark became the basis for the new decodin" and intelli"ence work at /;0[. (ith the cold war this became an important operation and Turin" continued to work for /;0[, althou"h his $anchester collea"ues were totally unaware of this. After his conviction, his security clearance was withdrawn. (orse than that, security officers were now etremely worried that someone with complete knowled"e of the work "oin" on at /;0[ was now labelled a security risk. 0e had many forei"n collea"ues, as any academic would, but the police be"an to investi"ate his forei"n visitors. A holiday which Turin" took in /reece in 1:<5 caused consternation amon" the security officers. Turin" died of potassium cyanide poisonin" while conductin" electrolysis eperiments. The cyanide was found on a half eaten apple beside him. An inGuest concluded that it was self9administered but his mother always maintained that it was an accident.
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Pavel Sa8%ilovih Ur7sohn )orn< # Fe' .4:4 in O"essa? U!raine? R%ssia 5ie"< .9 A%g .:26 in )at&s%rMer? Frane Pavel Ur7sohn is also known as Pavel Ur7son@ 0is father was a financier in Bdessa, the town in which avel !amuilovich was born. 0e came from a family descended from the siteenth century Rabbi $ ?affe. -t was a well9off family and %rysohn received his secondary education in $oscow at a private school there. -n 1:1< %rysohn entered the %niversity of $oscow to study physics and in fact he published his first paper in this year. ein" interested in physics at this time it is not surprisin" that this first paper was on a physics topic, and indeed it was, bein" on ;oolid"e tube radiation. 0owever his interest in physics soon took second place for after attendin" lectures by Luin and "orov at the %niversity of $oscow he be"an to concentrate on mathematics. %rysohn "raduated in 1:1: and continued his studies there workin" towards his doctorate. The authors of 4D6 write89 Luzin was a dynamic mathematician and it was he who persuaded *rysohn to stay on in order to study for a doctorate during 1:1:921. At this sta"e %rysohn was interested in analysis, in particular inte"ral eGuations, and this was the topic of his habilitation. 0e was awarded his habilitation in ?une 1:21 and, followin" this, became an assistant professor at the %niversity of $oscow. %rysohn soon turned to topolo"y. 0e was asked two Guestions by "orov and it was these which occupied him durin" the summer of 1:21. The first Guestion that "orov posed was to find a "eneral intrinsic topolo"ical definition of a curve which when restricted to the plane became ;antor's notion of a continuum which is nowhere dense in the plane. The second of "orov's Guestions was a similar one but applied to surfaces, a"ain askin" for an intrinsic topolo"ical definition. These were difficult Guestions which had been around for some time. -t was not that "orov had come up with new Guestions, rather he was "ivin" the bri"ht youn" mathematician %rysohn two really difficult problems in the hope that he mi"ht come up with new ideas. "orov was not to be disappointed, for %rysohn attacked the Guestions with "reat determination. 0e did not sit still waitin" for inspiration to strike, rather he tried one idea after another to see if it would "ive him the topolo"ical definition of dimension that he was lookin" for. A holiday with other youn" $oscow mathematicians to the villa"e of urkov, on the banks of the river Falyamy near to the town of olshev, did not stop him tryin" to find the Jri"htJ definition of dimension. [uite the opposite, it was a "ood chance for him to think in con"enial surroundin"s, and one mornin" near the end of Au"ust he woke up with an idea in his mind which he felt, even before workin" throu"h the details, was ri"ht. -mmediately he told his friend Aleksandrov about his inspiration. Bf course there was a lot of hard work after the moment of inspiration. )urin" the followin" year %rysohn worked throu"h the conseGuences buildin" a whole new area of dimension theory in topolo"y. -t was an ecitin" time for the topolo"ists in $oscow for %rysohn lectured on the topolo"y of continua and often his latest results were presented in the course shortly after he had proved them. 0e published a series of short notes on this topic durin" 1:22. The complete theory was presented in an article which Lebes"ue accepted for publication in the 0omptes rendus of the Academy of !ciences in aris. This "ave %rysohn an international platform for his ideas which immediately attracted the interest of mathematicians such as 0ilbert. %rysohn published a full version of his dimension theory in 1undamenta mathematicae. 0e wrote a ma#or paper in two parts in 1:25 but they did not appear in print until 1:2< and 1:2=. !adly %rysohn had died before even the first part was published. The paper be"ins with %rysohn statin" his aim which was89 <=
&o indicate the most general sets that still merit being called 9lines9 and 9surfaces9 ... -n fact %rysohn set out to do far more in this paper than to answer the two Guestions that "orov had posed to him. As ;rilly and ?ohnson write 4D689 5ot only did he see" definitions of curve and surface% but also definitions of n-dimensional 0antorian manifold and hence of dimension itself. &he dimension concept was% in fact% the centre of his attention. Althou"h %rysohn did not know of rouwer's contribution when he worked out the details of his theory of topolo"ical dimension, rouwer had in fact published on that topic in 1:15. 0e had "iven a "lobal definition, however, and this was in contrast to %rysohn's local definition of dimension. Another important aspect of %rysohn's ideas was the fact that he presented them in the contet of compact metric spaces. After %rysohn's death, Aleksandrov ar"ued that althou"h %rysohn's definition of dimension was "iven for a metric space, it is, nevertheless, completely eGuivalent to the definition "iven by $en"er for "eneral topolo"ical spaces. %rysohn visited /ttin"en in 1:25. 0is reports to the $athematical !ociety of /ttin"en interested 0ilbert and while in /ttin"en he learnt of rouwer's contributions to the area made in the paper of 1:15 to which we referred above. %rysohn spotted an error in rouwer's paper re"ardin" a definition of dimension while he was studyin" it in /ttin"en and easily constructed a counter9eample. 0e met rouwer at the annual meetin" of the /erman $athematical !ociety in $arbur" where both "ave lectures and %rysohn mentioned rouwer's error, and his counter9eample, in his talk. -t was an occasion which made rouwer be"in to think about topolo"y a"ain, for his interests had turned to intuitionism, the sub#ect of his talk at $arbur". -n the summer of 1:2* %rysohn set off a"ain with Aleksandrov on a uropean trip throu"h /ermany, 0olland and rance. A"ain the two mathematicians visited 0ilbert and, by C $ay, they must have left since 0ilbert wrote to %rysohn on that day tellin" him his paper with Aleksandrov was accepted for publication in )athematische Annalen 3see below7. This letter, "iven in 4116, also thanks %rysohn for caviar he had "iven 0ilbert, and epresses the hope that %rysohn will visit a"ain the followin" summer. They then met 0ausdorff who was impressed with %rysohn's results. 0e also wrote a letter to %rysohn which was dated 11 Au"ust 1:2* 3see 41167. The letter discusses %rysohn's metriation theorem and his construction of a universal separable metric space. The construction of a universal metric space, containin" an isometric ima"e of any metric space, was one of %rysohn's last results. Like 0ilbert, 0ausdorff epressed the hope that %rysohn would visit a"ain the followin" summer. Ean )alen writes in 4156 about their final mathematical visit which was to rouwer89 &his time 4*rysohn and Ale"sandrov6 visited Brouwer% who was most favourably impressed by the two +ussians. ,e was particularly ta"en with *rysohn% for whom he developed something li"e the attachment to a lost son. After this visit the two mathematicians continued their holiday to rittany where they rented a cotta"e. %rysohn drowned in rou"h seas while on one of their re"ular swims off the coast. %rysohn was not only an Jinseparable friendJ to Aleksandrov but the two collaborated on important publications such as ur &heorie der topologischen +ume published in )athematische Annalen in 1:2*. %rysohn's main contributions, in addition to the theory of dimension discussed above, are the introduction and investi"ation of a class of normal surfaces, metriation theorems, and an important eistence theorem concernin" mappin" an arbitrary normed space into a 0ilbert space with countable basis. 0e is remembered particularly for '%rysohn's lemma' which proves the eistence of a certai n continuous function takin" values + and 1 on particular closed subsets. After %rysohn's death rouwer and Aleksandrov made sure that the mathematics he left was properly dealt with. As van )alen writes 415689 Brouwer was bro"en hearted. ,e decided to loo" after the scientific estate of *rysohn as a tribute to the genius of the deceased. &ogether with Ale"sandrov he ac8uitted himself of this tas". ;rilly and ?ohnson write 4D689
0onsidering that he only had three years to devote to topology% he made his mar" in his chosen field with brilliance and passion. ,e transformed the subect into a rich domain of modern mathematics. ,ow much more might he there have been% had he not died so young<
Her8ann la%s H%go ;e7l )orn< : Nov .44 in El8shorn =near Ha8'%rg>? Shles*igHolstein? Ger8an7 5ie"< : 5e .: in Drih? S*it&erlan" Her8ann ;e7l was known as eter to his close friends. 0is parents were Anna )ieck and Ludwi" (eyl who was the director of a bank. As a boy 0ermann had already showed that he had a "reat talents for mathematics and for science more "enerally. After takin" his Abiturarbeit 3hi"h school "raduation eam7 3see 41=67 he was ready for his university studies. -n 1:+* he entered the %niversity of $unich, where he took courses on both mathematics and physics, and then went on to study the same topics at the %niversity of /ttin"en. 0e was completely captivated by 0ilbert. 0e later wrote89 ! resolved to study whatever this man had written. At the end of my first year ! went home with the 9ahlbericht9 under my arm% and during the summer vacation ! wor"ed my way through it without any previous "nowledge of elementary number theory or 'alois theory. &hese were the happiest months of my life% whose shine% across years burdened with our common share of doubt and failure% still comforts my soul. 0is doctorate was from /ttin"en where his supervisor was 0ilbert. After submittin" his doctoral dissertation ingulre !ntegralgleichungen mit besonder Ber(c"sichtigung des 1ourierschen !ntegraltheorems he was awarded the de"ree in 1:+D. This thesis investi"ated sin"ular inte"ral eGuations, lookin" in depth at ourier inte"ral theorems. -t was at /ttin"en that he held his first teachin" post as a privatdoent, a post he held until 1:15. 0is habilitation thesis @ber gewChnliche /ifferentialgleic"lungen mit ingularitten und die zugehCrigen Entwic"lungen will"(rlicher 1un"tionen investi"ated the spectral theory of sin"ular !turm9 Liouville problems. )urin" this period at /ttin"en, (eyl made a reputation for himself as an outstandin" mathematician who was producin" work which was havin" a ma#or impact on the pro"ress of mathematics. 0is habilitation thesis was one such piece of work but there was much more. 0e "ave a lecture course on Riemann surfaces in session 1:11912 and out of this course came his first book /ie !dee der +iemannschen 1lche which was published in 1:15. -t united analysis, "eometry and topolo"y, makin" ri"orous the "eometric function theory developed by Riemann. The book introduced for the first time the notion of a 4ot only did (eyl and his wife share an interest in philosophy, but they shared a real talent for lan"ua"es. Lan"ua"e for (eyl held a special importance. 0e not only wrote beautifully in /erman, but later he wrote stunnin" n"lish prose despite the fact that, in his own words from a 1:5: n"lish tet89 <:
... the gods have imposed upon my writing the yo"e of a foreign language that was not sung at my cradle. rom 1:15 to 1:5+ (eyl held the chair of mathematics at HMrich Technische 0ochschule. -n his first academic year in this new post he was a collea"ue of instein who was at this time workin" out the details of the theory of "eneral relativity. -t was an event which had a lar"e influence on (eyl who Guickly became fascinated by the mathematical principles lyin" behind the theory. (orld (ar - broke out not lon" after (eyl took up the chair in HMrich. ein" a /erman citien he was conscripted into the /erman army in 1:1< but the !wiss "overnment made a special reGuest that he be allowed to return to his chair in HMrich which was "ranted in 1:1=. -n 1:1C (eyl "ave another course presentin" an innovative approach to relativity throu"h differential "eometry. The lectures formed the basis of (eyl's second book +aum-eit-)aterie which first appeared in 1:1D with further editions, each showin" how his ideas were developin", in 1:1:, 1:2+, and 1:25. These later ideas included a "au"e metric 3the (eyl metric7 which led to a "au"e field theory. 0owever instein, auli, ddin"ton, and others, did not fully accept (eyl's approach. Also over this period (eyl also made contributions on the uniform distribution of numbers modulo 1 which are fundamental in analytic number theory. -n 1:21 !chrdin"er was appointed to Hurich where he became a collea"ue, and soon closest friend, of (eyl. They shared many interests in mathematics, physics, and philosophy. Their personal lives also became entan"led as $oore relates in 4<689 &hose familiar with the serious and portly figure of ;eyl at 6rinceton would have hardly recognised the slim% handsome young man of the twenties% with his romantic blac" moustache. ,is wife% ,elene Doseph% from a Dewish bac"ground% was a philosopher and literateuse. ,er friends called her ,ella% and a certain daring and insouciance made her the un8uestioned leader of the social set comprising the scientists and their wives. Anny 4chrCdingers wife6 was almost an exact opposite of the stylish and intellectual ,ella% but perhaps for that reason 4;eyl6 found her interesting and before long she was madly in love with him. ... &he special circle in which they lived in urich had enoyed the sexual revolution a generation before 4the *nited tates6. Extramarital affairs were not only condoned% they were expected% and they seemed to occasion little anxiety. Anny would find in ,ermann ;eyl a lover to whom she was devoted body and soul% while ;eyls wife ,ella was infatuated with 6aul cherrer. rom 1:2595D (eyl evolved the concept of continuous "roups usin" matri representations. -n particular his theory of representations of semisimple "roups, developed durin" 1:2*92=, was very deep and considered by (eyl himself to be his "reatest achievement. The ideas behind this theory had already been introduced by 0urwit and !chur, but it was (eyl with his "eneral character formula which took them forward. 0e was not the only mathematician developin" this theory, however, for ;artan also produced work on this topic of outstandin" importance. rom 1:5+ to 1:55 (eyl held the chair of mathematics at /ttin"en where he was appointed to fill the vacancy which arose on 0ilbert's retirement. /iven different political circumstances it is likely that he would have remained in /ttin"en for the rest of his career. 0owever 4C689 ... the rise of the 5azis persuaded him in 1:55 to accept a position at the newly formed !nstitute for Advanced tudy in 6rinceton% where Einstein also went. ,ere ;eyl found a very congenial wor"ing environment where he was able to guide and influence the younger generation of mathematicians% a tas" for which he was admirably suited. Bne also has to understand that (eyl's wife was ?ewish, and this must have played a ma#or role in their decision to leave /ermany in 1:55. (eyl remained at the -nstitute for Advanced !tudy at rinceton until he retired in 1:<2. 0is wife 0elene died in 1:*D, and two years later he married the sculptor llen Lohnstein Nr from HMrich. (eyl certainly undertook work of ma#or importance at rinceton, but his most productive period was without doubt the years he spent at HMrich. 0e attempted to incorporate electroma"netism into the "eometric formalism of "eneral relativity. 0e produced the first unified field theory for which the $awell electroma"netic field and the "ravitational field appear as "eometrical properties of space9time. (ith his =+
application of "roup theory to Guantum mechanics he set up the modern sub#ect. -t was his lecture course on "roup theory and Guantum mechanics in HMrich in session 1:2C92D which led to his third ma#or tet 'ruppentheorie und Nuantenmechani" published in 1:2D. ?ohn (heeler writes 4<<689 &hat boo" has% each time ! read it% some great new message. $ore recently attempts to incorporate electroma"netism into "eneral relativity have been made by (heeler. (heeler's theory, like (eyl's, lacks the connection with Guantum phenomena that is so important for interactions other than "ravitation. (heeler writes about meetin" (eyl for the first time in 4<<689 Erect% bright-eyed% smiling ,ermann ;eyl ! first saw in the flesh when 1:5C brought me to 6rinceton. &here ! attended his lectures on the Olie 0artan calculus of differential forms and their application to electromagnetism - elo8uent% simple% full of insights. (e have seen above how (eyl's "reat works were first "iven as lecture courses. This was a deliberate desi"n by (eyl 4<<689 At another time ;eyl arranged to give a course at 6rinceton *niversity on the history of mathematics. ,e explained to me one day that it was for him an absolute necessity to review% by lecturing% his subect of concern in all its length and breadth. 2nly so% he remar"ed% could he see the great lacunae% the places where deeper understanding is needed% where wor" should focus. $any other "reat books by (eyl appeared durin" his years at rinceton. These include Elementary &heory of !nvariants 31:5<7, &he classical groups 31:5:7, Algebraic &heory of 5umbers 31:*+7, 6hilosophy of )athematics and 5atural cience 31:*:7, ymmetry 31:<27, and &he 0oncept of a +iemannian urface 31:<<7. There is so much that could be said about all these works, but we restrict ourselves to lookin" at the contents of ymmetry for this perhaps tells us most about the full ran"e of (eyl's interests. ;oeter reviewed the book and his review beautifully captures the spirit of the book89 &his is slightly modified version of the Louis 0lar" anuxem Lectures given at 6rinceton *niversity in 1:<1 ... &he first lecture begins by showing how the idea of bilateral symmetry has influenced painting and sculpture% especially in ancient times. &his leads naturally to a discussion of 9the philosophy of left and right9% including such 8uestions as the following. !s the occurrence in nature of one of the two enantiomorphous forms of an optically active substance characteristic of living matter< At what stage in the development of an embryo is the plane of symmetry determined< &he second lecture contains a neat exposition of the theory of groups of transformations% with special emphasis on the group of similarities and its subgroups: the groups of congruent transformations% of motions% of translations% of rotations% and finally the symmetry group of any given figure. ... the cyclic and dihedral groups are illustrated by snowfla"es and flowers% by the animals called )edusae% and by the plans of symmetrical buildings. imilarly% the infinite cyclic group generated by a spiral similarity is illustrated by the 5autilus shell and by the arrangement of florets in a sunflower. &he third lecture gives the essential steps in the enumeration of the seventeen space-groups of two-dimensional crystallography ... 4 !n the fourth lecture he6 shows how the special theory of relativity is essentially the study of the inherent symmetry of the four-dimensional space-time continuum% where the symmetry operations are the Lorentz transformations7 and how the symmetry operations of an atom% according to 8uantum mechanics% include the permutations of its peripheral electrons. &urning from physics to mathematics% he gives an extraordinarily concise epitome of 'alois theory% leading up to the statement of his guiding principle: 9;henever you have to do with a structure-endowed entity% try to determine its group of automorphisms9. -n 1:<1 (eyl retired from the -nstitute for Advanced !tudy at rinceton. -n fact he described the ymmetry book as his 'swan son"'. After his retirement (eyl and his wife llen spent part of their time at rinceton and part at Hurich. 0e died unepectedly while in Hurich. 0e was walkin" home after postin" letters of thanks to those who had wished him well on his seventieth birthday when he collapsed and died. (e must say a little about another aspect of (eyl's work which we have not really mentioned, namely his work on mathematical philosophy and the foundations of mathematics. -t is interestin" to note what a lar"e number of the references we Guote deal with this aspect of his work and its importance is not only in the work =1
itself but also in the etent to which (eyl's ideas on these topics underlies the rest of his mathematical and physical contributions. (eyl was much influenced by 0usserl in his outlook and also shared many ideas with rouwer. oth shared the view that the intuitive continuum is not accurately represented by ;antor's set9 theoretic continuum. (heeler 4<<6 writes89 &he continuum ...% ;eyl taught us% is an illusion. !t is an idealization. !t is a dream. (eyl summed up his attitude to mathematics, writin"89 )y own mathematical wor"s are always 8uite unsystematic% without mode or connection. Expression and shape are almost more to me than "nowledge itself. But ! believe that% leaving aside my own peculiar nature% there is in mathematics itself% in contrast to the experimental disciplines% a character which is nearer to that of free creative art. 0is often Guoted comment89 )y wor" always tried to unite the truth with the beautiful% but when ! had to choose one or the other% ! usually chose the beautiful ... althou"h half a #oke, sums up his personality.
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Nor'ert ;iener )orn< 2+ Nov .4:6 in Col%8'ia? Misso%ri? USA 5ie"< .4 Marh .:+6 in Sto!hol8? S*e"en Nor'ert ;iener's father was Leo (iener who was a Russian ?ew. ecause Leo (iener was such a ma#or influence on his son, we should "ive some back"round to his education and career. Leo (iener attended medical school at the %niversity of (arsaw but was unhappy with the profession, so he went to erlin where he be"an trainin" as an en"ineer. This profession seemed only a little more interestin" to him than the medical profession, and he emi"rated to the %nited !tates havin" first landed in n"land. (e should note that throu"hout his education Leo was interested in mathematics and, althou"h he never used his mathematical skills in any #obs he held, it was a deep amateur interest to him all throu"h his life. Arrivin" in >ew Brleans in 1DD+, Leo tried his hand at various #obs in factories and farms before becomin" a school teacher in Fansas ;ity. 0e pro"ressed from bein" a lan"ua"e teacher in schools to becomin" rofessor of $odern Lan"ua"es at the %niversity of $issouri. (hile there he met and married ertha Fahn, who was the dau"hter of a department store owner. ertha, from a /erman ?ewish family, was 4C689 ... a small woman% healthy% vigorous and vivacious. !he #oined her husband in the boardin" house in ;olumbia, $issouri where their son >orbert was born in the followin" year. >ot lon" after >orbert's birth a decision was taken to split the $odern Lan"ua"es )epartment at the %niversity of $issouri into separate departments of rench and /erman. Leo was to #oin the /erman )epartment after the split but he lost out in some political manoeuvrin" so the family left ;olumbia and they moved to oston. There Leo brou"ht in money by takin" a variety of teachin" and other positions and eventually was appointed as an -nstructor in !lavic Lan"ua"es at 0arvard. This did not pay well enou"h to provide for his family, so Leo kept various other positions to au"ment his salary. 0e remained at 0arvard %niversity for the rest of his career, bein" eventually promoted to professor. As a youn" child >orbert had a nursemaid. (hen he was about four years old, a second child ;onstance was bornI (iener's second sister was born on 1:+1. 0e writes in 4C6 about his upbrin"in"89 ! was brought up in a house of learning. )y father was the author of several boo"s% and ever since ! can remember% the sound of the typewriter and the smell of the paste pot have been familiar to me. ... ! had full liberty to roam in what was the very catholic and miscellaneous library of my father. At one period or other the scientific interests of my father had covered most of the imaginable subects of study. ... ! was an omnivorous reader ... (iener had problems re"ardin" his schoolin", partly because the readin" which he had done at home had meant that he was advanced in certain areas but much less so in others. 0is parents sent him to the eabody !chool when he was seven years old and, after worryin" about which class he should enter, had him be"in in the third "rade. After a short time his parents and teachers felt he would be better suited to the fourth "rade and he was moved up a year. 0owever, he certainly did not fit into the school in either "rade and his teacher had little sympathy with so youn" a boy in the fourth "rade yet lackin" certain skills which would be epected the pupils at this sta"e in their education. 0e writes 4C689 )y chief deficiency was arithmetic. ,ere my understanding was far beyond my manipulation% which was definitely poor. )y father saw 8uite correctly that one of my chief difficulties was that manipulative drill bored me. ,e decided to ta"e me out of school and put me on algebra instead of arithmetic% with the purpose of offering a greater challenge and stimulus to my imagination.
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rom this time on (iener's father took over his education and he made rapid pro"ress for so youn" a child. 0owever, (iener had problems relatin" to his movements and was obviously very clumsy. This stemmed partly from poor coordination but also partly for poor eyesi"ht. Advised by a doctor to stop readin" for si months to allow his eyes to recover, he still had re"ular lessons from his father who now tau"ht him to do mathematics in his head. After the si months were up (iener went back to readin" but he had developed some fine mental skills durin" this period which he retained all his life. -n the autumn of 1:+5, at a"e nine, he was sent to school a"ain, this time to Ayer 0i"h !chool. The school a"reed to eperiment and to find the ri"ht level for (iener who was soon put into senior third year class with pupils who were seven years older than he was. The school only formed part of his education, however, for his father continued to coach him. 0e "raduated in 1:+= from Ayer at the a"e of eleven and celebrated with his ei"hteen year old fellow students 4C689 ! owe a great deal to my Ayer friends. ! was given a chance to go through some of the gaw"iest stages of growing up in an atmosphere of sympathy and understanding. -n !eptember 1:+=, still only eleven years old, (iener entered Tufts ;olle"e. !ocially a child, he was an adult in educational terms so his student days were not easy ones. Althou"h takin" various science courses, he took a de"ree in mathematics. (iener's father continued to coach him in mathematics showin" complete mastery of under"raduate level topics. -n 1:+: (iener "raduated from Tufts at a"e fourteen and entered 0arvard to be"in "raduate studies. Rather a"ainst his father's advice, (iener be"an "raduate studies in oolo"y at 0arvard. 0owever thin"s did not "o too well and by the end of a year a decision was taken, partly by (iener partly by his father, that he would chan"e topic to philosophy. 0avin" won a scholarship to ;ornell he entered in 1:1+ to be"in "raduate studies in philosophy. Takin" mathematics and philosophy courses, (iener did not have a successful year and before it was finished his father had made the necessary arran"ements to return to 0arvard to continue philosophy. ack at 0arvard (iener was stron"ly influenced by the fine teachin" of dward 0untin"ton on mathematical philosophy. 0e received his h.). from 0arvard at the a"e of 1D with a dissertation on mathematical lo"ic supervised by Farl !chmidt. rom 0arvard (iener went to ;ambrid"e, n"land, to study under Russell who told him that in order to study the philosophy of mathematics he needed to know more mathematics so he attended courses by / 0 0ardy. -n 1:1* he went to /ttin"en to study differential eGuations under 0ilbert, and also attended a "roup theory course by dmund Landau. 0e was influence by 0ilbert, Landau and Russell but also, perhaps to an even "reater de"ree, by 0ardy. At /ttin"en he learned that 4C689 ... mathematics was not only a subect to be done in the study but one to be discussed and lived with. (iener returned to the %nited !tates a couple of days before the outbreak of (orld (ar -, but returned to ;ambrid"e to study further with Russell. ack in the %nited !tates he tau"ht philosophy courses at 0arvard in 1:1<, worked for a while for the /eneral lectric ;ompany, then #oined ncyclopedia Americana as a staff writer in Albany. (hile workin" there he received an invitation from Eeblen to undertake war work on ballistics at the Aberdeen rovin" /round in $aryland. Takin" about mathematics with his fellow workers while undertakin" this war work revived his interest in mathematics. At the end of the war Bs"ood told him of a vacancy at $-T and he was appointed as an instructor in mathematics. 0is first mathematical work at $-T led him to eamine rownian motion. -n fact, as (iener eplained in 4C6, this first work would provide a connectin" thread throu"h much of his later studies89 ... this study introduced me to the theory of probability. )oreover% it led me very directly to the periodogram% and to the study of forms of harmonic analysis more general than the classical 1ourier series and 1ourier integral. All these concepts have combined with the engineering preoccupations of a professor of the )athematical !nstitute of &echnology to lead me to ma"e both theoretical and practical advances in the theory of communication% and ultimately to found the discipline of cybernetics% which is in essence a statistical approach to the theory of communication. &hus% varied as my scientific interests seem to be% there has been a single thread connecting all of them from my first mature wor" ...
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0e attended the -nternational ;on"ress of $athematicians at !trasbour" in 1:2+ and while there worked with r@chet. 0e returned to urope freGuently in the net few years, visitin" mathematicians in n"land, rance and /ermany. specially important was his contacts with aul L@vy and with /ttin"en where his work was seen to have important connections with Guantum mechanics. This led to a collaboration with orn. -n 1:2= (iener married $ar"aret n"emann, and after their marria"e (iener set off for urope as a /u""enheim scholar. After visitin" 0ardy in ;ambrid"e he returned to /ttin"en where his wife #oined him after completin" her teachin" duties in modern lan"ua"es at ?uniata ;olle"e in ennsylvania. Another important year in (iener's mathematical development was 1:51952 which he spent mainly in n"land visitin" 0ardy at ;ambrid"e. There he "ave a lecture course on his own contributions to the ourier inte"ral but ;ambrid"e also provided a base from where he was able to visit many mathematical collea"ues on the ;ontinent. Amon" these were laschke, $en"er and rank who invited him to make a visit, while he also met 0ahn, Artin and /del. (iener's papers were hard to read. !ometimes difficult results appeared with hardly a proof as if they were obvious to (iener, while at other times he would "ive a len"thy proof of a trivialit y. reudenthal writes 41689 All too often ;iener could not resist the temptation to tell everything that cropped up in his comprehensive mind% and he often had difficulty in separating the relevant mathematics neatly from its scientific and social implications and even from his personal experiences. &he reader to whom he appears to be addressing himself seems to alternate in a random order between the layman% the undergraduate student of mathematics% the average mathematician% and ;iener himself. )espite the style of his papers, (iener contributed some ideas of "reat importance. (e have already mentioned above his work in 1:21 in rownian motion. 0e introduced a measure in the space of one dimensional paths which brin"s in probability concepts in a natural way. rom 1:25 he investi"ated )irichlet's problem, producin" work which had a ma#or influence on potential theory. (iener's mathematical ideas were very much driven by Guestions that were put to him by his en"ineerin" collea"ues at $-T. These Guestions pushed him to "eneralise his work on rowian motion to more "eneral stochastic processes. This in turn led him to study harmonic analysis in 1:5+. 0is work on "eneralised harmonic analysis led him to study Tauberian theorems in 1:52 and his contributions on this topic won him the Ucher rie in 1:55. 0e received the prie from the American $athematical !ociety for his memoir &auberian theorems published in Annals of )athematics in the previous year. The work on Tauberian theorems naturally led him to study the ourier transform and he published &he 1ourier !ntegral% and 0ertain of !ts Applications 31:557 and 1ourier &ransforms in 1:5*. (iener had an etraordinarily wide ran"e of interests and contributed to many areas in addition to those we have mentioned above includin" communication theory, cybernetics 3a term he coined7, Guantum theory and durin" (orld (ar -- he worked on "unfire control. -t is probably this latter work which motivated his invention of the new area of cybernetics which he described in 0ybernetics: or% 0ontrol and 0ommunication in the Animal and the )achine 31:*D7. reudenthal writes in 41689 ;hile studying anti-aircraft fire control% ;iener may have conceived the idea of considering the operator as part of the steering mechanism and of applying to him such notions as feedbac" and stability% which had been devised for mechanical systems and electrical circuits. ... As time passed% such flashes of insight were more consciously put to use in a sort of biological research ... 40ybernetics6 has contributed to popularising a way of thin"ing in communication theory terms% such as feedbac"% information% control% input% output% stability% homeostasis% prediction% and filtering. 2n the other hand% it also has contributed to spreading mista"en ideas of what mathematics really means. (iener himself was aware of these dan"ers and his wide dealin"s with other scientists led him to say89 2ne of the chief duties of the mathematician in acting as an adviser to scientists is to discourage them from expecting too much from mathematics. !ome of (iener's publications which we have not mentioned include 5onlinear 6roblems in +andom &heory 31:
31:=*7. (e have mentioned above reudenthal's comments on (iener's poor writin" style. 0is most famous work 0ybernetics comes in for special criticism by reudenthal89 Even measured by ;ieners standards 90ybernetics9 is a badly organised wor" -- a collection of misprints% wrong mathematical statements% mista"en formulas% splendid but unrelated ideas% and logical absurdities. !t is sad that this wor" earned ;iener the greater part of his public renown% but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings. reudenthal, in 416, describes both (iener's appearance and his character89 !n appearance and behaviour% 5orbert ;iener was a baro8ue figure% short% rotund% and myopic% combining these and many 8ualities in extreme degree. ,is conversation was a curious mixture of pomposity and wantonness. ,e was a poor listener. ,is self-praise was playful% convincing and never offensive. ,e spo"e many languages but was not easy to understand in any of them. ,e was a famously bad lecturer. ) / Fendall writes 45689 As a human being ;iener was above all stimulating. ! have "nown some who found the stimulus unwelcome. ,e could offend publicly by snoring through a lecture and then as"ing an aw"ward 8uestion in the discussion% and also privately by proffering information and advice on some field remote from his own to an august dinner companion. ! li"e to remember ;iener as ! once saw him late at night in )agdalen 0ollege% 2xford% surrounded by a spellbound group of undergraduates% tal"ing% endlessly tal"ing.
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