Mathematicians I have known Michael Atiyah http://www.maths.ed.ac.uk/~aar/atiyahpg Trinity Mathematical Society Cambridge 3rd February, 2 !" T1 The Michael and +ily Atiyah Portrait Gallery 1ames Clerk Ma'well 2uilding, 3niversity o$ *dinburgh The #ortraits of mathematicians displayed in this collection have been #ersonally selected by us. They have been chosen for many di$ferent reasons, but all have been involved in our mathematical lives in one way or another; many of the individual te'ts to the gallery #ortraits e'#lain how they are related to us. First, there are famous names from the #ast ( starting with Archimedes ( who have built the great edi$ice of mathematics which we inhabit% This early list could have been more numerous, but it has been restricted to those whose style is most a##ealing to us. )e't there are the many 2teachers, both in Edinburgh and in Cambridge, who taught us at various stages, and who directly influenced our careers. The bulk of the #ortraits are those of our contemporaries, including some close collaborators and many Fields Medallists. +ily has a special interest in women mathematicians: they are well re#resented, both #ast and #resent% Finally we come to the ne't generation, our students% -f course, many of the categories overla#, with students later becoming collaborators and $riends. It was hardest to kee# the overall number down to seventy, to $it the gallery constraints! T2 !4 Classical T3 Leonhard Euler Studied in Basel% Worked in St% Petersburg and Berlin% F6S% 7is name ap#ears in all branches o$ mathematics: *uler characteristic, Euler8 +agrange e9uations, Euler:s constant% 7e gave the symbols to various im#ortant constants that we all use, such as e and % *uler’s solution of the Königsberg bridge problem was the beginning o$ topology% *uler introduced the famous in$inite #roduct that embodies the uni9ue $actorization of integers into prime factors, found the beauti$ul formula !>n2 ? >@, and the fundamental relation eicosA4Bi sinA) between the ex#onential function and trigonometry% 2lind towards the end of his li$e, Euler continued to work until the very end – an encouragement to the older generation of the present time. T4 Niels Henrik Abel 7is story o$ young genius, with a few years of creative li$e followed by an early death from tuberculosis, is iconic and is why the Abel Prize was founded in 2 2% 7is ideas on algebraic integrals were far in advance o$ his time, going well beyond the known theory of elli#tic functions% The general view8#oint he adopted is at the foundation of modern mathematics% He was the first to show that the general equation of degree 5 is not soluble by radicals% The brevity and eu#hony of his name has widened his a##eal, with abelian $unctions, integrals, grou#s, varieties and categories% Having your name become an adjective – without a capital – is the mark of real fame, as exem#li$ied earlier by the terms euclidean and newtonian% T5 Bernhard Riemann A student of Gauss, Riemann worked in Göttingen, and was a foreign member of the Royal Society% 7is name and work live on in Riemannian geometry, Riemann sur$aces, the 6iemann Hy#othesis, the Riemann86och Theorem, the Riemann87ilbert #roblem, and many more. 7is foundational work on di$$erential geometry was created as a sideline, to im#ress his doctoral examiners% 3nlike Euler his collected works take u# only one modest volume, but each chapter opens a new door in mathematics% 6iemann died of tuberculosis at the height of his powers, aged just 4 % T6 William Rowan Hamilton At 21, he was ap#ointed Royal Astronomer of Ireland, while still an undergraduate at Trinity College Dublin% Honorary F6S*% 7e made his name early by predicting conical refraction, ex#erimentally con$irmed a short while later% Now famous $or Hamiltonian mechanics, which became the paradigm for quantum mechanics a century later% 7amilton is famous in mathematics for his discovery of the quaternions, carving the equations on Broom 2ridge in Dublin where he had his brainwave in 18"3% He opened the door to non-commutative algebra and, in physics, to the role of the grou# S3A24% In one famous passage in an 18"@ pa#er, Hamilton notes that a first order di$$erential operator arising naturally in 9uaternions is the s9uare root of the +a#lace operator% He o#ines that this must have deep ap#lications in physics% 7is forecast was fully justi$ied since his operator was what Dirac rediscovered some 8 years later% T7 Solomon Lefschetz *ducated in France, moved to the United States in 1G C% Professor in Kansas and Princeton% Foreign Member RS% Trained as an electrical engineer, he lost both his hands in a lab accident in !G H% He became interested in the theory of higher dimensional algebraic varieties, following in the footsteps of Émile Picard. 7e develo#ed the topological tools needed for the pur#ose and #roved some $undamental theorems (later taken u# by Hodge and others4% In particular he developed intersection theory on mani$olds as a basis for algebraic geometry% The Lefschet= fi'ed point theorem became famous and led to many a##lications and generalizations, including the Weil conDectures and the Atiyah82ott theorem for elli#tic com#lexes% A towering presence in Princeton and great rival of another Solomon A2ochner4% Always claimed that his many years of isolation in Kansas had been ideal for his mathematics – no interru#tions% T8 Hermann Weyl /rofessor at Göttingen, Zürich and the Institute for Advanced Study, /rinceton% Foreign Member RS% 7is work on the representation theory of com#act Lie grou#s is well-known and his 1G!3 book on Riemann sur$aces laid the foundations for all later analytic geometry% 7e was interested in physics, particularly in the role of symmetry% He introduced gauge theory in an ef$ort to combine the equations of Einstein and Ma'well% He proved rigorously what physicists intuited, that the high frequencies of a quantum system converge to the classical limit% 7e was a su##orter o$ Hodge in his work on Harmonic Forms (correcting a technical error4, and later of Kodaira, whom he brought from Ja#an to /rinceton% Michael heard him at the 1GC" International Congress of Mathematicians in Amsterdam when he presented the Fields Medals to ;odaira and Serre. Michael had the unusual op#ortunity o$ writing the biographical memoir for Weyl A$or the US National Academy4 5 years a$ter his death, recti$ying an oversight% T9 2) The French School T10 Élie Cartan /rofessor in Nancy and Paris% Foreign Member RS% Cartan introduced di$$erential forms and the e'terior derivative, extending calculus to mani$olds% These are key ingredients in modern di$$erential geometry, and of de Rham cohomology% His work was particularly concerned with the geometric ap#lications of Lie grou#s, which are mani$olds with a grou# structure% 7e contributed to the definitive classi$ication of sim#le Lie grou#s% 7is work had a maDor in$luence on mathematical physics% Father of 7enri Cartan% T11 Henri Cartan /rofessor in Strasbourg and Paris% Foreign Member RS% 7is contributions were in the global theory that became possible a$ter the #ioneering work of Leray% Sheaf theory and cohomology were combined into a #ower$ul tool for the global theory of several com#lex variables% The purely $ormal side, developed in conDunction with Eilenberg, gave birth to Lhomological algebra”% Cartan had many brilliant students at the École Normale Su#érieure, notably 6ené Thom and Jean8/ierre Serre, both Fields Medallists% He was also active internationally, becoming the President o$ the International Mathematical 3nion% Cartan was the first French mathematician in contact with German colleagues a$ter World War II, in contrast to the ostracism shown a$ter World 5ar I% T12 Jean Leray /rofessor in Paris, Collège de France. Foreign Member RS% 7e started of$ with signi$icant contributions to the Navier-Stokes equations of $luid dynamics% But a quirk of fate diverted him in a di$$erent direction% Euring the war he was held in a 0erman prisoner of war cam# in Silesia. The cam# contained many scientists who were allowed to hold seminars% Leray was a$raid that, i$ the Germans realized he was an ex#ert on fluid dynamics, they might put him to military use. So he turned himself into “a useless topologistM% 7e then developed both s#ectral sequences and sheaves, the two most power$ul techni9ues of 2 th century topology% T13 Jean"Pierre Serre Studied at the École Normale Su#érieure in Paris under Henri Cartan% /rofessor at the Collège de France for 3F years, from 195@ on% Foreign Member 6S% 7e revolutionized both algebraic to#ology and algebraic geometry by using shea$ theory and s#ectral se9uences, both invented by +eray% He provided techni9ues to calculate the homoto#y grou#s of s#heres and success$ully used the Zariski topology as the basis for the cohomology grou#s of coherent sheaves, following the analytic a##roach of Henri Cartan% Serre worked subsequently in number theory and was very in$luential, with many fruit$ul ideas and conDectures% T14 Ale$ander %rothendieck 0rothendieck started research in functional analysis and solved many hard #roblems proposed by Dieudonné and Schwart=% He then moved into algebraic geometry, in the wake of Serre, and trans$ormed the subDect com#letely% He #roduced a new and elegant proof of the Hir=ebruch86iemann86och theorem, which was purely algebraic% For this he introduced K-theory, which has had an extensive li$e in other fields% Michael heard his first lectures on the subDect in 2onn in 1GCH and this motivated the use of K8theory in topology% 7is introduction of étale cohomology grou#s paved the way for the eventual #roof by Deligne of the famous Weil conDectures concerning the number of rational points of varieties over finite fields% An intense, power$ul and eccentric personality, he gave u# mathematics in !GH , and devoted himself to idealistic causes% T15 3) The Cambridge School T16 Paul Dirac +ucasian Professor of Mathematics at Cambridge 19328!G@G% F6S, Honorary F6S* and Nobel Prize.