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Home , Peter Hilton

Peter Hilton: Codebreaker and Mathematician (1923–2010) Jean Pedersen, Coordinating Editor

Peter Hilton was German. He then won a scholarship to Queen’s born in London on College, Oxford, where he read mathematics. When April 7, 1923, to he was eighteen, fate intervened for a second time. Elizabeth Freedman Peter began work at in January and a family physi- 1942 (see Copeland’s contribution below for more cian, Mortimer Hilton, details); he thereby avoided service in the military, whose office was in where, as Peter said at the time, “I would surely Peckham. His parents die—of boredom!” On Peter’s eightieth birthday, hoped that he would , Peter’s coauthor, wrote: become either a doc- “This young chap from Oxford tor (as his brother joined us in at Bletchley, Photo courtesy of Jean Pedersen. Sydney did) or a and soon made his mark. When he Peter Hilton at Santa Clara lawyer. Fate first in- moved on to work on Fish, he did University (c. 1985). tervened when he was more than that; he dominated his run over by a Rolls Royce at the age of ten. During a long hospital stay section... he was brilliant at his job he discovered that he could use the plaster cast on and enormous fun to be with.” his left leg, which reached all the way to his navel, Peter was awarded an M.A. at Oxford in 1948, as a sort of white board; he solved mathematical a D.Phil. from Oxford in 1950 for his thesis, Cal- 2 problems on it, erasing them each morning. This culation of the Homotopy Groups of An-Polyhedra, is when, in his words, “It came to me that I really and a Ph.D. from Cambridge in 1952. loved mathematics and thoroughly enjoyed doing He married Margaret (Meg) Mostyn on Septem- it. I recall even having unkind thoughts about ber 14, 1949; they had two sons, Nicholas and visitors who came to see if I was all right, as they Tim. Peter and Meg shared a love of theater, Meg would interrupt me when I was really enjoying professionally and Peter very much less formally. what I was doing.” Indirectly, Peter’s love of theater led to our He attended St. Paul’s School in Hammersmith, meeting. In 1977 I was asked to get George Pólya, where, being enlightened educators, they left him with whom I had been studying informally for to study what he wanted in the later years. With about ten years, and Peter together to do a pre- unexplained prescience about the unrest leading sentation at a joint meeting of the Mathematical up to World War II, he decided to teach himself Association of America (MAA) and AMS in Seat- tle. They wanted Hilton to tell about “How Not Jean Pedersen is professor of mathematics and com- to Teach Mathematics” and then have Pólya give puter science at Santa Clara University. Her email is [email protected]. “Some Rules of Thumb”. When I wrote to Peter, he said “No!” He thought it would be more interesting The advice from Gerald Alexanderson and Ross Geoghe- gan, the technical assistance of Bin Shao with the text, (and fun) if he simulated a thoroughly bad demon- and the expertise of Chris Pedersen with the pictures are stration of teaching mathematics, and he agreed gratefully acknowledged. to do that, instead, if I would be the moderator.

1538 Notices of the AMS Volume 58, Number 11 The lectures took place on August 14, 1977. A repeat performance, which was televised [1], took place in the fall of 1978 at the San Diego meeting of the National Council of Teachers of Mathematics. Peter then agreed to come to Santa Clara University to give a colloquium talk. When he saw my office and heard about my interest in geometry, we discovered we both had a strong interest in polyhedra (from totally different points of view), and, as they say, the rest is history! Our mutual interest lay in the mathematics involved with geometry. Peter got interested in an algorithm I had developed that concerned a systematic method of folding a straight strip of Photo courtesy of Meg Hilton. paper, say, adding-machine tape, that produced Meg, Peter, and son Nicholas (c. 2008). increasingly good approximations for any given rational multiple of π, at equally spaced points the facts then you would share my along an edge of the paper. We began working conclusions, and you persuaded seriously on it in January of 1981 when we were me that if I believed in your ver- both at the Eidgenössische Technische Hochschule sion of the facts then I would share (ETH) in Zürich. This naturally took us into number your conclusions. It was a kind of theory, polyhedral geometry, and combinatorics, mathematical approach; an agree- and it continued for over thirty years until the ment about the proofs, whatever time of Peter’s death. The relevant mathemat- the hypotheses and theorems. It was a lesson that has served me ics is chronicled in our most recent book, A well throughout my life, in con- Mathematical Tapestry: Demonstrating the Beauti- versations about everything under ful Unity of Mathematics [2]. In Section 16.6 we the sun.” wrote about “Pólya and ourselves—Mathematics, tea and cakes”, which describes, as the title sug- I was blessed with two great mentors, first gests, more about our interactions with each other George Pólya (for far too short a time) and then and with Pólya. Peter Hilton. Pólya introduced me to polyhedral An event that demonstrates the joie de vivre of geometry, the wonders of “looking for patterns”, collaborating with Peter involved the construction and the importance of posing problems. Peter of a figure for one of our articles. We needed a star showed me how to generalize and how to see con- 11 nections between various parts of mathematics. { }-gon (i.e., the top edge of the tape would visit 3 Both Pólya and Peter made the doing of mathemat- every third vertex of a bounding regular convex ics an adventure constantly filled with excitement. 11-gon). I had pulled the tape a little too tight and Together Peter and I coauthored 144 papers and got a star { 10 }-gon. When I reported this mishap 3 6 books, often giving joint lectures on our current to Peter he said cheerfully, “Don’t worry, we say in topic of interest. the beginning these are only approximations!” He My family was enriched by our friendship with never failed to see the humorous side of things. Peter’s family. I was excited to watch Meg perform Although Peter excelled at abstract thinking, (as Jim Dale’s mother) in the production of the he always applied mathematics to real life. He Broadway Tony-Award-winning play, A Day in the came to understand the principle of mathematical Death of Joe Egg (right after she won the Clarence induction at the age of seven when drying dishes Dewent award for her role in Molly). I have fond for his mother. He realized that the old trick of memories of Peter, Meg, Kent (my husband), and drying two plates at a time could, in theory, be me on our land-cruise tour of Alaska. I smile extended by induction to n plates. As for applying recalling when Kent, Peter, and I walked through mathematics to social situations, Sir Christopher some drizzly California redwoods on what Peter Zeeman said in his letter on Peter’s eightieth called a “lovely London day”, and later we played birthday: three-handed bridge at our house (where Peter “From you I also learned how to got the bid with an appallingly bad dummy). talk politics, without endangering I will always miss Peter, but I am grateful for friendship. You taught me that if the memories and the years of collaboration we we disagreed about some conclu- shared. sions then we should go back and examine our beliefs about the un- References derlying facts. You acknowledged [1] Hilton-Pólya DVD, recorded on VHS in 1978, dig- thatifyoubelievedinmyversionof itized 2011. Available from Media Services, Santa

December 2011 Notices of the AMS 1539 1540

Photo courtesy of Jean Pedersen. ainrprsfo eeasa h rn line, front the sit- at detailed generals as from such reports signals, uation top-level rang- from day, many ing each messages transmitted encrypted of military thousands German communi- The radio enemy cations. enjoyed to allies access her unprecedented and Britain 1939–1945 During Copeland Jack i he fsa,GnrlImy ae toured panel A instructed Ismay. Churchill General staff, done,” me of been to chief report his all had and have this priority they extreme that sure on “Make want letter: famous they the Churchill’s to was DAY” response THIS staff “ACTION more recruit [1]. to need Churchill, urgent Winston the Sir emphasizing leader, wartime Britain’s wrote to Welchman, Gordon and Milner-Barry, Stuart Alexander, Hugh Turing, Alan codebreakers, senior obituary.” often,” own his not to man listening is a of “that pleasure “It podium, the the much here. has took used he as as I then declared Park. Peter words Bletchley same at work the and described life he which his in me in lecture asked public role a Peter introduce key 2002 to In a operation. played of incredible Hilton this hours Peter at few codebreaker Park, a leading Bletchley a within As often transmitted. being hands, its Allied ended in information contents this up of the Much of vessels. supply inventories of and as such reports war, of weather minutiae important the to through nvriyo atruy e eln.Hsealis: email Computing, His of Zealand. History New the [email protected] Canterbury, for director of and Archive philosophy University Turing of the professor of is Copeland Jack [2] nOtbr14 oro lthe aksmost Park’s Bletchley of four 1941 October In etadJa eesn(nlf)wt Peter with left) (on Pedersen Jean and Kent eosrtn h euiu nt fMathematics Press, Unity Beautiful the Donmoyer, Demonstrating Sylvie by Hilton P. (US$25.00 handling). and 95053 postage CA + Clara, Santa University, Clara itni h aionardod (1996). redwoods California the in Hilton and .Pedersen J. ahmtclTapestry: Mathematical A ihillustrations with , . oie fteASVolume AMS the of Notices , nelgnefo u ees ucsflthat successful so were The 8 traffic. Hut U-boat from on based intelligence daily reroutings convoy the immediate; group was effect into his and Turing broke that finally 8, Hut in before arrival was months Peter’s few it a them; 1941, June around until however, the routed not, be read, and could known, convoys be remained be the would could U-boats submarines the Enigma the of positions U-boat by If used unbroken. Enigma of secure highly of the quantities German but form the services, large other by and Force reading transmitted Air being time being traffic this of Enigma danger by imminent were codebreakers Bletchley in The surrender. into was starved successfully Britain America—so Naval that raw North on other from and attack were oil, materials U-boats food, the Atlantic carrying of convoys North sinking the line 1941 front as In Enigma. simply the known 8”, organization “Hut the in Alexander nagrdb h re o ftect obe to city the of mob being ruthlessly.” armed taken troops the prevent by to endangered unfavourable Measures of contested. be case “Evacu- in [2]: development fighting Budapest without of ation defense the Guderian General regarding to 1944, Hitler from December order dated an detailing decrypt, this is by work conveyed the of fascination would and The horror, himself. he significance, Hitler by Occasionally signed messages lives. break even incal- of an number saved codebreakers culable fellow his and Peter up. him mathematical snatched read panel to The order work. in school at language the second- himself taught working had University Peter brilliant a German. of the knowledge with a at undergraduate interviewed mathematics 1941, year they of Oxford, end of the toward recruits, and suitable for searching universities British ngamcieadAa uigo t Helena St. on Turing Alan and machine Enigma ee okda rtaogieTrn and Turing alongside first at worked Peter yepsn h ealdtikn fteenemy, the of thinking detailed the exposing By . . . u fqeto.Eeyhueto house Every question. of out 58 tm,2005. stamp, Number , 11 the North Atlantic U-boats did not sight a single At first the broke Tunny messages convoy for twenty-three days following Turing’s purely by hand, using a method invented by first break. Turing in July 1942 It was in the heady period sparked by these known as “”. early successes that Peter joined Hut 8, on Janu- The following twelve ary 12, 1942 [3, Ch. 15]. His talent soon stood out, months saw approxi- and he was given the job of breaking “Offizier” mately 1.5 million [6] messages. Normally, the operators at the send- characters of cipher- ing and receiving ends of the Enigma link would text broken in this see the plaintext of the transmitted message, but way. As the Testery’s Offiziers were messages so sensitive that only chief mathematician, officers, not the machine operators themselves, Peter honed the unit’s were allowed to view the contents. Offiziers were cryptanalytical methods encrypted (and decrypted) twice, once by an offi- during this early phase cer and once by the usual Enigma crew, who saw of breaking Tunny. Photo courtesy of Meg Hilton. only ciphertext [4, pp. 14–15]. Alexander (Turing’s Thereafter, as the vol- A young Peter Hilton. successor as head of Hut 8) described Offiziers ume of Tunny traffic as the hardest of all Naval Enigma messages to grew and the number of operator errors—manna break [4, p.16], but break them Peter did, using to the codebreakers—became fewer, the Testery “cribs”—guessed words of the message. Peter was began to receive increasing help from high-speed a very adept guesser. analytic machinery, most notably the vast Colos- Toward the end of 1942 Peter’s skills were sus (from February 1944). Designed and built by needed elsewhere, and he was transferred from Tommy Flowers, Colossus was the first large-scale Hut 8 first to the Research Section and from there to electronic digital computer. By war’s end there the Testery, a section headed (naturally enough) by were ten Colossi operating around the clock in the Newmanry, a section headed by topologist Major Ralph Tester. The Testery’s single function Max Newman. The Newmanry was the world’s first was to break a new German teleprinter (teletype- electronic computing facility, and Peter took up writer) cipher codenamed “Tunny” by the British the role of liaising between it and the Testery. [5]. Tunny was quite unlike Enigma (although the “Life,” he said, “was as interesting as it could two are often confused in the literature). What the possibly be” [3, p. 194]. British called the Tunny machine was known to Using an algorithmic method devised by Tutte the Germans as the Schlüsselzusatz SZ40, a state- and tweaked by Newman and his mathematicians, of-the-art twelve-wheel cipher machine produced Colossus mechanically stripped away one layer in 1940 by the Lorenz Company. Enigma, on the of encryption from a Tunny message. The result, other hand, had three (or sometimes four) wheels known as a “de-chi”, still carried a second layer of and dated from the early 1920s. The Tunny ma- encryption, and the de-chi was passed on to the chine required only a single operator, whereas the Testery to be broken by human patience and inge- clumsier Enigma needed three, including a wireless nuity. The two sections, Testery and Newmanry, operator who tapped out the enciphered message worked hand in glove.Peterand his fellow breakers in Morse code. The operator of a Tunny machine in the Testery (Jerry Roberts, Peter Edgerley, Denis simply typed plain German at the teleprinter key- Oswald, Peter Ericsson, and others [3, Chs. 18 and board, and the rest was automatic. Morse was 21]) would chip away at the de-chi, using cribs not used; the encrypted output of the Tunny ma- to expose small sections of the plaintext. They chine went directly to air. The extent to which extended these short breaks by further guesswork the Lorenz engineers had succeeded in automat- until they had about thirty to eighty or so con- ing the processes of encryption and decryption secutive characters of “clear”—enough to work was striking; under normal operating conditions, out the settings of the wheels, so that the whole neither the sending nor the receiving operator message could be deciphered. Some of these short ever even saw the coded form of the message. breaks (usually all that the codebreaker saw of The British first intercepted Tunny messages in the plaintext) were imprinted on Peter’s memory June 1941, and, in January 1942, the great Bill for the rest of his life—“Ich bin so einsam” (I am Tutte single-handedly deduced the fundamental so lonely), from an operator on the Leningrad structure of the Tunny machine. The rest of the front, and “Mörderische Hitze” (murderous heat), Research Section joined him in his investigation, from the Italian front. The Testery’s “ATS girls” and soon the whole machine was laid bare, without (Auxiliary Territorial Service—the Women’s Army) any of them ever having set eyes on one. It was the performed the final stage of the decryption. They most remarkable feat of of the war. typed the often lengthy ciphertext into a British Tutte’s deductions broke open the entire Tunny replica of the Tunny machine, and plain German system. would emanate from the printer.

December 2011 Notices of the AMS 1541 1542

Photo taken by Marge Dodrill, courtesy of Meg Hilton. References thanked country. adequately his never by was Peter is that It shame sport). or a ser- business, for government, awarded local to regularly vices now Jerry honor least (an and very OBE the Hilton the at or Peter knighthoods like deserved Roberts Men his work. the for priceless whatsoever got recognition others public no Tutte “as received computer.” electronic 201], the p. creating si- for [3, credit in said listen and Peter to wealthy lence,” “had a Flowers become publicized engineer. even have celebrated or would patented he have Colossus, could his he in pounds— if 1,000 kept yet for he cheque a (which Flowers OBE and toolbox) an than received more Turing contribution nothing Ludicrously, victory. massive Allied their the little for to two received recognition as colleagues no much his or as and by Peter war yet D-Day years, the (the shortened have 1944 codebreakers Bletchley may the in of work France The landings). Northern of vasion in- Allied anticipated knowl- the for detailed counter-preparations providing concerning example, by strategy—for German of war, edge the the of changed course that intelligence contained Tunny decrypts field. the in generals the Army and Command the High between messages intelligence, of grade 2 rts esg eeec number reference message British [2] 1 etrt iso hrhl 14) in (1941), Churchill Winston to Letter [1] un ulk nga are nytehighest the only carried Enigma) (unlike Tunny est rs,20.Telte n Churchill’s Kew Archives, and National the letter in are The Ismay to 2004. minute Press, versity ouetrfrneDF /1.Tak to of Thanks copy a with 3/318. me message. this supplying for DEFE Archives Newman William National reference 9622); document (HP CX/MSS/T400/79 1/155). HW reference (document Copeland ee,digmteais(.1980). (c. mathematics doing Peter, (ed.), h seta Turing Essential The oie fteASVolume AMS the of Notices xodUni- Oxford , .J. B. hudg owr ihWieedi o the got I if I Whitehead that with decided work I to about go Whitehead. particularly should Henry him, and from the Oxford were heard Many I companion. anecdotes congenial very a him hntnyasm eir(lot5 ecn!,but a percent!), 50 had (almost he more senior was my He years together. ten bit than a quite out hung we and and principles Young underlying for general them. looking exploiting most School about the much Finishing me for taught “Hilton He the Topologists”. as book to the Duality became and Theory year that organized visiting beautifully lectures and was elegant His Peter year. the that for delighted absolutely was attitudes. Perelman, own J. my S. with resonated of Hilton works Marx and the the in and teenager a movies wit, as Brothers steeped was razor-sharp immediately I conversation me. his struck his visiting of charm, elegance was the modest and he His when there. Rochester) (from Cornell theorem), thesis. my generalized, in Hilton-Milnor role important (which, the an played [1] as which spaces known beautiful his loop became read on I student, attention paper graduate my to a came as first when, Hilton Peter name The Browder Bill ilBodri rfso fmteaisa Princeton at mathematics of is email professor His University. is Browder Bill [3] 5 h ulntr n cp fBeclysattack Bletchley’s of scope and nature full The [5] [4] 6 nomto rmJryRoberts. Jerry from Information [6] e a o ihhmmc ftetm htyear, that time the of much him with not was Meg I and year, following the Cornell to moved I visited I 1957, in when, man the met first I .Jc Copeland Jack B. otismve flcue yPtrHlo,Jerry web- photographs. Hilton, with Colossus Peter along others, by the and lectures Roberts, also of see movies contains [3]; first in was site print Tunny on in attack told Bletchley’s of Computing story of in The History is the Tunny” for at on Archive Report Turing dig- “General the A of 2)). (vol. facsimile 25/5 ital HW 1), (vol. HW25/4 “Gen- document reference Archives Jack Timms, (National Tunny” Geoffrey Tunny-breakers on when Report and eral by Michie, 500-page 2000, 1945 Donald a until Good, in declassified written revealed Government history not British was the Tunny on Comput- of History in the _naval_enigma Na- is for at 1945; typescript Archive ing Alexander’s (c. Turing A of the 25/1). Enigma, HW facsimile reference Naval digital document German Archives tional the on Alexander 2010. Hugh ed. 2nd 2006; Press, Computers University Codebreaking Park’s Bletchley http://www.AlanTuring.net/tunny_report http://www.colossus-computer.com oed vivre de joie http://www.AlanTuring.net/alexander . [email protected] rporpi itr fWork of History Cryptographic , tal. et , 2,wihIcm orefer to came I which [2], n nomlt htmade that informality and ooss h ert of Secrets The Colossus: 58 Number , Homotopy Oxford , which , 11 . . chance, which came when I got an NSF postdoc Now Meg was with him, which added dra- the following year. matically to the social scene. An accomplished Peter and I met again the summer of 1960 professional actress, her charm coupled with in Zürich at the Hopf birthday conference. He Peter’s made a great impact. had recommended to me a hotel run by the Peter’s mathematical interests overlapped mine Zürcher Frauenverein für Alkoholfreie Wirtschaft a great deal in the early days, and I remember (the Zürich Ladies Club for Alcohol-Free Hos- particularly his work with Berstein on the “co- pitality), which was very clean, pleasant, and H-spaces” (spaces with Lusternick-Schnirelmann inexpensivebutprohibitedalcoholonthepremises category 2) which were not suspensions [3], which (obviously). I was newly married, and we decided to emerged beautifully from the Eckmann-Hilton du- stay there to save money and to take our libations elsewhere. ality point of view, and later his work with Roitberg Peter acted as a guide for us at times, and I producing a new H-space of unexpected homotopy admired his fluency in German, his deep acquain- type [4]. tance with Zürich, and his general sophistication. The latter earned the nickname the “Hilton- My experiment imitating his smoking of the small Roitberg criminal” rather than “example” or other cigars called Stumpen went nowhere, and Ger- standard terminology. The only other similar man remained out of reach, but my admiration of terminology I know of is the famous “Eilen- Peter’s urbanity only increased. berg swindle”. These are the only “criminals” An inveterate traveler, he soon received the or “swindles” I know of in mathematics. honor of the following joke: An innocent tourist I left Cornell in 1963, and Peter left a few years looking for a hotel asks a passing mathematician later. “Where is the Ithaca Hilton?” and receives the We met again at his sixtieth birthday party in reply “Oh, he’s in Zanzibar this week.” 1983 in St. John’s, Newfoundland. He and I were Of his many destinations, Africa and Eastern asked to participate in a local television program Europe seemed frequent. In Romania, he made to discuss mathematics education. We disagreed the acquaintance of the Bucharest topology group, amicably for most of the program, and then, with consisting among others of Tudor Ganea, Israel exquisite timing at the very end of the program, Berstein, and Valentin Poenaru. Berstein, being Jewish, was allowed to emigrate, ostensibly to he came out with a statement that made steam Israel; Poenaru was allowed to attend the Stock- come out of my ears, but left no time for reply. holm International Congress of Mathematicians I realized that I was in the presence of a major and never returned. Berstein came to Cornell, Poe- league debater. naru went to Paris, but Ganea’s case was more I also attended his eightieth birthday celebra- difficult. As Peter told me, Ganea had to be “pur- tion in Binghamton, New York, in 2003. Peter chased” from the Ceausescu government by a gave a wonderful polished talk about his expe- consortium of mathematicians for a large sum of riences at Bletchley Park in World War II, which (hard) currency. was informative and moving and made a political After a year in Paris, Ganea accepted a high- point. salaried job in the United States (at Purdue) in I noticed that he frequently paused to refer to order to “buy himself back” from the consortium. a very small sheaf of notes in his hand. He left the Ganea, a gloomy but very witty person, would papers on the rostrum after the talk, and out of call Berstein each evening with his very funny curiosity I took a look. They were blank! It was a complaints. Berstein was of a very sunny and stage prop. lighthearted disposition, despite his wooden leg Needless to say my admiration of Peter was not (the result of a wound at age seventeen in the Red diminished by these occasions. A person like Peter army), and he would regale our group the next morning with Ganea’s latest observations. is a great rarity, and I feel very privileged to have An example: Cities are characterized by fluids: known him. Paris bywine andperfume, WestLafayette, Indiana, by milk and gasoline. References Peter returned to Cornell in 1962. Our topology [1] P. Hilton, On the homotopy groups of the union of group at Cornell in those years consisted of, spheres, J. London Math. Soc. 30 (1955), 154–172. besidesPeterandmyself,PaulOlum,RogerLivesey, [2] , Homotopy T heory and Duality, Gordon and Breach Science Publishers, New York-London-Paris Israel Berstein, Isaac Namioka, Casper Curjel, and (1965), x+224 pp. David Gillman, as well as a number of graduate [3] I. Berstein and P. Hilton, On suspensions and co- students, including Martin Arkowitz and Gerry multiplications, Topology 2 (1963), 73–82. Porter, who wrote theses in homotopy theory. [4] P. Hilton and J. Roitberg, On the classification There were several active seminars, and it was a problem for H-spaces of rank two, Comment. Math. very congenial and friendly group. Helv. 45 (1970), 506–516.

December 2011 Notices of the AMS 1543 Ross Geoghegan main reason he moved to Cornell. As he used to say: “I resigned to make way for an older man.” The first thing to know about Peter Hilton is that for I met Peter when I was a graduate student at half his life he was unusually young for whatever Cornell and he (when in residence, which first he was doing. He was recruited to Bletchley Park happened in my third year) was the best known for ’s codebreaking operation at the topology professor there. In fall 1968 he gave a age of eighteen. On the day of his arrival, Turing course on homotopy theory, rather in the style posed him a chess problem that he said he had of his elegant book Homotopy Theory and Duality. failed to solve, and Peter delivered the solution Thecourse was to meetfor three hours on Tuesday the next day. From then on, his place as a full mornings. On the first day he wrote down a participant was secure. list of those Tuesdays when his mathematical One of his activities would take him out of town. The miracle colleagues at was that in the remaining weeks he covered a Bletchley Park serious amount of mathematics. His handwriting was J. H. C. on the blackboard was tiny, and the material was Whitehead. Af- organized to perfection. Peter took much pride in ter the war the elegance of his presentations. Whitehead in- His dislike of administration was matched by vited Peter to a distrust of professional administrators. While go back with he was at Cornell, an article in the Ithaca Journal him to Ox- mentioned the then-provost of Cornell, Robert ford to do Plane, and described him as “Cornell’s number a doctorate in two man”. Peter wrote a letter to the editor asking topology. At “Which of our Nobel prizewinners is Number One?” this stage Peter While my main thesis work was in infinite- had had only dimensional topology, with David Henderson as Photo courtesy of Meg Hilton. one year of for- my advisor, I had a secondary project on which Beno Eckmann and Peter in Zürich mal university- Peter gave me feedback. And when Henderson (c. 1980). level mathe- was absent in Russia, Peter chaired my defense in matical train- 1970. So we knew each other from the late 1960s ing, and he had doubts about the idea: he told onward. But our paths rarely crossed until 1982, Whitehead that he didn’t know what topology was. when Peter joined my department at Binghamton “Oh don’t worry, you’ll love it,” was the reply. Be- as Distinguished Professor. By then any vestige of cause of the postwar housing shortage, Peter lived the student-professor relationship had gone, and in Whitehead’s home, ate dinner each evening with Peter and I became fast friends. By that time he the family, and learned topology, probably other was mainly doing number theory, combinatorics, mathematics, too, by osmosis. and polyhedral geometry, which Jean Pedersen But learn he did. In a letter written for Peter’s describes in her part of this tribute. eightieth birthday the topologist Sir Christopher That was just after his controversial tour of Zeeman, just two years younger than Peter, wrote: South African mathematics departments. It was a “Shaun [Wylie] was a lovely supervisor, who invari- time when “all right-thinking people” were boy- ably encouraged me and faithfully read everything cotting that country, but Peter took the view that I wrote. But he didn’t know much topology. It the mathematics community is worldwide and that was you who taught me topology without which shunning rather than influencing is not construc- I might never have got started. So I am eternally tive. For this he made enemies. I can still hear the grateful.” unprintable comments which a well-known visit- Peter’s career moved fast, and at an unusually ing mathematician made about Peter in my living young age he held the Chair of Mathematics room. (When the singer Paul Simon encountered at Birmingham. This was a mixed blessing, for in those days British universities allotted only similar criticism later, I thought of this.) one Chair to each subject, and the holder was My favorite Binghamton story about Peter hap- expectedto administerthe departmentfor life. The pened not long afterward, when a calculus student salary difference made it impossible to pass this came to his office for help with a math problem. administrative work on to lower-ranking people, Peter asked: “Is it the case that you do not un- and Peter disliked administration—which was the derstand what you are being asked to do, or is it the case that you do understand what you are Ross Geoghegan is professor of mathematics at Bingham- being asked to do but cannot do it?” And (as Peter ton University (State University of New York). His email explained in the Queen’s English—it really has to address is [email protected]. be spoken out loud):

1544 Notices of the AMS Volume 58, Number 11 “She answered: Well... like... kind the Hilton-Milnor theorem. A related, frequently of... both.” cited result in this same paper of Peter’s is the More, the abiding professional lesson I have “corrected left-distributive law”, expressing, for m learned from Peter is about judging the work and a space X, and elements α in πn(S ), β, γ in worth of mathematicians. I noticed in my years πm(X), an infinite sum expansion for (β + γ) ◦ α, at Cornell how charitable he was in dealing with in which the first two terms are β ◦ α and γ ◦ α and students and how positive he was in discussing the remaining terms involve Whitehead products the work of younger people. As a student I some- and Hilton-Hopf invariants. Among the many other times detected a strain of competitive negativity. noteworthy research papers published by Peter in Standing out against this “destructivism” was Pe- this time period, I restrict myself to mentioning ter, who was a celebrity mathematician and had two items: a joint paper with J. F. Adams on no need to prove himself. the chain algebra of a loop space, a forerunner Later, when he came to Binghamton and I had a to Adams’s work on the cobar construction; and chance to observe him as colleague rather than as a series of joint papers with Beno Eckmann on mentor, I understood his approach better. Peter duality (Eckmann-Hilton duality). believed that doing mathematics is intrinsically good and that the work of people, even if their achievements are not fashionable or herculean, should be praised and supported where possible. From time to time I would see surprisingly positive letters of recommendation written by Peter. I finally understood that he saw no good for his beloved subject in our tearing one another down. He had a mantra: “I would rather be a second-class person in a first-class discipline than a first-class person in a second-class discipline,” and he was not shy about identifying disciplines he did not consider first class. In short, the high standards Peter held himself to did not prevent him from emphasizing the Photo used with permission of Robin Wilson. positive in others. He was a first-class person in a Left to right: Emery Thomas, Michael Barratt, first-class discipline. Henry Whitehead, Ioan James, and Peter Hilton (c. 1955). Joe Roitberg

Peter and I first met at New York University in Peter’s passion for research was matched by early 1968. Peter was visiting the Courant Institute his passion for clear exposition. In fact, he him- of Mathematical Sciences at the time, and I had self regarded the task of making mathematical just completed my Ph.D. dissertation there, under ideas accessible to students as his highest call- MichelKervaire’s supervision. But before I proceed, ing. Among his contributions as an expositor allow me to backtrack. are three early, influential books, which were Peter was one of J. H. C. Whitehead’s most particularly important for my own mathematical brilliant students, some twenty years earlier, and development: Homology Theory: An Introduction had long been a leading homotopy theorist. His to Algebraic Topology, coauthored with Shaun most famous and most important contribution to Wylie and published in 1960 (predating Edwin Spanier’s book Algebraic Topology by six years), homotopy theory was his landmark 1955 paper which was for a long time virtually the only com- on the homotopy groups of the finite one-point prehensive, up-to-date text on algebraic topology; union of spheres, wherein he proved that any An Introduction to Homotopy Theory, published in such can be expressed as a 1953 (predating Sze-Tsen Hu’s book Homotopy direct sum of homotopy groups of spheres of Theory by six years), which included a useful various dimensions. Subsequently, John Milnor entree to Whitehead’s notion of CW-complex and generalized Peter’s result to the case of the finite also some of Peter’s earliest research, conducted one-point union of arbitrary suspension spaces. under Whitehead’s supervision; and Homotopy This more general result came to be known as Theory and Duality, published in 1965, which Joseph Roitberg is a professor in the department of math- summarized the recent results of his collabo- ematics and statistics at Hunter College and in the Ph.D. ration with Eckmann. I made good use of these program in mathematics at The Graduate School and texts after having become hopelessly hooked on University Center of the City University of New York. His algebraic and differential topology in 1964 as a email address is [email protected]. result of exposure to lectures on knot theory by

December 2011 Notices of the AMS 1545 Kervaire. The Hom-Ext version of the universal same localization (or Mislin) genus, that is, X and coefficient theorem for homotopy groups with Y are p-equivalent for all primes p; and work of coefficients, described in Homotopy Theory and Alexander Zabrodsky on the construction of new Duality, played a role in one of the results in my H-spaces—for one of our examples, X is the Lie Ph.D. dissertation, an odd counterexample to the group Sp(2), so that Y is a new H-space. Though classical Hurewicz conjecture. this fake Lie group Y cannot be homeomorphic to a topological group, thanks to the solution of Hilbert’s Fifth Problem, it was noted by James Stasheff that Y does have the homotopy type of a topological group. Thus began an intense collaboration on the themes initiated in our discussions at the Courant Institute, with Guido Mislin joining us in several of our joint projects. Our efforts were summarized in the 1975 monograph Localization of Nilpotent Groups and Spaces, which we began working on when Guido and I visited Peter at the Battelle Insti- tute Research Center in Seattle for several days in 1974. Homotopy theory, and, more generally, alge- Photo courtesy of Niloufer Mackey. braic and differential topology were experiencing Standing, Niloufer Mackey (Pi Mu Epsilon a long and spectacular growth period, begun in advisor), along with Peter Hilton, Jean the middle of the twentieth century—I think of Pedersen, Benjamin Phillips, Jonathan Hodge, this era as a golden age for topology, if I may and Jay Wood (department chair), during the be permitted a touch of hyperbole—and it was signing ceremony for Pi Mu Epsilon initiates at exhilarating to be (or so I imagined us to be) in the Western Michigan University. Peter and Jean thick of things. had just given a joint mathematical talk Peter and I continued to collaborate for several (March 21, 2002). more years on various topics in group theory and I return now to 1968. After completing my Ph.D. homotopy theory, our last joint publication ap- dissertation, I immersed myself in Peter’s research pearing in 1987. Of course, Peter’s ideas served at that time, so Peter’s arrival at NYU was very me well in much of my subsequent research. To fortuitous for me. Peter had demonstrated noncan- give two examples: in 2000, I published a paper cellation phenomena in the homotopy category by which, in particular, settled a question of Peter’s constructing examples of simply connected, finite on the Lusternik-Schnirelmann category; and in CW-complexes V, W such that V and W are not 2007, my former student Huale Huang and I pub- homotopy equivalent but, for a suitable sphere S, lished a paper on the genus of certain connective the one-point unions V ∨ S and W ∨ S are homo- covering spaces which required revisiting Peter’s topy equivalent. I introduced myself to Peter, and aforementioned foundational paper of 1955. As we proceeded to discuss possible dualizations of for Peter, he continued working in group theory his examples. We were not content, however, with and homotopy theory but also devoted a great deal a straightforward dualization, which leads to infi- of his attention to other areas of mathematics and nite CW-complexes. Eventually, with the aid of the mathematics education, which I will leave to more aforementioned “corrected left-distributive law”, knowledgeable colleagues to comment on. we were able to construct examples of simply con- Peter’s influence on my mathematical career nected, finite CW-complexes X,Y such that X and was decisive. However, I was far from the only Y are not homotopy equivalent but X × S3 and beneficiary of Peter’s mathematical and social ge- Y × S3 are homotopy equivalent, S3 denoting nius, which enabled him to carry on successful the 3-sphere. In fact, X and Y are the total and long-standing collaborations with many col- spaces of principal S3-bundles over spheres and leagues, too numerous to list here. His published so are closed, smooth manifolds, and the prod- works are, as already noted, models of clarity and uct spaces are actually diffeomorphic. We were eloquence of expression. So too were his lectures— pleased and excited about these examples, but, I fondly recall a beautiful talk he delivered to an as an unexpected bonus, our construction tied in undergraduate audience on “calculus using in- with certain cutting-edge developments in homo- finitesimals”, which underscored his highly tuned topy theory: localization theory in the homotopy sensitivity to the audience he was addressing. He category of CW-complexes, introduced by Den- had a lifelong commitment to the promotion of nis Sullivan for simply connected CW-complexes, excellence in mathematical teaching at all levels. then by A. K. Bousfield and D. M. Kan for nilpotent And he was vitally concerned with the develop- CW-complexes. For our examples X,Y are in the ment of mathematics world-wide, providing his

1546 Notices of the AMS Volume 58, Number 11 services wherever his extensive travels brought from his work in algebraictopology. They illustrate him. his clarity of exposition and his artful way of using My social interactions with Peter and his family the interplay between algebra and topology. were always highly pleasurable. Some highlights: the Hiltons hosting my daughter Daphne and me The Hilton-Milnor Formula at their home in Binghamton, prior to Daphne’s en- Let Sk1 ∨ ∨ Skm be a 1-connected wedge of rolling as an undergraduate at SUNY-Binghamton; spheres. Hilton computed in [7] the homotopy hosting Peter at our home; attending an NBA bas- groups of Sk1 ∨ ∨ Skm , showing that they can ketball game with Peter’s son Tim, while I was be expressed in terms of homotopy groups of visiting in Seattle; attending a New York City the- spheres, ater to see Peter’s wife Meg perform in the play Joe k1 km kw(π) Egg. My last contact with Peter was in August 2010, πn(S ∨ ∨ S ) ≅ πn(S ). when my wife Yael and I visited with Peter, Meg, w(π)

k km and their son Nick in Binghamton. While there, Here w(π) ∈ π∗(S 1 ∨ ∨ S ) runs over all we were shown Peter’s most recent joint venture basic Whitehead products, and the summand with Jean Pedersen, a 2010 book published by kw(π) πn(S ) is embedded into the sum via com- Cambridge University Press entitled A Mathemati- position with w(π) : Skw(π) → Sk1 ∨ ∨ Skm . The cal Tapestry: Demonstrating the Beautiful Unity of formula was generalized by Milnor [12] to the case Mathematics. This was a reminder that Peter was of a finite wedge of suspensions of connected CW - never one to rest on his laurels, that he was ever complexes. Hilton used it to derive the following active and productive. So, while we are deeply result (see [8] and [9]). Denote by P1 the set of saddened by Peter’s passing, we take solace in the homotopy types of 1-connected, finiteΣ (pointed) realization that he lived a full, rich life. CW -complexes, which are homotopy equivalent to suspensions. Consider this set as a monoid using Guido Mislin the wedge operation, and denote by Gr( P1) the corresponding Grothendieck group. He provesΣ the When I met Peter Hilton some forty-five years ago following: in Zürich, I was a student attending one of his Let X and Y be 1-connected finite CW -complexes classes, a course in homological algebra, with a of the homotopy type of suspensions. If [X] − [Y ] small group of other graduate students. He was is a torsion element in Gr( P1), then X and Y have open and approachable, which was an entirely new isomorphic homotopy groups.Σ experience for us, having only been exposed to a Germanic-distant kind of relationship between The Hilton-Hopf Invariants professors and students.We learnedfrom him how mathematicians think, how they tackle problems, In [7], Hilton proposed the following generalization present results, and design the right proof for a of the classical Hopf invariant r 2r−1 theorem. H : π2r−1(S ) → π2r−1(S ) = Z , r ≥ 2. Later, I had the privilege to collaborate with Writing Peter Hilton for many years, in a time before r r r r 2r−1 the convenience of email changed our lives. Pe- πn(S ∨ S ) = πn(S ) ⊕ πn(S ) ⊕ πn(S ) ter traveled a lot, so the exchange of letters was ⊕ π (S3r−2) ⊕ π (S3r−2) ⊕ , an ongoing challenge, as his whereabouts kept n n constantly changing. His responses were written he defines homomorphisms Hi , i ≥ 0, by compos- r r r on thin air-mail stationery, letters one had to ing the pinching map πn(S ) → πn(S ∨ S ) with open carefully so as not to cut through the text. the projection onto the (i + 3)rd factor in the sum r r His handwriting resembled print, his mathematics decomposition of πn(S ∨ S ): was crystal clear. Final drafts he completed while − H : π (Sr ) → π (S2r 1), crossing one of the oceans. Peter had a unique 0 n n r 3r−2 talent for rendering difficult mathematical con- H1,H2 : πn(S ) → πn(S ), . cepts transparent and easy to grasp. He applied He shows that H0 agrees with the classical Hopf his gift to further mathematical education on all invariant in case of n = 2r − 1. As an application, levels, and the older he got, the more he cared for Hilton obtains the following formula concerning the very young. We will remember him as a great the left distributive law for composition of ho- teacher, mathematician, and friend. r motopy classes. If α, β ∈ πr (X) and γ ∈ πn(S ), Hilton’s mathematical work covers a wide range then of topics. The examples which follow are chosen (α + β) ◦ γ =α ◦ γ + β ◦ γ + [α, β] ◦ H0(γ)

Guido Mislin is professor emeritus of mathemat- + [α, [α, β]] ◦ H1(γ) + [β, [α, β]] ics at the ETH in Zürich. His email address is ◦ + [email protected]. H2(γ)

December 2011 Notices of the AMS 1547 1548

Photo courtesy of Meg Hilton. nabtayctgre n[,5 6]. 5, [4, in categories arbitrary in saco- a is Σ map fEkanHlo.Te a ecaatrzdin way: characterized dual following be sense the the can in They notions Eckmann-Hilton. dual of of examples typical of homo- are concepts and The cohomology in topy. sequences triple dual ra sadiagram a is triad esrn h eito rmadtvt fthe of additivity from map induced deviation the measuring n aro spaces of pair a cohomology ing view- instance, generalizes For first, groups. homotopy resp. the variable groups, in second, functor a resp. as considered when which, spaces notions to as naturally such led a This module. other through projective factored factored the a it in be when nullhomotopic and can was module, map it injective if an con- through nullhomotopic was map as a notion, sidered one internal in properties; interesting duality exhibited which notions, two of worked definition Hilton suitable topy a Peter finding and on Eckmann together Beno 1955 In Duality Eckmann-Hilton hygnrlz h one oooyset homotopy to pointed the generalize They okfra nenldaiyi h homotopy the in duality frame- pointed a internal of consider category an to for them work leading notion topology, homotopy in ordinary the notions to just homotopy correspond these both that observed They cmn n itnwn nt nenlduality internal to on went Hilton and Eckmann X Y) (Y γ e n ee itnwt o i n new and Tim son with Hilton Peter and Meg ∗ X san is o aso oue.Te aeu with up came They modules. of maps for : → H π and → π cone r saei n nyi h aoia map canonical the if only and if -space Y n (X) Σ (X H a ih oooyinverse. homotopy right a has opspaces loop (X) ; saei n nyi h canonical the if only and if -space and → ) Y rnsnJk,i etl,1999. Seattle, in Jake, grandson π : a ethmtp inverse; homotopy left a has = n suspension X,α (X), X [ CW Σ X n → ,Y] Y X, H-spaces cmlxs(e 1 ,3]). 2, [1, (see -complexes ⊂ ntemdl category. module the in Y Y ֏ → n ulyto dually and , = samap a as α Z [X, hc ed to leads which , ◦ oie fteASVolume AMS the of Notices and ,(r γ, n , ] Y co-H-spaces X ≥ → homo- X ] Y [X, 2 ). path Y a , Y oayLegopbtsc that such but group Lie any to ing E References research Hilton’s of has many This in working. with articles. was dealt one been which all in in principles local-global context study the to had one pieces addressed be time now could theory e oasseai nesadn of understanding phenomena systematic lation technique a localization to The [10]. led there the book is the from spaces, of nilpotent topic and of localization groups lo- to on passing nilpotent chapter of algebraic purely calization elemen- a An with theory, and homotopy beginning doors. in Sullivan, localization new to D. approach tary opened Kan, Zabrodsky), D. A. being Dror-Farjoun, just E. Bous- K. field, A. then (including people ap- were several by to which developed theory. way techniques, homotopy natural in These a techniques in investigations. localization itself many ply lent for example point 10- The starting a the of was [11] Roitberg Joe and manifold dimensional example Hilton under- striking Peter The on of homotopy. to worked group up a people structure support which many complexes, finite 1960s standing the In H manifold manifold Hilton-Roitberg the enn htfreeyprime every for that meaning [10] [2] [1] [4] [9] [5] [8] [7] [3] [6] (p) Sae n Localization and -Spaces p 246 absolus, Groupes dualité. et .Eckmann B. 2991–2993. polyhedra, dra, categories, Primitive III. length, limits, Equalizers, II. 145 comultiplications, and Multiplications I. ficients, exactes, spheres, Hilton P. 187. 2555–2558. 150–186. .Hlo,G Mislin G. Hilton, P. tde o 5 ot-oln,1975. Spaces North-Holland, 15, and No. Groups Studies Nilpotent of tion n o h esebigo h resulting the of reassembling the for and , shmtp qiaett h correspond- the to equivalent homotopy is -localization 15) 2444–2447. (1958), 16) 227–255. (1962), ud Math. Fund. h rtedekgopo compact of group Grothendieck The polyhe- , compact of type homotopy the On , categories general in structures Group-like , categories general in structures Group-like , categories general in structures Group-like Coef- , dualité. et d’homotopie Groupes , Suites dualité. et d’homotopie Groupes , E .Lno ah Soc. Math. London J. eog othe to belongs ntehmtp ruso h no of union the of groups homotopy the On , .R cd c.Paris Sci. Acad. R. C. .R cd c.Paris Sci. Acad. R. C. ud Math. Fund. and Sp( nhmtp hoy fwhich of theory, homotopy in 61 .Hilton P. 2 16) 105–109. (1967), E ) o oooyequivalent homotopy not , and , (p) ah Ann. Math. 61 rbesi homotopy in Problems . 16) 199–214. (1967), rue d’homotopie Groupes , eu set genus E ah Ann. Math. .Roitberg J. E .R cd c.Paris Sci. Acad. R. C. p 30 sa xml.The example. an is × its 15) 154–172. (1955), S 58 n rm ta at prime one 3 150 p , , Number , ≅ Mathematics , -localization 246 246 Sp( 16) 165– (1963), noncancel- 151 G(Sp( ah Ann. Math. , Localiza- 2 (1958), (1958), (1963), ) × 2 S 11 )) 3 , , , [11] P. Hilton and J. Roitberg, On principal S3-bundles over spheres, Ann. Math. 90 (1969), 91–107. [12] J. Milnor, The construction FK, mimeographed lecture notes from Princeton University, 1956. Urs Stammbach Peter Hilton was one of the most important people in my life. He was someone with whom I could discuss anything, not just mathematics, but all aspects of life, politics, the state of the world, and the education of our children. When I was beginning work on my Ph.D. at the ETH in Zürich, Peter was regularly invited by Beno Eckmann to visit the Forschungsinstitut für Photo courtesy of Dominique Arlettaz. Mathematics. Peter gave numerous talks during Dominique Arlettaz and son Mathieu with these visits. Every one of impeccable quality. They Peter in the vineyards near Lausanne in May made a deep impressionon me. In those days there 1995. was a considerable distance between students and professors in Switzerland. Being a rather shy a few weeks after my arrival in the United States. student at the time, I was particularly affected by I was terrified; first, my English was rather rudi- this, and it was difficult for me to exchange even mentary at the time, and, second, I knew that the some ordinary words, or to ask questions, in this prospective audience would include many famous formal environment. Thus it was a great surprise people. It was possible that Hyman Bass, Gilbert to me when Peter asked me to come to his office Baumslag, Michel Kervaire, Wilhelm Magnus, and and tell him about my work. I was tense before others of similar standing would be present. Peter the meeting. But right from the start he put me at completely understood my psychological difficul- ease by addressing me in German. And Peter didn’t ties. He kindly calmed me down and supported begin by asking me tough mathematical questions; me in a way that instilled confidence. instead, he asked me to accompany him to buy During the first year in Ithaca I received a letter today’s copy of The Times. The ice was broken, from Peter, written in his characteristic tiny but and we had a long and interesting talk about very clear handwriting, in which he asked me mathematics afterwards. This encounter was the whether I would be willing to write a book with first of many, many more to come. I learned later him on homological algebra [1]. I needed to read that during those years The Times was Peter’s the letter several times before I comprehended must-have lifeline to information about the world. the full impact of his question. Only a couple of He once confessed that after several days without years after receiving my Ph.D., I was being asked The Times, he would have to read the newest to coauthor a book with one of the most important edition from cover to cover “to put the internal people in the field. coordinate system right again.” Writing the book took us over three years and After I completed my Ph.D., Peter was invited absorbed a huge amount of my time and energy. to spend a full year at the ETH, and during this This joint work was a unique experience for me. It time he gave a course on homological algebra. We is unbelievable how much I learned during these got somewhat closer, and during a conversation years. Each chapter of the book was first written he suggested that I should think of going to the by one of us. The manuscript was then sent to United States for a year or two and that perhaps the other and was corrected, criticized, sometimes would be a suitable place. It shortened, sometimes enlarged. Withmostparts of soon became obvious that this suggestion was the book, this process was repeated several times. actually a joint idea between Beno Eckmann and Some of the drafts were rejected altogether by the Peter Hilton (another example of their efficient other side and had to be completely rewritten. Of joint work!). course, Peter put my English into an acceptable Thus, in 1967, I had the opportunity to go to form; moreover, tomydismay, he regularlyspotted Ithaca and to spend two years in Cornell’s fine instances of mathematical sloppiness on my part. mathematics department. During my first year ThiswasahugelearningprocessasIreceived,from there Peter was away, visiting the Courant Insti- an expert, firsthand coaching on the art of writing tute in New York, where he invited me to give a mathematics. Here is an example: Commenting number of talks. The first of these took place only on my writing that such and such a theorem Urs Stammbach is professor emeritus of math- was false, Peter gently but firmly told me that ematics at the ETH in Zürich. His email address is a theorem could never be false, for a theorem, [email protected]. by definition, is true. Only the statement of the

December 2011 Notices of the AMS 1549 1550

Photo by Gert-Martin Greuel, courtesy of Mathematisches Forschungsinstitut Oberwolfach. ee.Wiigti okwsageteprec on experience by great levels! a accepted many was were book this suggestions Writing my “The Peter. book of title a most pleased immensely the when (in was a I categories” with algebra). book, homological abelian Peter on the to using to of reaction of chapter essay strategy in a wrote, long of I rather version instance early one some At acceptable were him. suggestions my to of to many experience that time) great see a that was at It email uncountable. no almost is was (there book the are writing fact? instances simple this many ignore that How literature the false. in there be can theorem h itn’ptgris ee intlk t and of it, like the one and didn’t Stammbachs ate the Peter between cat relationship dog, gerbils, the our pet Hiltons’ with when Hiltons’ friendly the the However, became even other! Rupert, days cat, each our few and that years a Lady, during happiest friends after the close of became time; We one lives. be our to of tiny out equally turned this conditions, an year two living and cozy had the kitchen, Despite apartment tiny bathroom. Irene, a The house wife, rooms, Heights. their small my of Cayuga annex and the the in me rent to offered opportunity the Meg, wife, his above went he duty. challenge, his beyond a and accepted When- Peter improvements. ever important many that to surprise led no this is It English. the the on and comment willing mathematics and be manuscript would entire he the yes, read but, to that, and do not text answered, would the he he No, through English. defective read still my to correct whether willing Peter be asked would I he proper. alge- theory group homological specialized to of bra more [2], applications of the theory collection on group material a be in would homology which on notes lecture Algebra nyasottm after time short a Only while exchanged we letters of number The uigorscn eri taa ee and Peter Ithaca, in year second our During .Heie tEkans9t birthday 90th Eckmann’s at Haefliger A. aeot eie opouesome produce to decided I out, came .Ekan .Hlo,J-.Sre and Serre, J.-P. Hilton, P. Eckmann, B. eerto nZrc n2007. in Zürich in celebration orei Homological in Course A oie fteASVolume AMS the of Notices oteSnFacsoArott eunt Cleveland, to return to the Airport get Francisco of to San way the end a to the for around At looking was met. over Peter group meetings contentious the days sometimes several and the discus- lively the was and sion in present, people were prominent reform and curriculum active mathematics Some on conference education. small met a first I at that Hilton mid-1970s, Peter the in Berkeley, in was It Alexanderson L. Gerald ihtetbcoitt edhmanwspl of supply new a him send to tobacconist the in with of arranged Peter brand When Stumpen. special so-called the a much. cigars, for Swiss liking very a developed Peter he Zürich Swiss pleased distinct a This developed had accent. he that told Germany, was in he German speaking when better—so that, and so better much become German his in spent Zürich, Peter periods extended the With house- us. with be a could which Meg as when days, especially enormously, Peter special enjoyed we very had always I were These and guest. Irene joint wrote Often we papers. and research; mathematical did otherwise; we and for mathematical year ideas, exchanged a We once continued. was least friendship deep (at Our weeks). regularly several ETH the visit to with well believe rather survived I us. he got that parenting. agree we in would Thus Tim training Tim. boy, on-the-job we teenage away some their were of Meg care and took Peter a Whenever for days. only few but strained, somewhat became Hiltons nea at lr nvriy i mi drs is address email His University. Clara Sci- in [email protected] Santa Professor Valeriote at the ence is Alexanderson L. Gerald be would References it that expected us time. good- of last said None the he us. 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Peter clearly be Customs, package U.S. the that have the to with not difficulty order any In intervals. regular at Stumpen [2] [1] n16 ertre oZrc.Ptrcontinued Peter Zürich. to returned we 1969 In ee’ atvstt üihwso eoEck- Beno on was Zürich to visit last Peter’s e ok 93 vii 1973, York, New Springer-Verlag, 359, Volume Mathematics, in Notes .Stammbach U. 1997. edition enlarged second Springer, Algebra mological .Hilton P. and , oooyi ru Theory Group in Homology .Stammbach U. rdaeTxsi Mathematics, in Texts Graduate , + 8 pp. 183 . , 58 orei ho- in Course A Number , Lecture , 11 where he then held the Louis D. Beaumont Chair opportunity to explore. Of course, an education at at Case Western Reserve. So I volunteered to drop Oxford and Cambridge opens some doors, too. him off at SFO on my way home. On the slow trip Beyond his erudition and his consummate good across the bridge to the airport in heavy traffic we taste, though, was the basic kindness that he had a stimulating conversation about the events showed to his colleagues and students. He had at the conference, and when I dropped him off I strong opinions, but he expressed them gently. felt that I had a friend in Cleveland. I saw many instances in which he generously Those were heady days in mathematics instruc- helped colleagues he did not know well, but tion, with the fallout from the 1963 Cambridge Re- he provided good advice and a willingness to port and the Bourbaki-influenced suggestions for champion work that they had done by directing reform of instruction having powerful advocates. them to appropriate journals. Thus he became The Cambridge Report had many mathematical a mentor to young colleagues who were not his luminaries as signers—Peter, of course, but also students or even people in his field. Creighton Buck, Andrew Gleason, Mark Kac, Ted I recall well the last time I had a chance to visit Martin, Ed Moise, Max Schiffer, and Pat Suppes, with Peter. He came to see me when I was in the among many others. It was provocative but in the hospital recovering from surgery resulting from a end probably not very influential, since it came to broken leg and hip. At that point we were both be viewed as utopian and unrealistic. For example, walking with comparable difficulty—mine due to “drill was to be eliminated from all grades”. I an accident, his due to the onset of a decline that recall hearing at about that time a talk by Peter resulted eventually in his recent death. But I had on category theory, for the benefit of teachers! In a great time visiting with him. It was the old Peter those days rigorous mathematics in the classroom we all loved, and his shuffling walk did not impede had many ardent supporters in the mathematical his ability to carry on a stimulating conversation community. with the usual flashes of wit and ability to spin It was about that time, however, that Peter met out a good anecdote. And he could still skewer Jean Pedersenof my department, and Jeanbrought the pompous and the hypocritical with his usual to the long collaboration he had with her another elegant choice of language. point of view. This followed from her work with Mathematics has lost one of its most articulate, her mentor, George Pólya, who advocated a more indeed eloquent, advocates. His kind does not participatory approach in the classroom based on come along very often, and mathematics is the his “guess and prove” philosophy. In pedagogy, worse for his departure from the scene. this seemed a collaboration doomed to failure be- cause their viewpoints were so disparate. It ended up, however, with each shifting more toward a common middle ground, and their joint work— approximately 140 papers and six books—ended up influencing mathematics instruction on at least four continents. Peter was an inveterate traveler, showing up at meetings in the United States, of course, but also all over Europe, as well as New Zealand, Australia, and South Africa. He was al- ways on the move. Whatever Peter talked about, it was fascinating. He brought to any conversa- tion an urbane, sophisticated viewpoint that was wonderfully appealing. He had, as often seen in people educated in Britain, a sharp wit, and, in Peter’s case, it could be used to encapsulate a really provocative idea in the form of an ironic quip. For example, I recall his pointing out in a lecture the irony of living in a society that actively supports driver education and teacher training! Peter was not only a world-class topologist and a spokesman for mathematics education, he was also an erudite and cultivated person who was at ease talking about music, literature, art, politics, just about anything. The fact that his wife Meg is an accomplished actor with a career spanning work in London and New York, regional theater, and festivals opened doors for him to the artistic world that most mathematicians do not have the

December 2011 Notices of the AMS 1551