Iron Rings, Doctor Honoris Causa Raoul Bott, Carl Herz, and a Hidden Hand

Iron Rings, Do ctor Honoris Causa Raoul Bot t, Carl Herz, and a Hidden Hand P . Rob ert Kotiuga Abstract. The degree of Do ctor of Sciences, honoris causa, wa s conferred on Raoul Bott b y McGill Univ ersit y in 1 987. Muc h of t he w ork to mak e t his happen was done b y Carl Herz. Some of the author’s pers ona l recollections of b oth professors are included, along with some contex t for the aw arding of this de gree and ample historical tangen ts. Some cultural asp ect s o ccurring in the addresses ar e elaborated on, pri marily , the Canadian engineer’s iron ring. This pap er also reprints b oth the con v ocation address of Raoul Bott and the presen tation of Carl Herz on that occasion. In troductio n Raoul Bott needs no intro duction in this volume. How ev er, repr in ting his address at the 1987 McGill conv oc a tion b oth gives some insight in to the e ffo r t to aw ard him an Honorar y Do ctorate in Mathematics from McGill, and a con text to develop some less than mathematica l themes, una shamedly from the point of view of an electrical engineer who e njo ys the historical asp ects o f his discipline. Early on I was tipp e d o ff that Carl Herz was b ehind the effor t, and the memory of Prof. Her z made me realize that I had to follow the trail like a hound. The result complements more technical presentations and anecdotes per taining to Montreal 1 . I a m gr ateful to Canda ce Bott for digg ing up her father’s co mmencemen t 2 address, and to Do minique Papineau of McGill Univ ersity who show ed up in Bosto n with a complete file pe r taining to the a w arding of the honorary degree from McGill’s archiv es. The supp ort of v arious McGill faculty who listened to me think aloud is m uch appreciated, na mely Peter Caine s , Jacques Hurtubise, Joa c him Lam bek, a nd Peter Russ ell. 1991 Mathematics Subje ct Classific ation. Pri mary 01A65 , 55-03; Secondary 15A04, 30D50. Key wor ds and phr ases. History and biography , algebraic top ol ogy , generalized inv erses. An edited and r eformatte d version of this pap er , wi th an additional photo, wil l app ear in a v olume dedicated to Raoul Bott[ K10 ] . The author hop es to expand on some asp ects of this preprint in future ve rsions. 1 F or anecdotes f rom B ott’s years in Montreal see [ T u ] and Candace Bott’s remarks in this v olume. 2 Although “conv o cation” and “commencemen t” hav e different meanings in general, in the con text of a university graduation ceremony they hav e the same meaning in Canada and in the USA respectively . F or the purp oses of this article, they are used i n tercha ngably . 1 2 P . R O BER T KOTIUGA Ho w I got to kno w Raoul Bott I can’t remember who fir st connected Raoul Bo tt to McGill Engine e ring in my mind; most likely it w a s Peter Caine s , Rob ert Hermann o r Carl Her z. How ever, I do r emem b er attending his talk in the Physics depa rtmen t at McGill in, I believe, 1982. These days, if the fact that I dr agged my wife to b e to the talk is mentioned, and tha t she cop ed with my enth usia sm with a s ense of humor, my kids will k indly remind me o f all my thrift y ideas for a go o d time! During those years I sp en t many hours “teething” on the bo ok of Bott and T u, and as a NSERC p ost-do c in the MIT Mathematics department in 1 985 I finally got the o pportunity to a udit a course given by Ra oul Bott. Not o nly was his mathematics entrancing, but we were b oth McGill engineers! During the first lecture he sp otted m y iro n ring. I int ro duced m yself afterwards, and we talked ab out a v ar iet y of things. Before long, I ma de a habit of a uditing ev e ry cour se he taugh t. Ab out a year later, he told me that he would b e the r ecipien t of an honora ry do ctorate from McGill — forty y ears after McGill wouldn’t hav e him as a gradua te student in the mathematics department (he was unwilling to co mplete a second undergr aduate degree in mathematics). Past histo ry aside, he s eemed genuinely honored, but he didn’t quite k no w who to share this news with. Reading the co n voca tion sp e ec h, I now se e it as his wa y to make p eace with histo ry a nd a means to repa c k age it cons tructiv ely for graduates forty y ears younger than himself. Stepping back from the cer emon y from decade s ago, the r e ader is invited to read his “Autobio graphical Sketc h”[ BoA ]. I clearly enjoy ed his lecturing style as well as the attention and discipline he demanded in the classr oom. It c e rtainly contrasted w ith the c o n voca tion sp eech . In the first lec ture, he encouraged students to ask questions and cla imed that he liked “stupid questions” b ecause he could answer them. He was very s erious a bout this a nd if we didn’t rea liz e it initially , we even tually lear ned that he put the bar very hig h, and even highe r for himself: His a nsw ers to questions were alwa ys mor e profound than the or iginal questions a nd he once walk e d out o f his own lecture in frustration b ecause he didn’t like it! That was dra matic, baffling , and unexpec ted — esp ecially since we knew he liked to think o n his feet. In this par ticular case, he reapp eared the nex t class , his arguments were imp eccably clear and e le gan t, and his credibility was not only resto red, it soar ed! Go ing b ey o nd the classro om, Bott’s men toring of gr a duate students is the source of leg end. This volume has plent y of testimony from his students, and the r eminiscences by Rob ert MacPhe r son [ MP ] testify to his high expec ta tions. The conv o cation s peech is v ery differ en t in that it demonstra tes his a bility to co nnect with an audience that ha s nev er seen him and most likely will never see him a gain. Before attempting to advise graduates, he warms up the audience by telling of the pivotal event in his pr ofessional life and saying: “I tell you all this only in par t as a jes t.” Only when the stage is set do es he give the essence o f his address credibly and in a few sen tences. The message app ears in a flash after saying “B ut m y time is up!”. The style mirrors his a pproach to g iving a collo quium talk. Bott enjoy ed c ultiv ating cer tain habits which were best left unmen tioned in the co n voca tion spee c h. F or instanc e , he lectured a t 8:30 a.m. in or der to hav e a flexible day after 1 0:00 a.m.! Graduate students felt this cramp ed b oth their style and sleeping habits, and this is where some o f Bott’s Old W orld sensibilities kick ed in when they dared to doze off in cla ss. He co uld tos s chalk and hav e it land on the table inches from the sleeping student’s face, s tartling them. He clearly relis he d IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 3 doing so. The memo r able line whic h would meet the sta rtled face changed with every suc c essiv e offense. F rom “didn’t wan t you to miss anything” to “how muc h is tuition at Harv a r d?” to “who pays your tuition?” In Old W o rld style, this was all for the benefit of the studen t and there was little ro om fo r self-preser v a tion. These days it isn’t easy for a professor to be r espected for doing this in a priv ate universit y where students can feel like pa ying customer s o nce the tuition bill is settled. Somehow Bott was consistently more mischiev ous than the students, a nd got aw ay with it. Clear ly he had extensive exp erience testing his teachers a nd this exp erience always gav e him the upp er hand in the cla s sroo m. As for the impact Bo tt’s lectures made on me, I’d b e tre a ding on thin ice if I tried to say why they w ere fantastic. Lo ring T u says that 3 Victor Guillemin, at a conference celebrating his 60th birthday , proudly announced that he to ok tw e lve courses from Bott, and to Loring’s chagrin, he could only list eleven. Clearly , I am in no po sition to sp eak with a utho r it y about Bott’s lectures! In m y case, I lov ed his lecture style, the lecture material, and I felt a definite kinship since I could ask ta ng en tially related questions after class a nd consistently g et profound answers. There is pe rhaps one p ersonal anecdo te I can add to the many that I’ve hear d. One day in class, after Bott explained the set-up of the Lefschetz fixed p oin t theor em in terms of the transverse intersection of the g raph of a map from a manifold to itse lf with the graph o f the iden tity map, he claimed that, by duality , the Lefsc hetz num be r could b e easily computed by picking a basis for int e gral cohomolo gy , pulling back by the appr opriate maps, taking wedge pro ducts with Poincar e duals and in teg rating. When he claimed that it reduced to basic matrix alg ebra inv olving the induced a uto morphisms on coho mo logy gro ups which an eng ineer could do, the class just didn’t make the t ype of ey e con ta ct he was hoping for. A t that p oin t he called me up to the boa rd and told me to fill in the details o f the calculation! As I (metho dically) wrapp ed up the calculation, he identified me as an engineer , emphas iz ed that budding top ologis ts sho uldn’t shy aw ay from such concrete calculations, and to ok satisfaction in the fact that he made his p oint . In my mind he reinfor ced the fact tha t Daniel Quillen’s thes is advisor could make us think functorially while, as a student of Ric ha rd Duffin, he could encourag e us to “think with our fingers” and to always maintain a bala nce b etw een the conceptual a nd the computational. Obviously , I’m enamored with Raoul Bo tt, a nd thrilled that he was the recipient of an ho no rary degr ee from McGill. Howev er, my purp ose he r e must be more fo cused. Specifically , in m y mind, a few key p oint s need ela boratio n: • Who was the driving force b ehind getting McGill to aw ard Bott a n ho n- orary degree in Mathematics forty years after he left McGill as a n engi- neer? Clearly , Bott w as deser ving, but it takes a kindred spirit to ov erco me the inertia o f a bureaucra cy a nd Carl Herz was suc h a kindred spirit. • In his conv o cation a ddress, Bo tt vividly descr ib es the pivotal moment at McGill when he decided to b ecome a mathematicia n. Ho wev er , ho w he w as going to do it was not at all clear a t the time. The wonderful and profound connections be tw een top ology and physics have b een studied intensely in recent decades, but what is needed is a hin t o f the pa th from “engineer - ing mathematics” to the mathematics Bott is known for[ Bo85 ],[ JB ]. In retrosp ect, it almost see ms that this path could ha ve b een more clear to 3 See “Reminiscences of W orking with Raoul Bott” in [ STY ] 4 P . R O BER T KOTIUGA Gauss and Maxwell than to mo dern sp ecialists. W e’ll s o on s ee, the c areer of Hermann W eyl pr o vides us with a p ersp ectiv e and some key insig h ts. • I’m fas c inated with Bott’s str uggle to reconcile o ld and new world sensi- bilities. I saw this enco ded in my int eractions with him and in the co n- vocation a ddress. I use the word “enco ded” in refer ence to the addr ess, bec ause his referenc e s to the uniquely C a nadian ir o n r ing hav e their r o ots in v arious Queb ec City bridge disasters, Rudy a rd Kipling’s po em “So ns of Ma rtha,” and asso ciated Biblical re ferences. These details a re required to fully deco de the message. Carl Herz as a gatew a y to hi story I can distinctly remember the day Carl Herz kno ck ed me off my feet. A t the time I was a gradua te student in E lectrical Engineering at Mc Gill and he was a feisty and famous professor o f Ma thematics. W e b egan to chat a fter some seminar in the E E depa rtmen t, and he asked me wha t I w a s doing for a thesis. He listened as I told him how I felt that the reformulation of Ma xw ell’s equatio ns in terms o f differential forms w as essential for the resolution of s o me key problems in computational electro ma gnetics. Specifica lly , most of the b oundary v alue problems in low frequency electromagnetics a moun ted to Ho dge theor y on ma nifolds with bo undary , with the p e r iods of harmo nic forms identified with the v ariables found in Kirhhoff ’s laws. F urthermore, I told him tha t the whole framew o rk has a v ariational setting whic h ca n b e discretized by app ealing to “ Whitney for ms” in or der to obtain a finite element discretiza tion with desira ble pr operties. T o me it was all o b vious if one rea d the pap ers written b y Dona ld Sp encer a nd his students in the 1950’s and int e rfaced them with Whitney’s “Geometric In teg ration Theory” . In retro spect, this was a natural connectio n given the work of J ozef Do dziuk and W erner Muller ’s pro of of the Ray-Singer conjecture, but it w as not apparent at the time. Carl listened, started pacing back a nd forth and I was b eginning to worry that he was going into a tra nce! I do n’t know wha t was going on in his mind, but I braced m yself for what could come out of his mo uth. I think I was standing in stunned silence when Carl s topped and asked me if I ever read Max well. Sheepishly , I told him that I read a g oo d deal of Maxwell and that every o ne in my field swears b y Maxw ell. He then asked me if I knew what a p eriphractic num b er was. When I expressed my ig norance, he wen t on to p oint out that Betti’s pap er was written a decade a fter Maxwell’s treatise, and that in Maxwell’s trea tise the first Betti num b er was ca lled a “cycloma tic num b er” - a term int r oduced by Kirchhoff, and still used in gra ph theory . He wen t on to tell me that the seco nd Betti n umber was called a “p eriphractic num b er”... I later found out that Maxwell b orrow ed the term from Listing 4 and that Listing was the p erson who coined the term top ology . In one swoo p Carl convinced me that Maxwell was often quoted but never rea d, and that if I wan ted to get to the origin of these top o logical ideas the o rigin w o uld be in some la ng uage other than English. Clearly , I was hu m bled — but I felt b etter when I loo k ed up “ periphra c tic” in the unabridged Oxford dictionar y and found that Maxwell’s treatise is the first and last use o f the word in the Englis h language! 5 4 See Breiten b erger [ Br ] in James [ J ] for an article on Johann Benedikt Listing and his bo ok. 5 It is irresistable to p oin t out the connection b et wee n M axw ell and M or se theory in this article about Bott. Listing[ L ] is credited as b eing the first to systematically obtain a cell decompositions IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 5 Besides b eing aw ed by Carl’s encyclop edic knowledge, there are t wo big less ons I have leaned ov er the years a nd which were initiated by my encounter with Car l and other mathematicians from his generation who “rea d the masters.” The fir st was that Maxwell ha d a profound exp erimen tal a nd theor etical knowledge, and that m uch of the inspira tion for his theoretica l work came fr om rea ding a nd corr espond- ing with Ger ma ns (Gaus s, Riemann, Kirchhoff, Claus ius , Helmholtz, L isting, ...). F urthermor e, it was the Germans who to ok Maxwell seriously when no one els e did- from Helmholtz’ studen t Hertz demonstrating radio wa ves, to B oltzman developing statistical mechanics, and to Einstein developing the log ical ph y sical conclusions of Maxwell’s theory . Contrast this with the situation in E ngland where Oliver Heav- iside was consider ed a self-educated eccentric who died in p ov erty des pite making brilliant co n tributions to Maxwell’s theory and b eing aw arded a n honorar y do ctor- ate from G¨ ottingen Universit y in 19 05. The other “Maxwellians” didn’t make it int o the limelight either . The se cond big lesson I lear ned from my enco unter with Car l is to never ignor e the Institute for Adv anced Study (IAS) in Pr inceton o r the pround influence of its t wo first founding p ermanen t member s : Hermann W eyl and Alb ert Einstein. Car l was a student of Salomo n Bo chner and thrived on all the mathematics e ma nating from the IAS. In retrosp ect, the world seems quite s mall. It was W eyl who in 1948 invited Bott to the IAS, it was W eyl who earlier got de Rha m, K odaira , and Sp encer to put Ho dge theory on a rigor ous fo o ting, and it was W eyl [ W ],[ Y ] a nd his close colleague Einstein who were the true curato r s of the developmen ts arising from Maxwell’s theory . It turns out that the “Whitney fo rms” that I was so fond of hav e their orig ins in a 1952 pa per o f Andr´ e W eil ca lled “ Sur les Theorems de de Rham.” Clearly , every part of the no vel mathematics I was using could b e tra ced back to the IAS; even if my application of these idea s to computational electromagnetics was unfore s een. If ever I was in denial ab out details, I could chec k in with Donald Spence r’s student, Rob ert Hermann, to verify facts 6 . If details w ere scarce in the literature, co n templating the influence o f Hermann W ey l co uld help bring things int o fo cus. Enough said a bout my int eractions with Carl Her z . T o apprecia te Ca rl Herz’ contributions to harmonic ana lysis a nd o ther fields of mathematics, as well a s the feist y c ha racter himself, thro ugh the eyes of his collea gues, the r eader is referr ed to other sources [ D ], [ H ]. Needless to say , when I realized that Car l Herz was behind the effor t to a ward Bott an honora ry do ctorate from McGill, it seemed like a big piece o f the puzzle fell in to place. He plays a central role on almost a ll corres p ondence with the universit y a dministration on the matter and in the end, he’s the one who prese nted Bott for the degree in June of 19 87. of 3-manifolds by tracking the change in top ology as lev el sets cross a critical point. Maxwell [ M70 ] then wrote a pap er citing Cayley and Lis ting. In hi s treatise[ M ], Maxwe ll uses the rudiments of Morse theory wi th the fact that a harmonic function cannot ac hieve a maximum or minimum in the i n terior of a region in order to mak e topological deductions. 6 The “Adv anced Calculus” text Nick erson, Spencer, and Steenro d [ NSS ] w as Princeton- inspired but wa s never published. How ev er, it initiated a wa ve of differential-form based multi- v ariable calculus texts in the 1960s. Altho ugh it is a very natural wa y to bring multiv ariable calculus to its r oots i n Physics, this wa ve of texts didn’t catc h on. Bott nev er wrote a text for such an undergraduate audience and so one can only hypothesize ab out how he wo ul d hav e int egrated Kir c hhoff ’s laws with Ho dge theory and Maxwell’s equations. In retrospect, it to ok a couple of decades to get things right and ultimately , the b ooks that reache d out to engineers and ph ysi cists most effectiv ely w er e wri tt en b y Bott’s close colleagues [ B S ], [ F ]. 6 P . R O BER T KOTIUGA The con v oca tion p resent a tion of Carl Herz Mr. Chancellor, I have the honour of pr esen ting to you, in order that you may confer on him the deg ree of Do c to r o f Sciences, honor is ca usa, Professo r Raoul Bott. Raoul Bott was b orn in Hunga ry , but his universit y e ducation up to the M. Eng . W as at McGill. He received his B. E ng. from McGill in 1945 . After a s hort stin t in the infantry , he contin ued his studies in electrical engineer ing at McGill. The immediate p ostw ar per iod s a w a great demand for mathema tics tea c hers, and Bott taught calculus her e while studying for his master’s degree. In addition he to ok some courses fro m Professor Gilson, then Chair of the Depa rtmen t of Mathematics. Nevertheless, he remained a studen t of electrical engineer ing un til he left McGill to go to Carnegie T ech for his do ctorate. Electrical e ng ineering has a close a ffiliation with what mig h t be viewed as an abstruse branch of mathematics, algebraic top ol- ogy , P rofessor Bo tt’s sp ecialty . One has only to recall that “Betti nu m ber s”, the fundamen tal numerical inv a r ian ts of top ology , a re named for an Italian ele ctrical engine e r , and one can read Ja mes Clerk Maxwell for pro found insight s into the s ub ject. At an even more primitive level, cir c uit theor y has always b een a source of go od problems for top ologists. Bott’s earliest work w a s ra ther a l- gebraic. The Bott–Duffin Theo rem (1949) on circuit synthesis was describ ed by a reviewer thus: “This proof of the realizability of the driving point imp edance without the use of transfor mers is one of the most interesting developmen ts in netw ork theory in re- cent y ear s.” It contin ues to be a muc h-cited result. This w or k came shortly after Bott ha d o btained a D.Sc. in mathematics. After the do ctorate, Raoul Bott wen t to the Ins titute for Ad- v anced Study in Princeton. He was at the Institute during 19 49– 1951 and retur ne d in 1 955–1957 . He joined the faculty of the Uni- versit y of Michigan in 19 51 wher e he rema ined unt il 1959 when he was invited to his present academic ho me, Ha r v a rd, where he is William Caspar Graustein Profess or of Mathematics. Professo r Bott’s seminal co n tributions to ma thematics a re to o extensive for me to do justice to them her e. Mos t of his early ideas s eem to ha ve drawn their inspira tion from the Calculus of V ariatio ns in its global version k no wn as “Morse T heo ry”. Bott applied Mor se Theory in an unexp ected and s tr iking w ay . Over a lo ng p erio d he, to gether with his v arious co llabora tors, worked out the top ology of Lie groups and s ymmetric spaces. One must men tion the Bott Perio dicit y Theore m which brought some order to the chaos of ho motop y theory . He w ent on to study fixed p oint theorems and their application to other branches of mathemat- ics including differential equations . Most recently Bott has bee n working o n applicatio ns of top ology and geometry to the Y ang- Mills equations in quantum field theor y . IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 7 F or his ac hievement s, Bott was aw arded the V eblen P rize of the American Mathematical So ciet y in 1964. In addition to his purely scientific a ccomplishmen t, Rao ul Bo tt stimu lates all those who are abo ut him. He is one of the b est and most exc iting exp ositors of mathematics I have had the privileg e to listen to. Mr, Cha ncelor, McGill can take grea t pride in honoring this year, as it did last 7 , ano ther of its graduates who stand in the forefront of mathematics of the tw entieth ce ntury . The E lev enth Day of June, Nineteen Hundred a nd Eight y- seven Carl Herz Professo r of Mathematics and Statistics. Given m y encounters with Prof. Herz, his corresp ondence with the McGill administration, and his encyclop edic brea dth, it is clear that he play ed a central role in the case for the ho norary degree . The masterful presentation of Carl Her z shows how a broad p ersp ectiv e can lead to a reor ganization of knowledge that lets the likes o f Paul Dirac a nd Eugene Wigner move from Enginee r ing to P h ysics , and the likes of Raoul Bott, Solomo n Lefschetz, J ohn Milnor, and Dona ld Spe ncer mov e from Engineering to Mathematics. O n the o ther hand, since Pro f. Herz alwa ys enjoy ed an ar g umen t (in the v er y best sense of the word!), I’ll take the liber t y to make a qualifica tio n and p e rhaps an eleb oration. The qua lification I might add is that Enr ico Betti was clear ly not an electr ical engineer but a mathematician. Indeed, Betti made co n tributions to bo th Ela sticit y theory and Elec tr omagnetism, and Maxw ell does indeed cite Betti’s work in his treatise, but he w a s a mathematician. Betti, lik e the entire s c ho ol o f Ita lian Alge- braic Geometry , was highly influenced by Riemann and top o logical ideas. How ever, the level of rig or in 19th century Italy was lax by mo dern standa rds and so his in- fluence on current mathematical resear c h ma y seem fa r r emo ved. I revere Car l’s resp ect for historical detail, and I’ll refra in from ca lling his labeling Betti as an Electrical Engineer as a mistake. Rather I’d say Betti, like Gauss, Riemann, and Vito V olterr a, had broad interests, and that Pro f. Herz suppressed the p edan t in himself and to ok some lice nse in his int erpretation of history . The hidden hand of Hermann W eyl The role of Her mann W eyl in getting mathematics off the gr ound in the ea rliest days of the IAS is now well do cumen ted [ B ]. What I find fascinating is the first and fateful encounter betw een Her mann W eyl and Ra o ul Bo tt. The enco unter has a lot to do w ith interplay b e t ween electrical circuit theory , the ea rly days o f algebr a ic top ology , and the p erception of top ology . The presentation of Car l Herz leav es out a lot of detail, m uch as a movie bas ed on a bo ok has to forgo a lot of detail. Given the e ncyclopedic knowledge of Carl Her z, it is tempting to specula te on what he could have put into a longer presentation. Bott told the story of his first encounter with Herma nn W eyl many times, em- phasizing differe nt asp ects and different amounts o f detail. See for example[ Bo 88 ]. I like the following rendition of the basic fac ts. During his gr ad student days a s a 7 Raoul Bott, Jim Lam b ek and Louis Nirenberg all graduated from McGill in 1945 and Niren- berg was a warded an honorary do ctorate f rom McGill i n 1986. 8 P . R O BER T KOTIUGA student of Richard Duffin at Carnegie T ech, B o tt play e d a large role in orga nizing the depar tmen t collo quium. Being fluent in German, Hunga r ian, a nd Slov akian he would ha ve an edge over other g rad studen ts in terms of “chatting up” foreign-b orn visitors. When Hermann W eyl visited, they w er e intro duced, and Bo tt immedi- ately b egan to tell W e y l of his thesis work[ BDu ]. Ther e are interesting a spects of his thesis whic h predate bo th the Bott–Duffin Syn thesis proce dure and W ang algebras [ D59 ]. O ne key asp ect is the “impeda nce po tential” and how it defines a generalized inv erse o f a matrix. Of co urse the Mo ore–Penrose axioms for a genera l- ized inv ers e were only formulated in the 1950 s and so Bo tt do es not use the term. 8 Early on, he a nd Duffin c alled it a “cons trained inv ers e”[ BDu ], and in ex p ository talks Bott later describ ed it in terms o f orthogona l pro jectio ns in a complex (i.e. Ho dge theory ). It tur ns out that his impeda nce p otential is a deter minan t which is intimately r e lated to what graph theoris ts call a “Ma trix-tree for m ula” – a r esult that go es back to Kirchhoff and was used in Maxwell’s tre a tise. The logar ithmic deriv ative of the imp edance potential with resp ect to branc h imp edances gives a generalized inv erse. When Bott expla ined the forma lism and asso ciated results to Hermann W eyl, W eyl grasp ed that Bott-Duffin synthesis was indeed a contribution to netw ork synthesis, but that the connection b etw een Ho dge theory a nd K ir c h- hoff ’s laws was not. He pointed Bott to some pa p ers connecting K ir c hhoff ’s laws to top ology which he wrote in the ea rly 1920 s 9 . Needless to say , Bott was invited to the IAS, but Bott felt a bit deflated ab out the Ho dge theoretic a spect and that W eyl saw it concretely in Kirchhoff ’s work. T o b e fair to Bott, we hav e to ask wh y these pap ers o f Her mann W eyl were so obscure. Gottingen was very closely tied to the technological asp ects of Maxwell’s theory , and so why w ere these tw o pap ers as obscure as Maxwell’s periphra ctic nu m ber s? What w a s the p oin t W eyl was trying to make? T o g iv e so me ins igh t, a digressio n is in or der. In the Winter of 2005 I sp e nt a month in the math department a t the ETH in Zurich while on sabbatica l. When I a r riv ed, my host g a ve me a c ho ice o f offices: a hu ge office with a stunning view of Zurich belong ing to a colleague on Sabbatical, or a very s mall empty office in the back of the building wher e “pure mathematicians hide their g ue s ts.” I told my hos t that I wan ted the fr eedom to “spread out,” and that I felt more comfortable in the small back o ffice. He was p erplexed but obliged. It turns out that my coz y office was next to that of Beno Eckmann. On the centennary of E instein’s golden year, Zuric h ce lebrated Eckmann a s the las t per son in the city who had p ersonal contact with Einstein! Since Hermann W eyl was the the head of the ETH mathematics department in the 1920s, I natur ally wan ted to pick Beno’s brain for anecdotes. Not w a n ting to mess with his w o rk habits, I pla nned to chat him up while he w a s a sitting targ et. In the hallwa y outside our o ffice s w a s a high-tech espres s o machine and ev ery morning Beno would take a bre ak to sit and enjo y an espresso outside our offices. The first day , I “co inciden tally” joined him a nd he related w onderful anecdotes from 1950- 1955, after Hermann W eyl retired from the IAS study , resettled in Zuric h, 8 See Chapter 2 and App endix A of Ben-Israel and Grevill e’s b ook [ B IG ] for an exp osition that puts the Bott-Duffin constrained inv erse in the con text of generalized inv erses, and for putting the “M o ore” of “Mo ore-Pe nr ose” in historical p erspective . 9 More precisely , W eyl’s papers dealt wi th Kir c hhoff ’s laws [ W23 a ] and combinatorial topol- ogy [ W23b ]. IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 9 and frequented the department. (I was sufficently impressed that when I retur ned to B o ston, I co n tacted the editors of the Notices of the AMS and a year later the a necdotes a ppeared in print[ EWZ ]). The next day I r esolv e d to a sk Beno a question which I didn’t think an y liv ing pers o n c o uld a nsw er . Little did I know that he had written a pap er on the sub ject [ E ] and ha d a definite o pinio n on every nu ance I could ask him to elab orate on! The conv ersa tion wen t something lik e this: RK: Be no, there is something I rea lly do n’t understand ab out Her mann W eyl. BE: What is it? RK: W ell, in his collected works, ther e are ar e tw o paper s a bout electrical cir cuit theory and topo logy dating from 1922/3 . They are written in Spanish and published in an obs c ure Mexican mathematics jour na l. They are also the o nly pap ers he ever wrote in Spanish, the only pap ers published in a r elativ ely obscure place, and just ab out the only expositor y paper s he ever wrote on alg ebraic topolo gy . It w ould seem that he didn’t wan t his colleag ues to r ead these pap ers. BE: Exactly! RK: What do you mea n? BE: Because topo logy was not resp ectable! RK: Why w a s toplogy not resp ectable? BE: Hilbert! RK: Hilb ert? BE: Just loo k at his 23 pro blems from 1 900. Do you see a n ything to do with combinatorial g roup theory or topo logy? No ! RK: Why? BE: Poincar ´ e! 10 RK: What did Hilb ert think of Poincar´ e’s work on toplog y? BE: Poincar ´ e would write a h ug e pap er on Analysis Situs. Half of it would b e completely wrong! So, he’d write a no ther huge pa per try ing to co rrect the first, but it would b e half wro ng ! And so he’d write a thir d pa per, but it w o uld b e half wrong. And so on... deuxieme c o mplemen t, troisieme, quatrie me, cinquieme,.. and in the end what did we g et? Dubious results and conjectur es! Hilb ert didn’t think this was mathema tics! RK: So w hy did Hermann W eyl write these pap ers? BE: He wan ted to take sto ck of the honest res ults and reorg anize them using a more mo dern abstract algebraic a pproach. Emmy No ether a nd others w ere doing int e resting things in a lgebra and he ha d a need to wr ite these pape r s for himse lf. These pa pers also contain s ome new results like the signature of a 4-d manifold. Beno wen t o n to p ortray Hilber t a s a bit of a reactiona ry figure, ar ound which Hermann W eyl had to tip-to e. How ever, if W eyl wan ted an opp ortunit y to mov e things forward, it came in 19 30 when W eyl succeede d Hilb ert up on his r e tiremen t from G¨ ottingen. Although he was only at the helm fro m 193 0 un til he fled the Nazis in 1933 , during this v ery br ie f time German topo logy flow er ed in the ha nds of Emil Ar tin, Kurt Reidemeister, and others. Accor ding to Beno Eckmann, W e y l made a historic decision in 193 0 which was highly cont rov er sial at the time, but ultimately vindica ted: he app oin ted Heinz Hopf, a young res e a rc her a nd relatively unseasoned, a s his successo r at ETH. 10 Sark aria[ S ] has given a m o dern executiv e summary of Poincar ´ e’s work i n T opology . 10 P . R O BER T KOTIUGA If Beno’s histor ic al p ersp e c tiv e is taken super ficially , there is a temptation to susp ect there was s ome lasting disag reemen t betw ee n W eyl and Hilber t. Ho wever, one only needs to r ead W eyl’s master ful s ummary of Hilbert’s work [ W44 ] to realize that b oth men held themselves to the hig he s t standards. In a sense, every time Hilber t o r Poincar´ e dug their heels in, W eyl found a n o pp ortunity to move mathematics forward. Algebraic top ology may b e one example a nd the contin uum hypothesis may b e another ; p erhaps the b est exa mple is the fact that Kur t G¨ odel was amo ng the first four hir e s at the IAS, but unemploy able in Eur ope. What did Carl know? W e can only sp eculate. I can only say that he is one of the many peo ple who impresse d up on me the imp ortance of having a his to rical per spective when reco nciling a lgebraic top ology with its applicatio ns. F rom History to Bott’s reconciliatio n with it The historic al details discussed so far predate 1 952. The next eight years would usher in the revolution in homo top y theory bro ugh t on by Serre’s thesis, CW c o m- plexes which tie Morse theory to homotopy theory , Bott perio dicity , generalized cohomolog y theor y such as K -theory and Rene Tho m’s cob ordism theory , a nd the reformulation of generalized cohomology theories in terms of sp ectra. Beno Eck- mann p oin ted out to me that in the 195 0s, those in Zurich who dismissed Her mann W eyl as an old man w er e fav o rably stunned by his summary of the w o rk o f Kunihiko Ko daira and Jean-Pierr e Serre on the o ccasion o f their b eing aw ar ded the Fields Medal in 1954 [ W54 ]. Nonetheless, co n tras ting W eyl’s presentation o f the work of the tw o Fields medalists, it is apparent that he was challenged b y the homotopy theoretic world Bott had entered into, e ven if he did a lot to unlea s h the ho motop y theoretic per spective. E nough sa id; it is time to leav e threads of mathematical history a nd exp erience a nother view of history: The con v oca tion a ddress o f R a oul Bott Mr. Chancellor, Mr. Chairman of the Board– m y dear fellow graduates: Congratula tions to you- class o f ’87! Y ou lo ok splendid! I think you wash more be hind the ears than your American c o usins at Harv ard do . It is nice to get a degree, isn’t it? O f course you only had to work ha rd for four years or so to get yours, while it too k me ov er forty years to g et mine. And pr esumably you hav e pa id for yours, while I am paying for mine at this moment by b eing her e on this platform, making a foo l of myself. But, there is rea lly nothing like one’s first degree. And what I lo ved esp ecially ab out my Bachelor of Eng ineering was that an iron ring (from a falle n bridge) came with it. I hop e this tra dition contin ues, so that at least you e ngineers, ca n co n trive – as I did – to display it on every o ccasion. It is a marvelous wa y of starting a conversation and at the sa me time lets o ne know that you have “gradua ted.” So my fir st admonition to you is: “Flaunt your de- gree in fro n t of the whole world!” F o r a few weeks enjoy it to the hilt! The real world will rein you in so on enoug h. IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 11 Of course the p eople who enjoy your deg ree most are your parents. So by a ll means— here co mes my second admonition — Get yourselves so me children, in time for degree- ha rv e s ting when you are still in your forties ! (That wa y y ou migh t a ls o hav e time to repay the lo ans b efore you die.) But let me tell y ou no w a little bit a bout the g oo d old days, just to k eep so me sort o f his torical p ersp ectiv e in a so ciety , whose customs change at such a rate that the last forty years mo s t prob- ably represe n t tw o hundred uninflated ones. First of all I must tell you that, be autiful as your Campus is today , it used to b e even more so in 1 941. There w er e lawns to stretch out on, there was even a tennis court b y the Redpath library! There w a s so m uch space and such a fine line of propo rtion! And there were no skyscr apers! (On the other hand, the area around McGill w a s very rundown. And I see that our “gr easy sp oon” has now flow ered in to a pizza joint.) Classes were small and some of my professo rs wore rob es, a s we ar e now, to teach in. They billow ed and flow ed delightfully with each step. Thes e gowns were usually torn and co mpletely cov er e d with dus t; s till they added to the p erformance. I remember that later when I had my o wn calculus class to teac h – the v etera ns had returned in huge num b ers in the fall of 1945 and the Math. Department had presse d a lowly enginee r into serv ice to meet the demand – my dear friend and mentor, Pr of. McLennan – the So crates o f our campus – len t me his w e ll- w eather ed gown. “T r y it in your class” , he s aid, with a twinkle in his eye. W ell, the cla ss of course guffawed at first, but then a ctually s ettled down to work in a more business lik e manner than usual. Possibly it was this ba llet-lik e a spect of the lectures that kept me going to classes very diligently in the b eginning. How ever, this epo ch of m y life ca me to an end in short o rder after one of my ro ommates in our b oarding house on Duro c her called me in for a serious talk. Elwoo d Henneman w as his name and he co n tinues as a dear friend and co lleague a t Harv ard. Elwo o d, w ith the full authority o f a first year medical studen t and a Harv ard B.A. w ar ned me of the danger of being addicted to classe s . “Never b ecome a slav e to them,” he declared; “do the bulk of your thinking on your own!” This p oint of view made immediate sense ; and therea fter it was safest to lo o k for me in the Mus ic r oom. Usually in the company of my dear friend, W alter O dze, and muc h later on a lso with my wife to b e. How muc h this had to do with m y falling gra des I don’t know, but in any case I did manage to always to “b eat the Dean” a s we used to say . Do you know wha t I mean? (But it sounds like go od fun in any case, do esn’t it?) W ell, one of the endear ing pro cedures of o ur Alma Mater at that time was that they didn’t divulge our final g rades until August – I think. Then s uddenly your name was print ed in the Gazette – if you pass ed, that is – and with an 12 P . R O BER T KOTIUGA asterisk if you flunked one co ur se, etc. On the same day the names of all the passing students were listed on a billb oard in linear order of merit. Those who had failed w er e not o n the list, and at the bo ttom of this terr ifying do cumen t ca me the sig nature of the dean! Hence the expression o f b eating the Dean if one got through. But sp eaking o f Deans and advic e, let me tell you ab o ut one McGill Dean who in his own inimitable wa y gave me the b est advice o f my life. These w e r e the war y e ars and in ’4 5 right a fter graduatio n, I joined up in the Cana dian infan try and was being trained for co m bat in Ja pan. After three mo n ths in ba sic tr aining the atomic bomb w a s dropp ed on Hir oshima a nd Naga s aki, the war ended abruptly a nd my fellow recruits a nd I ther eb y suddenly and miraculously reprieved – in this unbeliev able and terrible manner. Of course, the o ne great adv antage of being in the army is that one ha s no career pro blems wha ts o ever! Hence the doubts I had ab out my vo cation in engineering were completely s ubmer ged b y m y efforts to keep out of the Serg ean t’s hair. But when in Octob er I found m y self ba c k in the Eng ineering Depa r tmen t, where they had v ery kindly let me return for a Master’s Degree on – a s you can imag ine – very short notice, the o ld doubts flared up again and I w as in a quanda ry ab out what to do. It was sometime in ‘46 then that I presented myself at Dea n Thompson’s o ffice a nd asked him whether he could see his w ay to putting me thro ugh medical school. (On the Jewish s ide of m y family they alwa ys did say: “ c hutzpah he do es not lack”.) And Dean Thompson was quite encour a ging at first. “W e need scientifically tra ined do ctors”, he s a id. “But”, he contin ued, “first tell me a little ab out yourself ”. It was at this p o in t that our int e rview sta rted to g o sour. No, I never enjoy ed Biology muc h. No I hated dis secting frogs. Ala s, Botany bo red me and I ha d little use for Chemistr y! After this sorr y litan y , Dr. Thompson surveyed me a nd the situation for a while, pip e in ha nd, and los t in thought. “Is it mayb e that you wan t to do go o d for humanit y”, he said at last. I hemmed and haw ed in my s eat, but b efore I ha d time to say anything he ca me out with: “ Because they make the lousiest do ctors!” W ell, that was it for me – and you must admit tha t it explains a lot of things, do e sn’t it? In an y case , I go t up and a s I wen t to the do or, I thought to myself: well you (explicative deleted) if that is how the land lies, then I will simply do wha t I like b est: “I will bec ome a Mathematician. Put that in your pip e and smoke it!” I tell you a ll this only in pa rt a s a jest. I w o uld also like it to be a word o f e nc o uragement to tho s e of y ou who, degree in hand, still are not quite certain of your path. May you als o b e blessed with a co unselor with such dia gnostic skills and such a knack for putting you o n the right cour s e. But my time is up! Still I ca nnot resist a serio us word. Over m y McGill da y s the W ar hu ng like an everpresen t black cloud, IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 13 subtly affecting every a s pect o f our lives. F or you in the nuclear age the cloud is, thank Go d, fa r ther aw ay , but p oten tia lly m uch, m uch darker. These things you will hav e to live with and somehow hop e to co nquer. But for this ro ad I know of no b etter advice than m y friend Elwoo d’s - “Do your own thinking ”. In our mo re immedia te lives we are also b eset to day mor e than ever be fo re, with show, with image, with ja rgon; a nd here again – to pick one’s way through this qua gmire, there is no b etter exhortation than: Be your own man; be your own woman. F o r then, I am confiden t, you will never confuse fashion with substa nce, hero es with the p eople who depic t them o n the tub e, computers with p eople, Science with virtue, or wealth with happiness. Y es, may Go d bless you and may you b e joyfully and pro duc- tively yourselves - but may you also be ultimately ser v a n ts of a larger and an all-encompass ing benign world view. And that is really no more than what I take to be the co rrect reading o f Dr . Thompson’s advice to me fort y y ea rs ago. Only remember tha t the concerns of your ge ne r ation must even tra nscend thos e off our hu man family . They mu st em brac e ev er y aspect of life itself on this deeply troubled, but magnificent and ma gical planet of ours. Raoul Bott On Iron Rings and other asp ects of the Co n v o cation Address Bott’s conv o cation addr ess sets the stage for my fascination with his struggle to reconcile old and new world sensibilities. He was clear ly the same age as my parents and like my parents, the disr uption of his adolesce nc e by Hitler, Stalin, and the event s in the E urope of his youth was a tra umatic exp erience and a profound education ev e n if they didn’t view it that wa y at the time[ BoD ]. His refugee exp erience w a s a stark c o n tras t to the wa y kids g rew up in North America during the de c a des after WWI I. This was ev iden t in his sense of humor and in the way he handled those who did not choo se their words co rrectly . The testimonies o f his student s in this volume a ttest to this. The w ar years and the economic turmoil that preceded it left him w ith little tolerance for the danger ous comfor ts of se lf- preserv ation. The Engineer’s iron ring a long with its uniquely Canadian origin s eem to frame so me of his advice on r esponsibility and indep enden t thinking. Reading the conv o cation s peech, and re c alling our interaction after the first clas s I audited at Harv ard, it is apparent that Bott had a muc h more pro found appreciation of, and r espect for, the iron ring than did students he was addressing . One ca nnot do justice to the to pic here 11 , but it is us e ful to connect a few key ideas to even ts a nd sensibilities of other times. Engineering is full of trade-offs, and the stor y of the iron ring is ab out the in- terface b etw een technical trade-offs, ethics and ambit ion. O ne eng ineering trade-off is b et ween the theor etical effort that go es into designing something without making a physical mo del, and the willingnes s to build prototypes and make mistakes. If one were to design a pa per c lip, one would make many prototypes in or de r to s e e “what works.” On the o ther hand, if one w ere building a bridge, o ne would like to 11 These da ys one has to lo ok for s ta inless steel if one w ants to s pot an “Iron Ri ng” —the iron of the early rings used to b e eaten a wa y by sweat and wa s so on replaced. 14 P . R O BER T KOTIUGA av oid disasters, and the developmen t o f a theoretical mo del with predictive prop- erties is in or der. In the cas e of bridge building, espec ially new desig ns, o ne is v ery cautious beca use public confidence is paramo un t. How ever, there hav e bee n ma n y bridge disaster s and con trary to wha t one mig h t naively exp ect, they usually do not inv olve new des ig ns! They usually involv e refined desig ns which take in to account that earlie r desig ns were ov erly cautious, to o costly , and less than ambit io us. The temptations involv ed a r e quite universal a nd are not restricted to bridges; one can send up spa ce shutt les routinely without an accident, but when one decides tha t the rules for launch a r e overly cautious in light o f an opp ortunity to make some “State of the Union Address ” sp ectacular, stra nge things ca n happ en — just like when the detailed prop erties of O-rings were purp osely ig nored in the lead- up to the Challe ng er disa s ter. Similarly , the dismis sing of foa m impacts during launch a s routine in the lead-up to the Columbia Shuttle disas ter underlines the vigilance re- qired to ma k e complicated things work. These days we hav e “financia l eng ineering” and computer mo dels so predictive that there is a temptatio n to lose tra c k of un- derlying a ssumptions and to consider the re g ulation of inv es tors as unimagina tiv e and cum b ersome- here a gain, strange things can happ en when ambition trumps regulation. Engineering disaster s cr eate tea c hable momen ts and they ar e very well do cumen ted when the sta k es are high. In the case of bridges, the sp ectacular dis- asters hav e b een studied and categor iz e d, a nd scholarly b o oks such as the one by Petroski[ P ] hav e b een written. Chapter three of Henry Petroski’s b o ok details the Queb ec City bridge disas - ter(s) that lead to the iron ring worn by Cana dian Engineer s. It has a lot of detail on the New Y ork based constructio n firm, the details o f the bridge, the ignoring of warning sig ns and the 7 5 peo ple k ille d in the first disaster . A key ingredient in this August 1 907 disaster was an a ttempt to re de s ign the bridge during co nstruction in order to ensure it broke a world r ecord. There was a second disaster in 19 16 during the c onstruction of the redesigned bridge whic h killed 13 w o rk e rs. In all, a total of 89 workers were k ille d in the co nstruction o f the bridge. The completion of the 1800 fo ot span o f the Queb ec City bridge in 1917 made it the large st ca n tilever brige in the world and vindicated the concept of the cantilev e r bridge for a mix of rail and automobile traffic. How ever, worldwide, no other ma jor cantilev er br idge was co mpleted unt il the 193 0 s. T o this day the Queb ec City bridge has the longest span of any c a n tilever bridge – o ther bridge designs ar e used for longer spans. The original ir on rings were made of the colla psed bridge’s iro n as a reminder of the stu- pidities engine e rs a re capable of, and as a reminder of the engineer’s resp onsibilit y to so ciety – so on after the rings were made of stainles s steel. Rudyard Kipling w as the r ecipien t of the 1907 Nobel Prize in literature a nd lived in Bratleb oro V ermont for a few years in the 1890s . It was in V er mon t, b et ween Queb ec City and the home of the bridge’s architect in New Y ork, that he wrote “The J ungle Bo ok.” It was also in 190 7, following the first Queb ec Cit y bridge disaster, that he wro te a p o em called “ The Sons o f Mar tha”. 12 It is inspire d by the Gosp el of Luke (1 0:38-42) and for ms the basis of the o riginal iron ring cer emon y which K ipling was comissio ned to write. The o riginal Canadian c e remon y w as called “The Ritual of the Calling o f an Engineer” and it was first per formed in 1922 . These days the o riginal ceremony with its Biblical r eferences is considered noninclusive. 12 See, for instance: ”Ki pling: A Selection of H i s Stories and Poems” by John Beecroft, (in t wo vo lumes), Doubleday 1956. In V ol. I I, The Sons of Martha appears on p 451. IR ON RINGS, DOCTOR HONORIS CAUSA RA OUL BOTT,CARL HERZ, AND A HIDDEN HAND 15 The iro n ring ceremony , first per formed in the United States in 19 7 0, is centered around “The obligation of the engineer ” which is devoid of Biblical references. 13 In Bott’s commencement addres s he is clea rly refer ring to the the orig inal ceremo n y and I recommend a rea d through Kipling ’s po em to a ppr eciate the iro n ring as Bott would have b een in tr oduced to it. This concludes m y musings on a r efugee as an engineering student in Mon- treal, his metamorp osis into a top ologist, and a new world suc c ess story who was ultimately aw arded an hono rary do ctorate by his alma mater. References [B] S. Batterson, Pursuit of Genius. A. K. Peters, W ellesl ey , MA, 2006. [BIG] A. Ben-Israel and T. Grevill e Gener alize d Inverses; The ory and Applic ations 2 nd ed., Springer-V erlag, 2003. [BDu] R. Bott and R. J. Duffin O n t he A lge br a of Networks T rans. AMS, 74 (1), Jan. 1953, 99–109. [BoA] R. Bott Autobio gr aphic al Ske tch , in: R aou l Bott Collected Papers V ol. 1. [BoD] R. Bott The D iosze ger Y e ars , in: Raoul Bott Collected Papers V ol. 1 [Bo85] R. Bott O n T op olo g y and Other Things , Notices of the AM S, 32, (1985), 152-158. [Bo88] R.Bott On Induc e d R epr esentations Pro ceedings of Symposia in Pure and Applied M ath- ematics, AMS (1988), 1-13. [Br] E. Br eite n berger Johann Bene dikt Li sting , pp. 909-924 of I. M. James Hi st ory of T op olo gy . [BS] P . Bam berger and S. Stern b erg A c ourse in mathematics for students of physics , in tw o v ols., Camb. U. Press, 1990. [D] S. W. Drury , The M athematica l Work of Carl S. He rz , in: S. W. Drur y Eds Harmonic Analysis and Numb er The ory; p ap ers in Honour of Carl S. Herz, Pr o c ee dings of a Confer e nc e on Harmonic Analysis and N umb er The ory, April 15-19, 1996, McGil l University, Montr e al, Canada , CMS Conf. Pro c., V ol. 21., AMS 1997. [D59] R. J. Duffin An analysis of the Wang A lge br a of networks T rans. AMS, 53 (10), 1959, 114-131. [E] B. Eckmann, Is algebr aic top olo gy a r esp e c table field? , i n: Beno Ec kmann, Mathematical Surve y Lectures, 1943-2004. Springer-V erlag, 2006 [EWZ] B. Ec kmann, Hermann Weyl i n Zuric h 1950-1955 , Notices of the AMS, 53 (10), No v. 2006, 1222-1223. [F] T. F rank el, The Ge ometry of Physics: An Intr o duction 2 nd ed., Camb . U. Press, 2004. [H] Carl Herz 1930-1995 , Notices of the AMS, 43 (3), July 1996, 768–771. [J] I. M . James, History of T op olo gy North-Holland, Elsevier, 1999. [JB] R. Bott and Al lyn Jac kson, Interview with R aoul Bott Notices of the AM S, 48 (4), Apr il 2001, 374-382. [K10] P . R. Kotiuga, editor, A Celebr ation of the Mathematic al L e gacy of R aoul Bott , CRM Pro cee di ngs and Lecture Notes, V ol. 50, AMS, 2010. [L] J. B. Li sting, De r Census r¨ aumlicher Complex o der V er al lgemeinerung des Euler’schen Satzes von der p olye dern 1862. [M] J. Clerk Maxwell, A T r e atise on Ele ct ricity and Magnetism in t wo volumes. Dov er Publica- tions, 1954; reprint of the 3 r d ed. Clarendon Press, 1891. [M70] J. Clerk Maxwell, O n hil ls and dales , London Edin burgh Dublin Philos. Mag. J. Sci. 40(269) , 421-427 (1870). [MP] R. M acPherson, Intr o duction to V olume 2 , in: Raoul Bott Collected Papers V ol. 2 [NSS] H. K . Nick erson, D. C. Spencer and N. E. Steenro d, A dvanc e d Calculus , pp ix+540, V an Nostrand 1959. 13 Kipling l iv ed af te r “the da ys of woo den ships and iron men”, and i n the p eak of the English Empire. Of the 89 work ers kill ed in the tw o Q ueb ec City br idge disasters, it appears that 33 were Moha w k steel work ers for m the Kahnaw ake reserve just outside of Mont real, creating 24 wi do ws and numerous fatherless children. The Mohawk work ers were well adapted to heights and were the “high tec h” work ers of their time. I hav e yet to see this asp ect ari se in the con text of a modern iron ring cremon y , or in the world vi ew of Kipli ng’s time. 16 P . R O BER T KOTIUGA [P] H. Petroski, Engineers of Dr e ams; Gr ea t Bridge Builders and the Sp anning of Americ a , Al fred Knopf, NY 1995. [S] K. S. Sark ar ia, The T op olo g ic al Work of He nri Poinc ar e , pp. 123-168 of I. M. James History of T op olo g y . [T u] L.W. T u, The Life and Works of R aoul Bott Notices of the AMS, 53 (5), Ma y 2006, 554-570. [W] H. W eyl, Sp ac e-Time-Matt er Do ver Publications, 1950; repri n t of the 1921 English translation of the 4 th German ed.(1920). [W23a] H. W eyl, R ep artic ion de c orrient e en una r e d c onductor a (Intr o duc c ci on al analisis c om- binatorio) , Revista Matematica Hispano-Americana 5, 153-164 (1923). Engli sh translation: George W ashington Universit y Logistics Research Pro ject (1951). [W23b] H. W eyl, A nalisis sit us c ombinatorio , R evista M ate m atica Hispano-Americana 5, p43(1923 ) and Ana lisis situs c ombinatorio (c otinuacion) Revista Matematica Hispano- Americana 6, p1-9 and 33-41(1924). [W44] H.W eyl, David Hi lb ert and his mathematic al work Bull. AMS, vol. 50, 612-654 (1944). [W54] H.W eyl, A ddr ess of the Pr esi dent of the Fields Committe e 1954 , Pro c. ICM (1954) [Y] C. N. Y ang, Hermann Wey l’s Contribution t o Physics . In: Hermann W eyl 1885-1985, Springer-V erlag 1986. [STY] S.-T. Y au ed. The F ounders of Index The ory , 2 nd ed., In ternational Press, 2009. Dep ar tment of Electrical and Computer Engineering , Boston University, Boston, MA 0221 5 Curr ent addr ess : Departmen t of Electrical and Computer Engineering, Boston Universit y , Boston, MA 02215 E-mail addr ess : prk@bu.ed u