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Celebratio Mathematica

Wolfgang Haken

Wolfgang Haken: A biographical sketch

by Ilya Kapovich

Wolfgang Haken1 was born in Ber­lin on June 21, 1928. His fath­er was a phys­i­cist work­ing for the Ger­man Pat­ent Of­fice and his moth­er was a house­wife. His two young­er broth­ers died in 1927 of scar­let fever, and Wolfgang grew up as a single child. Dur­ing his child­hood in Ber­lin, Wolfgang de­veloped an early in­terest in Math­em­at­ics. At the age of 4 he made what he thought was his first im­port­ant math­em­at­ic­al dis­cov­ery: that count­ing should start with 0 rather than with 1, and that 0 is the first nat­ur­al num­ber. He tried to con­vince his fath­er to pat­ent this fact but did not suc­ceed at the time.

Wolfgang’s moth­er died in Au­gust 1939, sev­er­al days be­fore the start of World War II, and the fam­ily re­mained in Ber­lin for most of the war. In 1944, at the age of 15, Wolfgang was draf­ted to serve in an an­ti­air­craft bat­tery. He was soon trans­ferred from Ber­lin to Des­sau and from there to Soest, where he met the end of the war. After the war, Wolfgang first worked as a farm hand, and passed a high school GED ex­am in 1946. In the sum­mer of 1946 he star­ted his un­der­gradu­ate stud­ies at the Uni­versity of Kiel. At the time Kiel had only two math­em­at­ics fac­ulty mem­bers: a Pro­fess­or of Math­em­at­ics and a Pro­fess­or of Geo­metry, which were re­garded as dif­fer­ent sub­jects when the uni­versity was foun­ded in 17th cen­tury. The Pro­fess­or of Math­em­at­ics at Kiel was Karl-Hein­rich Weise; most of Haken’s math­em­at­ics classes were taught by Weise. In the Fall of 1947, while Haken was still an un­der­gradu­ate stu­dent, he at­ten­ded a to­po­logy course by Weise. In this course Weise stated sev­er­al fam­ous open prob­lems in to­po­logy, in­clud­ing the Poin­care Con­jec­ture, the Four Col­or Prob­lem and the Un­knot Prob­lem. This ex­per­i­ence marked the start of Haken’s in­terest in to­po­logy. He re­ceived a pre­dip­loma (roughly equi­val­ent to the Bach­el­or of Sci­ence in the west­ern sys­tem) in phys­ics and math­em­at­ics at Kiel in 1948. He then star­ted his doc­tor­al stud­ies in Math­em­at­ics at Kiel, with Weise as his thes­is ad­visor. Haken ob­tained his PhD from Kiel in 1953, with the dis­ser­ta­tion en­titled “Ein to­po­lo­gis­cher Satz uber die Ein­bettung \( (d - 1) \)-di­men­sionaler Man­nig­faltigkeiten in \( d \)-di­men­sionale Man­nig­faltigkeiten”. Haken met his fu­ture wife, Irmgard,2 at Kiel in 1950 where she was also study­ing math­em­at­ics as an un­der­gradu­ate stu­dent. They got mar­ried in 1953. Later Irmgard also re­ceived a PhD in Math­em­at­ics, and also with Weise as the ad­visor, in 1959.

After get­ting his PhD, Haken ob­tained a job at Siemens in Mu­nich as an elec­tric­al en­gin­eer, where he worked on design­ing mi­crowave devices un­til 1962. The first three of six chil­dren of Wolfgang and Irmgard were born dur­ing this peri­od: Armin in 1957, Dorothea in 1959, and Lip­pold in 1961.

In 1956 Haken sus­tained a near fatal ac­ci­dent while moun­tain climb­ing in the Ger­man Alps. He fell over 30 feet and re­mained in a coma for sev­er­al days. The ac­ci­dent sig­ni­fic­antly dam­aged Wolfgang’s foot but did not dampen his en­thu­si­asm for the out­doors.

While at Mu­nich, Haken con­tin­ued do­ing math­em­at­ic­al re­search in com­bin­at­or­i­al to­po­logy. He solved the long-stand­ing Un­knot Prob­lem by pro­du­cing an al­gorithm for de­cid­ing wheth­er a knot dia­gram rep­res­ents the un­knot. Haken’s solu­tion, at over 100 pages long, was even­tu­ally pub­lished in Acta Math­em­at­ica in 1961 in a pa­per titled “The­or­ie der Nor­malflächen [1]. The a pa­per star­ted Haken’s work on de­vel­op­ing the the­ory of “nor­mal sur­faces”, which came to play cru­cial role in 3-man­i­fold to­po­logy. The solu­tion of the Un­knot Prob­lem got Haken’s work no­ticed by sev­er­al math­em­aticians in the United States. Ral­ph Fox, a to­po­lo­gist at Prin­ceton, had his gradu­ate stu­dents go over Haken’s proof in de­tail, and, some­what to Fox’s sur­prise, they found the proof to be cor­rect. Bill Boone, a group the­or­ist at UIUC, also be­came in­trigued by Haken’s pa­per. At the time Boone was work­ing on top­ics re­lated to the un­solv­ab­il­ity of the word prob­lem for fi­nitely presen­ted groups and he un­der­stood that there were close con­nec­tions between al­gorithmic prob­lems in group the­ory and al­gorithmic prob­lems in low-di­men­sion­al to­po­logy. Since by then it was known that the word prob­lem for fi­nitely presen­ted groups is, in gen­er­al, un­de­cid­able, Boone ex­pec­ted the Un­knot Prob­lem to be un­de­cid­able as well. There­fore Haken’s proof came as a con­sid­er­able sur­prise to him. Just 6 weeks after the pub­lic­a­tion of Haken’s pa­per in Acta, Boone in­vited Haken to come for a year to UIUC. Haken came to Urb­ana-Cham­paign with his fam­ily in 1962 and spent the 1962–63 aca­dem­ic year at UIUC as a Vis­it­ing Pro­fess­or. Mahlon Day, who was the Math De­part­ment Head at UIUC then, sug­ges­ted to Haken that, when pre­par­ing for the vis­it to Illinois, Haken ob­tain a U.S. im­mig­rant visa (which was re­l­at­ively simple to do at the time). Haken fol­lowed this ad­vice, which made it easi­er for him to settle per­man­ently in the U.S. later. Dur­ing his year at Illinois, Haken ap­plied for and ob­tained a tem­por­ary mem­ber­ship at the In­sti­tute for Ad­vanced Study at Prin­ceton. He spent two years, 1963–1965, at Prin­ceton. In 1965 Haken joined the fac­ulty at the De­part­ment of Math­em­at­ics at UIUC as a ten­ured Pro­fess­or. The last three chil­dren of Wolfgang and Irmgard were born in the U.S.: Ag­nes in 1964, Rudolf in 1965 and Armgard in 1968.

Haken’s 1961 pa­per on the Un­knot Prob­lem ex­plored in de­tail the prop­er­ties of in­com­press­ible sur­faces in a 3-man­i­fold and in­tro­duced the no­tion of what came to be called a Haken 3-man­i­fold, defined as a com­pact prime and \( \mathbb{P}^2 \)-ir­re­du­cible 3-man­i­fold which con­tains a two-sided in­com­press­ible sur­face. The no­tion of a Haken man­i­fold be­came cent­ral in the de­vel­op­ment of 3-man­i­fold to­po­logy over the next 50 years, par­tic­u­larly in Thur­ston’s geo­met­riz­a­tion pro­gram. In a 1962 pa­per Haken proved that Haken man­i­folds ad­mit a “hier­archy”, that is, they can be cut in­to 3-balls along in­com­press­ible sur­faces [2]. Haken pub­lished sev­er­al more pa­pers on the to­po­logy of 3-man­i­folds in the 1960s and 1970s [3], [4], [5], [6], [7], [8], [9], [10], and his work in this area re­mains widely in­flu­en­tial. Mo­tiv­ated by Haken’s work, in 1968 Wald­hausen for­mu­lated a ques­tion which came to be known as the “Vir­tu­al Haken Con­jec­ture” [e1]. The con­jec­ture states that every com­pact, ori­ent­able, ir­re­du­cible 3-man­i­fold with in­fin­ite fun­da­ment­al group ad­mits a fi­nite cov­er which is a Haken man­i­fold. The Vir­tu­al Haken Con­jec­ture in­formed much of the sub­sequent de­vel­op­ment of 3-man­i­fold to­po­logy and geo­met­ric group the­ory. The con­jec­ture was re­cently proved by Agol [e2], with a sub­stan­tial por­tion of the proof re­ly­ing on the work of Wise, and of Kahn and Marković.

In 1970 Haken began to work on the Four Col­or Prob­lem, which fas­cin­ated him ever since the 1947 to­po­logy course by Weise and a sub­sequent 1948 lec­ture at Kiel by Hein­rich Heesch. In 1976 Haken and Ken Ap­pel (who was also a pro­fess­or at UIUC then) solved the Four Col­or Prob­lem. Their proof in­cluded a sub­stan­tial com­puter-aided com­pon­ent and marked the first time that a ma­jor math­em­at­ic­al res­ult of this level of im­port­ance was solved with the help of a com­puter. Ap­pel and Haken pub­lished an an­nounce­ment of their proof in the Bul­let­in of the AMS in 1976 [11]. The proof it­self was pub­lished in a series of pa­pers in the Illinois Journ­al of Math­em­at­ics in 1977 [12], [13], [14], [15]. In­ev­it­ably, the proof gen­er­ated much dis­cus­sion and con­tro­versy in the math­em­at­ic­al com­munity and was to a large ex­tent re­spons­ible for the birth of Com­pu­ta­tion­al and Ex­per­i­ment­al Math­em­at­ics as sig­ni­fic­ant dir­ec­tions in mod­ern math­em­at­ic­al re­search. Ap­pel and Haken pub­lished an ex­ten­ded ver­sion of their proof, ad­dress­ing some of the ques­tions and cri­ti­cisms that were raised in the mean­time, as a 700\( + \) page book in 1989 [16]. Shortly after Ap­pel and Haken an­nounced their proof in 1976, the UIUC De­part­ment of Math­em­at­ics put the phrase “Four Col­ors Suf­fice” on its of­fi­cial post­mark, which re­mained in use un­til mid-1990s; see Fig­ure 1.

Figure 1.  The “Four Colors Suffice” postmark used by the UIUC Department of Mathematics after the Appel–Haken proof.

Wofgang Haken de­livered an in­vited ad­dress at the In­ter­na­tion­al Con­gress of Math­em­aticians in Hel­sinki in 1978. In 1979, Haken and Ap­pel shared the Fulk­er­son Prize from the Amer­ic­an Math­em­at­ic­al So­ci­ety for their solu­tion of the Four Col­or Prob­lem.

Haken re­mained a pro­fess­or of the UIUC De­part­ment of Math­em­at­ics un­til his re­tire­ment in 1998. He was also a mem­ber of the Uni­versity of Illinois Cen­ter for Ad­vanced Study from 1993 to 1998. While at UIUC, Haken was a thes­is ad­visor for 7 PhD stu­dents: Richard Re­m­pel (1973), Thomas Os­good (1973), Mark Dugo­pol­ski (1977), Howard Burkom (1978), Robert Fry (1979), and Scott Brown (1995).

The “Sat­urday hike” is a de­light­ful UIUC tra­di­tion go­ing back to 1909 and hav­ing a long as­so­ci­ation with the math­em­at­ics de­part­ment; the hike was for many years led by the late Joseph Leo Doob. From 1960s and through the present day, Wolfgang Haken has been a con­stant par­ti­cipant of the Sat­urday hike, as have been many oth­er mem­bers of the Haken clan. Wolfgang’s wife Irmgard was an in­form­al lead­er of the hike from 1993 to 2005, and con­tin­ued to come to the hike in sub­sequent years, while her health al­lowed. Sadly, Irmgard passed away on April 4, 2017, while this art­icle was in pro­duc­tion.

Figure 2.  Wolfgang Haken and Ian Agol in Altgeld Hall 314 in May 2014.

Dur­ing a winter hike some 14 years ago, a group of hikers, in­clud­ing Wolfgang and this au­thor, walked for sev­er­al miles along a creek, crossed an old bridge and walked on the oth­er side of the creek back to the point right across the creek from our base-camp. While the rest of us looked at the camp and con­tem­plated an ar­du­ous walk back, Wolfgang said “Well, we’ll just have to wade through”, then slowly but surely moved in­to waist-deep freez­ing wa­ter and crossed to the oth­er side. Thus we saw how a Vir­tu­al Haken dealt with this real life prob­lem, and we fol­lowed him.

Much of Wolfgang’s cur­rent sci­entif­ic in­terests con­cern think­ing about fun­da­ment­al prob­lems in cos­mo­logy.

Three of Wolfgang Haken’s six chil­dren live in the Urb­ana-Cham­paign area. Rudolf Haken, a renowned mu­si­cian and a com­poser, is a pro­fess­or of vi­ola in the Uni­versity of Illinois School of Mu­sic. Lip­pold Haken designs elec­tron­ic mu­sic­al in­stru­ments and equip­ment and owns a com­pany man­u­fac­tur­ing a unique “Con­tinuum Fin­ger­board”. He is also a Lec­turer at the De­part­ment of ECE at the Uni­versity of Illinois, and con­ducts re­search re­lated to sound. Armgard Haken re­ceived B.S. and M.S. in bio­logy from UIUC, and is cur­rently a pro­gram man­ager at the Beck­man In­sti­tute, man­aging activ­it­ies and grants re­lated to can­cer re­search.

Haken’s eld­est son, Armin, ob­tained a PhD in Math­em­at­ics from UIUC in 1994, spe­cial­iz­ing in com­plex­ity the­ory and prob­lems re­lated to the­or­et­ic­al com­puter sci­ence. He is cur­rently a soft­ware en­gin­eer in San Fran­cisco. Dorothea Blostein, née Haken, re­ceived a PhD in com­puter sci­ence from UIUC in 1987 and is cur­rently a Pro­fess­or in the School of Com­put­ing at Queen’s Uni­versity in King­ston, Ontario. Ag­nes Debrun­ner, née Haken, re­ceived a B. Sci. in An­im­al Sci­ence from UIUC. She is a lead­er in the U.S. un­der­wa­ter hockey com­munity and lives near Den­ver, Col­or­ado.

As of this date, Wolfgang and Irmgard have 13 grand­chil­dren.

In 2014, in hon­or of Haken’s re­search con­tri­bu­tions, the UIUC De­part­ment of Math­em­at­ics es­tab­lished a gradu­ate stu­dent award called the Wolfgang Haken Prize in Geo­metry and To­po­logy.

Works

[1] W. Haken: “The­or­ie der Nor­malflächen: Ein Iso­topiekri­teri­um für den Kre­isk­noten” [The­ory of nor­mal sur­faces: An iso­top­ic cri­terion for the cir­cu­lar knot], Acta Math. 105 : 3–​4 (1961), pp. 245–​375. MR 141106 Zbl 0100.​19402 article

[2] W. Haken: “Über das Homöomorphiep­rob­lem der 3-Man­nig­faltigkeiten, I” [On the ho­mo­morph­ism prob­lem for 3-man­i­folds, I], Math. Z. 80 : 1 (December 1962), pp. 89–​120. MR 160196 Zbl 0106.​16605 article

[3] W. Haken: “On ho­mo­topy 3-spheres,” Ill. J. Math. 10 : 1 (1966), pp. 159–​178. Re­prin­ted in Ill. J. Math. 60:1 (2016). MR 219072 Zbl 0131.​20704 article

[4] W. Haken: “Trivi­al loops in ho­mo­topy 3-spheres,” Ill. J. Math. 11 : 4 (1967), pp. 547–​554. MR 219073 Zbl 0153.​25703 article

[5] W. Haken: “Al­geb­ra­ic­ally trivi­al de­com­pos­i­tions of ho­mo­topy 3-spheres,” Ill. J. Math. 12 : 1 (1968), pp. 133–​170. MR 222902 Zbl 0171.​22302 article

[6] W. Haken: “Some res­ults on sur­faces in 3-man­i­folds,” pp. 39–​98 in Stud­ies in mod­ern to­po­logy. Edi­ted by P. J. Hilton. MAA Stud­ies in Math­em­at­ics 5. Pren­tice-Hall (Engle­wood Cliffs, NJ), 1968. MR 224071 Zbl 0194.​24902 incollection

[7] W. W. Boone, W. Haken, and V. Poénaru: “On re­curs­ively un­solv­able prob­lems in to­po­logy and their clas­si­fic­a­tion,” pp. 37–​74 in Con­tri­bu­tions to math­em­at­ic­al lo­gic (Han­nov­er, Ger­many, Au­gust 1966). Edi­ted by H. A. Schmidt, K. Schütte, and H.-J. Thiele. Stud­ies in Lo­gic and the Found­a­tions of Math­em­at­ics 50. North-Hol­land (Am­s­ter­dam), 1968. MR 263090 Zbl 0246.​57015 incollection

[8] W. Haken: “Vari­ous as­pects of the three-di­men­sion­al Poin­caré prob­lem,” pp. 140–​152 in To­po­logy of man­i­folds (Athens, GA, 11–22 Au­gust 1969). Edi­ted by J. C. Cantrell and C. H. Ed­wards. Markham (Chica­go), 1970. MR 273624 Zbl 0298.​55002 incollection

[9] W. Haken: “Con­nec­tions between to­po­lo­gic­al and group the­or­et­ic­al de­cision prob­lems,” pp. 427–​441 in Word prob­lems: De­cision prob­lems and the Burn­side prob­lem in group the­ory (Irvine, CA, Septem­ber 1969). Edi­ted by W. W. Boone, R. C. Lyn­don, and F. B. Can­nonito. Stud­ies in Lo­gic and the Found­a­tions of Math­em­at­ics 71. North-Hol­land (Am­s­ter­dam), 1973. Con­fer­ence ded­ic­ated to Hanna Neu­mann. MR 397736 Zbl 0265.​02033 incollection

[10] W. Haken: “Some spe­cial present­a­tions of ho­mo­topy 3-spheres,” pp. 97–​107 in To­po­logy con­fer­ence (Blacks­burg, VA, 22–24 March 1973). Edi­ted by H. F. Dick­man, Jr. and P. Fletch­er. Lec­ture Notes in Math­em­at­ics 375. Spring­er (Ber­lin), 1974. MR 356054 Zbl 0289.​55003 incollection

[11] K. Ap­pel and W. Haken: “Every planar map is four col­or­able,” Bull. Am. Math. Soc. 82 : 5 (September 1976), pp. 711–​712. MR 424602 Zbl 0331.​05106 article

[12] K. Ap­pel and W. Haken: “Every planar map is four col­or­able, I: Dis­char­ging,” Ill. J. Math. 21 : 3 (1977), pp. 429–​490. A mi­crofiche sup­ple­ment to both parts was pub­lished in Ill. J. Math. 21:3 (1977). MR 543792 Zbl 0387.​05009 article

[13] K. Ap­pel, W. Haken, and J. Koch: “Every planar map is four col­or­able, II: Re­du­cib­il­ity,” Ill. J. Math. 21 : 3 (1977), pp. 491–​567. A mi­crofiche sup­ple­ment to both parts was pub­lished in Ill. J. Math. 21:3 (1977). MR 543793 Zbl 0387.​05010 article

[14] K. Ap­pel and W. Haken: “The class check lists cor­res­pond­ing to the sup­ple­ment to ‘Every planar map is four col­or­able. Part I and Part II’,” Ill. J. Math. 21 : 3 (1977), pp. C1–​C210. Mi­crofiche sup­ple­ment. Ex­tra ma­ter­i­al to ac­com­pany the sup­ple­ment pub­lished in Ill. J. Math. 21:3 (1977). MR 543794 article

[15] K. Ap­pel and W. Haken: “Mi­crofiche sup­ple­ment to ‘Every planar map is four col­or­able. Part I and Part II’,” Ill. J. Math. 21 : 3 (1977), pp. 1–​251. Mi­crofiche sup­ple­ment. Sup­ple­ment to the two part art­icle pub­lished as Ill. J. Math. 21:3 (1977) and Ill. J. Math. 21:3 (1977). A class check list was also pub­lished as Ill. J. Math. 21:3 (1977). MR 543795 article

[16] K. Ap­pel and W. Haken: Every planar map is four col­or­able. Con­tem­por­ary Math­em­at­ics 98. Amer­ic­an Math­em­at­ic­al So­ci­ety (Provid­ence, RI), 1989. With the col­lab­or­a­tion of J. Koch. MR 1025335 Zbl 0681.​05027 book