Celebratio Mathematica

Andrew A. Ranicki

Andrew Alexander Ranicki, PhD
30 December 194821 February 2018

by Dusa McDuff

An­drew Ran­icki (born 30 Decem­ber 1948, Lon­don, and died 21 Feb­ru­ary 2018, Ed­in­burgh) spent his early years with his par­ents in Po­land and Ger­many. Later, after a few years at school in Can­ter­bury, he went to Trin­ity Col­lege, Cam­bridge, where he gradu­ated with a BA in 1969, and a PhD in 1973 un­der the su­per­vi­sion of Frank Adams and An­drew Cas­son. A Re­search Fel­low at Trin­ity Col­lege dur­ing 1972–77, he spent the next five years at Prin­ceton Uni­versity, be­fore tak­ing up a Lec­ture­ship in the De­part­ment of Math­em­at­ics at Ed­in­burgh Uni­versity in 1982. He worked at Ed­in­burgh for the rest of his ca­reer, be­ing ap­poin­ted as Pro­fess­or of Al­geb­ra­ic Sur­gery 1995–2016.

An­drew wrote sev­en in­flu­en­tial books on vari­ous as­pects of sur­gery the­ory, as well as some sev­enty re­search pa­pers; he or­gan­ised many con­fer­ences both in Ed­in­burgh and in Ger­many, and su­per­vised el­ev­en PhD stu­dents. He (co)ed­ited twelve col­lec­tions of pa­pers on a wide vari­ety of top­ics in al­geb­ra­ic to­po­logy, as well as work­ing on the ed­it­or­i­al boards of many journ­als. He was very eru­dite and cul­tur­ally lit­er­ate, and cur­ated an ima­gin­at­ive and in­form­at­ive web site (now archived by the Uni­versity of Ed­in­burgh). Among oth­er hon­ours, he was awar­ded the Cam­bridge Uni­versity Smith prize in 1972, was elec­ted a Fel­low of the Roy­al So­ci­ety of Ed­in­burgh in 1992, and won the Ju­ni­or White­head Prize in 1983 and Seni­or Ber­wick prize in 1994, both from the Lon­don Math­em­at­ic­al So­ci­ety.

When An­drew was pro­moted to full pro­fess­or he be­came pro­fess­or of Al­geb­ra­ic Sur­gery. This title was some­what whim­sic­al; it had to be ap­proved by the Ed­in­burgh Med­ic­al School, whose an­noy­ance greatly amused An­drew. However it was also very apt: al­geb­ra­ic sur­gery was ac­tu­ally a very sig­ni­fic­ant part of An­drew’s con­tri­bu­tion to math­em­at­ics.

Sur­gery the­ory was ini­ti­ated by Ker­vaire and Mil­nor and de­veloped by Browder, Novikov, Sul­li­van and Wall. It es­tab­lished a meth­od for clas­si­fy­ing high-di­men­sion­al com­pact man­i­folds (a very ba­sic and im­port­ant class of to­po­lo­gic­al spaces), based on found­a­tion­al tools from geo­met­ric to­po­logy, in­clud­ing cobor­d­ism the­ory, vec­tor bundles, and trans­vers­al­ity. The res­ult­ing “sur­gery ex­act se­quence” [e1], ex­presses the clas­si­fic­a­tion in terms of ho­mo­topy the­ory and cer­tain al­geb­ra­ic­ally defined sur­gery ob­struc­tion groups.

One of the main themes of An­drew’s work is the ex­plor­a­tion of the bor­der­land between geo­metry and al­gebra. An early achieve­ment [1] was to show that Wall’s sur­gery ob­struc­tion groups could be defined as the cobor­d­ism groups of al­geb­ra­ic Poin­caré com­plexes over a suit­able ring. This was the ori­gin of a dra­mat­ic re­for­mu­la­tion of sur­gery. An­drew con­tin­ued to de­vel­op and ap­ply the rel­ev­ant al­gebra throughout his whole ca­reer, in a series of lengthy pa­pers and books. This new the­ory, called al­geb­ra­ic \( L \)-the­ory, cul­min­ated in his 1992 book Al­geb­ra­ic \( L \)-The­ory and To­po­lo­gic­al Man­i­folds [2] which iden­ti­fies the geo­met­ric sur­gery ex­act se­quence en­tirely in al­geb­ra­ic terms. Many col­leagues re­gard this as his most in­flu­en­tial work.

The flex­ib­il­ity of al­geb­ra­ic Poin­caré cobor­d­ism, which is defined over every ring with in­vol­u­tion, was es­sen­tial for es­tab­lish­ing Wall’s “Main ex­act se­quence” (1976) to ef­fect­ively cal­cu­late the sur­gery ob­struc­tion groups. This meth­od de­pends on the al­geb­ra­ic \( L \)-the­ory of loc­al­iz­a­tion and com­ple­tions of rings with in­vol­u­tion to re­duce cal­cu­la­tions from in­teg­ral group rings to those over sim­pler rings.

Al­geb­ra­ic \( L \)-the­ory has a large num­ber of strik­ing ap­plic­a­tions, most of which were ob­tained by An­drew him­self, or in col­lab­or­a­tions with col­leagues and his PhD stu­dents. A brief list of top­ics in­dic­ates the breadth of these ap­plic­a­tions:

  • Fi­nite­ness ob­struc­tions and tor­sion in­vari­ants for chain com­plexes (1985–87)
  • Al­geb­ra­ic split­ting for \( K \)- and \( L \)-the­ory of poly­no­mi­al rings \( A[t,t^{-1}] \) (1986)
  • Sur­gery trans­fer maps for fibre bundles (1988, 1992)
  • Chain com­plexes over ad­dit­ive cat­egor­ies (1989)
  • Lower \( K \)- and \( L \)-the­ory; con­trolled to­po­logy (1992, 1995, 1997, 2003, 2006)
  • Fi­nite dom­in­a­tion and Novikov rings (1995)
  • Bor­d­ism of auto­morph­isms (1996)
  • Circle-val­ued Morse the­ory and Novikov ho­mo­logy (1999, 2000, 2002, 2003)
  • Non­com­mut­at­ive loc­al­iz­a­tion (2004, 2006, 2009)
  • Blanch­field and Seifert al­gebra in high-di­men­sion­al knot the­ory (2003, 2006)
  • The geo­met­ric Hopf in­vari­ant and sur­gery the­ory (2016)

As well as de­vel­op­ments that arose from his lively in­ter­ac­tions with col­leagues and cowork­ers such as Mi­chael Weiss, Wolfgang Lück, Erik Ped­er­sen, Jim Dav­is, and Ian Hamb­leton, An­drew’s work has the po­ten­tial to in­spire sig­ni­fic­ant new dir­ec­tions.

Den­nis Sul­li­van points out that \( L \)-the­ory, which is based on the Poin­caré du­al­ity prop­erty of man­i­folds, provides a sys­tem­at­ic way to in­cor­por­ate du­al­ity in­to a vari­ety of al­geb­ra­ic set­tings, and this might help make vari­ous long­stand­ing con­jec­tures about man­i­folds fi­nally ac­cess­ible.

Every­one who met An­drew en­joyed his wit, his in­ex­haust­ible store of lit­er­ary an­ec­dotes, and his vivid per­son­al­ity. This was per­haps in large part due to his fam­ily back­ground: his fath­er Mar­cel Reich-Ran­icki, after a har­row­ing es­cape from the Warsaw ghetto with his wife, even­tu­ally be­came the dom­in­ant lit­er­ary crit­ic in Ger­many. Mar­cel was a lar­ger than life celebrity — he was some­times called the “Pope of Ger­man lit­er­at­ure” — and An­drew had a pres­ence and laugh to match.

Iain Gor­don, re­flect­ing on An­drew’s con­tri­bu­tions to de­part­ment life, writes as fol­lows:

An­drew was an ir­re­press­ible spir­it in the School of Math­em­at­ics who in­spired reg­u­lar vis­its to Scot­land from the very best to­po­lo­gists, of­ten un­der the ban­ner of Scot­tish To­po­logy which he helped to run since its found­a­tion in 1981. In ad­di­tion, he ar­ranged many pres­ti­gi­ous sem­inars, col­loquia and events all across the dis­cip­line, fea­tur­ing great math­em­aticians from throughout the world, in­clud­ing in the last few years Fields Medal­lists Sir Mi­chael Atiyah, Gerd Falt­ings, Timothy Gowers and Ed Wit­ten. His house was Ed­in­burgh’s math­em­at­ic­al liv­ing room, filled with laughter, gos­sip, and re­search high­lights. In de­part­ment­al life, he taught a vari­ety of courses mostly in al­gebra, geo­metry, num­ber the­ory and to­po­logy, nearly al­ways ex­pos­ing the stu­dents to quad­rat­ic forms and giv­ing them glimpses of the rich math­em­at­ic­al world he in­hab­ited. He was a con­stant ad­voc­ate for the use of tech­no­logy in the School, from email through the in­ter­net and onto so­cial me­dia, best il­lus­trated by his mo­nu­ment­al and mul­ti­far­i­ous webpage which is now a per­man­ent re­source for math­em­aticians all over the world.

The fol­low­ing trib­ute was writ­ten by his stu­dent Car­men Rovi.

It has been al­most a year since An­drew left us, but try­ing to write about him, I feel I lack the words to ex­press the whirl­wind of emo­tions that his memory brings to me. How was An­drew as an ad­visor? An­drew was kind-hearted, gen­er­ous, un­der­stand­ing and above all, he was en­thu­si­ast­ic. His en­thu­si­asm for to­po­logy and in par­tic­u­lar for sur­gery the­ory was a power­ful beacon for any­one work­ing in the area and es­pe­cially for his stu­dents. He was able to com­mu­nic­ate this en­thu­si­asm and im­press onto us his el­eg­ant un­der­stand­ing of math­em­at­ics.

It is true, and every­one who talked to An­drew will have ex­per­i­enced this at some point, that he would some­times fall asleep. It is also true that he would then wake up and ask an in­ter­est­ing ques­tion or make an in­sight­ful com­ment. The point is that be­ing with An­drew it was im­possible to get bored. His sense of hu­mour and hearty laugh com­bined with a breadth and depth of know­ledge not just in math­em­at­ics, but also in oth­er areas like mu­sic or lit­er­at­ure, was al­ways a source of mo­tiv­a­tion.

An­drew’s lec­tures were al­ways ex­cit­ing. Many times he would add a joke or a pic­ture that would help fix an ab­stract no­tion in stu­dents’ minds. One of his fa­vor­ites was the pic­ture of the Rev­er­end Robert Walk­er “Skat­ing on Dud­ding­ston Loch” by Rae­burn, one of Scot­land’s best-known paint­ings. Look­ing very closely at this pic­ture one can see that the min­is­ter is draw­ing a fig­ure eight on the ice with his skates. An­drew would use this to give an ex­ample of the ab­stract no­tion of an ele­ment in the fun­da­ment­al group of the wedge sum of two circles.

When think­ing about how An­drew nur­tured his stu­dents, the beau­ti­ful poem by Seamus Heaney Saint Kev­in and the black­bird comes to mind. The poem nar­rates the story of how a black­bird lays her eggs in Saint Kev­in’s hand while he is pray­ing and how he pa­tiently pro­tects them un­til the young are hatched and fledged and flown. And just like Saint Kev­in’s pray­er, “to la­bour and not to seek re­ward” was also part of An­drew’s philo­sophy: so much ef­fort in put­ting to­geth­er in­ter­est­ing pro­jects for his stu­dents, so many hours spent guid­ing us to­wards our goals and listen­ing to our at­tempts…he did all this in his self­less, gen­er­ous man­ner that we will al­ways treas­ure in our memory.

An­drew’s gen­er­os­ity of spir­it, com­bined with his wife Ida’s amaz­ing abil­ity to con­jure up mag­ni­fi­cent ve­get­ari­an meals and ima­gin­at­ive aper­itifs, cre­ated con­vivi­al gath­er­ings of laughter and great good cheer. These of­ten took place in their sur­pris­ingly sunny garden, a whole world cre­ated by Ida of green beauty, mys­ter­i­ous dense un­der­growth and bright blooms, with in­vit­ing little spaces to sit and chat. Their house was also a mar­velously wel­com­ing space for many math­em­aticians at all levels, eager stu­dents and es­tab­lished prize-win­ners, their friends and fam­il­ies in­cluded. In later years, a gift from Mar­cel (pro­ceeds from the book Mein Leben about his life) al­lowed them to build a won­der­ful stone con­ser­vat­ory be­side their house, dec­or­ated by a bust of Mar­cel, a stone frog by the to­po­lo­gist Frank Quinn, and many flowers. One mem­or­able oc­ca­sion was the 80th birth­day party for the dis­tin­guished math­em­aticians Fritz Hirzebruch and Sir Mi­chael Atiyah, with their renowned Sig­na­ture For­mula baked in bread by Ida and placed along the full length of the table.

An­drew and Ida had one child Carla, who grew up in Ed­in­burgh. In mar­ry­ing Ida, An­drew also gained two teen­age step-chil­dren, Matt and Alice, who stayed in Amer­ica when An­drew and Ida moved to Ed­in­burgh. Even­tu­ally An­drew had four Amer­ic­an grand­chil­dren, Emily, Christina, Mi­chael, and Thomas, whom he loved and who all loved their “Grandrew”. Later his daugh­ter Carla also had a child, his much be­loved grand­son Nico Mar­cel, who greatly en­riched his fi­nal years.

Many thanks to Iain Gor­don, Ian and Taida Hamb­leton, Anna Mc­Duff, Erik Ped­er­sen, Carla Ran­icki, Car­men Rovi and Den­nis Sul­li­van for their con­tri­bu­tions to this trib­ute.


[1] A. A. Ran­icki: “Al­geb­ra­ic \( L \)-the­ory, IV: Poly­no­mi­al ex­ten­sion rings,” Com­ment. Math. Helv. 49 (1974), pp. 137–​167. Parts I and II were pub­lished in Proc. Lond. Math. Soc. 27 (1973). Part III was pub­lished in Al­geb­ra­ic \( K \)-the­ory III (1973). MR 414664 Zbl 0293.​18021 article

[2] A. A. Ran­icki: Al­geb­ra­ic \( L \)-the­ory and to­po­lo­gic­al man­i­folds. Cam­bridge Tracts in Math­em­at­ics 102. Cam­bridge Uni­versity Press, 1992. Re­prin­ted in pa­per­pack in 2008. MR 1211640 Zbl 0767.​57002 book