The Great­est Math­e­mati­cian You’ve Never Heard Of

Robert Lang­lands’s work may have no prac­ti­cal ap­pli­ca­tions — and he doesn’t mind at all

The Walrus - - CON­TENTS - by Vi­viane Fair­bank

The Great­est Math­e­mati­cian You’ve Never Heard Of Robert Lang­lands’s work may have no prac­ti­cal ap­pli­ca­tions—and he doesn’t mind at all 40

One morn­ing in late May, in an au­di­to­rium at the Uni­ver­sity of Oslo, Robert Lang­lands, one of the most in­flu­en­tial math­e­ma­ti­cians in the world, de­liv­ered a lec­ture to a crowd of dis­tin­guished col­leagues. They had gath­ered to mark the eighty- one-year- old’s re­ceipt of the Abel Prize — what many con­sider the equiv­a­lent of the No­bel Prize for math­e­mat­ics. Sharpie in hand, Lang­lands paced back and forth, look­ing mostly to­ward the floor, and be­gan speak­ing about the field of re­search that bears his name. “The ques­tion is,” the Cana­dian math­e­mati­cian told the as­sem­bled re­searchers, pro­fes­sors, and stu­dents, “in its present form, do all as­pects of it en­joy my ap­proval? And the an­swer is no.” Some in the crowd chuck­led. “If some­one could just re­move my name from that part and give it an­other name, I’d ap­pre­ci­ate it very much.” For the rel­a­tively se­date world of academia, this was the equiv­a­lent of an ac­claimed direc­tor ac­cept­ing an Os­car for best movie only to take the stage and lam­baste the movie’s se­quel, di­rected by some­one else. Near the front of the au­di­to­rium sat a man named Ed­ward Frenkel — a Rus­sian-born, Cal­i­for­nia-based math­e­mati­cian who has spent

much of his ca­reer ex­pand­ing on and pop­u­lar­iz­ing Lang­lands’s work. In the 1960s, Lang­lands dis­cov­ered a re­la­tion­ship be­tween two fields gen­er­ally con­sid­ered to be sep­a­rate: num­ber the­ory (the study of in­te­gers) and har­monic anal­y­sis (the study of con­tin­u­ous phe­nom­ena such as waves made by the os­cil­la­tion of a gui­tar string). His work — which es­tab­lished a strong and un­ex­pected bridge be­tween the realm of whole num­bers and the realm of con­tin­u­ous func­tions —cre­ated the pos­si­bil­ity for a new way of think­ing about con­nec­tions in math­e­mat­ics al­to­gether. Lang­lands’s ob­ser­va­tions are so sig­nif­i­cant they have in­spired hun­dreds of schol­ars, many of whom have spent their ca­reers re­fin­ing and prov­ing them, and launched whole new ar­eas of re­search. One of these new fields—in which Frenkel is a lead­ing scholar—ex­pands upon Lang­lands’s orig­i­nal ob­ser­va­tions by ex­tend­ing them to ge­om­e­try, which, in turn, in­tro­duces the pos­si­bil­ity of ap­pli­ca­tions in other dis­ci­plines. To­gether, the var­i­ous lines of in­ves­ti­ga­tion (the so-called clas­sic Lang­lands Pro­gram and the ex­panded geo­met­ric and phys­i­cal Lang­lands pro­grams) com­prise one of the most am­bi­tious math­e­mat­i­cal re­search projects in ex­is­tence — one over which Lang­lands him­self has no con­trol. The gap be­tween Lang­lands and Frenkel, who teaches at the Uni­ver­sity of Cal­i­for­nia, Berke­ley, is shaped by their dif­fer­ent tech­ni­cal ap­proaches to pure math­e­mat­ics and also by their mo­ti­va­tions: some of Frenkel’s work deals with the po­ten­tial “use­ful­ness” of math­e­mat­ics, which Lang­lands isn’t so con­cerned with. This is most ap­par­ent in ap­pli­ca­tions of the geo­met­ric (but not the clas­sic) Lang­lands Pro­gram to the­o­ret­i­cal physics—which, with its cap­ti­vat­ing no­tions of space-time ex­plo­ration and quest to un­der­stand the build­ing blocks of the uni­verse, can cap­ture the imag­i­na­tion more than es­o­teric math­e­mat­i­cal ini­tia­tives do. The two math­e­ma­ti­cians have also come to sym­bol­ize a greater chasm in the aca­demic realm: the dif­fer­ence be­tween fid­dling with ab­stract con­cepts for the pure en­joy­ment of the ex­er­cise and do­ing it to solve con­crete prob­lems in the real world . As uni­ver­si­ties shift away from their tra­di­tional role as cen­tres for pure re­search and in­creas­ingly favour more prag­matic and ap­plied dis­ci­plines (less phi­los­o­phy, more en­gi­neer­ing), the ques­tion of what pur­pose math­e­mat­ics serves—or whether it needs to serve one—has very real con­se­quences for its prac­ti­tion­ers. Against this back­drop, the var­i­ous in­ter­pre­ta­tions of the Lang­lands pro­grams have be­come a case study for de­bates about the jus­ti­fi­ca­tion of ab­stract sci­ence. In 2013, Canada’s then min­is­ter of sci­ence and tech­nol­ogy, Gary Goodyear, told the CBC that his de­part­ment was ask­ing sci­en­tists to “ap­pre­ci­ate the busi­ness side of sci­ence. . . . Knowl­edge that is not taken off the shelf and put into our fac­to­ries is ac­tu­ally of no value.” And so, says Michael Har­ris, an Amer­i­can num­ber the­o­rist, pure math­e­ma­ti­cians some­times par­tic­i­pate in a col­lec­tive myth to se­cure their sur­vival: they speak about their re­search as im­mi­nently ap­pli­ca­ble. The pres­sure many math­e­ma­ti­cians feel to make the re­sults of their re­search ei­ther as ro­man­tic as black holes or as prac­ti­cal as stock bro­ker­age has be­gun to rid that re­search of its free­dom.

Robert lang­lands grew up in the Lower Main­land of British Columbia in the 1930s and ’40s. Bored in class, he thought of drop­ping out for a time. Af­ter a high-school teacher en­cour­aged him to con­tinue his stud­ies, Lang­lands at­tended the Uni­ver­sity of British Columbia, then Yale. Now a pro­fes­sor emer­i­tus at the In­sti­tute for Ad­vanced Study—he oc­cu­pies one of the for­mer of­fices of Al­bert Ein­stein—lang­lands spends his time in Prince­ton and some­times Mon­treal. His speech is slow and cal­cu­lated, and he has the kind of harsh, dead­pan hu­mour that is par­tic­u­lar to cer­tain self-aware in­tel­lec­tu­als; he will of­ten sug­gest to an in­ter­viewer what they re­ally meant to ask or should have asked in­stead. (From Lang­lands’s re­ply at the end of one printed in­ter­view with a UBC grad­u­ate stu­dent: “I have al­ready ex­hausted both my­self and you by an­swer­ing your very few ques­tions at great length. You will be grate­ful that you asked no more. So am I.”) Frenkel, whose looks in­spired one news­pa­per to call him “the world’s sex­i­est spokesman for math­e­mat­ics,” is a much more nat­u­ral emis­sary to the out­side world: he has a soft voice that in­spires com­fort, and he tends to tell his au­di­ences that they have good “en­ergy,” that math­e­ma­ti­cians are a “fam­ily,” that he speaks “from the heart.” In 2013, Frenkel pub­lished a best­selling gen­eral-in­ter­est book about math­e­mat­ics called Love & Math: The Heart of Hid­den Re­al­ity. The book tells the story of Frenkel’s life as a young Jewish man in Moscow, and it ex­plains, in (rel­a­tively) sim­ple terms, the Lang­lands Pro­gram and Frenkel’s own work on it. In the book, Frenkel uses fa­mil­iar metaphors to coax his read­ers into the world of math: find­ing ir­ra­tional num­bers is like adding cubes of sugar to a cup of tea; quan­tum du­al­ity is like the re­la­tion­ship be­tween pota­toes and onions in his mother’s borscht recipe; the Lang­lands Pro­gram is like a gi­ant jig­saw puzzle. Frenkel has writ­ten columns for the New York Times and has been a guest on the Col­bert Re­port. He costarred in his own semierotic movie about the beauty of math­e­mat­ics, and he is prone to telling re­porters things like “I will show you the ec­stasy of math­e­mat­ics.” So when a jour­nal­ist writes about the Lang­lands Pro­gram, they gen­er­ally don’t in­ter­view Lang­lands; they in­ter­view Frenkel.

Lang­lands’s work has formed the ba­sis of one of the most am­bi­tious math­e­mat­i­cal re­search projects in ex­is­tence.

“Math­e­ma­ti­cians have mis­er­ably failed, as a pro­fes­sion, in terms of com­mu­ni­cat­ing with the gen­eral au­di­ence,” says Frenkel. “But also within math­e­mat­ics we fail com­pletely in com­mu­ni­cat­ing with each other. . . . In my mind, it’s our obli­ga­tion to share.” Lang­lands is not much in­clined to this kind of sales ef­fort. If math­e­mat­ics is a “uni­ver­sal” lan­guage, as many pop­u­lar sci­ence books claim (2+2=4, whether it’s writ­ten in English or Rus­sian), then it is also one of the least ac­ces­si­ble — there is an over­whelm­ing num­ber of di­alects, rules, and ex­cep­tions. Dif­fer­ent fields of study within math­e­mat­ics use dif­fer­ent ap­proaches and vo­cab­u­lary, and it’s rare for a re­searcher in one area to fully ab­sorb work in an­other. Lang­lands is con­sid­ered an ex­cep­tional tal­ent be­cause of his pro­fi­ciency in sev­eral dis­ci­plines. The work he de­vel­oped in 1967 is so so­phis­ti­cated that, still to­day, only a few peo­ple can com­pre­hend it. (Lang­lands was up for the Abel Prize at least three times be­fore it was ac­tu­ally awarded to him; the schol­ars who nom­i­nated him think it was in part be­cause the com­mit­tee had a hard time un­der­stand­ing his work.) His­tor­i­cally, pure math­e­ma­ti­cians have al­most al­ways em­braced the fu­til­ity of their most ab­stract in­quiries. They would of­ten de­scribe their re­search as the un­tar­nished pur­suit of beauty or uni­ver­sal truth. The num­ber the­o­rist G. H. Hardy wrote in 1940 that “real math­e­mat­ics . . . must be jus­ti­fied as arts if it can be jus­ti­fied at all.” That kind of en­gage­ment with the philo­soph­i­cal un­der­pin­nings of math­e­mat­ics is less com­mon now, and to­day even some pure math­e­ma­ti­cians will talk read­ily about prac­ti­cal ex­ten­sions of their work. Those who are in­ter­ested in com­puter sci­ence or the­o­ret­i­cal physics, for in­stance, also ar­gue that while the ab­stract might not be im­me­di­ately ap­pli­ca­ble, it will al­most al­ways de­velop into some­thing use­ful — in the same way that non-eu­clidean ge­om­e­try even­tu­ally helped Ein­stein de­velop his the­ory of gen­eral rel­a­tiv­ity and num­ber the­ory is now in­stru­men­tal to pass­word se­cu­rity. For Lang­lands, sci­en­tific pur­suits such as ge­ol­ogy or bi­ol­ogy help us “deal with the world and the uni­verse as they are”; the Lang­lands Pro­gram, on the other hand, is “only im­por­tant in­so­far as the math as such is im­por­tant.”

When he started do­ing re­search in the ’60s, Lang­lands was sim­ply in­ves­ti­gat­ing po­ten­tial ex­ten­sions of class field the­ory, a branch of pure math­e­mat­ics. The most ob­vi­ous ap­pli­ca­tions of his work are in other ar­eas of of­ten equally es­o­teric math­e­mat­ics. An­drew Wiles, the num­ber the­o­rist who, in 1995, pub­lished a proof of Fer­mat’s Last The­o­rem—then in the Guin­ness Book of World Records

De­bates about the Lang­lands Pro­gram and its “use­ful­ness” have be­come a case study in the jus­ti­fi­ca­tion of ab­stract sci­ence.

as the “most dif­fi­cult math­e­mat­i­cal prob­lem”— solved it with help from the Lang­lands Pro­gram. Most math­e­ma­ti­cians also agree that the Lang­lands Pro­gram could help find a proof for the Rie­mann Hy­poth­e­sis, prob­a­bly the most fa­mous un­solved math­e­mat­i­cal prob­lem (about the dis­tri­bu­tion of prime num­bers). These prob­lems are just as ab­stract as Lang­lands’s own work, how­ever, which means his re­search pro­gram as it was orig­i­nally con­ceived has lit­tle rel­e­vance to ev­ery­day life. As Lang­lands bluntly puts it, his work “could dis­ap­pear without any trace, and the world would go on as it was be­fore.” (The most ba­sic—yet still painfully in­ac­ces­si­ble—ex­pla­na­tion of Lang­lands’s cal­cu­la­tions is that he re­lated Galois groups, which are num­ber field ex­ten­sions, to au­to­mor­phic forms and rep­re­sen­ta­tion the­ory of al­ge­braic groups.) In the 1980s, math­e­ma­ti­cians around the world, in­clud­ing Frenkel, started work­ing on the geo­met­ric Lang­lands Pro­gram: a re­lated but par­al­lel field of study that ex­plores some ap­pli­ca­tions of Lang­lands’s find­ings. Ge­om­e­try and num­ber the­ory had long been con­sid­ered sep­a­rate dis­ci­plines: ge­om­e­try comes mostly from the Greek tra­di­tion, while al­ge­bra orig­i­nated largely in the Mid­dle East dur­ing the golden age of Is­lam. Even af­ter René Descartes fa­mously merged the two sub­jects in his 1637 La Géométrie, they rarely over­lapped: al­ge­bra was con­sid­ered a pure sub­ject, while ge­om­e­try, be­cause of its var­i­ous prac­ti­cal uses, was of­ten treated as a kind of “mixed” pur­suit — what we would now call ap­plied math­e­mat­ics. A va­ri­ety of con­nec­tions be­tween ge­om­e­try and num­ber the­ory have de­vel­oped since, but few are as ex­ten­sive as what the geo­met­ric Lang­lands Pro­gram is hop­ing to con­struct. The ex­pan­sion of the geo­met­ric pro­gram also par­al­leled many de­vel­op­ments in the world of the­o­ret­i­cal physics, which sets out to ex­plain the most fun­da­men­tal par­ti­cles and forces in na­ture; in part be­cause of this progress, physi­cists and math­e­ma­ti­cians have been col­lab­o­rat­ing more in the past decade than ever be­fore. In the 1970s, physi­cists de­vel­oped a the­o­ret­i­cal frame­work that at­tempts to ex­plain all of the known forces of the uni­verse at once. Called the Stan­dard Model, it has been in­cred­i­bly suc­cess­ful in pre­dict­ing phe­nom­ena (in­clud­ing the Higgs bo­son, the fa­mous “God par­ti­cle,” which could be re­spon­si­ble for all the mass of the uni­verse) that have since been con­firmed to ex­ist. What the model does not do, how­ever, is ac­count for the forces of grav­ity or dark mat­ter (a mys­te­ri­ous sub­stance—if it even is a sub­stance—that is thought to make up a large part of the uni­verse). In ef­forts to ad­dress this and other is­sues, re­searchers have ex­panded on the Stan­dard Model by in­vent­ing the prin­ci­ple of su­per­sym­me­try, which pre­dicts a spe­cific re­la­tion­ship be­tween par­ti­cles. But de­spite nu­mer­ous costly ex­per­i­ments, that re­la­tion­ship has

never been shown to ex­ist. “This is a lit­tle bit of a cri­sis,” one math­e­mati­cian said re­cently. The­o­ret­i­cal physi­cists tack­ling this prob­lem to­day work with a cou­ple of dif­fer­ent types of the­o­ries (“su­per­sym­met­ric gauge the­o­ries” and “su­per­sym­met­ric quan­tum field the­o­ries”) that at­tempt to de­scribe the con­nec­tions be­tween par­ti­cles in math­e­mat­i­cal terms. And that, says Uni­ver­sity of Toronto math­e­mati­cian Joel Kam­nitzer, is where the geo­met­ric Lang­lands Pro­gram can help (though, he con­cedes, “I can’t say I com­pletely un­der­stand why”). Though it can be as ab­stract as pure math­e­mat­ics, the­o­ret­i­cal physics has found a much more es­tab­lished place in the pub­lic con­scious­ness, in part be­cause of its en­tic­ing vis­ual el­e­ments and its of­fer of ad­ven­ture. Sci­ence fic­tion has ex­posed even the most math averse to ideas about black holes, worm­holes, and nth di­men­sions—what could be more com­pelling than the prospect of trav­el­ling the uni­verse and com­ing back only one day older?—and enor­mous amounts of money are poured into in­ves­ti­gat­ing phe­nom­ena such as dark mat­ter. We seem to be much more will­ing to face the ab­stract when it can fuel our imag­i­na­tion of the fan­tas­ti­cal. This may ex­plain why the out­side world has been so much more in­ter­ested in Lang­lands’s work since Frenkel came along. The­o­ret­i­cal physics starts with con­jec­ture that can lead to ex­per­i­men­ta­tion and re­fine­ment; math­e­mat­ics, on the other hand, is based on ax­ioms and proofs, and re­jects any kind of spec­u­la­tion. This

Lang­lands spent ev­ery morn­ing, seven days a week, for five years work­ing on the pa­per he de­liv­ered in Oslo. It is writ­ten en­tirely in Rus­sian and ded­i­cated in large part to re­for­mu­lat­ing the geo­met­ric pro­gram cham­pi­oned by Frenkel. This new pa­per is an at­tempt to shift the field to­ward a more tra­di­tional ap­proach: it pro­poses a new math­e­mat­i­cal ba­sis for the geo­met­ric the­ory that re­lates more closely to Lang­lands’s own con­jec­tures by us­ing sim­i­lar tools to the ones he used in the ’60s—in the process, restor­ing his work back to its orig­i­nal arith­metic pu­rity. Be­fore giv­ing his talk, Lang­lands sent por­tions of the pa­per to var­i­ous math­e­ma­ti­cians fa­mil­iar with his re­search. Sev­eral months later, he hasn’t heard back from any­one who’s been able to un­der­stand it in its en­tirety. “If he wants more peo­ple to pay at­ten­tion to it,” Arthur says, “why on earth would he write it in Rus­sian?” Some peo­ple have the­o­rized that it was tar­geted at means that “physi­cists and math­e­ma­ti­cians have set­tled on a fine di­vi­sion of labour in which the for­mer com­plain about the finick­i­ness of the lat­ter, and the lat­ter com­plain about the slop­pi­ness of the for­mer,” the­o­ret­i­cal physi­cist Sabine Hossen­felder wrote in her re­cent book Most math­e­ma­ti­cians work­ing on the Lang­lands Pro­gram have no idea what the­o­ret­i­cal physi­cists are work­ing on, and the ex­tent to which the Lang­lands Pro­gram could be use­ful to physics is still un­known. Con­nect­ing the geo­met­ric the­ory to physics can, none­the­less, make it more en­tic­ing. Pop­u­lar cov­er­age of the geo­met­ric and phys­i­cal Lang­lands pro­grams makes it sound like there’s a wide-sweep­ing col­lab­o­ra­tion be­tween sci­en­tists, where the edges of ab­stract math­e­mat­ics and physics have fi­nally con­joined—in an ideal pla­tonic con­ver­gence of truths—such that the se­crets of the uni­verse will be un­cov­ered in math­e­mat­i­cal terms. Still, a math­e­mati­cian who claims that a project like the Lang­lands Pro­gram has “any phys­i­cal sig­nif­i­cance,” Lang­lands says, is in­evitably “go­ing to dis­ap­point peo­ple.”

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