36 MENTALJLOSS

In 2002, a reclusivessian genius ed

Grigori relman put an enore than 100years of

suffering in t mathematicalcommunity: He so ed emost difficult ma rIm

f e 20th cen ry- ep. are Conjecture. Its sircall had lured ge rati s fmathematicians t · t 11 alaves.At first, its simplici

would seduce th m, andt ey'd b com co ·ncet e ans er was n areBut asyears passed, they'd be leftwith nothing to show for theirlives' toil but e d ends. Bythe time Grigori Perelmanproed the Conj re, thesolution was 0 $1million.

I

i I

I 'I

THE MAN BEHIND THE MADNESSIn 1885, all of Europe was talking about HenriPoincare, a 30-year-old genius who'd mathematicallyproven why the solar system holds together. When ahole appeared in his calculations, he plugged it upby essentially inventing chaos theory: Kings weretripping over themselves to make him a knight·and Sweden gave him a small fortune in prizemoney. To this day; Poincare holds the recordfor the most physics Nobel Prize nominations,though he never actually won one.

But his most legendary achievement wassomething no one noticed until much, much later.At the turn of the century: Poincare invented anentirely new field called algebraic topology; andtoday, it's one of the most complicated and vibrantbranches of mathematics. Think of it as a twistedversion of geometry, in which shapes stretch, bend,and fold inside out. Poincare's goal was to classifyobjects by identifying their basic form, much thesame way botanists classify new species of plants.In the process of creating topology, Poincare tossedout a conjecture that seemed to be true. It wasa side note to a larger problem, and he figuredhe'd work out the details later. Little did he know;

his side note would become one of the greatestchallenges in the mathematical world.

THE VICTIMSPoincare's conjecture seemed simple enough. Itclaimed that any object without a loop is essentiallya sphere. Think of a knife made out of Play-Doh.Without punching a hole in it or closing a loop,can you squish it into a ball? Yes, of course. Nowpicture a pair of Play-Doh scissors. No matterhow hard you try, you can't crush it into a ballwithout closing up the finger holes. It's impossible.Poincare believed that objects like the knife wererelated to spheres, while objects with holes andloops in them were not.

Poincare thought the conjecture would be easyto prove, and he even published a solution. Butthen, he saw a flaw in his work and retracted it.After his death in 1912, the question lay dormantfor decades, until an Oxford professor named J,H.C.Whitehead rediscovered it in the late 1930s. J,H.C.(known to his students as "Jesus, he's confusing")also published a solution. But he, too, found amistake and retracted it. However, his work sparkedinterest in the problem. By the 1950s, the PoincareConjecture was one of the best-known challengesin the math community:

That's when two Princeton students, EdwinMoise and Christos Papakyriakopoulos (commonlyknown as Papa), decided to try their hands at it.Moise in particular looked like the guy to do it.Young and brash, he liked to announce his next bigproblem like a batter calling his shot. Twice thatincluded one of the toughest problems in topology;and twice he returned with the solution. Then, heset his sights on Poincare.

Papa was vastly different. A self-taught politicalrefugee from Greece, he was famous for his odd,obsessive nature. Legend has it that when hecame to Princeton, he checked into a motel and

MENTALFlOSS.COM 37

TO THE VICTOR GONONE OF THE SPOILSThrough all of this, some mathematicians devotedthemselves to disproving Poincare. In fact, duringthe 1950s, a man named R.H. Bing would spendtwo weeks trying to prove the Conjecture wastrue and then two weeks trying to prove it was

it had always been assumed that problems wereeasier to solve in dimensions that we can visualize.Smale broke new ground by solving a problemin higher dimensions before the lower ones, andtoday; it's common practice. Mathematicians say theextra dimensions .give them room to manipulateimaginary objects.

Smale's discovery inspired the math world, and anew generation of Don Quixotes started sharpeningtheir lances. Another ray of hope came in 1982,whenmathematician Michael Freedman managed to scoopup Poincare in the fourth dimension. Both he andSmale received the Fields Medal. math's equivalentto the Nobel Prize, just for their partial proofs. Andyet, the question of the third dimension-the onlyone that had actually interested Henri Poincare-stillremained. Technically speaking, without the thirddimension, mathematicians were no closer to theanswer than they'd been in 1904.

"When Moise rea edbest effo were

never goin to solveine '1 s -ritke. He never did

e smath researcain and spents last few years-tiquing poe~'1

never checked out. He nevereven unpacked his bags. Hesimply fell into a routine thathe followed every day; downto the minute, which alwaysincluded a midday nap on topof his desk.

Throughout the 1950s, thetwo geniuses dueled with eachother over Poincare. Papawould announce a proof, andMoise would shoot it down.Then Moise would announce a proof, and Papawould shoot it down. This went on for yea ,whileneither man worked on almost anything els

Eventually; Moise cracked. One day; he implyturned away from math altogether. hae1Freedman, a topologist who works for Mic ,soft,describes this phenomenon as getting "wre ·ked"by a problem. He says many mathematicians t 'veon the knowledge that they are smart enoug tosolve almost anything, given enough time. enMoise realized his best efforts were never go gto solve Poincare, his spirit broke. He never .dserious math research again and spent his last fyears critiquing poetry. Papa kept working on t eproblem for 25 years, swearing he wouldn't marf}luntil he'd solved the Conjecture. In 1976, he died 0

stomach cancer, still a bachelor.

HOPE IN A HIGHER DIMENSION, ~ After dozens of mathematicians had devoted their~ careers to the Poincare Conjecture, a breakthroughI. came in 1960,when a young hotshot named Steven

Smale made the first tangible headway into the

I problem. Smale decided not to worry about objects inthe three-dimensional or even the four-dimensionaluniverse. Instead, he proved that the Conjecture wastrue in the fifth dimension and higher. Until then,

I·1

38 MENTAL_FLOSS

MORE!• Math Explains the"Beer Goggles" Effect, Why Vampires areMathematically Impossiblementalfloss.com/more

a Ricci flow. The math community started tobuzz that he'd resolved the underlying issuesof the problem. Eventually; one mathematicianasked Perelman directly if his paper answered thePoincare Conjecture. Never one to be long-winded,Perelman wrote back, "That is correct."

Because Perelman hadn't taken the normal stepsof running his ideas past colleagues and publishingin a referenced journaL it took some time toverify this claim. It didn't help that his paper only

included the barest essentialsto prove the Conjecture. Ittook six experts two years tofill in the gaps that Perelmanhad seen as self-evident.

By 2006, his proof had heldup to every attack, and Perelmanwas offered a Fields Medal. Butby that time, Perelman wasso disillusioned with the fieldof mathematics that withouta flourish, he turned it down,becoming the first person inhistory to refuse the award.

Also without explanation, Perelman never claimed the$1 million prize from the Clay Mathematics Institutein Cambridge, Mass.

Today; Poincare's side note may only beapplicable to the most obscure physics problems.But mathematicians expect that, like most theoreticalbreakthroughs, the effects will eventually diffuse tothe rest of science. After all, when Newton firstunveiled calculus, experts said only a few people onEarth could possibly understand it. Now;it's taught toteenagers and used in everything from engineeringto statistics. Whether Grigori Perelman likes it ornot, someday he may find himself honored in highschool textbooks, and the Poincare Conjecture maybecome as easy to understand as gravity; ~

i~'t1J~ "

"l.'lI

"ll,Il II ~

l Il /I IIl {l I

lI',

false. Neither effort panned out. Regardless, mostmathematicians believed the solution to Poincarewas out there-somewhere.

Finally; in 2002, Grigori Perelman, a recluseliving with his mother in 8t. Petersburg, posteda short paper on a math Web site. The reticentPerelman never once mentioned Poincare in hisessay; but the few people who read it understoodits implications. The paper addressed one of thebiggest obstacles that had blocked mathematiciansfrom proving the Conjecture. Whenever they'd triedreducing certain shapes to their most basic forms,little irregularities kept popping up like painfulburrs. To smooth out the rough spots, Perelmanapplied a type of mathematical sandpaper called

," {

/ /,, \"l;'- \,

" '. "."'/ '..", "i •""/' "" !" .""'/'" ",,'" I.•' !

{ I \II

\I

\I

\

MENTAlFLOSS.COM 39