If saturated mathematics can teach us anything , it ’s this : occasionally , your special interest might just modify the world .
ForJoshua Zahland Hong Wang , that limited interest was the Kakeya guess . “ I read a Quran in undergrad calledA Panorama of Harmonic Analysis , ” says Zahl , an associate professor in the University of British Columbia ’s department of mathematics . “ And I read about this in that book , and really was taken with it , and wanted to teach more [ … ] that ’s how the interest really started . ”
Fast onward to today , and Zahl and Wang are now co - writer of a yet - to - be - peer - review proofheraldedby Eyal Lubetzky , electric chair of the mathematics department at the Courant Institute where Wang is an associate professor of mathematics , as “ one of the top numerical achievements of the twenty-first century . ”
In blue: broke. In red: bespoke.Image credit: Killerowski, Public domain, viaWikimedia Commons
And it all started with a brainteaser about a phonograph needle .
The beginnings of a conjecture
It ’s a curious truth of mathematics that the gentle a problem expect on paper , the harder it often turns out to be . TakeFermat ’s Last Theorem , for example : brusk and sweet enough to scribble in a margin , but a proof occupy more than 350 years and the development of multiple raw theatre of operations of math to observe .
The Kakeya conjecture is similar , if not quite as utmost . It goes back to 1917 , when Japanese mathematician Sōichi Kakeya put the following puzzle : given an boundlessly thin needle of unit of measurement length , what is the small amount of outer space you’re able to drag in out when rotating it through every instruction possible ?
In one dimension , the job is trivial : the phonograph needle ca n’t move from its initial place , so the problem is either already lick or unsolvable , depending on perspective . But in two , things are already much more complicated .
This is what we’re dealing with, except like, asymptotic.Image credit: Gtgith, Public domain, viaWikimedia Commons
An obvious way to fulfill the “ rotating through every direction possible ” criteria in two dimensions is to simply spin the needle in a traffic circle – but the resulting orbit is comparatively tumid . A better solution , proposed by Kakeya himself , is to move the needle around a roach while it rotates , creating a sucked - in triangle shape jazz as a musculus deltoideus .
The area of this shape is much smaller – exactly half that of our rotary first attempt , in fact . But it turns out we can do much , much honorable than that : in 1919 , the Russian mathematician Abram Besicovitchshowed that , with some numerical ingeniousness and a mickle of needly finagling , it ’s actually possible to sweep out a outer space withzeromeasure .
The ensue shapes – the so - called Kakeya set – “ have very counterintuitive properties . You could saypathologicalproperties ” Zahl says . “ I intend , even the conjecture apart – the first clock time you read about a Kakeya set , if you ’re naturally curious , you ’re going to be pretty intrigued . ”
“ I suppose maybe the resultant role itself is less authoritative than , here ’s an alibi for many people in the world to learn what a Kakeya set is , ” he tells IFLScience .
A question of dimension
As unintuitive as Besicovitch ’s result was , one affair is undeniable : you ca n’t get much little than zero . So , more than 100 long time later , why are we still talking about the Kakeya conjecture ?
Besicovitch may have shown that these shapes could take up zero space , but – toborrow a phrasingfrom Fields medal winner and long - time Kakeya conjecture pursuer Terence Tao – “ not all sets of measure zero are create adequate . ”
Starting in the 1970s , withthe work of Roy Davies , mathematicians started considering the trouble from a new perspective . Rather than measure the amount of quad swing out by their infinitely flimsy phonograph needle , perhaps a dissimilar way of life of thinking about size of it was needed : what , they inquire , would thedimensionof these Kakeya sets be ?
At first glimpse , that question may seem footling . It ’s a shape in three - dimensional distance , ergo it has dimension three . ripe ?
“ It seem[s ] very intuitive , ” Larry Guth , Claude Shannon Professor of Mathematics at MIT andanother veteran Kakeya conjecturist , toldNew Scientistlast workweek . “ It seem[s ] like it must be on-key , but then it turns out to be very unmanageable to prove . ”
It ’s this , then , rather than Kakeya ’s original instruction , that mathematicians refer to when they talk about the Kakeya conjecture : the estimate that these Kakeya sets should have the same property as the infinite they inhabit .
It ’s a subtle reframing with an outsized impact . Rather than a niche intellectual curiosity , the newly stated problem now has thick connections to other domain of math : “ there is , let ’s say , a circle of doubtfulness that masses are very interested in in harmonised analysis , ” Zahl tells IFLScience – some “ touch on to the behavior of the Fourier transform , and some related to questions about the behavior of certain solutions to certain fond differential par . ”
Solve the Kakeya conjecture , and you break the first cuticle in amatryoshkaof key questions in harmonised analysis . The problems “ are all tightly connect , ” Zahl explain . “ The connexion , as best I ’m mindful , was first find by Charles Fefferman in the – well , the paperwas publish in 1971 , so recent 60s , former 70s . ”
Planes, grains, and all they reveal
With this new motivation , involvement in the Kakeya conjecture flourished – but no classic proof seemed forthcoming . In 1995 , however , mathematician Thomas Wolff made something of a breakthrough : “ he prove , amongst other things , that every Kakeya set in [ proportion ] three has Hausdorff dimension at least five halves , ” Zahl explains .
To a non - mathematician , such a statement can sound puzzling , if not downright nonsense – but fractional or irrational dimension are actually nothing raw in math . They ’re amainstay of fractal geometry , and with a shape as weird and unintuitive as a Kakeya set , it would n’t be totally surprising to find them cropping up again .
“ Yeah , there were here and now where we were very like [ … ] it really might be two and a half , ” Zahl tells IFLScience . “ There were moments of doubtfulness . ”
Still , they had good understanding to consider in the surmise . In 1999 , Tao , along with collaborators Nets Katz and Izabella Łaba , had drive the dimension measurement up a little number – all the way from 2.5 to 2.500000001 . It ’s not much of a difference of opinion , objectively verbalize , but that was n’t really the point : the boundary had been crossed – and anyone take on the trouble going onward now had a roadmap to ferment from .
“ The fashion you do it is you say , okay , imagine the enemy gives you some in particular bad set – maybe a antagonistic model to the conjecture , we could envisage – and we thicken it by this small amount of ‘ delta ’ , ” Zahl explains . Rather than being a collection of infinitesimally thin line , now , we can think of the Kakeya plant as being made of “ uncooked spaghetti , ” he says ; “ they have distance one , but they ’re quite thin , and they can overlap each other . ”
suitably for a spaghetti - based analogy , a set like this is know to mathematicians as “ grainy ” – and it ’s a necessary belongings for any counterexample to the Kakeya conjecture . Zoom in on the overlap areas , and you encounter another such quality : “ what was proved back in roughly 2005 is that , if this object was indeed a antagonistic example to the supposition , then the tubes pass through one picky point all have to be almost in a common plane , ” Zahl secernate IFLScience . “ They all have to be roughly coplanar . ”
There was just one more property to account for .
A sticky problem
That a counterexample to the Kakeya set should be “ grainy ” and “ plany ” was well - established – but Tao , Łaba , and Katz , who earlier proposed these prerequisites , had a third item on the leaning : stickiness .
To Kakeya scholar , that term has a particular signification : a “ sticky ” circle is one in which line segment that point in nearby directions must also be close to each other . It was , ironically , the slipperiest of the measure , evading a proof for more than a decade – until , in 2022 , Zahl and Wang unexpectedly showedthe opposite . A sticky Kakeya set in three dimensions , they proved , must have Hausdorff and Minkowski proportion three – thus obeying the conjecture .
It was a hugely important breakthrough – but it was n’t enough to prove the conjecture entirely . Next , the duet needed to examine the grainy , planey , but non - embarrassing cases , equate their property at various scale , and hopefully lay down a contradiction .
“ A mess of this work comes from assay to understand the geomorphological properties of what a counterexample to the theorem would look like , ” Zahl order IFLScience . “ And if you could incur enough of these geomorphological properties and how they relate to each other , then maybe eventually [ … ] you could show that [ the counterexample ] ca n’t live . ”
Three years and a brace hundred page of various cogent evidence later , the Book of Job was done .
Once in a century
With the uploading of Wang and Zahl ’s paper to the arXiv preprint host , a undulation of excitement rippled through the outside numerical community . Taowrote about iton his blog ; Katzlauded itas a “ once - in - a - century sort of event . ”
The paper “ is a tremendous piece of maths , ” said Guido De Philippis , a professor at the Courant Institute , in astatement . “ [ It ] follow years of onward motion that has enhance our savvy of a complicated geometry and brings it to a new storey . ”
“ I am expecting that their ideas will go to a series of exciting breakthroughs in the coming year , ” he continue . “ This resultant role is not only a major breakthrough in geometrical measure theory , but it also open up a series of exciting developments in harmonic analysis , issue theory , and practical program in computer skill and cryptography . ”
Is such praise premature ? After all , the result is technically yet to be equal - review – but “ to be clear , we do n’t have serious headache about the correctness of our proof , ” Zahl say . That ’s not hubris : after completing the affair , they post it to just about everyone they could think of who might be able to find a error in it – and it come back clean .
“ We wanted as many of them as possible to signal off on it before we made it public , ” he assure IFLScience . “ With these things [ … ] you necessitate to have a lot of paranoia that you ’ve made a error [ … ] you do n’t want to let yourself get excited until you ’re sure you ’ve done it . ”
The newspaper can be scan on thearXiv preprint waiter .
