What is it that French people do better than all the others? If you would take polls, the top three answers might be: love, wine and whining.
Šta to Francuzi rade bolje od svih drugih? Ako biste sproveli istraživanje, najčešća tri odgovora bi možda bila: ljubav, vino i žaljenje.
(Laughter)
(Smeh)
Maybe. But let me suggest a fourth one: mathematics. Did you know that Paris has more mathematicians than any other city in the world? And more streets with mathematicians' names, too. And if you look at the statistics of the Fields Medal, often called the Nobel Prize for mathematics, and always awarded to mathematicians below the age of 40, you will find that France has more Fields medalists per inhabitant than any other country.
Možda. Ali, dozvolite mi da predložim i četvrti: matematika. Da li znate da Pariz ima više matematičara od bilo kog drugog grada u svetu? I više ulica sa imenima matematičara, takođe. I ukoliko pogledamo statistiku Fildsove medalje, često nazivane Nobelovom nagradom za matematičare, koja se uvek dodeljuje matematičarima ispod 40 godina, otkrićete da Francuska ima više dobitnika ove medalje po glavi stanovnika od bilo koje druge države.
What is it that we find so sexy in math? After all, it seems to be dull and abstract, just numbers and computations and rules to apply. Mathematics may be abstract, but it's not dull and it's not about computing. It is about reasoning and proving our core activity. It is about imagination, the talent which we most praise. It is about finding the truth. There's nothing like the feeling which invades you when after months of hard thinking, you finally understand the right reasoning to solve your problem. The great mathematician André Weil likened this -- no kidding -- to sexual pleasure. But noted that this feeling can last for hours, or even days.
Šta nam je tako privlačno u matematici? Naposletku, deluje da je dosadna i apstraktna; samo brojevi, računanje i pravila koja treba primeniti. Matematika je možda apstraktna, ali nije dosadna i nije u vezi sa proračunima. U vezi je sa razmišljanjem i dokazivanjem naše suštinske aktivnosti. U vezi je sa maštom, talentom koji cenimo više od svega. U vezi je sa nalaženjem istine. Nema ničega boljeg od osećaja koji te prožme kada, posle meseci teškog razmišljanja, konačno razumeš kako da rešiš svoj problem. Veliki matematičar, Andre Vejl, je uporedio to - bez šale - sa seksualnim zadovoljstvom. S tim da ovaj osećaj može da traje satima ili čak danima.
The reward may be big. Hidden mathematical truths permeate our whole physical world. They are inaccessible to our senses but can be seen through mathematical lenses. Close your eyes for moment and think of what is occurring right now around you. Invisible particles from the air around are bumping on you by the billions and billions at each second, all in complete chaos. And still, their statistics can be accurately predicted by mathematical physics. And open your eyes now to the statistics of the velocities of these particles.
Nagrada može biti velika. Skrivene matematičke istine prožimaju sav naš fizički svet. Do njih se ne može doći putem čula, ali se mogu uočiti kroz matematička sočiva. Zatvorite oči na trenutak i mislite o tome šta se dešava trenutno oko vas. Nevidljive čestice iz vazduha se sudaraju sa vama, milijarde i milijarde svake sekunde, potpuno haotično. Ali, ipak, njihova statistika se može tačno predvideti putem matematičke fizike. Otvorite oči sada da vidite statistiku brzina ovih čestica.
The famous bell-shaped Gauss Curve, or the Law of Errors -- of deviations with respect to the mean behavior. This curve tells about the statistics of velocities of particles in the same way as a demographic curve would tell about the statistics of ages of individuals. It's one of the most important curves ever. It keeps on occurring again and again, from many theories and many experiments, as a great example of the universality which is so dear to us mathematicians.
Poznata Gausova kriva u obliku zvona, ili zakon grešaka - odstupanja koja se odnose na srednjestatističko ponašanje. Kriva pokazuje statistiku brzina čestica na isti način kao što i demografska kriva pokazuje statistiku godina uzrasta pojedinaca. To je jedna od najvažnijih krivih u istoriji. Ponavlja se iznova i iznova u mnogim teorijama i eksperimentima, kao divan primer univerzalnosti koja je nama, matematičarima, tako draga.
Of this curve, the famous scientist Francis Galton said, "It would have been deified by the Greeks if they had known it. It is the supreme law of unreason." And there's no better way to materialize that supreme goddess than Galton's Board. Inside this board are narrow tunnels through which tiny balls will fall down randomly, going right or left, or left, etc. All in complete randomness and chaos. Let's see what happens when we look at all these random trajectories together.
O ovoj je krivoj poznati naučnik Francis Galton rekao: „Da su znali za nju, Grci bi je obogotvorili. To je viši zakon iracionalnosti.“ Nema boljeg načina za materijalizovanje te visoke boginje nego kroz Galtonovu tablu. Unutar ove table su uski tuneli kroz koje će male loptice padati nadole nasumično, idući desno, ili levo, ili levo itd. U potpunoj nasumičnosti i haosu. Da vidimo šta se dešava kada pogledamo sve ove nasumične putanje zajedno.
(Board shaking)
(Tabla se trese)
This is a bit of a sport, because we need to resolve some traffic jams in there. Aha. We think that randomness is going to play me a trick on stage.
Ovo je pomalo poput sporta, zato što se moramo rešiti saobraćajne gužve ovde. Aha. Mislimo da će nasumičnost da me prevari na sceni.
There it is. Our supreme goddess of unreason. the Gauss Curve, trapped here inside this transparent box as Dream in "The Sandman" comics. For you I have shown it, but to my students I explain why it could not be any other curve. And this is touching the mystery of that goddess, replacing a beautiful coincidence by a beautiful explanation.
Evo je. Naša visoka boginja iracionalnosti. Gausova kriva, zarobljena ovde unutar ove providne kutije poput Sna u stripu „Sandman“. Vama sam je pokazao, ali svojim studentima objašnjavam zašto to ne može biti neka druga kriva. I tu se približavamo misteriji te boginje, zamenjujući prelepu slučajnost prelepim objašnjenjem.
All of science is like this. And beautiful mathematical explanations are not only for our pleasure. They also change our vision of the world. For instance, Einstein, Perrin, Smoluchowski, they used the mathematical analysis of random trajectories and the Gauss Curve to explain and prove that our world is made of atoms.
Sva nauka je takva. A prelepa matematička objašnjenja nisu tu samo zbog našeg zadovoljstva. Ona takođe menjaju naš pogled na svet. Na primer, Ajnštajn, Perin, Smolučovski, koristili su matematičku analizu nasumičnih putanja i Gausovu krivu da objasne i dokažu da je naš svet izgrađen iz atoma.
It was not the first time that mathematics was revolutionizing our view of the world. More than 2,000 years ago, at the time of the ancient Greeks, it already occurred. In those days, only a small fraction of the world had been explored, and the Earth might have seemed infinite. But clever Eratosthenes, using mathematics, was able to measure the Earth with an amazing accuracy of two percent.
To nije prvi put da je matematika iz korena menjala naš pogled na svet. Pre više od 2 000 godina, u vreme antičke Grčke, to se već dogodilo. U to vreme, samo mali procenat sveta je bio otkriven, a Zemlja je možda delovala kao da je beskonačna. Ali, umni Eratosten, koristeći se matematikom, bio je u stanju da izmeri Zemlju sa sjajnom preciznošću od dva procenta greške.
Here's another example. In 1673, Jean Richer noticed that a pendulum swings slightly slower in Cayenne than in Paris. From this observation alone, and clever mathematics, Newton rightly deduced that the Earth is a wee bit flattened at the poles, like 0.3 percent -- so tiny that you wouldn't even notice it on the real view of the Earth.
Evo ga još jedan primer. Godine 1673. Žan Rišer je primetio da se klatno malčice sporije ljulja u Kajenu nego u Parizu. Iz samo ove observacije i uz lukavu matematiku, Njutn je pravilno zaključio da je Zemlja malo ravnija na polovima, negde oko 0,3 procenta - toliko malo da to ne možete ni primetiti na pravoj slici Zemlje.
These stories show that mathematics is able to make us go out of our intuition measure the Earth which seems infinite, see atoms which are invisible or detect an imperceptible variation of shape. And if there is just one thing that you should take home from this talk, it is this: mathematics allows us to go beyond the intuition and explore territories which do not fit within our grasp.
Ovakve priče pokazuju da uz matematiku možemo da idemo dalje od naše intuicije, da izmerimo Zemlju koja deluje kao beskonačna, da vidimo atome koji su nevidljivi ili da detektujemo nevidljive varijacije oblika. I ako biste izdvojili samo jednu stvar koju treba da zapamtite iz ovog govora, to je ovo: matematika nam dopušta da idemo dalje od naše intuicije i da istražujemo teritorije koje ne možemo da zamislimo.
Here's a modern example you will all relate to: searching the Internet. The World Wide Web, more than one billion web pages -- do you want to go through them all? Computing power helps, but it would be useless without the mathematical modeling to find the information hidden in the data.
Evo modernog primera koji je blizak svima: pretraživanje interneta. Svetska mreža, više od milijardu veb-stranica - da li biste pretraživali kroz sve njih? Računarska moć pomaže, ali bi bila beskorisna bez matematičkog modelovanja kojim se nalazi informacija skrivena u podacima.
Let's work out a baby problem. Imagine that you're a detective working on a crime case, and there are many people who have their version of the facts. Who do you want to interview first? Sensible answer: prime witnesses. You see, suppose that there is person number seven, tells you a story, but when you ask where he got if from, he points to person number three as a source. And maybe person number three, in turn, points at person number one as the primary source. Now number one is a prime witness, so I definitely want to interview him -- priority. And from the graph we also see that person number four is a prime witness. And maybe I even want to interview him first, because there are more people who refer to him.
Hajde da prođemo kroz mali zadatak. Zamislite da ste detektiv koji radi na slučaju zločina, i imate mnogo ljudi koji imaju svoju verziju toga šta se dogodilo. Koga biste prvo intervjuisali? Razuman odgovor: glavne svedoke. Vidite, pretpostavimo da vam osoba broj sedam ispriča priču, ali kada je upitate odake ona to zna, ona vam kaže da je njen izvor osoba broj tri. I možda osoba broj tri dalje istakne osobu broj jedan kao primarni izvor. Sada je osoba broj jedan glavni svedok. tako da definitivno hoću nju da intervjuišem - prioritet. I iz grafika možemo da pročitamo da je osoba broj četiri takođe među glavnim svedocima. I možda ja mogu i nju prvo da intervjuišem, zato što ima više osoba koje upućuju na nju.
OK, that was easy, but now what about if you have a big bunch of people who will testify? And this graph, I may think of it as all people who testify in a complicated crime case, but it may just as well be web pages pointing to each other, referring to each other for contents. Which ones are the most authoritative? Not so clear.
U redu, to je bilo lako, ali šta ako imate gomilu ljudi koji treba da svedoče? Ovaj grafik može da se razume kao da su to svi svedoci ovog komplikovanog kriminalnog slučaja, ali to isto tako mogu biti i veb-stranice koje ukazuju jedna na drugu, čiji sadržaj usmerava sa jedne na drugu. Koje stranice su najautoritativnije? Nije sasvim jasno.
Enter PageRank, one of the early cornerstones of Google. This algorithm uses the laws of mathematical randomness to determine automatically the most relevant web pages, in the same way as we used randomness in the Galton Board experiment. So let's send into this graph a bunch of tiny, digital marbles and let them go randomly through the graph. Each time they arrive at some site, they will go out through some link chosen at random to the next one. And again, and again, and again. And with small, growing piles, we'll keep the record of how many times each site has been visited by these digital marbles.
Pristupimo „Pejdž ranku“, jednom od kamena temeljaca Gugla. Ovaj algoritam koristi zakone matematičke nasumičnosti da automatski odredi najrelevantnije veb-stranice, na isti način kako smo koristili nasumičnost u eksperimentu Galtonove table. Onda, propustimo kroz ovaj grafik gomilu malih, digitalnih klikera i pustimo ih da idu nasumično kroz grafik. Svaki put kad su na nekom sajtu, izaći će sa tog sajta i preći na drugi koristeći se nekim nasumičnim linkom. I ponovo, i ponovo, i ponovo. I malim, rastućim gomilama, merićemo koliko puta su ovi digitalni klikeri posetili svaki sajt.
Here we go. Randomness, randomness. And from time to time, also let's make jumps completely randomly to increase the fun.
Krećemo. Nasumičnost, nasumičnost. I s vremena na vreme, da bi bilo zabavnije, preskačimo sasvim proizvoljno.
And look at this: from the chaos will emerge the solution. The highest piles correspond to those sites which somehow are better connected than the others, more pointed at than the others. And here we see clearly which are the web pages we want to first try. Once again, the solution emerges from the randomness. Of course, since that time, Google has come up with much more sophisticated algorithms, but already this was beautiful.
I pogledajte ovo: iz haosa se pojavljuje rešenje. Najviše hrpe odgovaraju onim sajtovima koji su na neki način bolje povezani od drugih, koji češće od drugih upućuju na druge stranice. I ovde jasno vidimo koje veb-stranice hoćemo prvo da isprobamo. Opet, rešenje se pojavljuje iz nasumičnosti. Naravno, od tada, Gugl je smislio mnogo sofisticiranije algoritme, ali je već tada ovo bilo prelepo.
And still, just one problem in a million. With the advent of digital area, more and more problems lend themselves to mathematical analysis, making the job of mathematician a more and more useful one, to the extent that a few years ago, it was ranked number one among hundreds of jobs in a study about the best and worst jobs published by the Wall Street Journal in 2009.
Ipak, to je samo jedan problem iz milion problema. Sa razvojem digitalne oblasti, sve više i više problema se može podvrgnuti matematičkoj analizi, čineći posao matematičara sve korisnijim, do tog nivoa da je pre par godina, bio broj jedan na listi među stotinama poslova u istraživanju o najboljim i najgorim poslovima koje je „Vol strit džurnal“ objavio 2009. godine.
Mathematician -- best job in the world. That's because of the applications: communication theory, information theory, game theory, compressed sensing, machine learning, graph analysis, harmonic analysis. And why not stochastic processes, linear programming, or fluid simulation? Each of these fields have monster industrial applications. And through them, there is big money in mathematics. And let me concede that when it comes to making money from the math, the Americans are by a long shot the world champions, with clever, emblematic billionaires and amazing, giant companies, all resting, ultimately, on good algorithm.
Matematičar - najbolji posao na svetu. To je zbog primena: komunikaciona teorija, informaciona teorija, teorija igara, kompresija signala, mašinsko učenje, teorija grafova, harmonijska analiza. I zašto da ne, stohastički procesi, linearno programiranje ili simulacija dinamike tečnosti? Svaka od ovih oblasti ima ogromnu primenu u industriji. I kroz primenu, matematika donosi mnogo novca. I da potvrdim da su, kada je reč o zarađivanju od matematike, Amerikanci daleko najbolji svetski šampioni u tome, sa mudrim, realnim milijarderima i sjajnim, ogromnim kompanijama, a svi počivaju, na kraju krajeva, na dobrim algoritmima.
Now with all this beauty, usefulness and wealth, mathematics does look more sexy. But don't you think that the life a mathematical researcher is an easy one. It is filled with perplexity, frustration, a desperate fight for understanding.
Sada, uzimajući u obzir svu ovu lepotu, praktičnu primenu i bogatstvo, matematika doista izgleda privlačnije. Ali, nemojte misliti da je život matematičara istraživača lak. Ispunjen je zbunjenošću, frustracijom, očajnom borbom da se nešto razume.
Let me evoke for you one of the most striking days in my mathematician's life. Or should I say, one of the most striking nights. At that time, I was staying at the Institute for Advanced Studies in Princeton -- for many years, the home of Albert Einstein and arguably the most holy place for mathematical research in the world. And that night I was working and working on an elusive proof, which was incomplete. It was all about understanding the paradoxical stability property of plasmas, which are a crowd of electrons. In the perfect world of plasma, there are no collisions and no friction to provide the stability like we are used to. But still, if you slightly perturb a plasma equilibrium, you will find that the resulting electric field spontaneously vanishes, or damps out, as if by some mysterious friction force.
Da vam predstavim jedan od najboljih dana u mom matematičkom životu. Ili bolje, jednu od mojih najboljih noći. U to vreme, bio sam u Institutu za primenjeno istraživanje u Prinstonu - koji je mnogo godina bio dom Albertu Ajnštajnu i koji je možda najsvetije mesto za istraživanje matematike u svetu. Te noći sam radio i radio na jednom teškom dokazu, koji je bio nepotpun. Reč je bila o razumevanju paradoksalnog svojstva stabilnosti plazmi, koje su suštinski skupina elektrona. U savršenom svetu plazmi, nema sudara i nema trenja koji bi obezbedili stabilnost na koju smo navikli. Ipak, ako se malčice naruši ekvilibrijum plazme, primetiće se da električno polje koje se pojavilo kao posledica toga spontano nestaje i gasi se, kao da je pod dejstvom neke tajanstvene sile trenja.
This paradoxical effect, called the Landau damping, is one of the most important in plasma physics, and it was discovered through mathematical ideas. But still, a full mathematical understanding of this phenomenon was missing. And together with my former student and main collaborator Clément Mouhot, in Paris at the time, we had been working for months and months on such a proof. Actually, I had already announced by mistake that we could solve it. But the truth is, the proof was just not working. In spite of more than 100 pages of complicated, mathematical arguments, and a bunch discoveries, and huge calculation, it was not working. And that night in Princeton, a certain gap in the chain of arguments was driving me crazy. I was putting in there all my energy and experience and tricks, and still nothing was working. 1 a.m., 2 a.m., 3 a.m., not working. Around 4 a.m., I go to bed in low spirits. Then a few hours later, waking up and go, "Ah, it's time to get the kids to school --" What is this? There was this voice in my head, I swear. "Take the second term to the other side, Fourier transform and invert in L2."
Ovaj paradoksalni efekat pod nazivom Landauovo prigušenje jedan je od najvažnijih u fizici plazme i otkriven je kroz matematičke ideje. Ipak, nedostajalo je potpuno matematičko razumevanje ovog fenomena. I zajedno sa svojim bivšim studentom i glavnim saradnikom, Klementom Muoom, tada u Parizu, radili smo mesecima i mesecima na tom dokazu. U stvari, ja sam već greškom objavio da možemo to da rešimo. Ali, istina je bila da dokaz jednostavno nije funkcionisao. Bez obzira na to što je postojalo više od 100 stranica komplikovanih matematičkih pretpostavki, kao i niz otkrića i ogromni proračuni, dokaz nije funkcionisao. I te noći u Prinstonu, određena nelogičost u lancu pretpostavki me je izluđivala. Uložio sam svu svoju energiju, iskustvo i trikove, a ipak ništa nije funkcionisalo. Jedan ujutru, dva ujutru, tri ujutru; ne funkcioniše. Oko četiri ujutru, ležem u rđavom raspoloženju. Onda, nekoliko sati kasnije, budim se i mislim: „Ah, vreme je da odvedem decu u školu - “ Šta je ovo? Čuo sam glas u glavi, kunem se. „Prebaci drugi član na drugu stranu, Furijeova transformacija i invertuj na L2.“
(Laughter)
(Smeh)
Damn it, that was the start of the solution!
Dođavola, to je bio početak rešenja!
You see, I thought I had taken some rest, but really my brain had continued to work on it. In those moments, you don't think of your career or your colleagues, it's just a complete battle between the problem and you.
Vidite, mislio sam da sam se odmorio, ali je moj mozak u stvari nastavio da radi na tome. U tim trenucima, ne misliš na svoju karijeru ili na kolege; to je prosto potpuna bitka između problema i tebe.
That being said, it does not harm when you do get a promotion in reward for your hard work. And after we completed our huge analysis of the Landau damping, I was lucky enough to get the most coveted Fields Medal from the hands of the President of India, in Hyderabad on 19 August, 2010 -- an honor that mathematicians never dare to dream, a day that I will remember until I live.
Imajući to u vidu, ne škodi ni kada se dobije unapređenje kao nagrada za naporan rad. I pošto smo završili našu ogromnu analizu Landauovog prigušenja, bio sam dovoljno srećan da dobijem najpoželjniju Fildsovu medalju koju mi je uručila predsednica Indije, u Hajderabadu 19. avgusta 2010. - čast o kojoj se matematičari i ne usuđuju da sanjaju, dan koji ću pamtiti dok sam živ.
What do you think, on such an occasion? Pride, yes? And gratitude to the many collaborators who made this possible. And because it was a collective adventure, you need to share it, not just with your collaborators. I believe that everybody can appreciate the thrill of mathematical research, and share the passionate stories of humans and ideas behind it. And I've been working with my staff at Institut Henri Poincaré, together with partners and artists of mathematical communication worldwide, so that we can found our own, very special museum of mathematics there.
O čemu misliti u takvoj prilici? Ponos, zar ne? I zahvalnost mnogim saradnicima koji su učinili to mogućim. I zato što je to bio zajednički poduhvat, potrebno ga je podeliti ne samo sa saradnicima. Verujem da su svi u stanju da cene uzbuđenje matematičkog istraživanja i da strastveno dele priče o ljudima i idejama iza njih. Moje osoblje pri Institutu Anri Poenkare i ja smo radili, zajedno sa partnerima i umetnicima matematičke komunikacije širom sveta, na osnivanju sopstvenog, veoma specijalnog muzeja matematike tamo.
So in a few years, when you come to Paris, after tasting the great, crispy baguette and macaroon, please come and visit us at Institut Henri Poincaré, and share the mathematical dream with us.
Tako, kroz par godina kada dođete u Pariz, posle isprobavanja sjajnog, hrskavog bageta i makaruna, molim vas dođite i posetite nas u Institutu Anri Poenkare i sanjajte matematički san zajedno sa nama.
Thank you.
Hvala.
(Applause)
(Aplauz)