Interviews with Three Fields Medalists Interviewed by Vicente Muñoz and Ulf Persson AndreiOkounkov,TerenceTao,andWendelinWernerreceivedFieldsMedalsatthe2006 International Congress of Mathematicians in Madrid, Spain (Grigory Perelman was also awarded a medal but declined to accept it). These interviews with Okounkov, Tao, and Werner were conducted via email by Vicente Muñoz and Ulf Persson in the fall of 2006 and originally appeared in the December 2006 issue of the European Mathematical Society Newsletter. Andrei Okounkov from achieving the true goal of science, which is to Muñoz & Persson: How did you get interested in understand our world. mathematics? M & P: So, you wouldn’t say that competition is Okounkov: The most important part of be- the best way to do mathematics? coming a mathematician is learning from one’s Okounkov: I think it is a mistake that compe- teachers. Here I was very fortunate. Growing up in tition is actively promoted on every level of math. Kirillov’s seminar, I had in its participants, espe- While kids take solving puzzles perhaps a bit too cially in Grisha Olshanski, wonderful teachers who seriously,grown-upsplacetheultimatevalueonbe- generously invested their time and talent into ex- ing the first to prove something. A first complete plaining mathematics and who patiently followed proof, while obviously veryimportant, is only a cer- my first professional steps. I can’t imagine be- tainstageinthedevelopmentofourknowledge.Of- coming a mathematician without them. So it must ten, pioneering insights precede it and a lot of cre- be that in this respect my professional formation ative work follows it before a particular phenome- resembleseverybodyelse’s. non maybeconsideredunderstood. It isthrilling to What was perhaps less usual is the path that bethefirst,butaclearproofisforallandforever. led me to mathematics. I didn’t go through spe- M&P:How do you prefer to work on mathemat- cial schools and olympiads. I came via studying ical problems? Alone or in collaboration? economics and army service. I had a family before Okounkov: Perhaps you can guess from what I papers. As a result, my mind is probably not as said beforethat I like to work alone, I equally like to quick as it could have been with an early drilling freely share my thoughts, and I also like to perfect in math. But perhaps I also had some advantages mypapersandtalks. over my younger classmates. I had a broader view There may well be alternate routes, but I person- of the universe and a better idea about the place ally don’t know how one can understand something of mathematics in it. This helped me form my without both thinking about it quietly over and over own opinion about what is important, beautiful, and discussing it with friends. When I feel puzzled, promising,etc. I like long walksor bike rides.I like to be alone with It also made mathematics less competitive for my computer playing with formulas or experiment- me. Competition is one of those motors of human ing with code. But when I finally have an idea, I can’t society that will always be running. For example, we wait to share it with others. I am so fortunate to be arehaving this interview becauseof the outcome of able to share my work and my excitement about it a certain competition. But I believe it distracts us withmanybrilliantpeoplewhoareatthesametime wonderfulfriends. Vicente Muñoz is professor of mathematics at the And when it comes to writing or presenting, Universidad Autónoma de Madrid. His email ad- dress is [email protected]. Ulf shouldn’t everyone make an effort to explain? Persson is professor of mathematics at Chalmers Uni- Wouldn’titbeashameifsomethingyouunderstood versity of Technology, Sweden. His email address is were to exist only as a feeble neuron connection in [email protected]. yourbrain? March 2007 Notices of the AMS 405 M & P: Do you prefer to solve problems or to Imeanbothnumericandsymbolicwork.Somepeo- develop theories? ple can manage without dishwashers, but I think Okounkov: I like both theory and problems, proofs come out a lot cleaner when routine work is but best of all I like examples. For me, examples automated. populate the world of mathematics. Glorious emp- This brings up many issues. I am not an ex- ty buildings are not my taste. I recall my teacher pert, but I think we need a symbolic standard to Kirillov saying that it is easier to generalize an make computer manipulations easier to document examplethan to specialize a theory. Perhaps he did and verify. And with all due respect to the free not mean this 100 percent seriously, but there is market, perhaps we should not be dependent on a certain important truth in those words. Under- commercial software here. An open-source project standing examples links with ability to compute. could, perhaps, find better answers to the obvi- Great mathematicians of the past could perform ous problems such as availability, bugs, backward spectacular computations. I worry that, in spite compatibility, platform independence, standard of enormous advances in computational methods libraries, etc. One can learn from the success of and power, this is a skill that is not adequately T X and more specialized software like Macaulay2. emphasized and developed. Any new computation, E I do hope that funding agencies are looking into exact or numeric, can be very valuable. The ability this. to do a challenging computation and to get it right M & P: The age of the universalists is gone. is animportantmeasure ofunderstanding, justlike Nowadays mathematics is very diverse and people theabilitytoproveis. M &P: Much of your work has deep connections tend to get mired in subspecialities. Do you see any to physics. Does that mean that you find it essential remedy to this? Okounkov: that mathematics is related to the natural world, Mathematics is complex. Specializa- or that you would even think of it as subservient to tion, while inevitable, doesn’t resolve the problem. the other natural sciences? Mathematics is a living organism; one cannot sim- Okounkov: When I said “our world” earlier I plychopitup.Sohowdowebothembraceandresist didn’t mean just the tangible objects of our every- specialization? day experience. Primes are as real as planets. Or, We can be better neighbors. We shouldn’t build in the present context, should I say that celestial high fences out of sophisticated words and a “you bodies are as real as primes? Throughout their wouldn’t understand” attitude. We should explain history, natural sciences were a constant source what we know in the simplest possible terms and of deep and challenging mathematical problems. minimal generality. Then it will be possible to see Let’s not dwell now on the obvious practical impor- what grows in the next field and use the fruits of tance of these problems and talk about something yourneighbor’slabor. else, namely the rich intuition that comes with Good social contact makes good neighbors. Ef- them. This complex knowledge was derived from fective networks are hard to synthesize but they a multitude of sources by generations of deep may be our best hope in the fight against frag- thinkers. It is often very mathematical. Anyone mentation of mathematics. I, personally, wouldn’t looking to make a mathematical discovery needs a get anywhere without my friends/collaborators. I problem and a clue. Why not look for both in natural think there is a definite tendency in mathematics to sciences? work in larger groups, and I am certain this trend Thisdoesn’tmakemathematicsasubordinateof willcontinue. other sciences. We bring, among other things, the power of abstraction and the freedom to apply any Terence Tao tools we can think of, no matter how apparently un- M & P: When did you become interested in mathe- related to the problem at hand. Plus what we know matics? wereallydoknow.Sowecanbuildonfirmerfounda- Tao: tions, hence higher. And look—mathematics is the As far as I can remember I have always en- tallest building on campus both in Princeton and in joyed mathematics, though for different reasons at Moscow. different times. My parents tell me that at age two I M&P:There is a common view of the public that wasalreadytryingtoteachotherkidstocountusing computers will make mathematicians superfluous. numberandletterblocks. Do you see a danger in that? And in particular M & P: Who influenced you to take the path of what is your stand on computer-assisted proofs? mathematics? Something to be welcomed or condemned? Tao: I of course readabout greatnamesin math- Okounkov: Computers are no more a threat to ematics and science while growing up, and perhaps mathematicians than food processors are a threat had an overly romanticizedviewof how progressis to cooks. As mathematics gets more and more made; for instance, E.T. Bell’s Men of Mathematics complex while the pace of our lives accelerates, we had an impact on me,even though nowadays I real- must delegate as much as we can to machines. And izethatmanyofthestoriesinthatbookwereoverly 406 Notices of the AMS Volume 54, Number 3 dramatized. But it was my own advisors and men- Tao: Itdependsontheproblem.SometimesIjust tors, in particular my undergraduate advisor Garth want to demonstrate a proof of concept, that a cer- GaudryandmygraduateadvisorEliStein,whowere tain idea can be made to work in at least one sim- thegreatestinfluenceonmycareerchoices. plified setting; in that case, I would write a paper as M & P: What was your feeling when you were short and simple as possible and leave extensions told about being a medalist? andgeneralizationstoothers.InothercasesIwould Tao: I had heardrumorsof mygetting the medal wantto thoroughly solve a major problem, and then a few months before I was officially notified—which I would want the paper to become very systematic, meant that I could truthfully deny these rumors be- thorough, and polished, and spend a lot of time fo- foretheygotoutofhand.Itwasstillofcourseagreat cusing ongetting the paper justright.