Andrew Wiles - The Abel Prize interview 2016

professor wise please accept our warm congratulations for having been selected as the Arbor Prize laureate in 2016 to be honest the two of us we expected that we would have had this interview already several years ago you earn Fame not only among mathematicians for a now eyesight you're stunning proof of Fermat's Last Theorem by way of the modularity conjecture for lept occurs opening a new era in number theory this proof goes already back to 1994 which means that we and you had to wait for more than 20 years before it earned you the Arbour pricey nevertheless you are the youngest Arbor laureate so far in the time after you finished your brief proof of Fermat's Last Theorem you had to undergo many interviews I know which makes our task quite difficult how on earth are we to find questions that you haven't answered several times before what we should try to do our very best but nevertheless in particular in view of our spectators from Norwegian TV we have to start at the very beginning and that is in fact in Latin so now I try to cite in Latin no l'm in infinitum ultra quadratum put a statin indoors Houston nominees fastest Nvidia which means it is impossible to separate any power higher than the second into two light powers and then he continues who you slay the most solemn me reveal insanity take see Hank Magnus exhibit has non Cabarete which means I've discovered a truly marvelous proof of this which this margin is too narrow to contain so this remark was written by the French lawyer and amateur mathematician Pierre otama in his copy of the ofon tous item Attica somewhere in the 1630s he said certainly didn't expect at the time that he would keep mathematicians professionals and amateurs alike busy for centuries in trying to unearth the proof so could you please give us a short account on some of the series attempts towards proving Fermat's Last Theorem until you embarked on your journey and why was such a simple-minded question so attractive and why were attempts to answer it so productive in the development of number theory so the first serious attempt to solve it was presumably by farmer himself but unfortunately we know nothing about it except for what he explained about his proof in the specific cases of N equals 3 and N equals 4 that he he showed that you can't have the sum of two cubes being another cube or the sum of two fourth powers being a fourth power so he did this by a beautiful method which he called infinite descent which is was a new idea a new way of presenting mathematics anyway in arithmetic and he explained these proof several times to his colleagues in letters and he wrote it also in the margin was big enough for some of it but then after this was published by his son after his death it lay dormant for a while but was picked up by Euler and others later who tried to find this truly marvelous proof and they failed and it became quite dramatic in the early 19th century the people thought they could solve it with some discussion the French Academy and Co she jumped in and said he thought he could too and so on but in fact it transpired that a German mathematician coma had actually written a couple of papers where he explained that the fundamental problem was what's known as the fundamental theorem of arithmetic that is in our normal number system any number can be factorized in essentially one way so you take the number like 12 it's 2 times 2 times 3 there's no other way of breaking it up but in the system of numbers that you want to use to try and solve this Fermo problem you actually use systems of numbers where this doesn't hold and every attempt that was made to do it failed because of this failure of unique factorization Kombat analyzed this in incredible detail he came out with most beautiful results but the end product was that he could solve it for many many cases for example less than n equals a hundred he solved it for all numbers except 37 59 and 67 but it didn't finally solve it his method actually was the same that Verma had tried the method of infinite descent but in these new number systems with these new number systems he was using spawned number theory algebraic number theory as we see it today so it involved developing other number systems where you might try and solve equations instead of just solving it with ordinary integers or rational numbers and attempts at fam'ly carried on but somewhat petered out in the 20th century no one came up with a fundamentally new idea until second half of the 20th century number theory moved on considered other questions and then in 1985 gerhard frey german mathematician came up with a stunning new idea where he took a hypothetical solution to the famine and rewrote it so that it made what's called an elliptic curve and he showed or she suggested that this elliptic curve would have very peculiar properties and he conjectured that this you couldn't really have such an elliptic curve well American mathematician kind of ribbit then actually demonstrated that using this Frye curve that any solution to firma would contradict another well-known conjecture called the modularity conjecture this conjecture had been proposed in a weak form by taniyama and refined by Shimura was first real evidence for it came from Andre V who made it possible to check that new conjecture the modularity conjecture in some detail and a lot of evidence of being amassed showing that this modularity conjecture should certainly be true so at that point mathematicians could see yes fam'ly is going to be true moreover there has to be a proof of it because what had happened is that this modularity conjecture was not a problem that mathematics could just put to one side and go on for another 500 years it was right there was a roadblock right in the middle of mathematics it was a very very central problem whereas FEM I you could just leave it aside and forget it almost forever this you could not forget it so at that point when I heard rivet had done this that moment I knew okay this problem can be solved and I'm going to try could I interject the hello about the speculation about firm ass proof so-called proof do you think that he had the same ideas le'me had the wrong I mean assuming that the cyclotomic you know integers hyper unique factorization no I don't think sourdough the error might be in there somewhere it's very hard to understand on drove a wrote about this because all the other problems he considered were to do with curves that were genus 0 or genus 1 and suddenly he's writing down a curve that's higher genus how is he going to think about it so I think when I was trying this myself when I was a teenager I I try I put myself in fellas frame mine not because there was anything else I could do I just thought his mathematics from the 17th century I was capable of understanding later mathematics probably not at that point and it seemed to me that everything he did came down to something about quadratic forms and I thought maybe that would be a way to try and think about it of course I never succeeded but there's nothing else that suggests he he he fell into this trap with unique factorization and in fact from the point of view of quadratic forms he understood sometimes the ways unique factorization and sometimes there wasn't so he understood that difference okay in his own context so I think it's very unlikely that was the mistake because I mean in the same book by under the way that you mentioned if Emma had this a cube minus a square equal to two has only in has only essentially one solution namely X equal to 3 and y equal to plus -5 yes and I think that and Rivera speculates that the Maya times looked at ring Z plus Z square root of 2 against that unique factorization in that so Cedric - - yeah he artists yes so okay yes he used the unique factorization but the way he did it was in terms of quadratic forms okay and I think he also looked at quadratic forms corresponding to 0-6 where there's not unique factorization so I think he understood that was my impression at the time what he understood the difference okay so we have already started on your personal mathematical education I'm curious to know I mean you were apparently already interested in mathematical puzzles as a quite young boy have you any thoughts about where this interest come freaking from is it family or any other influence that you can pinpoint well I just enjoyed mathematics when I was very young but I I at the age of 10 I was evidently looking through library shelves devoted to mathematics and I will pull out books and then at one point I pulled out the book by et bell which describes on the cover describes this problem in the wall scale prize and the romantic history of this problem I was completely captivated by it what was the other things in that book that fascinated you this book by Eric temple bar and it's entirely about that one equation really and it's actually quite wordy so that is less mathematics in some sense than you might think so I think it was more the equation but then one nice found this equation and I think I must have found other elementary books on number theory and learnt about congruence and I see solving conferences and sound and other things like firm I did and you did this sort of work besides of your ordinary school career yes I don't think my school work was too taxing from that one okay so was it already clear for you at that time that you had an extraordinary mathematical talent I think I had certainly mathematical attitude and and obviously loved to do it but I don't think I felt I was unique and I don't believe I was in that school I think there are others who I had just a strong acclaim to be future mathematicians and some of them have become mathematicians and it was already clear for you very early on that you would study mathematics and getting into a mathematical career no I don't I don't think I really understood you could spend your life doing mathematics I think that only came later but certainly studying it as long as I could I'm sure I've as far as my horizon was it involved mathematics yes you then started to study mathematics as a student in Oxford and back in 1971 is that right can you tell us a little bit about how that worked out were there any particular teachers and in particular areas that were particularly important for you so before I went to college actually in high school one of my teachers had actually had a PhD in number theory and he gave me a copy of ardian right and I'd or also found a copy of Davenport's higher arithmetic and these two books I found very very inspiring in terms of number theory so you were in track before you started yeah I was on track before and in fact to some extent I felt college was a distraction because I had to do all these other things applied maths logic and all these things and I just wanted to do non theory and you weren't allowed to do not a theory in your first year and you didn't really get down to it to your third year so which you were not interested in geometry as much as soon as in algebra or a number theory now time early in algebra I mean I was happy to learn these other things but but I really was most excited about the number theory and so I I think I my teachers enabled me to take extra classes in number theory but but there wasn't that much on offer I think at one point I even decided all the years of Latin I'd had to do at school I should put it to good use and try to read some of their mind the original but I decided that was actually that was too hard because even if you've translated the Latin the way they wrote in those days was oh yeah wasn't him algebraic symbols very much yeah it was quite difficult yes so it must have been a relief when you then went to Cambridge to start studying really number theory with John Coates that's right as your supervisor so I went there you had a year preliminary year in which you just studied a range of subjects and then I I could do a special paper and John Coates was not yet at Cambridge but I think he helped me all maybe over the summer after it he I met him and then I started working with him right away and that was just wonderful that the transition from undergraduate work where you're just reading and studying the transition to research was that was the real break yeah this is wonderful and ginger it was John Coates I assume who initiated you to work really on elliptic curves absolute and Eva Sava to Syria and so on he had some wonderful ideas and generous to share them with me and you told John Coates that you were interested in the thermal problem I gather I probably did and he probably warned me now it's really true that there hadn't really been any new idea since the 19th century people were trying to refine the old methods and yes the were refinements but it didn't look like the refinements and the solution we're getting going to converge it was just too hard that way right but at that time when you started working with John Coates you had no idea that this elliptic curve should turn out to be really the cushy no it's how a wonderful coincidence and and the strange thing is that the two things in a way that are most prominent in Fairmont that we remember his work on elliptic curves on this I mean for a number theorist this equation you meant mention Y squared plus 2 equals x cubed elliptic curve right right yes same thing could you spare a few seconds to explain what an elliptic curve is and how it is divided the theory is developed by means of Eva's AMA theory yeah so for a number theorist the life of elliptic curve started with Falmer as equations of the form y squared equals a cubic in X and the cubic in X would have rational coefficients and then the problem is to find the rational solutions to such an equation and fam'ly what famine otis was besides the kind of result you get if you look at integer solutions sometimes you can prove there are very few and sometimes even say exactly what they are in contrast with rational solutions you could sometimes start with one or even two and use them to generate infinitely many others and yet sometimes as actually it's not obvious Disney have to care for the case N equals three of them are theorem is in fact an elliptic curve in disguise sometimes you could show there are no solutions so you could have infinitely many and you could have none so this was already apparent to ephemeral and so people studied these equations and then at the turn of the 20th century while Caray actually [Music] realized that you could study these well sorry I'm jumping ahead we should say in the early 19th century once studied these equations in complex numbers and then árbol himself comes in at this point studying elliptic functions and they were very well understood in terms of doubly periodic functions in the early 19th century but that's the complex solutions solutions to the equation in complex numbers the solutions to the equations in rational numbers were studied actually pipe on Curie and in there was what's now known as the more delve a group was proved by model and inve in the early 20s that the rational points on elliptic curve form of finitely generated abelian group that is you from famous language you could start with a finite number of solutions and using those generate all the solutions using what he called the cordon tangent process so that that group okay you now know the structure is a very beautiful algebraic structure but that doesn't actually help you find the solutions so no one really had any methods for finding the solutions many good methods until the conjectures of the late 50s which emerge from vergence one didn't die so there are two ways of formulating it one is somewhat analytic and one is in terms of what's called the teacher average group but basically the teacher Frey which group gives you the obstruction to using to finding an algorithm for finding these solutions and the birch minute and dire conjecture tells you there's actually an analytic method for analyzing this so-called th average group and if you combine all this together ultimately it should give you an algorithm and you worked on ready on vergence minute and daya when you were a graduate student together with John Coates yes that's exactly what he proposed working on and we got the first results in this analytic link between solutions and what's called the l function of the elliptic curve in certain special families of elliptic curves this was complex multiplication so these were they left occurs with complex multiplication what's this theory the first result concerning this birch suing it entire conjecture I think it was the first one that treated a family of cases rather than an individual case which there was a lot of numerical data on individual cases but this was the first infinite family and this was all with the rational numbers right yes you should just mention that this conjecture divergence winners and dire conjecture is one of the millennium clay prize yes which if somebody could solve it would earn the somebody 1 million dollars right that's right I think it's appealing because partly because its roots are in firm are just like the Fairmont problem so it's another elementary two-state problem concerned with equations of in this case very low degree which we can't master and which I initiated so I think it's a very appealing problem do you think it's in reach somehow that we have the tools somehow that if somebody's daring enough or do we have another I'd have to wait another three hundred years I don't suppose it's three hundred years but I'm afraid there I don't think it's the easiest of the millennium problem so I think it's still still we're lacking some things which whether the tools are all here now they may be me now there's always the problem with these really difficult problems it may be that the tools simply aren't there no I don't believe anyone in the 19th century could have solved them are certainly not in the way it was eventually solved it there was just too big a gap in mathematical history you have to wait another 100 years for the right pieces to be in place so you can never be quite sure with these problems whether there are accessible to your time that's in fact what makes it so challenging if you've got the intuition for what can be done now and what can't be done now you mentioned a teacher for a rich group yeah and in that connection the Selmer group also pops up and said Morocco's was a norwegian mathematicians right and i think it was castles that named this group that removes the selma group could you say in a few words what the selma group habits related is they people it is technical but on the other hand I can probably say so basically the Selma group what you what you're trying to do is to find the rational solutions on the elliptic curve and the method is you take the points on the elliptic curve suppose you've got some and you can if you like generate extensions from those so when I say generate extensions you can take roots of those points on the elliptic curve just like taking the nth root of five or the cube root of two you can do the same thing on elliptic curve you can take the nth root of a point so all points that times n give you the point you started with they generate certain extensions of the number field you start with a set of the number field Q so you can put a lot of restrictions on those extensions and the selma group is basically the smallest set of extensions you can get putting on all the obvious restrictions and so you've got the group of points they generate some extensions that's too big you don't want all extensions you cut that down as much as you can using local criteria so using p-adic numbers you cut it down as much you can that's called the Selma group and the difference essentially between the group generated by the points and the Selma group is the Tait shaffer image group so the thra which group gives you the error term if you like in trying to get the points from the Selma group I see I see but the settlers paper he of course looked at the equation the Daiya phantom equation 3x cubed plus 4y cubed plus 5z cube equal to 0 and it showed that it had no just a trivial solution in the integers yes mod M it had non-trivial solution for all n yes and and well Easter why did castle call it the I mean in insanity there's quite a subtle relationship between is so what happens is your actually looking at one elliptic curve which in this case I forgotten exactly which curve it comes out to be but it would be equivalent to X cubed plus y cubed plus 60 Z cubed equals zero and that's an elliptic curve and the HRH group involves all those involves looking at other ones like it in other word for example 3x cubed plus 4 y cubed plus 5 y cubed equals 0 which is not technically an elliptic curve it's a genus 1 curve but it's not only lived to curve precisely because it has no points and the thra which group one other way of describing it is in terms of these curves for the genus 1 but don't have points and by assembling these together you can make the teacher for average group and that's reflected in the selma group it's too intricate to explain in words but it's another point of view i gave it in the more modern terminology in terms of extensions the older terminology was in these these twisted forms ok so what you proved in the very end is by now called well form a special part of the modularity conjecture in order to somehow trying to explain it one has to start with modular forms and how modular forms can be used to generate elliptic curves could give us a short account on this topic elliptic curve we've described as an equation y squared equals x cubed plus ax plus B and the a and B we're assuming to be rational numbers so as I said the beginning of the 19th century you could describe the complex solutions to this so you can actually describe these very very nicely in terms of the Weierstrass p function in elliptic functions but what we want is we want actually find a completely different uniformization of this which captures the fact that the a and B irrational numbers so it's like a parameterization just of the rational elliptic curves and because it captures the fact that it's defined over the rationals it gives you much better hold on solutions over the rationals than the elliptic functions do which somehow only sees really sees the elliptic on the complex structure and the place it comes from are modular forms or much of the curves so to describe modular functions first so we're used to functions which satisfy the relation of being invariance under translation every time we write down a Fourier series we have a function which is invariant under translation modular functions are ones which are invariant under the action of a much bigger group usually a subgroup of s l2 of the integers a vessel to Z so you would ask for a function in one complex variable usually on the upper half plane which satisfies f of z f of z is the same as f of a Z plus B over C Z plus D or more generally is that times the power of C Z plus D so these are called modular functions were extensively studied in the nineteenth century and they hold the key to this arithmetic and the way they hold the key is perhaps the simplest way to describe it is because we have an action of SL 2z on the upper half plane by this action Z goes to AZ plus B over Caesar plus D we can look at the quotient H mod this action and you can give that the structure of a curve in fact it's naturally got the structure of a curve over the rational numbers if you take a subgroup of s L to Z which is what's called a congruent subgroup so it's defined by saying the C value being divisible by n then you call the curve a modular curve of level m and the modularity conjecture asserts that every elliptic curve over the rationals is actually a quotient of one of these modular curves for some number n so it gives you a pram uniformization of elliptic curves by these other entities these modular curves which on the face of it might seem you're losing because this is a higher genus curves more complicated but it actually has a lot more structure because it's a moduli space and that's something very very powerful tool and that's a very powerful tool and you have function theory you have a lot of tools for that but did Tanya the young japanese mathematician that first sort of conjecture or suggested this thing did he have this his conjecture was more vague I guess his was more vague so he didn't pin it down to a function invariant under the modular group I forget exactly what he did was uninvent under some kind of group but I forget exactly what kind of group he was predicting but it wasn't as precise as that I think it was originally in Japanese so it probably wasn't circulated as widely as it might have been guys I think it was just notes after a conference in Japanese it was an incredibly audacious conjecture at the time wasn't it apparently yes but then it caught caught the attention of I mean you told part of the story of get out fire in particular who then came up with a conjecture relating the massless theorem with the modularity conjecture God Frye showed that if you take a solution to a famil a to the P plus B to the P equal C to the P you make an elliptic curve y squared is X X minus a to the P X plus beta P then the discriminant of that curve would end up being a perfect peeth power and if you think about what that means if you assume the this modularity conjecture and you have to assume something slightly stronger as well then it forces this elliptic curve actually to have the end that I spoke about equal to 1 but H mod SL to z is a curve of genus 0 it has no elliptic curve quotients so there wasn't anything there so that in a sense was the point of departure for your for your story I mean followed up of course by the by the work of certain Ribit who made this more clear so my shortly summarize the story that then follows so it has been told by you many times it was even on a BBC feature so you had moved to the u.s. to the United States first to Harvard and then to Princeton University becoming professor there and when ribbits result was out use devoted to started to devote all your research time to working on a real attempt to solve the modularity conjecture for semi-stable curves if I understand it correctly and then follow seven years of really hard work in isolation while you were working as a professor at Princeton while you were raising small kids and then a proof seems to be accomplished by 1993 and your work culminates in the series of three talks at the Isaac Newton Institute in Cambridge back in England and as a result you are celebrated by your peer mathematicians and even the word press takes an interest in your results which is happens very rarely for a mathematical result but then when your result is scrutinized by in total six referees for one of the very respected journal it turns out there is a subtle gap in one of your arguments and you are sent to back to your desk and then it took 14 months of hard and frustrating and sometimes really hard work in the hand hair and a hero a fight in a sudden flash of insight you find out that you can combine some of your previous attempts with new results to some hour circumvent the problem that occurred and well it turns out that this is just enough to proof to prove what you really needed in order to get this part of the Madureira conjecture that then encompassed famous Last Theorem what a relief that must have been so well would you like to give a few comments to the story and some additions or yes so I in my turn to my own work I when I became a professional mathematician I worked with Kurtz I realized I really had to stop working on far more because it was time-consuming and I could see the last hundred years had done nothing and I saw others even very distinguished mathematicians had come to grief on it but this revelation actually when Fry came out with his result I was bit skeptical that the ser part of the conjecture was going to be true but when rivet proved it then I okay now this is it and it was yeah a long hard struggle but it was it's in some sense it's irresponsible to work on one problem to the exclusion of everything else but it is the way I I tend to do things and I did feel with this problem whereas film I was very narrow I mean it's one equation which whose solution may or may not help with anything else the setting of the modularity conjecture was one of the big problems in number theory and and there was was a great thing to work on anyway so it just was a tremendous opportunity and I think when you're working on something like this it takes many years to really build up the intuition to see what kinds of things you need what kinds of things a solution will depend on and just to see it's almost like discarding everything that you can't use that won't work until your mind is so focused that even making a mistake you you see you've seen enough that you'll find another way to the end funnily enough the the mistaken argument people have worked on that aspect of the argument and quite recently they've actually shown that you can produce arguments very like that in fact for every neighboring case it's the unique point that it doesn't seem to work for and there's no real explanation so the same kind of argument I was trying to use using Euler systems and so on has been made to work in every surrounding case that really extraordinary so no wonder that it escaped your attention in the first place you described I guess this quest for the proof of the modularity theorem that something like going into a mansion and could you describe how the hobbit feels like and - yeah I think I think the when you take on a problem where it's not you're trying to remove some condition that somebody had on a theorem or you're trying to extend something I started this off really in the dark I had no prior insights how modularity conjecture might work how you might approach it I mean one of the troubles with that problem it's little like the Riemann hypothesis but perhaps even more so at that one is you didn't even know what branch of mathematics the answer was coming from I mean there are three ways of formulating the problem one is geometric one is arithmetic and one is analytic and there were analysts with whom I wouldn't understand their techniques at all they were trying to solve this problem so I think I was a little lucky because my natural instinct was the arithmetic and I went straight for the arithmetic route but I could have been wrong I mean in some sense the the previous the only known cases of this modularity conjecture are the cases of complex multiplication and that proof is analytic completely analytic so I partly Adam necessity I suppose and partly because that's what I knew so I could try it straight away I went straight for an arithmetic approach and I was I found it very useful to think about it in the way that I've been studying he was our theory so with John Coates I'd applied he was our theory to elliptic curves being his idea to use that and then when I went to Harvard I'd learned about mazes work where he'd he'd been studying the geometry of modular curves using a lot of the modern machinery but there were certain ideas and techniques I could drawn from that that I realized after a while I could actually use to make a beginning some kind of entry into the problem because before you started on the modularity theorem you published a joint paper with very may sir yes may main theory and 7li Messala yes shortly in this case you needed over the the rationals right that was over the rationals I did it with him and then I'd worked myself on extending it to case of the total so that turned out to be very very useful when you then starting with the it did it gave me a starting point I wasn't obvious at the time but when I thought about it for a while I realized that there might be a way to start it there and this is the result you extend it to two real fields right I mean no room yes I'd already done that before well okay let me be careful so I basically done totally reals but when I started working on modularity conjecture I I realized I was going to need some time and I wasn't going to be publishing very much so my publication process of these results on miiverse our Theory slowed down somewhat covering my tracks a little bit you covered your tracks so that people who use it well it wasn't disguised from other people it was that I was I was supported by universities and plan you know financing foundations that expected results not just attempts so if I published all mine yeah things at once yeah besides I I I really it was hard to distract myself from thinking about modularity to finish off preparing these results for publication so most of that work was was already I I'm going to read you a quotation uh-huh the ramparts are erased all around but enclosed in its last read out the problem defends itself desperately who will be the fortunate genius who will lead the assault upon it or force it to capitulate there must be ET banishment I don't know it's not it is actually from Jean Etienne Monte claw the east wall the mathematic that's the first book in history of mathematics written in the 18th late 18th century ah and it refers to the solvability or unsolvable ax T or the quintic equation ok and the general quintic equation which of course Abel is when he was 21 years old here in Oslo here in Oslo and he worked in complete isolation as far as mathematics yes things is concerned in fact this was Abel's obsession I mean he really delved into this and he also got that full start he has to he proved thought yet prove that you could actually solve the quintic i radicals then he discovered his his mistake and he finally found the proof right well this problem was almost 300 years old very very famous well if we now move fast forward 200 years the same quotation could be used about the pharma problem which was three hundred fifty years old on which you sold in I mean it's a very parallel story in many ways do you have any comments on on this yes and in some sense I do feel that our ball and then gawah were marking the transition in algebra from these equations which which are solvable in some very simple way two equations which can't be solved by radical CERN but this is an algebraic break with the quintic and in some ways the the whole trend in number theory now is the transition from basically a billion possibly solvable extensions to in in solvable extensions how do we do the arithmetic of in solvable extensions now the modularity conjecture I believe it was solved because we've moved on from this original abelian situation to a non abelian situation and we developing tools the modularity and so on which are non abelian tools so it's the same transition in number theory that he was making an algebra it's the same transition in number theory which which provides the tools for solving this equation so I think it is very parallel yes it is a very very ironic story because our bill then it was 21 years old he visited Copenhagen to visit professor Dragan who was the leading mathematician in in Scandinavia at the time and he writes a letter to his mentor in Oslo hombu where he had write some three theorems about the Fermo theorem I one of them is not easy to prove actually he doesn't give any proof and of course that's just a parenthesis today because I mean that wouldn't lead anywhere but in the same letter he says that he's desperate we cannot understand why he tries to solve it to divide the lemniscate arc in in n equal pieces and he gets an equation of degree N squared and he can't understand why N squared should be only n because they're only n points and of course that is because you have the double you know periodicity so when you get got back to Oslo he realized then here he's sort of inverted from the elliptic functions and if you think about it the thing that he did on the firma turn out to be completely I mean not valuable but the thing that he did on the elliptic functions turn out to be a tool of course I would had no idea this was going to have anything to do with arithmetic so this story sort of tells that mathematics develops in mysterious were in sentence yes so based on your experience could you describe a little bit this interplay that we were already discussing between this hard and persevere and work on your desk on the one side on the other hand sometimes sudden flashes of insight ayats that come out in a more relaxed atmosphere and what's difficult to see whether I can come from but making it clear that you must have worked unconsciously further on what you have been doing with your with your pencil before I think what you do is you get to a situation where you know a theory so well and maybe there's more than one theory that you know so well and you've seen every angle you've tried lots of different routes and you can just see you connecting five dots if they're sort of 5 places away from each other you can't do if they're one thought away it's automatic and somehow you just have developed such an intuition and understand these objects so well that you can see where they have to fit together and this final insight is not something you rationally think out and it's this tremendous amount of work in this preparatory stage where you have to understand all the details maybe some examples and then when you've developed all this then you let the mind relax and then at some point maybe you go away and do something else for a little bit you come back and suddenly it's all clear why did you not do that and this is something then the mind has I remember this this in a trivial example of this in non mathematical terms I remember once someone showed me some script and it was some kind of gothic script and I couldn't make head or tail of it and I was trying to understand a few letters and I gave up and then I came back half an hour later I could read the whole thing and the mind somehow does this for you and we don't quite know how but we do know what you have to do to set up the conditions where it will happen did this whole thing we discussing on me reminds me about one story about Abel he came to Berlin and he lived with some Norwegian friends who were not mathematicians but one of his friends told that I would typically wake up during the night light a candle and write down ideas that he woke up with so so I mean apparently his mind was work yeah yeah I do that except I don't feel the need to write it down when I wake up with it because I know I won't forget it well I'm terrified as if I have an idea and I'm about to go to sleep that I would not wake up with that idea so then I have to write it but okay so at the end of the day are you most of all thinking in formulas or geometric pictures or what is it you're shaking your head it's not really geometric I think I think it's patterns and I think it's it's just parallels between situations I've seen elsewhere and the one I'm facing now or just in a perfect world what's it all pointing to what what are the ingredients that ought to go into this proof what am I not using that I have sometimes it's just desperation I assemble every piece of evidence I have and that's all I've got and I've got to work with that and there's nothing else right I often feel that doing mathematics it's like you're a squirrel and you're being chased and there's these or even that was some some nuts at the top of a very tall tree but there are several trees you don't know which one and what you do you run up one and then you think oh no doesn't look good this one you go down of another one and you spend your whole life going up and down these trees but you only go up to 30 feet now if someone told you the rest of the trees that it's not up them you've only got one tree left you just keep going and you find it so in some sense it's ruling out the wrong things that it's really crucial and if you just believe in your intuition and you stick with your one tree you will find it yeah I talking about more general thing in mathematics there's a citation for a year of the police klein he said that mathematics develops as old results are being understood and illuminated by new methods and insights from this new problems naturally arise and Hilbert said something like good problems are the lifeblood of mathematics you subscribe to these things I certainly agree with with helmets yes good problems of the life plan of mathematics I think you can see in our century this I mean for me obviously modularity conjecture but the whole Langlands program which when i conjecture these problems give you a very clear focus on what we should try and achieve now the very conjectures some curves that will find out crimes and then varieties over finite fields these these problems somehow concentrate the mind and also simplify our goals and mathematics otherwise we can get very very spread out and we're not sure what's of value what's not a value do we have as good problems today as we had been Hilbert gave you say I think so yes I think the Riemann hypothesis is the single greatest problem that I understand and it's sometimes hard to think exactly why that is but I do believe it it's actually solving that will help solve some of these other problems and then of course I have a very personal attachment to the virtual dark injections course of course it's sort of intuition can feel as I mean Hilbert thought that the Riemann hypothesis would be sold within his lifetime and there was another problem he thought would never be sold in a time which yeah gelfand you know the some so our intuition can be wrong and that's right I'm not surprised he felt that way I mean one does feel the Riemann hypothesis is such a such a clear statement and we have the analogues in function fields sort of understand why it's true there we ought to be able to translate it of course many people have tried and wares and lawyers but I would still myself expect it to be solved before the birds from the dark injunction are interested in other Sciences apart from mathematics I'd say somewhat but not again I'm not going to devote these are things I do to relax so I don't like them to be too close to mathematics so if it's something like animal behavior or or [Music] astrophysics or something from a qualitative point of view yes I I certainly enjoy reading about those or know what machines are capable of in sort of popular science but I'm not going to spend my time learning details of string theory that would be too much I'm too focused to be willing to do that not that I wouldn't be interested about just this is my job choice I would like to thank you very much for this wonderful interview and this is first of all on behalf of the two of us but moreover also on behalf of the Norwegian the Danish and the European mathematical society thank you so much thank you ladies you

Channel: The Abel Prize

Views: 164,481

Keywords: abelprize, abelprisen, the abel prize, Andrew Wiles, andrew wiles fermat's last theorem, andrew wiles interview, Interview, fermat's last theorem proof, fermat last theorem proof, Abel, Sir Andrew Wiles, Proof, Fermat's last theorem, Fermath's teorem, Fermats theorem, Fermat's last teorem, Kummer’s new number systems, André Weil, number theory, John Coates, elliptic curves, Swinnerton-Dyer, Birch, Selmer group, Modularity Conjecture, Taniyama, Iwasawa theory

Id: cWKAzX5U85Q

Length: 59min 2sec (3542 seconds)

Published: Thu May 11 2017

Reddit Comments

[u/percyjackson44]

Whilst watching this, I kept feeling like Andrew was virtually born to prove FLT. It seemed like it was too perfect of a coincidence that he would be a master in the specific field that the proof would arise from.