Andrew Wiles talks to Hannah Fry

[Music] I'll try that good evening ladies and gentlemen and welcome to the Science Museum I'm Mary Archer chairman of the Science Museum and we're thrilled to play host this evening to the University of Oxford's mathematical Institute which has kindly chosen the museum as the first non-oxford venue for its series of public lectures on mathematics and our talk this evening is of course by none other than Sir Andrew Wiles who is the Royal Society research professor of mathematics at the mathematical Institute this is a fairly rare public appearance by Andrew who has had to guard his time jealously since he made international headlines in 1994 I think it was so Lawrence Bragg who said he knew no recipe for certain success in research but one recipe for certain failure was a full engagement book but in 1994 Andrew of course reported that he'd established the truth proved the truth of Fermat's Last Theorem more properly called fermat's last conjecture which had been formulated some three hundred and fifty years earlier after his lecture Sir Andrew will be joined by mathematician and broadcaster dr. Hannah fry of University College London who is a good friend of the science museum group she presented Britain's greatest invention from our national collection center near Swindon this year and Hannah you were also one of the faces associated with our tomorrow's world partnership with the BBC the Royal Society the Open University and the Wellcome Trust mathematics of course puts the M into stem and building stem capital in individuals and society is one of the core missions of the science museum group I've just had the pleasure of showing Andrew round mathematics the Winton gallery which in the one year since it has opened has welcomed 1.2 million visitors to a gallery on the second floor of the Science Museum called mathematics and that I think speaks of the public appetite to understand this amazing subject better Wonderlab our interactive gallery has a new mathematics show - called prime time which looks at the remarkable impact on everyday life of mathematical thinking and of course at prime numbers and how they're useful in a wide range of areas such as encryption and then thanks to Professor Marcos de Soto also of the mathematical Institute who's one of the science museums advisers illuminating India exhibition has as its centerpiece a leaf from the Backshall a manuscript kindly lent to us by the Bodley own library which shows a very early use of the symbol of a circle for zero alongside the other numerals in the decimal system so at the Science Museum we embrace mathematics we seldom can do so at the level we're going to enjoy this evening and to take us into the rest of the evening it's now my pleasure to introduce professor Sir Martin bright sand who's Whitehead prefer of pure mathematics and head of Oxford's mathematical Institute thank you well generally I'd like to thank Mary and everybody here at the Science Museum for hosting us here this evening it's a wonderful venue it's a fabulous institution which we all respect very much and we're looking forward to working on many further projects with the Science Museum as Mary said this is an adventure outside Oxford for our public lecture series this public lecture series has been going for about four years we have a about six lectures spread throughout the year and has proved be fantastically successful and it was originally young the vision that grew out of conversations but was driven by Professor Alan Gary Ailey who's here somewhere this evening and made a reality by daraa Lombard who's made he and his team made this evening a reality and it's been quite a revelation to us and as Mary Lou did to this a surprisingly large appetite in public which i think is underestimated in many circles for engaging in the great adventure of modern mathematics our public lectures sell out incredibly quickly and people have a real appetite not just for some showmanship with with it with mathematical elements but really for delving into what mathematics is and what it's doing for society today I'd also like like to extend a particular welcome to our alumni and friends who are here this evening Oxford colleges have always been very good about nurturing a sense of lifelong belonging amongst their alumni and departments less so and we're very keen in Oxford that our alumni and friends should have a sense of lifelong belonging to the extended family of Oxford mathematics that doesn't just mean a flow of information or was asking for support it also means trying to engage in events like this where we offer some intellectual stimulation to that part of your brain which drew you to Oxford math Dick's in the first place so a particular welcome through all alumni and Friends so what what so what's the intellectual stimulation on offer this evening was the rare treat first of all of seeing Andrew Wiles give a 30-minute lecture on elliptic curves and I know you're all chomping at the bit so I shall delay long from that and then after andrew has given his lecture and he will do a question and answer session for about 30 minutes with Hanna Frey and then you'll have a chance to ask some questions at the end of that um Hannah many of you probably know is a distinguished author and broadcaster popularizer of mathematics which is also an active mathematician herself she's the lecturer in the mathematics of cities which sounds fascinating at the Center for spatial analysis at UCL those of you who do know as an author and appreciate it let me say that there's a chance to get her to sign your book at the end of this lecture so she'll be signing copies of her new book the mathematics of love very interesting seem mathematics is everywhere so the mathematics of love and that that's the sort of thing we will probably get into do one we do lots of public lectures on that sort of thing unexpected out mathematics reaching into society touching all aspects of modern life but we've also found a very genuine appetite for a different type of public lecture and that's more the one we're having this evening where people are genuinely curious to sort of lift the veil on what it is that people engaged in the struggle with the fundamental problems at the core of mathematics what's their life really like what's this sort of relentless and often frustrating struggle with the great problems of mathematics looking for truth and beauty trying to serve mankind well what's that really like can one get some flavor of it by listening to all the great proponents of the art and tell you something about what they're thinking about now and we're very privileged to have Andrew Wiles here this evening who's really no one exemplifies the values of that vocation more than he does I could start listing Andrews achievements and honors at this point but that would take up far too much of the evening so I won't we just say so oh yeah I got to know Andrew and Princeton in the early nineties when we were both there and it was the time when he was working on the solution to Fermat's Last Theorem and many of you will know the drama of that situation and its ultimate positive resolution and to watch Andrews struggling with this huge problem ultimately conquering this theorem under the glare of the international media was the most remarkable thing I've seen in my professional life it was really a quite extraordinary time and a quite extraordinary achievement um he is quite rightly by far the most celebrated mathematicians of modern times were immensely proud to have him as a colleague in Oxford mathematics and I won't delay his lecture any longer ladies and gentlemen please welcome Sir Andrew Wiles well thank you very much Martin for that introduction so uh as you observed I am NOT going to talk about famous last theorem that of course is the equation that firm R is best known for but it wasn't the only problem he left us and I I want to talk a bit about another problem which he really initiated the modern study of C solving equations goes back a very long time as you probably know the ancient Greeks solved some kinds of equations basically equations quadratic style equations and if you take equations in one variable that takes up the history of 13th 14th 15th century Italian Renaissance mathematics and was only completed in the early 19th century but what I'm going to talk about is actually solving equations in two variables so equations in two variables so here I've given a typical example the equation y squared is X cubed plus ax plus B so here a and B a rational numbers that just means fractions and you have to find solutions x and y in rational numbers or in fractions as well well as I said equations in two variables started with the Greeks and they solved equations where the maximum exponent was two so if you have equations like Y squared equals x squared plus ax plus B they can solve equations like that but I'm afraid it's very embarrassing to say this but any equations of higher degree like the one I've written and that's only going up two cubes we haven't solved at all there is no solution for any general class or equations of degree higher than than two so the Greeks dealt with the case where the exponent was two but in the last two thousand years we've we've made some progress but we haven't got there and the second half of the twentieth century I think we actually did a lot but we still couldn't quite master these equations so I want to talk about the next what seems like the next hardest and what progress has been so like all the best things in mathematics the modern story started with Vemma and what he did he he didn't publish mathematics in fact nothing was published under his name no journals he wrote letters to friends and to colleagues especially to English mathematicians giving challenges and two of the things he mentioned and he later explained in notes in his own work in his own books where the following equations which I've put here so y squared is X cubed minus 2 and Y squared is X cubed minus 4 and he he showed that the only solutions in integers that means whole numbers not fractions but whole numbers are the ones I've listed wise plus or minus 5x is 3 for the first and then the fourth solutions for the second one that's an incredibly simple statement and I defy any of you to go home and try and write out a proof it is really really tricky although it doesn't require any more mathematics than you would learn prio level very very simple pre GCSE and yet it's incredibly ingenious well this equation solving these kinds of equations in integers actually has been solved only in the second half of the 20th century but solving it in rational numbers has not been done what famish erred interestingly is that sometimes these equations have infinitely many solutions sometimes they have none and we don't know how to tell the difference we have an idea but we don't know he also bequeath there's something else which was fascinating about these equations and I want to describe that now so this is an equation this is a graph of an equation y squared is X cubed plus 17 now it's actually the graph of the real solution so about solutions in real numbers that's the kind of graph you would normally plot if you're doing high school mathematics what firm I observed was that if you take any two solutions to this so you take the solution P is 4/9 so that's X is 4 and Y is 9 which is a solution 9 squared say t1 4 cubed is 64 64 plus 17 is 81 and similarly Q is a solution with X is 2 and Y is 5 you draw the straight line through those two points and it will hit the which is y squared is X cubed plus 17 this is this curved one one more time at a point R and what he observed is that this one more point has to be rational it has to be made up of fractions so the fact that minus 2 minus 3 are both rational numbers that is not chance that always happens that was the observation of Fermo and interestingly we know it was an observation of fela because Newton thanked him for this observation much later so this process we call taking a chord so you take a chord and it meets the curve again one of the tricks of mathematicians is they say well what happens if you actually make T and Cuba the same you just take a tangent here then it counts as if it is two points and the same thing happens if you take a tangent at a rational point it meets the curve again at another rational point so firm I was able in this way to show you can build more rational points from ones that you've started with so more fractional solutions if you start with one and he observed that sometimes by pursuing this chord and tangent process you could generate infinitely many solutions sometimes on the other hand you would get into a loop and it would go back so you go P to Q to R then you might take R twice take a tangent you did something else and then go back again and you get back to the points you started with but sometimes you don't and you get infinitely many okay why do you get rational solutions for the third point well this is it's very Elementary now I'll just explain why so if you take Y squared is X cubed plus ax plus B so you think of a and B as fixed rational numbers and then when we do algebra we we write a line as - why is MX plus C so again M and C irrational numbers if you try and find out where those two equations are both satisfied that will be the points where the straight line meets the meets the curve you're solving you substitute for y you're solving MX plus C squared is X cubed plus ax plus B and if you know that two of the solutions of that cubic equation irrational then it's very easy to see the third is rational because the some of the solutions is actually M Squared so that's an easy thing to check so if two solutions are rational then so is the third but Arthur infinitely many and if there are how do we find them well this question laid low for a couple of hundred years and then in 1901 Parker a great topologist founder of other branches of mathematics actually raised the question how many points can you generate this way how many do you need to start generating all the points and it was proved by ma Dell in 1922 that you can get all the rational solutions all the rational points by starting with a fixed finite number of them using this process that family introduced by taking chords between two points or tangents at any point and then taking the other point that meets the curve so this was a great step forward and it now enables us to refine the question how many generators do we need to generate all the points and how do we find the generators so the answer to this question we don't know yet but we've got partway there and has a rather surprising surprising feature that is I've been talking about finding solutions in integers or in fractions and rational numbers but what mathematicians do is they they look at it in some completely different context so very often we look at solving it in the real numbers or solving it in the complex numbers well that's rather natural by now although it wasn't in the fifteen century it is by now but for this one you do something more peculiar and that is you look first at mod P arithmetic so let me remind you what mod p arithmetic is so you take a prime so seven is a prime number then mod seven arithmetic you add two to six that makes eight but you always subtract off a multiple of seven so you get back into the range zero to six so two plus six is the same as one mod seven the difference is divisible by seven two times six which should be 12 don't know how it comes to save 1 1 7 I'm sorry 2 times 6 is 12 which should be 5 mod 7 oh it does ok sorry yes it doesn't say the same on my notes ok 2 times 6 who's five months have very good hour in the equations support you can solve equations mod seven as well x squared is 2 mod 7 has solutions 3 & 4 because 3 times 3 is 2 mod 7 and 4 times 4 is 16 take off 2 multiples of 7 you get to month 7 so you can talk about equations mod 7 and the first step in trying to understand solving in rationals and integers strangely enough is to think about the equations in mod P arithmetic so for each prime P what you do is you let NP be 1 plus the number of solutions of Y squared equals x cube plus ax plus B mod P so you try and count the number of solutions in mod P arithmetic of the equation you really want to understand okay well that's a reasonable thing it doesn't take long you just try all the different X's all the different Y's and you count up how many of them give you inequality in mod P arithmetic but then you do something which is much more distressing even to a mathematician you take 3 over n 3 that's fine multiply it by 5 over n 5 that's some double of solutions mod 5 arithmetic then times 7 over N 7 and you multiply those together over all Prime's P now they're infinitely many Prime's P that's goes back to Euclid and well supposing we could do that we expect that this product that's product means multiplying them all together should go to 0 if the equation y squared X X Q plus xB plus P has infinitely many solutions and should go to something nonzero if it has only finitely many solutions well that's our criterion for whether or not it has infinitely many solutions it's a very bizarre one and its origins I can't really explain to you which is a good thing because first we can't prove it and second even worse we can't even I mean I said take the limit but there's no reason that limit should exist anyway how would we multiply together in fluently many things this is just hopeless so people tested Burton's winneth and I tested this on a computer and want the early days of use of computers in the late 50s and it seemed that that as they got more and more primes this was getting to give these results it seemed that but it wasn't it wasn't necessarily going to be too helpful but then they were convinced to reformulate it in the following way so this looks more like modern mathematics and I apologize but I hope I can explain some of it so you make a function of s where you take this product over Prime's P and a product just means you multiply them together so it's just like we did before of some strange creature here one - and that's a number P is the prime P to the minus s + pizza 1 - 2 s to the minus 1 AP is the number 1 plus P minus the NP that was counting solutions mod P which we did before ok now if you compute this function at the value 1 just formally so this doesn't actually mean anything except I'm just putting one in for s here you get 1 minus a P P inverse plus P to the minus 1 all the minus ones bit complicated you simplify it you get back to this product of P over N P okay well that's that's what mathematicians do so they start with something that they couldn't really work out this product of P over N P this multiplying together all these things for different primes then they make a function out of it which seems like it gives the same same kind of answer but it's still hopeless because it's still taking a product of things that we don't know converge okay but actually we've made progress because this is a function we think of this as a function of s with s now any complex number so we've used the mod p arithmetic we're now going to use complex numbers as well use s of complex numbers and this function le s actually makes sense if we use complex numbers okay this may seem like a strange result but this actually absolutely fundamentals of the subject it actually was the key result that I had to prove to solve fair Mao's Last Theorem this is what lay behind it was showing this this is the the part of the taniyama-shimura conjecture that proves Fermat's Last Theorem okay now now that we know that this le s makes sense we can now make of actual conjecture which is actually defined that II has infinitely many rational solutions if and only if this function takes the value 0 at 1 so this was this is a part of the Bertrand it entire conjecture which is one of the millennium prize problems if you've heard of those problems if you get a million dollars if you can actually solve a slightly more difficult version of this but it's this is a key piece of it so it was formulated in the 1960s and between 1977 actually I worked on this and my thesis and the 1990s when it was finally proved this theorem says well one direction is known that if this function is non zero at 1 then it the elliptic curve the equation only has finitely many rational solutions we don't know how to go in the other direction ok well this I just wanted to show you a real modern mathematical problem this is in some sense the arithmetic problem that takes over from famous Last Theorem it's got its roots in farmer's work and it's one of the Millennium problems this can be interpreted in a special case in a way that describes aksheehan even old a problem this problem is a thousand years old and the problem is if you take a right angle triangle with rational length sides does there exist such a triangle whose area is n for n some given number like 1 2 5 and so on can you solve that well in fact you can solve some of them and you can't others for example there is no such triangle with area 1 you cannot find a right-angled triangle whose sides have rational numbers a news area is one that's actually equivalent to the case N equals 3 of famous last theorem so it was solved by fam'ly in self what's it got to do with these elliptic curves that I described well it's actually a very easy calculation and this is something you could do II the equation y squared is X cubed minus N squared X has in definitely many solutions if and only if there exists a triangle with a B and C fractions of rational numbers and with area N and one direction I've just given you the proof you can just write it down so you take a is x squared minus M squared over Y and so on and you find that actually this right angle is are the lengths of a right-angled triangle and the area is in and you can go back from this to this as well so solving this cubic equation y squared is X cubed minus N squared X this elliptic curve is related to these right angled triangles if you want area 157 it would take you a long time even with modern computers to find this solution and yet this is the simplest solution this solution is found by taking that elliptic curve that I said Y squared is X cubed minus 157 squared X you take that elliptic curve and you you can't do by trial and error they're simply too big but we have some very ingenious techniques dating from the 1980s which in some cases will find you a solution and that's what was done with this particular equation so this was not found by trial and error it was actually found using our methods for solving elliptic curves just in very special examples it doesn't always work we don't know how to do it in general but in that particular case it works but this problem of in general determining which right-angled triangles exist to give areas of different numbers we don't have any answer to that yet okay well those this problem is the Bertrand and I can this problem of solving these equations as I said is part of a greater a greater problem that number theorists mathematicians have worried about solving equations in rationals and integers and as I said since the Greeks we haven't actually found another set of equations we can really say we've we've solved so what I wanted to do is tell you something very recent just to tell you what mathematicians do so I told you it's not clear with these elliptic curves how many generators you need do you need a lot infinitely many how many generators do we need using the chord and tangent process to give you all the points so sometimes I said if you start with a point you could get a closed loop and then you let's just forget about those ones that's actually they're quite easy to find so we call the rank the minimum number of generators that you need to generate all the solutions to an elliptic curve so when I was a student everybody thought that you could have arbitrarily high rank that is you could get right down cubic equations elliptic curves which required a very very large number of generators but then over the years so in 1938 the highest rank anyone had found so that's where you find you need three generators was 1938 you found three 1975 they found seven 1986 was 14 1994 is 21 and so on so everyone thought ok there's no bound on this it's going to go on forever we're going to need more and more generators as we write down more complicated cute elliptic curves and then someone some group of people decided to start applying probably a theory to this now this is a strange kind of probability you're used to tossing a dice as probability or maybe using a pack of cards it's it's a small finite number of things you can test the probability of but what you do when you're testing the probability where there are infinitely many objects it's not so clear and the kind of probability they use for this I'm not going to begin to explain it because it's rather complicated instead of tossing a dice you're kind of tossing mathematical objects and you're giving them a weighting according to how many symmetries they have so you don't consider all the ways that dice can come down as equal it depends on the amount of symmetry involved so it's it's a bit controversial but people applied this model and this is I have to say somewhat in frustration that we can't do the budget when it entire conjecture so people have to look for other things they can do and this is the thing they tried and with their model they came up with the prediction that actually there is a maximum rank so it's completely upended what we believe before and I have to say the number theory community is divided on whether this is plausible or not whether we should listen to this probability model so they predict that there's only finitely many with rank in more than 22 that is where you need more than 22 generators I my mind is open on this and I don't know but I wanted to tell you something that was just from last year that's got people excited but of course if you can solve the birch to entire conjecture that's much better and you get a million dollars as well thank you very much [Applause] thank you very much marvelous Andrew thank you so we have a Q&A now and then you will have your chance to ask Andrew walls your own questions and towards the end so get thinking of the kind of things that you want to ask him and but I wanted to start off and you mentioned the Millennium problems there at the end I for people who don't know about them could you tell us a little bit more because of course feminists are cerium is not the only famous unsolved problem in mathematics no it's not so in the year 2000 people wanted to celebrate the year partly because in 1900 famous German mathematician Hilbert produced a list of problems of which we've solved a fair number and in 2000 we wanted to do something similar produce a list of problems but it's a list of problems which have been unsolved for a while not new problems but just ones that we thought summed up some of the big challenges in mathematics one of them has been solved so it was solved and soon after they were set up actually at 2003 but there are six left six million dollars on a six million dollar problems yes do you have a sense of which one will be next to be solved so the most famous problem in mathematics is without doubt what's called the Riemann hypothesis it's it says something about the way prime numbers are distributed but it says is about a function that was introduced by Riemann and I don't have any real feel for which one would be next but I think if I had to bet I'd bet on that one okay get down the bookies now everyone and other I mean these these problems span sort of the breadth of of mathematical research are there are there areas that you wish you'd had time to study more deeply in your career or are you sort of very single-minded about I confess I'm I was a big it's number theory from the time I was 10 years old and I I've never found anything else and mathematics appealed quite as much as that does that mean when you were doing your undergrad then there were some areas of math that you felt slightly weaker in the number theory it's a cheeky question to ask and it's definitely true in fact there isn't very much number theory in undergraduate mathematics and I would sneak off to the library and try and read fair Moffat family had this really irritating habit of writing in Latin though it was required to learn Latin to get into Oxford it wasn't at the standard that I could read fair master not effectively how's your Latin now minimal in terms of the way that the mathematicians describe their subject it always strikes me that if you I have only ever been acquainted with with maths at school hearing my petitions use words like you know the thrill of discovering Fermat or the beauty and elegance of equations they feel like slightly strange words to be using about about the subject and for you though could you tell us a little bit about why you find mathematics such a romantic subjective well I think there's two things ever one is the romance of this particular story which captivated me that fam'ly wrote down this problem in in a copy of a book of Greek mathematics it was only found after his death by his son and then it reached the wider world and then so many people tried it and failed and those great dramas in the French Academy when people claimed it and it was wrong and so on so that had particular romantic story and a very personal one for me but why is why do we talk about beauty in mathematics and arrogance I mean it's it's hard to explain in any terms what beauty and elegance are and paintings are in music and so on but I think perhaps it's easiest divide talk in terms of if you go to a capability Brown Lansky garden you walk through on the path and the way designs it is that suddenly you come out into an area and suddenly everything is clear and you see a building behind the trees and you see a new landscape and it's this surprise element of suddenly seeing everything clarified and and beautiful that that we feel it's mathematicians so I think it there's an element of it's beautiful the first time and still beautiful again but you shouldn't stare at it non-stop because like with paintings or music it'll it'll fade if you're if you're just standing in front of it forever you need to keep walking through the girl s so why do people get put off by it if that's the experience that the professional mathematicians have when you know doing their subjects why do people get put off well I I think the biggest handicap from mathematics my impression is that certainly was my experience in the u.s. that it's if you're young you really need someone who cares about mathematics one likes it teach you in your first steps and unfortunately it's quite rare to at least it was there to have maths teachers before you reach the age of 10 or 11 who actually trained in mathematics and wanted to be teaching mathematics I think what happens is that mathematics is a very useful subject people go off and do many other things with it and the weren't enough left as teachers so the teachers tended to be recruited from other subjects or even from sports or something like that and they didn't care about mathematics and that got passed on so I think that's the stumbling block I think most young people children do have a real appetite for mathematics and it does appeal to them but you really need to learn it from someone who who enjoys it and shows you enjoyment and after the age of 10 or 11 it's often too late you know does the public perception of Malthus it's kind of contributes to that to that the way that people give up I mean I'm thinking here about how you know mathematicians or mathematics is sort of portrayed in you know in the media or on film I read for instance that you're not particularly a big fan of the film Good Will Hunting is that correct yeah so Good Will Hunting is a problem for many mathematicians because the idea is you're born with it and then it's easy okay there are some things you're born with that might make it easier but it's never easy and not for mathematicians I mean mathematicians struggle with mathematics even more than the general public does that's what they need to understand we really struggle is hard we could go to a seminar by someone else in the department and be completely lost and just struggle but we're used to it so we learned how to adapt to that struggle but no so I think there's no one who it's really easy for some people have worked so hard at it that they convey this impression that it's easy but it wasn't easy the first time so do you think you need a natural attitude to to become a mathematician then well okay so there's being a professional mathematician where you do research or just being competent in let's say being competent okay so being competent I think you have to be born with some intelligence that's obviously variations but I don't think it's that exceptional but I think the qualities that make a good research mathematician are not so much technical ones but ones of character you need particular kind of personality that will struggle with things will focus won't give up and so on so when you go to those seminars of colleagues and I mean I know when I go to seminars of colleagues I often come away having on not understood about 90% of what's going on what's what's your what's your process how do you how do you sort of approach a new bit of mathematics new challenge okay so new approaching new mathematics if it's really outside my field and it's something I need to learn it's always much better to learn from another person who does know that subject so as I said before about having a good teacher it's the same for us that if you have someone who can teach it well who knows the subject it's much better so one-on-one is best in a small seminar is second best and the worst possible way to learn it is from a book or a journal bottom of the pile you sort of mentioned there that you know mathematicians are kind of used to used to struggling and that's quite an I guess an easy statement to make but how do you shield yourself against you know being discouraged when you're finding it very difficult well I think it's the same as in other parts of life I think you will get discouraged you learn from experience that you make it through and you'll get disappointments but you know there's time heals these things and sleep heals these things you replacement therapy you know you do something else takes away the day yeah of course but um well is that are your views on sort of discouragement and and feeling discouraged colored by the fact that you eventually were successful of course with her massage theorem I mean would you would you feel the same way I mean because there are other mathematicians who have worked for years on problems unsuccessfully do you think your view is is sort of colored by your experience I don't know but I I'm sure it must be obviously if you know that it worked out in the end and you accept it you accept the setbacks and yes obviously I've had a positive resolution so I'll be more more benign about these periods of being stuck but even in smaller province where I haven't solved them I just accept it's part of being a mathematician this being stuck I mean whether from being in high school and being stuck on a homework problem to being stuck on an exam question it's the same thing it's just a question of scale I mean if you're stuck for five years on the problem some people can't make the transition from being stuck for you know for a few hours to being stuck for a few years and that's even true with people who become PhD students they've been very very good top of the world literally in in mathematics competitions or in education and their homeworks whatever and they may try and make the transition to being a research mathematician they can't cope with being stuck for more than 24 hours I mean there are some great stories about people you know given a thesis problem and 24 hours later they come back and say no I can't do it I want another one and there's one famous story about someone and apparently went off and became is quite famous as writing textbooks but this really happens well if we say frustrated at not being able to get the answers right away right I mean it particularly happens to very gifted people I mean you mustn't be too good if you're too good you get used to solving everything very quickly and you know if you're not quite that good you a little more used to this being stuck but I guess when it comes to Fermat's Last Theorem in particular I mean you were stuck for a very long time well perhaps that's mildly unfair it took a long time to get towards that to the end of the proof were there ever any points during that period where it sort of felt like a chore where we're being stuck you know was discouraging no I think I think I always felt I'd gained insights into the problem I'd yes I was stuck for very long periods but it wasn't like I was back to zero every time I felt I was making progress and maybe I wouldn't get the whole thing but I was getting somewhere and no I I didn't I didn't feel like giving up I didn't feel that was stuck forever and I think my anxiety wasn't actually so much in wasting my time or being stuck in a loop or something like that it was first I didn't literally have enough time our lifetimes are are limited and and also though not until later on I was I would be worried that I was in general in mathematics like put in all this effort and actually someone else does it a better way quicker you don't want to put that much energy into something and I could take failing to get it but it would be very difficult to take you know someone comes up with a quick solution did you always believe that it was possible day that proving it was possible yes so I tried it as a teenager and young person but then I realized when I became a professional mathematician that actually those methods which were basic in 19th century methods had been exhausted and everyone that had tried everything basically so I I did put it away for a while but what happened was in an 86 Gerhard fry made a connection with these actually with these elliptic curves with this result I I mentioned about this l-series Elia vas making sense for all complex numbers made that connection and once that connection was made I knew it was part of mainstream mathematics and it wasn't a problem that would ever go away it was had to be solved we couldn't go round it couldn't leave it behind but there were some of your peers at the time who didn't think it was it was possible even after gahard Freud's work came out that's right I I think many of the yes my voice actually I'm always quite encouraged when people say something like you can't do it that way always feel that's a real hint of the right way to do it um and how I mean cuz you are at Princeton at the time where you were regularly how did you manage to keep your work secrets from well the University I mean your bosses presumably wanted to make sure that you were actually doing some work yes well we have this great system called tenure which protects you a little bit but what I had was I actually had some some results I'd been working on and I was just a little slower in publishing them and I strung it out over a few years and people said ah he's gone off the boil and things like that I knew I I knew it would be an issue eventually but I managed for that length of time did you sort of you know in the evenings or whatever imagine at that moment of solving it and sort of the looks on your colleagues faces if you like no I think it's worse than that I think there are times when you think you have solved everything but you know you realize that there's a problem there's a gap do you ever look back at the truth now not really no I mean I I have the outline of the proof in my mind I could reconstruct it but the details no not really now as I say it I mean it is very nice going back out for your early work because some sense you forget what you've done and you're really impressed by this smart young man who were reversed who did it but you wonder how you did but I don't spend any time really doing that I mean this was solving Fermat's Last Theorem happened quite early on really annual in your career as a professional mathematician were things that you worked on after that ever quite as rewarding No quite simply no I don't think they could have been I mean okay if I if I could solve the Riemann hypothesis but you know that'd be nice if that would be nice yeah but it's it's not quite my specialty anyway so okay I'm gonna go throw to the audience in a second but I guess it would be nice to just get some some words of advice if you like from you four students who are coming through who are perhaps doing their a levels now what what would you say to your 17-year old self if you if you had a chance to well I think I probably did it the right way I think what I would say is yes try these impossible problems while you're in high school while you're an undergraduate actually the time to stop doing them is when you're starting your career so as a graduate student as a you know junior faculty and so on then I think you have to be have to be responsible career-wise otherwise you could just spend those ten years trying an impossible problem and you have nothing to show for it and that would be a professional mistake but once you once you're settled and you've got your job and everything yes thank you well that's that's the point of tenure is so that you can try these things otherwise of course everyone's just going to chase the the easy thing that you do you know looks good and keep producing things but it's good to play with mathematical ideas when you're younger absolutely I've wasted a lot of time when I was a child trying to solve these these impossible problems and I don't think it was a in the long run it wasn't a waste of time at all it gave me the idea of what research is and it gave me the the taste for it I think it was very beneficial and my last question I think in terms of I mean I do a lot of work trying to engage more people in mathematics how do you think mathematics should be viewed how do you want the world to view our subject well I think mathematics there's two roles really so to me I'm solving these these equations I've talked about it to me it's very beautiful I'm passionate about solving them and always have been then I'd like for people to be able to to see it I mean to experience it if they can but just to to see what it is we find so appealing about it I mean in some ways the so it's it's it's something that's immutable I mean people talk about other universes and everything i I just can't imagine any other kind of mathematics it's somehow it's the most permanent thing there is on the other hand it's the language of science and it's incredibly useful you know more and more as the world goes on I mean it's it's you're so employable if you do it and it's wonderful that it can be applied to medicine to sort of reducing queueing times for cars or whatever it is I mean you know for securing your your internet communications your credit card whatever it is it's underlies everything in the world so it's tremendously useful but at the same time well I care about even more it's just seeing this this beautiful edifice that somehow as I say I think the most permanent thing there is wonderful great point I think okay so there are I think some people wondering rounded microphones if you put your hands up you're no question okay we'll go let's get someone on the edge first so if we go to you first you one and then we'll get another mic up to you in the middle of F two and then three epic so I first wanted to ask you whether you have read gaussians first number theory book but then you said the books are the first thing to learn mathematics from so I wonder whether you have to try things first and then read or read first and try then I mean but maybe also say whether you have read houses this quiz it's known as arithmetic in translation I presume so I I haven't to one say the question channels are so there's question is whether I've read Gauss's famous book on number theory called discursive Sione's Mathematica and the answer is I've looked at sections of it but I I would look at it I certainly didn't read it cover-to-cover know I'm not someone who goes back and and studies books in great detail I find it very difficult and a little bit off-putting our second questionnaire oh you've got one ready fantastic you said that you thought that the 19th century methods had been used that looked at so is that there was nothing left there does that mean that you think that Fermat didn't actually have a proof well the question is whether I think Fermo actually had a proof so he certainly wouldn't have had the 19th century methods I mean he was 17th century and it had to be more simple than that so when I was young I tried to do it based on the kinds of things he started using quadratic forms and so on I think the probability is almost zero but it is conceivable it's just conceivable but I can't see how it how it could work third question there we go I had exactly the same question yeah it seems unnecessary to ask the second time so think of a different question okay off the top of my head your progress aren't Fermat's Last Theorem there were ups and there are downs if you were to graph us against time what function would best sum up that graph you wanted a question off-the-cuff I'm sorry okay so I think there are perhaps if you start put a flagpole in the beginning a flagpole at the end there'd be three flag poles in between and each one's higher than the one before but the rope that joins them is sort of sinks it was a particularly big sag on the last one that will be roughly their function but you went back over the same ground tried the same tricks several times right while you were while you were working yes so I find that mathematically you have to go back and try the same things again because often it's it's just a minor variation minor variant on what you've tried before that actually works and you're just missing one little piece of information or one one extra idea it's a little like evolution no evolution works by making mistakes and in some sense mathematics I feel is a bit the same you you have to make these mistakes too because you have to try all these different things and lots of them are going to be wrong but eventually one is right other questions okay let's try the site okay so we've got one there perfect and then we'll go to just long and then three other mess I'm Andrew thank you so much I just wanted to ask if you think maths is addictive certainly number theory I think the answer is obviously completely yes I think number theory in particular there even among professional mathematicians there are quite a few started in other branches of mathematics it's very rare for them to move out of number theory once they've gone there they they've often migrated to number theory but they very rarely migrate away from it surrender I've got I'm a secondary school math teacher so my question might be slightly banal for this for this audience but I promised my colleagues I'd ask you to solve a dispute we have in the star from about the way we teach square roots that whether or not the the square root of a positive number is always a positive number and only a positive and a negative when it is the root of an equation or is the square root of that number always the positive M Square and negative root well I think you you have to decide I mean it's you can check you can choose it's a matter of terminology really so it's up to you to decide which is the standard terminology not gonna stop your dispute today I'm not gonna solve this dispute okay I'm curious to know whether the resolution of fermat's last conjecture as many practical applications in the practical world so the actual equation itself doesn't seem to be that useful but the techniques used to solve it are useful and we always expect will become more useful there's often a lag time between the mathematics that's done and the utility of it and the lag time is probably higher in number theory than in most things though in the last 40 years it's it's gone down to much smaller lengths of time now because it's used in security and cryptography and so on a great deal so the utility in those fields would I think is already it's already there really I think we've got time for one more round okay so if we go one you there too and then anyone else three there we go just done here oh okay what's each month you go first yeah hi I'm also a secondary school math teacher and I was interesting what you're saying about sort of spending a long time on a problem not being successful at a problem and how that helped you and I wonder I kind of think maybe exam culture isn't great for that do you do you think it would be a positive thing if we were to do away with school exams I don't know if it's practical to do with school exams but there is this question of whether you should encourage people to do mathematics via competitions and clearly some people really like that but some are completely put off by it I'm on the side of being put off by it I never did competitions as a child and certainly you want a venue where they can do mathematics in a more collaborative and less competitive way this question is also on the teaching theme you mentioned the importance of having a great maths teacher when you're younger do have any thoughts on how we can encourage more mathematicians to go into teaching yes pay them more who is your math teacher when when you were at school was there one that sort of stood out that inspired you well there were two so there was one in junior school mrs. Briggs who was a wonderful teacher and there was one in my high school who would actually done a PhD in number theory and gave me some books and so on he was great inspiration did you speak to them after you sold them as a theorem I did meet my high school teacher I think I met both yes nice moment for them okay the alarms gone off saying we have to finish but I did promise one more question which was up there yes perfect if we get microphone to this gentleman oh okay I'm sorry okay okay just fascinated by that the idea of being stuck on a problem for a long long period of time and then having an insight in terms of bringing the freshness in terms of things that help you gain that fresh approach to the same problem with it was there anything common to sort of how you put yourself in a mindset to look at the same thing in a different way after such a long period of time so I think what the way it seems to work is you work very very intensively on on these math problems so that you know everything that's been done you you try out everything and your conscious mind kind of runs out of ideas and then it's often said in mathematics it's the three B's that help bus bath and bed that when your mind is relaxed somehow your subconscious takes over and pieces it together so I can't say anything useful except at some point you let your mind relax and and something gets put together in your mind for no prescription sorry now technically we should get off but the thing is I'm in charge so I think I don't want to rob you of your chance to Android's question so if we can get a microphone over here that's good Oh was it okay great it's all solved no one need to tell me off then okay in which case then all that remains is to say an enormous thank you to Sir Andrew Wells and thank you all for coming very much thank you [Applause]

Channel: Oxford Mathematics

Views: 117,892

Rating: 4.9254537 out of 5

Keywords: Oxford Mathematics Public Lectures, Andrew Wiles, Hannah Fry, Elliptic Curves, Maths Lecture, Math Lecture

Id: uQgcpzKA5jk

Length: 68min 47sec (4127 seconds)

Published: Mon Dec 04 2017

Reddit Comments

[u/TauBone]

There are some things you're born with that might make [math] easier but it's never easy and not for mathematicians. I mean mathematicians struggle with mathematics even more than the general public does. That's what they need to understand. We really struggle. It's hard. We could go to a seminar by someone else in the department and be completely lost and just struggle, but we're used to it so we learned how to adapt to that struggle. [...] I think there's no one who it's really easy for some people have worked so hard at it that they convey this impression that it's easy but it wasn't easy the first time.

Couldn't have said it better myself.

[u/y45y564]

Idk if anyone is interested in jumping in to this particular question

---

Question : Why to people get put off by mathematics

Timestamp to question and response : https://youtu.be/uQgcpzKA5jk?t=2510

[u/-tp-]

I found the Q&A in the latter part of the video interesting. Wiles talks about math competitions, what it takes to be a mathematician, books being the worst way of learning mathematics.

EDIT: His talk about elliptic curves was also cool! I've seen little to no number theory in my undergrad so far.

[u/velcrorex]

3rd Q is from Dara O'Briain! I recognized that voice right away!

[u/velcrorex]

I enjoyed the lecture, the conversation, and the Q&A. Thanks for sharing.

[u/ntc1995]

Amazing talk, especially the QA, mathematicians can certainly relate to what Wiles was talking about. It’s a series of getting stuck over and over and over again. Until you get stuck again.