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Good evening, ladies and gentlemen. I'll try that.

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Good evening, ladies and gentlemen, and welcome to the Science Museum.

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I'm Mary Archer, chairman of the Science Museum.

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And we're thrilled to play host this evening to the University of Oxford Mathematical Institute,

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which has kindly chosen the museum as the first known Oxford venue for its series of public lectures on mathematics.

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And our talk this evening is, of course, by none other than Sir Andrew Wiles,

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who is the Royal Society Research Professor of mathematics at the Mathematical Institute.

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This is a fairly rare public appearance by Andrew, who has had to guard his time jealously since he made international headlines in 1994.

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I think it was Sir Lawrence Bragg who said he knew no recipe for certain success in research.

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But one recipe for certain failure was a full engagement book.

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But in 1994, Andrew, of course, reported that he'd established the truth, proved the truth of Fermat's Last Theorem,

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more properly called Fermat's Last Conjecture, and which had been formulated some 350 years earlier.

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After his lecture, Sir Andrew will be joined by mathematician and broadcaster Dr. Hannah Fry of University College London,

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who's a good friend of the Science Museum Group.

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She presented Britain's greatest invention from our national collection centre near Swindon earlier this year.

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And Hannah, you were also one of the faces associated with our Tomorrow's World Partnership with the BBC,

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the Royal Society, the Open University and the Wellcome Trust.

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Mathematics, of course,

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puts the M into STEM and building STEM capital in individuals and society is one of the core missions of the Science Museum Group.

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I've just had the pleasure of showing Andrew Round Mathematics.

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The Winton Gallery, which in the one year since it has opened,

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has welcomed 1.2 million visitors to a gallery on the second floor of a science museum called Mathematics.

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And that, I think, speaks of the public appetite to understand this amazing subject better.

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Wonderlab. Our interactive gallery has a new mathematics show too called Prime Time,

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which looks at the remarkable impact on everyday life of mathematical thinking.

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And of course, at prime numbers and how they're useful in a wide range of areas such as encryption.

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And then thanks to Professor Marcus de Soto, also of the Mathematical Institute,

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who's one of the Science Museum's advisors Illuminating India exhibition,

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has as its centrepiece a leaf from the back Charlie manuscript kind lent to us by the Bodleian Library,

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which shows a very early use of the symbol of a circle four zero alongside the other numerals in the decimal system.

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So at the Science Museum, we embrace mathematics.

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We seldom can do so at the level we're going to enjoy this evening.

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And take us into the rest of the evening. It's now my pleasure to introduce Professor Sir Martin Bridson, who's Whitehead,

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professor of pure mathematics and head of Oxford's Mathematical Institute.

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Thank you. Thank you, Mary, for that very nice introduction.

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Well, generally, I would like to thank Mary and everybody here at the Science Museum for hosting us here this evening.

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It's a wonderful venue. It's a fabulous institution which we all respect very much.

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And we're looking forward to working on many further projects with the Science Museum.

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As Mary said, this is an adventure outside Oxford for our public lecture series.

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This public lecture series has been going for about four years.

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We have about six lectures spread throughout the year and it's proved to be fantastically successful.

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It was originally the vision that grew out of conversations but was driven by Professor Alan Gabrielli,

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who's here somewhere this evening and made a reality by Darren Lombard, who's made he and his team made this evening reality.

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And it's been quite a revelation to us. And as Mary alluded to this, a surprisingly large appetite in public,

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which I think is underestimated in many circles for engaging in the great adventure of modern mathematics.

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Public lectures sell out incredibly quickly, and people have a real appetite, not just for some showmanship with with mathematical elements,

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but really for delving into what mathematics is and what it's doing for society today.

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And I'd also like to extend a particular welcome to our alumni and friends who are here this evening.

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And Oxford Colleges have always been very good about nurturing a sense of lifelong belonging amongst their alumni and departments, less so.

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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.

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And that doesn't just mean a flow of information or asking for support.

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It also means trying to engage in events like this, where we offer some intellectual stimulation to that part of your brain,

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which drew you to Oxford Mathematics in the first place. So a particular welcome to all alumni and friends.

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So what? So what's the intellectual stimulation on offer?

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This evening was the rare treat, first of all, of seeing Andrew Wiles give a 30 minute lecture on Elliptic Curves,

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and I know you're all chomping at the bit, so I shan't delay long from that.

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And then after Andrew has given his lecture, he will do a question and answer session for about 30 minutes with Hannah Fry,

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and then you'll have a chance to ask some questions at the end of that.

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I'm Hannah.

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Many of you probably know is a distinguished author and broadcaster populariser of mathematics, but she's also an active mathematician herself.

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She's the lecturer in the Mathematics of Cities.

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Sounds fascinating. At the Centre for Spatial Analysis at UCL.

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And those of you who do know as an author and appreciate it.

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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,

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The Mathematics of Love. Very interesting.

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So mathematics is everywhere. So the mathematics of love.

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And that's the sort of thing we all we get it to do one, we do lots of public lectures on that sort of thing,

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unexpected about mathematics, reaching into society, touching all aspects of modern life.

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But we've also found a very genuine appetite for a different type of public lecture,

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and that's more of the one we're having this evening where people are genuinely curious to lift the veil on

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what it is that people engaged in the struggle with the fundamental problems at the core of mathematics.

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What's their life really like?

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What's the sort of relentless and often frustrating struggle with the great problems of mathematics looking for truth and beauty,

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trying to serve mankind? What's that really like?

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Come on, get some flavour of it by listening to what the great proponents of the art and tell you something about what they're thinking about now.

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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.

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And I could start listing Andrew's achievements and honours at this point, but that would take up far too much of the evening.

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So I won't. Let me just say, I. I got to know Andrew in Princeton in the early nineties when we were both,

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and it was the time when he was working on the solution to Fermat's Last Theorem.

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And many of you will know the drama of that situation and its ultimate positive resolution.

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And to watch Andrew struggling with this huge problem and ultimately conquering this theorem under the

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glare of the international media was the most remarkable thing I've seen in my professional life.

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It was really a quite extraordinary time and a quite extraordinary achievement.

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And he is, quite rightly by far the most celebrated mathematician of modern times.

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We're immensely proud to have him as a colleague in Oxford Mathematics, and I won't delay his lecture any longer.

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Ladies and gentlemen, please welcome Sir Andrew Wiles. Well, thank you very much, Martin, for that introduction.

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So as you observed, I am not going to talk about Fermat's Last Theorem.

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That, of course, is the equation that Verma is best known for.

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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.

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The modern study of C solving equations goes back a very long time.

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As you probably know, the ancient Greeks solved some kinds of equations, basically equations, quadratic style equations.

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And if you take equations in one variable.

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That takes up the history of 13th, 14th, 15th century Italian Renaissance mathematics and was only completed in the early 19th century.

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But what I'm going to talk about is actually solve equations in two variables.

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So equations in two variables. So here I've given a typical example.

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The equation y squared is x, q plus x plus B,

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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.

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Well. As I said, equations in two variables started with the Greeks.

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And they solved equations where the maximum exponent was two.

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So if you have equations like y squared equals x squared plus x plus b, they can solve equations like that.

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But I'm afraid it's very embarrassing to say this.

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But any equations of high degree like the one I've written, and that's only going up to cubes we haven't solved at all.

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There is no solution for any general class of equations of degree higher than than two.

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So the Greeks dealt with the case where the exponent was two.

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But in the last 2000 years we've, we've made some progress, but we haven't got there.

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And the second half of the 20th century, I think we actually did a lot, but we still couldn't quite master these equations.

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So I want to talk about the next what seems like the next hardest and what progress has been.

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So like all the best things in mathematics, the modern story started with a farmer and what he did, he didn't publish mathematics.

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In fact, nothing was published under his name. No journals.

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He wrote letters to friends and to colleagues, especially to English mathematicians, giving challenges.

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And two of the things he mentioned and he later explained in notes and his

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own work and his own books were the following equations which I've put here.

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So y squared is x cubed minus two, and y squared is x cubed minus four.

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And his he showed that the only solutions in integers that means whole numbers, not fractions, but whole numbers are the ones I've listed.

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Y is plus or minus five. X is three for the first and then the fourth solutions for the second one.

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That's an incredibly simple statement, and I defy any of you to go home and try and write out a proof.

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It is really, really tricky, although it doesn't require any more mathematics than you would learn prior level.

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Very, very simple. Pretty GCSE and yet it's incredibly ingenious.

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Well, this equation, solving these kinds of equations and integers actually has been solved only in the second half of the 20th century.

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But solving it in rational numbers has not been done.

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What family showed, interestingly, is that sometimes these equations have infinitely many solutions.

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Sometimes they have none. And we don't know how to tell the difference.

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We have an idea, but we don't know. He also bequeathed to something else which was fascinating about these equations.

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And I want to describe that now.

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So this is an equation. This is a graph of an equation.

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Y squared is X cubed plus 17. Now it's actually the graph of the real solution.

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So that solutions in real numbers, that's the kind of graph you would normally plot if you're doing high school mathematics.

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What Fermi observed was that if you take any two solutions to this, so you take the solution t is for nine.

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So that's X is four and Y is nine, which is a solution nine squared 81 for cube 264, 64 plus 17 is 81.

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And similarly, Q is a solution with X is to and Y is five.

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You draw the straight line through those two points and it will hit the curve, which is y squared is x, q plus 17.

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This is this curved one one more time. At a point are.

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And what he observed is that this one more point has to be rational.

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It has to be made up of fractions. So the fact that minus two, minus three are both rational numbers, that is not chance.

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That always happens. That was the observation of FMR.

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And interestingly. We know it was an observation of failure because Newton thanked him for this observation much later.

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So this process we call taking a chord.

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So you take a chord and it beats the curve again.

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One of the tricks of mathematicians is they say, well, what happens if you actually make P and Q the same?

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You just take a tangent here, then it counts as if it.

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It's 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.

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So Pharma was able in this way to show you can build more rational points from ones that you've started with.

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So more fractional solutions. If you start with one.

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And he observed that sometimes by pursuing this cord and tangent process, you could generate infinitely many solutions.

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Sometimes, on the other hand, you would get into a loop and it would go back.

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So you go, Pete.

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Q two of and you might take ah twice, take a tangent, you had something else and then go back again and you get back to the points you started with.

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But sometimes you don't and you get infinitely many. Okay.

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Why do you get rational solutions for the third point?

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While this is it's very elementary. Now I'll just explain why.

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So if you take y squared is x Q plus x plus P, so you think of A and B as fixed rational numbers.

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And then when we do algebra, we, we write a line, a straight line as is X plus C.

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So again, aman c irrational numbers. If you try and find out where those two.

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Equations are both satisfied. That will be the points where the straight line meets the.

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Meets the curve you're solving, you substitute for Y, you're solving X plus C squared is x Q plus x plus B.

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And if you know that two of the solutions of that cubic equation of rational, then it's very easy to see.

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The third is rational because the sum of the solutions is actually m squared.

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So that's an easy thing to check.

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So if two solutions are rational, then so is the third, but author infinitely many.

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And if there are, how do we find them? Well.

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This question sort of laid low for a couple of hundred years.

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And then in 1901, quackery, great apologist, founder of other branches of mathematics actually raised the question.

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How many points can you generate this way? How many do you need to start generating all the points?

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And it was proved by model in 1922 that you can get all the rational solutions,

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all the rational points by starting with a fixed, finite number of them,

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using this process that Fahmi introduced by taking chords between two points or tangents at any point,

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and then taking the other point that meets the curve.

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So this was a great step forward.

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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?

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So the answer to this question, we don't know yet. But we've got partway there and it has a rather surprising, surprising feature, that is.

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I've been talking about finding solutions in integers or in fractions and rational numbers.

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But what mathematicians do is they they look at it in some completely different context.

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So very often we look at solving it in the real numbers or solving it in the complex numbers.

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Well, that's rather natural by now, although it wasn't in the 15th century.

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It is by now. But for this one, you do something more peculiar.

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And that is you look first at modern arithmetic.

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So let me remind you what my prismatic is. So you take a prime.

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So seven as a prime number, then mod seven arithmetic, you add 2 to 6.

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That makes eight, but you always subtract off a multiple of seven.

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So you get back into the range 0 to 6. So two plus six is the same as one mod seven.

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The difference is divisible by seven. Two times six, which should be 12.

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I don't know how it comes to say 117. I'm sorry.

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Two times six is 12, which should be five, not seven. Oh, it does.

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Okay. Sorry. Yes, it doesn't say the same on my notes.

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Okay. Two times six. This five months have a very good hour in the equations.

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So you can solve equations mod seven as well. X squared is two, mod seven has solutions three and four because three times three is two mod seven.

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And four times 416 take off two multiples of seven, you get two months seven.

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So you can talk about equations, mod seven and the first step in trying to understand solving in rationals and integers.

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Strangely enough. Is to think about the equations in modern arithmetic.

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So for each prime p what you do is you let np be one plus the number of solutions of y squared equals excu plus x plus p mod p.

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So you try and count the number of solutions in mod p arithmetic of the equation you really want to understand.

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Okay. Well, that's a reasonable thing. It doesn't take long.

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You just try all the different axes, all different wise, and you count up how many of them give you inequality in modern arithmetic?

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But then you do something which is much more distressing even to a mathematician.

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You take three over and three that's fine. Multiply it by five over n five.

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That's the number of solutions.

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Modify arithmetic, then type seven over n seven and you multiply those together over all primes, another infinitely many primes.

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P that's goes back to Euclid.

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And well, supposing we could do that, we expect that this product, that's product means multiplying them all together should go to zero.

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If the equation y squared x x Q plus XP plus p has infinitely many solutions.

201
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And should go to something nonzero if it has only finite remaining solutions.

202
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Well, that's our criterion for whether or not it has infinitely many solutions.

203
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It's a very bizarre one. And its origins, I can't really explain to you.

204
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Which is. Good thing, because first we can't prove it.

205
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And second, even worse, we can't even. I mean, I said take the limit, but there's no reason that limit should exist.

206
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Anyway. How could we multiply together infinitely many things. This is just hopeless.

207
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So people tested boats and spread it.

208
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And I tested this on a computer in the early days of the use of computers in the late fifties.

209
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And it seemed that that as they got more and more primes, this was getting to give these results.

210
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It seemed that. But it wasn't. It wasn't necessarily going to be too helpful.

211
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But then they were convinced to reformulate it in the following way.

212
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So this looks more like modern mathematics and I apologise, but I hope I can explain some of it.

213
00:24:48,750 --> 00:24:56,820
So you make a function of S where you take this product over primes P and a product just means you multiply them together.

214
00:24:56,970 --> 00:25:04,680
So it's just like we did before of some strange creature here, one minus, and that's a number.

215
00:25:05,040 --> 00:25:09,570
P is the prime P to the minus. S plus pizza. One minus two S to the minus one.

216
00:25:11,550 --> 00:25:18,210
AP is the number one plus P minus the SNP that was counting solutions mod p, which we did before.

217
00:25:20,440 --> 00:25:28,839
Okay. Now if you compute this function at the value one just formally so this doesn't actually mean anything

218
00:25:28,840 --> 00:25:38,319
except I'm just putting one in for s here you got one minus ap inverse plus three to the minus one.

219
00:25:38,320 --> 00:25:41,740
All the minus ones bit complicated. You simplify it.

220
00:25:42,070 --> 00:25:49,170
You get back to this product of P. Okay.

221
00:25:49,180 --> 00:25:52,389
Well, that's that's what mathematicians do.

222
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So they start with something that they couldn't.

223
00:25:56,670 --> 00:26:04,500
Really work out this product of P over and P, this multiplying together all these things for different primes, then they make a function out of it.

224
00:26:05,660 --> 00:26:12,090
Which seems like it gives the same. Same kind of answer, but it's still hopeless because.

225
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It's still taking a product of things that we don't know converge.

226
00:26:18,900 --> 00:26:22,560
Okay, but actually we've made progress because this is a function.

227
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We think of this as a function of S. With S now any complex number.

228
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So we have used the mod prismatic. We're now going to use complex numbers as well use s as complex numbers.

229
00:26:37,430 --> 00:26:44,250
And. This function alias actually makes sense if we use complex numbers.

230
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Okay. This may seem like a strange. Result.

231
00:26:50,430 --> 00:26:54,700
But it's actually. Absolutely fundamental to the subject.

232
00:26:55,270 --> 00:27:00,070
It actually was the key result that I had to prove to solve Fermat's Last Theorem.

233
00:27:00,310 --> 00:27:04,580
This is what lay behind it was showing this.

234
00:27:04,600 --> 00:27:12,060
This is the. The part of the tiny Amazon or a conjecture that proves Fermat's Last Theorem.

235
00:27:13,470 --> 00:27:18,400
Okay. Now, now that we know.

236
00:27:19,620 --> 00:27:24,120
That this alias makes sense. We can now make of actual conjecture,

237
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which has actually defined that it has infinitely many rational solutions if and only if this function takes the value zero at one.

238
00:27:36,330 --> 00:27:38,430
So this was, um.

239
00:27:38,970 --> 00:27:45,120
This is a part of the boat's finished and I conjecture, which is one of the Millennium Prize problems if you've heard of those problems.

240
00:27:46,030 --> 00:27:52,739
So you get $1,000,000 if you can actually solve a slightly more difficult version of this.

241
00:27:52,740 --> 00:27:55,890
But it's this is a key piece of it.

242
00:27:57,390 --> 00:28:10,950
So it was formulated in the 1960s. And between 1977, actually, I worked on this in my thesis and the 1990s when it was finally proved.

243
00:28:10,950 --> 00:28:22,469
This theorem says, While One Direction is known that if this function is non zero at one, then it the elliptic curve.

244
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The equation only has financially many rational solutions. We don't know how to go in the other direction.

245
00:28:32,320 --> 00:28:39,219
Okay. Well, this. I just wanted to show you a real modern mathematical problem.

246
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This is, in some sense, the arithmetic problem that takes over from Fermat's Last Theorem.

247
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It's got its roots in famous work, and it's one of the millennium problems.

248
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This can be interpreted in a special case.

249
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In a way that describes actually an even older problem.

250
00:29:03,360 --> 00:29:06,660
This problem is a thousand years old. And the problem is.

251
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If you take a right angle triangle with rational length sides doesn't exist.

252
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Such a triangle whose area is an.

253
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For an some given number, like 1 to 5 and so on.

254
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Can you solve that while in fact you can solve some of them and you can't.

255
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Others, for example, there is no such triangle with Area one.

256
00:29:36,330 --> 00:29:42,809
You cannot find a right angle triangle whose sides have rational numbers and whose area

257
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is one that's actually equivalent to the case and equals three of Fermat's Last Theorem.

258
00:29:51,540 --> 00:29:58,800
So it was solved by Femi himself. What's it got to do with these elliptic curves that I described?

259
00:29:58,830 --> 00:30:02,970
Well, it's actually a very easy calculation. And this is something you could do.

260
00:30:04,630 --> 00:30:08,770
E the equation y squared is x cubed minus n squared.

261
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X has infinitely many solutions if and only if there exists a triangle with A, B and C, fractions are rational numbers and with area and.

262
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And One Direction. I've just given you the proof. You can just write it down.

263
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So you take a is X squared, minus N squared over Y and so on.

264
00:30:28,280 --> 00:30:34,280
And you find that actually this right angled is all the lengths of a right angle

265
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triangle and the area is n and you can go back from this to this as well.

266
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So solving. This cubic equation y squared is x cubed minus n squared x.

267
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This elliptic curve is related to these right angled triangles.

268
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If you want Area 157, it would take you a long time, even with modern computers, to find this solution.

269
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And yet this is the simplest solution. This solution is found by taking that elliptic curve that I said y squared is x cubed -157 squared x.

270
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You take that elliptic curve and you. You can't do it by trial and error.

271
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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.

272
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And that's what was done with this particular equation. So this was not found by trial and error.

273
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It was actually found using our methods for solving elliptic curves just in very special examples.

274
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It doesn't always work. We don't know how to do it in general, but in that particular case, it works.

275
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But this problem of of in general determining which right angled triangles exist to give areas of different numbers,

276
00:32:16,780 --> 00:32:21,259
we don't have any answer to that yet. Okay.

277
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Well, those this problem is the boats from the dock injection.

278
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This problem of solving these equations, as I said, is part of a greater a greater problem that number theorists,

279
00:32:34,340 --> 00:32:38,150
mathematicians of worried about solving equations and rationals and integers.

280
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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.

281
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So what I wanted to do was tell you something very recent, just to tell you what mathematicians do.

282
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So I told you, it's not clear with these elliptic curves how many generators you need.

283
00:33:02,960 --> 00:33:07,610
Do you need? A lot. Infinitely many.

284
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How many generators do we need using the Gordon Townsend process?

285
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To give you all the point.

286
00:33:14,670 --> 00:33:21,890
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 funds.

287
00:33:22,160 --> 00:33:32,390
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.

288
00:33:33,480 --> 00:33:40,590
Two Elliptic Curve. So when I was a student.

289
00:33:43,810 --> 00:33:51,000
Everybody thought. That you could have arbitrarily high rank that is you could get.

290
00:33:53,320 --> 00:34:00,420
Write down cubic equations, elliptic curves. Which required a very, very large number of generators.

291
00:34:03,190 --> 00:34:07,720
But then over the years. So in 1938, the highest rank anyone had found.

292
00:34:07,750 --> 00:34:12,760
So that's where you find. You need three generators was 1938.

293
00:34:12,760 --> 00:34:19,780
You found three, 1975, they found seven. 1986 was 14, 1994 was 21 and so on.

294
00:34:20,740 --> 00:34:24,480
So everyone thought, okay, there's no bound on this. It's going to go on forever.

295
00:34:24,490 --> 00:34:29,830
We're going to need more and more generators as we write down more complicated elliptic curves.

296
00:34:31,820 --> 00:34:38,660
And then someone, some group of people decided to start applying probability theory to this.

297
00:34:39,500 --> 00:34:44,600
Now, this is a strange kind of probability you're used to tossing a dice as.

298
00:34:45,750 --> 00:34:49,030
Probability or maybe using a pack of cards.

299
00:34:49,390 --> 00:34:53,730
It's a small, finite number of things you can test the probability of.

300
00:34:55,050 --> 00:34:59,130
But what do you do when you're testing the probability? Whether infinitely many objects.

301
00:35:00,810 --> 00:35:05,700
It's not so clear. And the kind of property they use for this.

302
00:35:05,700 --> 00:35:11,090
I'm not going to begin to explain it because it's rather complicated.

303
00:35:11,100 --> 00:35:19,260
Instead of tossing a dice, you're kind of tossing mathematical objects and you're giving them a weighting according to.

304
00:35:20,360 --> 00:35:28,190
How many cemeteries they have. So you don't consider all the ways A dies can come down as equal?

305
00:35:29,810 --> 00:35:38,870
It depends on the amount of symmetry involved. So it's it's a bit controversial, but people applied this model.

306
00:35:38,870 --> 00:35:43,519
And this is, I have to say, somewhat in frustration that we can't do The Bachelor and I conjecture.

307
00:35:43,520 --> 00:35:47,840
So people have to look for other things they can do. And this is the thing they tried.

308
00:35:49,010 --> 00:35:55,250
And with their model, they came up with the prediction that actually there is a maximum rank.

309
00:35:56,780 --> 00:36:00,649
So it's completely upended what we believe before.

310
00:36:00,650 --> 00:36:08,870
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.

311
00:36:11,580 --> 00:36:17,320
So they predict that that's only finite. Many with rank more than 22.

312
00:36:17,340 --> 00:36:22,950
That is where you need more than 22 generators. I have my mind is open on this.

313
00:36:22,950 --> 00:36:31,950
And I don't know. But I want to tell you something that is just from last year that's got people excited.

314
00:36:32,880 --> 00:36:38,220
But of course, if you can solve the budget around the entire conjecture, that's much better and you get $1,000,000 as well.

315
00:36:38,940 --> 00:36:47,860
Thank you very much. Thank you very much.

316
00:36:49,060 --> 00:37:01,959
Marvellous talk, Andrew. Thank you. So we have a Q and A now and then you will have your chance to ask and you ask your own questions towards the end.

317
00:37:01,960 --> 00:37:09,340
So get thinking of the kind of things that you want to ask him. And I wanted to start off, you mentioned the millennium problems there at the end.

318
00:37:09,520 --> 00:37:13,419
I. For people who don't know about them, could you tell us a little bit more?

319
00:37:13,420 --> 00:37:18,310
Because, of course, as I said, I am is not the only famous unsolved problem in mathematics.

320
00:37:18,400 --> 00:37:22,240
No, it's not so. In the year 2000.

321
00:37:22,900 --> 00:37:26,500
People wanted to celebrate the year, partly because in 1900,

322
00:37:27,040 --> 00:37:33,760
famous German mathematician Hilbert produced a list of problems of which we've solved a fair number.

323
00:37:34,090 --> 00:37:37,240
And in 2000, we wanted to do something similar.

324
00:37:38,240 --> 00:37:43,160
Produce a list of problems, but it's a list of problems which have been unsolved for a while.

325
00:37:43,880 --> 00:37:50,870
Not new problems, but just ones that we thought summed up, uh, some of the big challenges in mathematics.

326
00:37:51,320 --> 00:37:58,850
One of them has been solved, so it was solved and soon after they were set up, actually, in 2003.

327
00:37:59,870 --> 00:38:04,340
But there are six left. $6 million on a $6 million problem?

328
00:38:04,410 --> 00:38:08,570
Yes. Do you have a sense of which one will be next to be solved?

329
00:38:09,050 --> 00:38:14,840
So the most famous problem in mathematics is without doubt what's called the Riemann hypothesis.

330
00:38:16,400 --> 00:38:24,950
It's it says something about the way prime numbers are distributed, but it says it's about a function that was introduced by Raymond.

331
00:38:25,940 --> 00:38:33,890
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.

332
00:38:34,010 --> 00:38:38,300
Okay. Get down the bookies now, everyone and all that.

333
00:38:38,600 --> 00:38:43,010
I mean, these these problems span sort of the breadth of of mathematical research.

334
00:38:43,280 --> 00:38:49,399
Are there other areas that you wish you'd had time to study more deeply in your career?

335
00:38:49,400 --> 00:38:57,590
Or are you sort of very single minded about. I confess I'm I was addicted to number theory from the time I was ten years old.

336
00:38:57,590 --> 00:39:05,220
And I've never found anything else in mathematics that appealed quite as much as that does not mean when you were doing your undergrad.

337
00:39:05,240 --> 00:39:10,940
Then there were some areas of maths that you felt slightly weaker in than the number theory.

338
00:39:11,600 --> 00:39:15,200
It's a cheeky question to ask, and it's definitely true.

339
00:39:15,380 --> 00:39:18,610
In fact, there isn't very much number theory in undergraduate mathematics,

340
00:39:18,620 --> 00:39:26,450
and I would sneak off to the library and try and read them apart from I have this really irritating habit of writing in Latin.

341
00:39:26,960 --> 00:39:31,130
And so it was required to learn Latin to get into Oxford.

342
00:39:31,130 --> 00:39:35,840
It wasn't at the standard that I could read Fair Master, not effective.

343
00:39:35,900 --> 00:39:40,730
How is your Latin now? Minimal. Yes, indeed.

344
00:39:42,200 --> 00:39:46,340
In terms of the way that that mathematicians describe their subject,

345
00:39:47,120 --> 00:39:53,569
it always strikes me that if you have only ever been acquainted with maths at school hearing,

346
00:39:53,570 --> 00:40:00,500
mathematicians use words like, you know, the thrill of discovering Fermat or the beauty and elegance of equations.

347
00:40:00,650 --> 00:40:04,280
They feel like slightly strange words to be using about about the subject.

348
00:40:04,610 --> 00:40:12,960
And for you, though, could you tell us a little bit about why you find mathematics such a romantic subject in.

349
00:40:14,400 --> 00:40:15,630
Well, I think there's two things.

350
00:40:15,780 --> 00:40:26,660
One is the romance of this particular story, which captivated me that from I wrote down this problem in in a copy of a book of Greek mathematics.

351
00:40:26,670 --> 00:40:31,799
It was only found after his death by his son. And then it reached the wider world.

352
00:40:31,800 --> 00:40:39,900
And then so many people tried it and failed. And those great dramas in the French Academy when people claimed this and it was wrong and so on.

353
00:40:41,790 --> 00:40:45,960
So that had a particular romantic story and a very personal one for me.

354
00:40:46,740 --> 00:40:51,209
But why is why do we talk about beauty and mathematics and elegance?

355
00:40:51,210 --> 00:40:59,780
I mean, as. It's hard to explain in any terms what beauty and elegance are in paintings or in music and so on.

356
00:40:59,780 --> 00:41:01,700
But I think perhaps it's.

357
00:41:03,210 --> 00:41:13,020
Easiest way to talk in terms of if you go to a capability brown landscape garden you walk through on the path and the way he designs it is

358
00:41:13,020 --> 00:41:21,900
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.

359
00:41:22,440 --> 00:41:31,739
And it's this surprise element of suddenly seeing everything clarified and beautiful that that we feel that's mathematicians.

360
00:41:31,740 --> 00:41:38,910
So I think there's an element of it's beautiful the first time and still beautiful again,

361
00:41:38,910 --> 00:41:43,170
but you shouldn't stare at it non-stop because like with paintings or music,

362
00:41:43,290 --> 00:41:49,199
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 garden.

363
00:41:49,200 --> 00:41:56,520
Yes. So why do people get put off by, if that's the experience that the professional mathematicians have when,

364
00:41:56,790 --> 00:41:59,970
you know, doing their subject, why do people get pissed off?

365
00:42:00,940 --> 00:42:04,690
Well, I think the biggest handicap for mathematics.

366
00:42:05,290 --> 00:42:12,470
My impression is. That's. And certainly it was my experience in the U.S. that.

367
00:42:13,790 --> 00:42:22,670
It's if you're young, you really need someone who cares about mathematics and likes it to teach you and your first steps.

368
00:42:23,960 --> 00:42:35,360
And unfortunately. It's quite rare to at least it was there to have maths teachers before you reached the age

369
00:42:35,360 --> 00:42:40,820
of ten or 11 who actually trained in mathematics and wanted to be teaching mathematics.

370
00:42:41,540 --> 00:42:46,990
I think what happens is that mathematics is a very useful subject.

371
00:42:47,000 --> 00:42:52,550
People go off and do many other things with it and there weren't enough left as teachers.

372
00:42:53,000 --> 00:43:02,149
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.

373
00:43:02,150 --> 00:43:08,320
And that got passed on. So I think that's the stumbling block.

374
00:43:08,340 --> 00:43:15,480
I think most young people, children do have a real appetite for mathematics, and it does appeal to them.

375
00:43:16,770 --> 00:43:21,660
But you really need to learn it from someone who who enjoys it and shows you that enjoyment.

376
00:43:22,350 --> 00:43:25,860
And after the age of ten or 11, it's often too late. Yeah.

377
00:43:26,280 --> 00:43:30,929
Does the public perception of mathematics kind of contribute to that, too?

378
00:43:30,930 --> 00:43:32,650
That the way that people are?

379
00:43:32,660 --> 00:43:41,550
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.

380
00:43:42,120 --> 00:43:46,350
I read, for instance, that you're not particularly big fan of the film Good Will Hunting.

381
00:43:46,500 --> 00:43:56,100
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.

382
00:43:57,180 --> 00:44:01,920
Okay. There are some things you're born with that might make it easier, but it's never easy.

383
00:44:02,490 --> 00:44:08,430
And not for mathematicians. I mean, mathematicians struggle with mathematics even more than the general public does.

384
00:44:08,430 --> 00:44:12,180
That's what they need to understand. We really struggle.

385
00:44:12,180 --> 00:44:19,170
It's hard. We could go to a seminar by someone else in the department and be completely lost and just struggle.

386
00:44:19,470 --> 00:44:22,830
But we're used to it, so we learn how to adapt to that struggle.

387
00:44:23,580 --> 00:44:28,260
But, you know, so I think.

388
00:44:29,960 --> 00:44:32,450
There's no one who it's really easy for.

389
00:44:32,450 --> 00:44:39,410
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.

390
00:44:39,680 --> 00:44:43,850
So do you think you need a natural aptitude to become a mathematician then?

391
00:44:44,880 --> 00:44:53,260
Well. Okay. So that's being a professional mathematician where you do research or just being competent and say being competent.

392
00:44:53,380 --> 00:44:55,690
Okay. So being competent, i. I.

393
00:44:56,830 --> 00:45:03,250
I think you have to be born with some intelligence that's obviously variations, but I don't think it's that exceptional.

394
00:45:04,260 --> 00:45:10,440
But I think the qualities that make a good research mathematician are not so much technical ones, but ones of character.

395
00:45:10,830 --> 00:45:17,430
You need a particular kind of personality that will struggle with things, will focus, won't give up, and so on.

396
00:45:18,000 --> 00:45:24,299
So when you go to these seminars of colleagues and I mean, I know when I go to seminars of colleagues,

397
00:45:24,300 --> 00:45:28,260
I often come away having not understood about 90% of what's going on.

398
00:45:28,560 --> 00:45:35,430
What's what's your what's your process? How do you how do you sort of approach a new bit of mathematics or a new challenge?

399
00:45:37,220 --> 00:45:40,190
Okay. So new approaching new mathematics.

400
00:45:41,450 --> 00:45:51,350
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.

401
00:45:51,800 --> 00:45:55,340
So as I said before about having a good teacher,

402
00:45:55,340 --> 00:46:02,810
it's the same for us that if you have someone who can teach it well, who knows the subject, it's much better.

403
00:46:03,170 --> 00:46:11,180
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.

404
00:46:12,830 --> 00:46:20,360
But with the file, he sort of mentioned there that, you know, mathematicians are kind of used to used to struggling.

405
00:46:20,990 --> 00:46:23,990
And that's quite an, I guess, an easy statement to make.

406
00:46:24,020 --> 00:46:30,860
But how do you shield yourself against, you know, being discouraged when you're finding it very difficult?

407
00:46:32,420 --> 00:46:37,520
Well, I think it's the same as in other parts of life.

408
00:46:37,520 --> 00:46:40,960
I think you will get discouraged.

409
00:46:40,970 --> 00:46:47,760
You learn from experience that you make it through and you'll get disappointments.

410
00:46:47,780 --> 00:46:52,990
But, you know, the time heals these things and sleep heals these things.

411
00:46:53,000 --> 00:47:01,760
And you replacement therapy, you know, you do something else that takes away the pain of the day.

412
00:47:01,770 --> 00:47:03,540
Yeah, of course. But.

413
00:47:04,610 --> 00:47:13,819
Well, is that are your views on sort of discouragement and feeling discouraged at college by the fact that you eventually were successful,

414
00:47:13,820 --> 00:47:18,200
of course, with them therapy. I mean, would you would you feel the same way?

415
00:47:18,980 --> 00:47:23,270
I mean, because there are other mathematicians who have worked for years on problems unsuccessfully.

416
00:47:24,170 --> 00:47:29,870
Do you think your view is sort of coloured by your experience? I don't know, but I'm sure it must be.

417
00:47:31,670 --> 00:47:37,250
Obviously, if you know that it worked out in the end, then you accept it.

418
00:47:37,790 --> 00:47:42,940
You accept the. The setbacks. Yes.

419
00:47:42,950 --> 00:47:51,140
Obviously, I've had a positive resolution. So I'll be more more benign about these periods of of being stuck.

420
00:47:51,560 --> 00:47:55,209
But. Even in smaller problems where I haven't solved them.

421
00:47:55,210 --> 00:47:59,080
I just accept it's part of being a mathematician as being stuck in weather.

422
00:48:00,000 --> 00:48:06,660
I know from being in high school and being stuck on a homework problem to being stuck on an exam question, it's the same thing.

423
00:48:06,670 --> 00:48:14,590
It's just a question of scale. I mean, if you're stuck for five years on a problem, some people can't make the transition from being stuck for,

424
00:48:15,490 --> 00:48:22,600
you know, for a few hours to being stuck for a few years. And that's even true with people who have become Ph.D. students.

425
00:48:22,930 --> 00:48:27,280
They've been very, very good top of the world, literally.

426
00:48:28,090 --> 00:48:35,860
And in mathematics, competitions are an education and the homework is whatever.

427
00:48:36,910 --> 00:48:39,910
And they make try and make the transition to being a research mathematician.

428
00:48:39,940 --> 00:48:42,960
They can't cope with being stuck for more than 24 hours.

429
00:48:42,970 --> 00:48:47,500
I mean, there are some great stories about people, you know, given a thesis problem.

430
00:48:47,500 --> 00:48:50,590
And 24 hours later, they come back and say, no, I can't do it.

431
00:48:50,590 --> 00:49:00,249
I want another one. And there's one famous story about someone and apparently went off and became quite famous as writing textbooks.

432
00:49:00,250 --> 00:49:06,100
But this really happens. We're so frustrated at not being able to get the answers straight away.

433
00:49:06,190 --> 00:49:09,639
Right. I mean, it particularly happens to very gifted people.

434
00:49:09,640 --> 00:49:14,770
I mean, you mustn't be too good. If you're too good, you get used to solving everything very quickly.

435
00:49:14,770 --> 00:49:19,840
And, you know, if you're not quite that good, you little more used to this being stuck.

436
00:49:20,980 --> 00:49:25,330
But I guess when it comes to Fermat's Last Theorem particular, I mean, you were stuck for a very long time.

437
00:49:26,140 --> 00:49:31,660
Well, perhaps that's slightly unfair. It took a long time to get towards that to the end of the proof,

438
00:49:31,960 --> 00:49:40,900
with ever many points during that period where it sort of felt like a chore, where where being stuck, you know, was discouraging.

439
00:49:42,030 --> 00:49:48,470
I'm. No, I think I think I always felt I'd gained insights into the problem.

440
00:49:48,520 --> 00:49:51,639
But, yes, I was stuck for very long periods.

441
00:49:51,640 --> 00:50:01,620
But. It wasn't like I was back to four zero every time I felt I was making progress and maybe I wouldn't get.

442
00:50:02,780 --> 00:50:05,940
The whole thing. But I was getting somewhere.

443
00:50:06,540 --> 00:50:14,540
And, no, I didn't. I didn't feel like giving up.

444
00:50:14,540 --> 00:50:20,970
I didn't feel I was stuck forever. And. I think my anxiety.

445
00:50:22,270 --> 00:50:28,959
Wasn't actually so much in wasting my time or being stuck in a loop or something like that.

446
00:50:28,960 --> 00:50:33,250
It was, first of all, I didn't literally have enough time.

447
00:50:34,600 --> 00:50:40,870
You know, our lifetimes, though, are limited and.

448
00:50:43,110 --> 00:50:46,410
And also, though not until late on I was,

449
00:50:46,950 --> 00:50:54,900
I would be worried that I was in general the mathematics I put in all this effort and actually someone else does it a better way quicker.

450
00:50:56,760 --> 00:51:04,320
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.

451
00:51:06,120 --> 00:51:14,690
You know, someone comes up with a quick solution. Did you always believe that it was possible, though, that proving it was possible?

452
00:51:14,770 --> 00:51:19,870
Yes. So I tried it as a teenager and young person.

453
00:51:19,870 --> 00:51:25,179
But then I realised when I became a professional mathematician that actually those methods,

454
00:51:25,180 --> 00:51:31,270
which were basically 19th century methods, had been exhausted and everyone that had tried everything basically.

455
00:51:31,570 --> 00:51:37,950
So I did put it away for a while. But what happened was in 86, get out free.

456
00:51:37,950 --> 00:51:41,979
I made a connection with these actually with these elliptic curves.

457
00:51:41,980 --> 00:51:53,170
With this result I, I mentioned about this L series Ali of asks making sense for all complex numbers made that connection.

458
00:51:53,710 --> 00:51:59,340
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.

459
00:51:59,340 --> 00:52:03,310
It was had to be solved. We couldn't go round it, couldn't leave it behind.

460
00:52:03,940 --> 00:52:09,220
But there was some of your peers at the time who didn't think it was it was possible even after Jihad Frye's work came out.

461
00:52:09,850 --> 00:52:20,850
That's right. I think many of them, yes. I've always actually I'm always quite encouraged when people say something like, you can't do it that way.

462
00:52:21,400 --> 00:52:31,500
I always feel that's a real hint of the right way to do it and how I mean, because you were at Princeton at the time where you were working on it.

463
00:52:31,770 --> 00:52:35,610
How did you manage to keep your work secret from?

464
00:52:35,940 --> 00:52:41,640
Well, the university I mean, your bosses presumably wanted to make sure that you were actually doing some work.

465
00:52:42,210 --> 00:52:46,040
Yes. Well, we have this great system called tenure, which protects you a little bit.

466
00:52:46,090 --> 00:52:49,680
But what I had was I actually had some.

467
00:52:50,720 --> 00:52:55,280
Some results I'd been working on, and I was just a little slower in publishing them,

468
00:52:55,280 --> 00:53:01,940
but I strong it out over a few years and people said, Ah, he's gone off the boil and things like that.

469
00:53:01,950 --> 00:53:10,190
And I knew I. I knew it would be an issue eventually, but I managed for that length of time.

470
00:53:10,820 --> 00:53:20,150
Did you sort of, you know, in the evenings or whatever imagined that moment of solving it and sort of the looks on your colleagues faces, if you like?

471
00:53:20,390 --> 00:53:21,770
No, I think it's worse than that.

472
00:53:21,770 --> 00:53:31,190
I think there are times when you think you have solved it, but you know, you realise on the moment there's a problem, there's a gap that goes on.

473
00:53:31,760 --> 00:53:35,959
Do you ever look back at the proof now? Not really, no.

474
00:53:35,960 --> 00:53:41,110
I mean, I have the outline of proof in my mind. I could reconstruct it, but the details?

475
00:53:41,120 --> 00:53:52,400
No, not really. Now, as I say it, I mean, it is very nice going back out for your early work because.

476
00:53:53,520 --> 00:53:58,820
Sometimes you forget what you've done and you're really impressed by this smart young man who runs it.

477
00:53:59,460 --> 00:54:02,470
You did it, but you wonder how you did.

478
00:54:02,490 --> 00:54:05,910
But I don't spend any time really doing that now.

479
00:54:06,720 --> 00:54:12,810
I mean, this was solving Fermat's Last Theorem happened quite early on, really, in your in your career as a professional mathematician.

480
00:54:13,050 --> 00:54:17,370
Were things that you worked on after that ever quite as rewarding?

481
00:54:17,400 --> 00:54:21,090
No. Quite simply, no.

482
00:54:21,990 --> 00:54:29,060
I don't think they could have been. I mean. Okay, if I. I could solve the Riemann hypothesis, but you know, that would be nice.

483
00:54:29,180 --> 00:54:33,650
That would be nice. Yeah, but it's. It's not quite my speciality anyway, so.

484
00:54:34,430 --> 00:54:34,830
Okay.

485
00:54:34,880 --> 00:54:44,090
I'm going to get through 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.

486
00:54:44,330 --> 00:54:48,860
For students who are coming through who are perhaps doing their A-levels now,

487
00:54:49,250 --> 00:54:54,350
what what would you say to your 17 year old self if you if you had a chance to.

488
00:54:56,600 --> 00:55:00,130
Well, I think. I probably did it the right way.

489
00:55:00,140 --> 00:55:06,890
I think what I would say is, yes, try these impossible problems while you're on high school, while you're an undergraduate.

490
00:55:07,940 --> 00:55:17,419
Actually, the time to stop doing them is when you're starting your career as a graduate student, as a, you know, junior faculty and so on.

491
00:55:17,420 --> 00:55:22,100
Then I think you have to be have to be responsible career wise.

492
00:55:22,490 --> 00:55:28,940
Otherwise you could just spend those ten years trying an impossible problem and you have nothing to show for it.

493
00:55:29,480 --> 00:55:32,960
And that would be a professional mistake.

494
00:55:33,800 --> 00:55:38,090
But once you once you settled and you've got your job and everything gets thrown into it.

495
00:55:39,620 --> 00:55:42,919
Well, that's that's the point of tenure so that you can try these things.

496
00:55:42,920 --> 00:55:51,350
Otherwise, of course, everyone's just going to chase the the easy thing that you do, you know, looks good and keep producing things.

497
00:55:51,890 --> 00:55:55,390
But it's good to play with mathematical ideas when you're younger. Absolutely.

498
00:55:55,400 --> 00:56:01,870
I've wasted a lot of time when I was a child trying to solve these, uh, these impossible problems.

499
00:56:01,880 --> 00:56:05,860
And I don't think it was a in the long run, it wasn't a waste of time at all.

500
00:56:05,870 --> 00:56:11,210
It gave me the idea of what research is, and it gave me the.

501
00:56:12,730 --> 00:56:15,670
The taste for it. I think it was very beneficial.

502
00:56:16,780 --> 00:56:22,600
And my last question, I think in terms of I mean, I do a lot of work to engage more people in mathematics.

503
00:56:22,900 --> 00:56:27,550
How do you think mathematics should be viewed? How do you want the world to view a subject?

504
00:56:29,480 --> 00:56:34,040
Well, I think mathematics. There's two roles.

505
00:56:34,040 --> 00:56:37,650
Really? So. To me.

506
00:56:38,220 --> 00:56:41,910
I'm solving these these equations. I've talked about it.

507
00:56:42,060 --> 00:56:45,750
To me, it's very beautiful. I'm passionate about solving them and always have been.

508
00:56:45,750 --> 00:56:49,470
And I'd like for people to be able to.

509
00:56:51,160 --> 00:56:58,240
To see it. I mean, to experience it if they can, but just to see what it is we find so appealing about it.

510
00:56:58,660 --> 00:57:04,270
I mean, in some way, it's the third. It's it's something that's immutable.

511
00:57:04,270 --> 00:57:07,120
I mean, people talk about other universes and everything.

512
00:57:07,480 --> 00:57:14,950
I just can't imagine any other kind of mathematics that somehow it's the most permanent thing there is.

513
00:57:18,590 --> 00:57:26,150
On the other hand, it's the language of science and it's incredibly useful, you know, and more and more as the world goes on, I mean, it's.

514
00:57:27,300 --> 00:57:29,520
It's you're so employable if you do it.

515
00:57:29,520 --> 00:57:38,840
And it's wonderful that it can be applied to medicine to sort of reducing queuing times for cars or whatever it is.

516
00:57:38,850 --> 00:57:47,819
I mean, you know, for securing your your Internet communications, your credit card, whatever it is that underlies everything in the world.

517
00:57:47,820 --> 00:57:58,290
So it's tremendously useful. But at the same time, what I care about even more is just seeing this this beautiful edifice that somehow,

518
00:57:58,290 --> 00:58:02,549
as I say, I think the most permanent thing there is. Wonderful.

519
00:58:02,550 --> 00:58:07,470
Great point. I think. Okay, so there are, I think some people wandering around with microphones.

520
00:58:07,710 --> 00:58:11,370
If you put your hands up, you want a question? Okay, we'll go. Let's get someone on the edge first.

521
00:58:11,370 --> 00:58:18,560
So if we go to you first, you one, and then we'll get another mike up to you in the middle there have two and then three have it.

522
00:58:20,120 --> 00:58:25,550
So I first wanted to ask you whether you have read Coulson's first number theory book,

523
00:58:25,940 --> 00:58:28,940
but then you said the books are the worst thing to learn mathematics from.

524
00:58:28,970 --> 00:58:34,160
So I wonder whether you have to try things first and then read or read first and try then.

525
00:58:34,430 --> 00:58:40,460
I mean, but maybe also say whether you have read this because it's known as arithmetic in translation, I presume.

526
00:58:41,330 --> 00:58:44,900
So I haven't to once say the question again.

527
00:58:45,560 --> 00:58:52,580
So the question is whether I've read Gauss's famous book on number theory called Disquisitions mathematically.

528
00:58:52,940 --> 00:58:57,380
And the answer is, I've looked at sections of it, but I.

529
00:58:58,600 --> 00:59:01,850
I would look at it. I certainly didn't read it. Cover to cover.

530
00:59:01,870 --> 00:59:07,360
No, I'm not someone who goes back and and studies books in great detail.

531
00:59:07,740 --> 00:59:15,070
I find it very difficult and a little bit off putting. Um, second questioner.

532
00:59:15,100 --> 00:59:24,790
Oh, you've got one. Very fantastic. And you said that you thought that the 19th century methods had been, uh, used, looked at.

533
00:59:24,790 --> 00:59:30,250
So there was. There was nothing left that. Does that mean that you think that Fermat didn't actually have a proof?

534
00:59:31,960 --> 00:59:38,820
Well, the question is whether I think Fama actually had a proof, so he certainly wouldn't have had the 19th century methods.

535
00:59:38,830 --> 00:59:42,580
I mean, he was 17th century, and it had to be more simple than that.

536
00:59:43,060 --> 00:59:51,190
So when I was young, I tried to do it. Based on the kinds of things he studied using quadratic forms and so on.

537
00:59:53,580 --> 00:59:58,200
I think the probability is almost zero. But it is conceivable.

538
00:59:58,290 --> 01:00:06,060
It's just conceivable. But I can't see how it how it could work.

539
01:00:08,610 --> 01:00:15,720
Third question. There we go. I had exactly the same question and it seems unnecessary to ask it a second time.

540
01:00:16,200 --> 01:00:21,770
So think of a different question. Okay. Off the top of my head, your progress on Fermat's Last Theorem.

541
01:00:21,780 --> 01:00:27,030
There were ups and there were downs. If you were to graph against time, what function would best?

542
01:00:27,030 --> 01:00:31,470
Some of that graph. You wanted the question after cosine.

543
01:00:31,500 --> 01:00:35,340
I'm sorry. Right. I like that. Yeah. Okay.

544
01:00:35,340 --> 01:00:45,570
So I think there are perhaps if you start put a flagpole in the beginning, a flagpole at the end,

545
01:00:45,960 --> 01:00:49,740
there'll be three flagpoles in between and each one's higher than the one before.

546
01:00:51,900 --> 01:01:01,680
But the rope that joins them is sort of think there's a particularly big snag on the last one, but that will be roughly the function.

547
01:01:02,640 --> 01:01:07,270
But you went back over the same ground, tried the same tricks several times, right.

548
01:01:07,320 --> 01:01:10,860
While you were working? Yes. So I find.

549
01:01:11,840 --> 01:01:20,720
The mathematically, you have to go back and try the same things again because often it's it's just a minor variation,

550
01:01:21,290 --> 01:01:27,860
minor variant on what you've tried before that actually works and you're just missing one little piece of information or one.

551
01:01:31,060 --> 01:01:34,750
One extra idea. It's a little like evolution.

552
01:01:35,320 --> 01:01:41,650
Now, evolution works by making mistakes, and in some sense, mathematics, I feel, is bit the same.

553
01:01:41,660 --> 01:01:43,510
You have to make these mistakes too,

554
01:01:45,400 --> 01:01:52,510
because you have to try all these different things and lots of them are going to be wrong, but eventually one is right.

555
01:01:53,770 --> 01:01:59,130
Other questions? Okay, let's try this side. Okay. So we've got one there. Perfect. And then we'll go to just long and then three out on that.

556
01:02:00,450 --> 01:02:05,160
I'm Andrea. Thank you so much. I just wanted to ask if you think maths is addictive.

557
01:02:06,600 --> 01:02:11,130
Well, certainly number theory, I think the answer is obviously completely yes.

558
01:02:13,590 --> 01:02:22,680
I think number theory in particular that even among professional mathematicians, there are quite a few who started in other branches of mathematics.

559
01:02:23,010 --> 01:02:26,610
It's very rare for them to move out of number theory once they've gone there.

560
01:02:27,960 --> 01:02:32,550
They they often migrate to number theory, but they very rarely migrate away from it.

561
01:02:34,620 --> 01:02:40,169
Christine. Yes, perfect. Thank you. A stranger I've got I'm a secondary school maths teacher.

562
01:02:40,170 --> 01:02:44,010
So my my question might be slightly banal for this for this audience,

563
01:02:44,010 --> 01:02:52,800
but I promised my colleagues I'd ask you to solve a dispute we have in the staff from about the way we teach square roots,

564
01:02:53,490 --> 01:03:02,600
that whether or not the the square to the positive number is always a positive number and only a positive and a negative one,

565
01:03:02,610 --> 01:03:08,460
it is the root of an equation. Or is the square root of that number always the positive square, a negative root?

566
01:03:09,580 --> 01:03:13,290
Well, I think you you have to decide.

567
01:03:13,300 --> 01:03:16,370
I mean, you can choose. You can choose.

568
01:03:17,620 --> 01:03:23,560
It's a matter of terminology, really. So it's up to you to decide which is the standard terminology.

569
01:03:24,040 --> 01:03:29,920
It's not going to solve your dispute today. I'm not going to solve this dispute now and not affect.

570
01:03:31,460 --> 01:03:46,990
Oh, sorry. I'm curious to know whether the resolution of Fermat's last conjecture has many practical applications in the practical world.

571
01:03:48,010 --> 01:03:53,050
So the actual equation itself doesn't seem to be that useful.

572
01:03:53,710 --> 01:04:02,010
But the techniques used to solve it. Are useful and we always expect will become more useful.

573
01:04:02,700 --> 01:04:07,950
There's often a lag time between the mathematics that's done and the utility of it,

574
01:04:08,310 --> 01:04:12,090
and the lag time is probably higher in number theory than in most things.

575
01:04:12,540 --> 01:04:20,940
Though in the last 40 years, it's it's gone down to a much smaller length of time now because it's used in.

576
01:04:21,060 --> 01:04:24,600
And security and cryptography and so on. A great deal.

577
01:04:25,470 --> 01:04:29,020
So. The utility in those fields.

578
01:04:29,650 --> 01:04:33,100
What I think is already it's already there.

579
01:04:33,100 --> 01:04:36,850
Really. Think we've got time for one more round.

580
01:04:36,850 --> 01:04:43,810
Okay, so if we go one, you there two and then anyone else three does the same.

581
01:04:44,020 --> 01:04:47,380
Whereas one can't remember that we get just done here.

582
01:04:48,860 --> 01:04:53,130
Oh. Okay. We'll see each month. You go first. Yeah. Hi.

583
01:04:53,430 --> 01:04:59,950
I'm also a secondary school maths teacher and I was interested what you're saying about, um,

584
01:05:00,240 --> 01:05:05,940
sort of spending a long time on a problem, not being successful a problem and how that helped you.

585
01:05:07,410 --> 01:05:11,370
And I wonder, I kind of think maybe exam culture isn't great for that.

586
01:05:11,790 --> 01:05:16,530
Do you do you think it would be a positive thing if we were to do away with school exams?

587
01:05:18,800 --> 01:05:25,470
Oh, I don't know if it's practical to do with school exams, but there is this question of whether.

588
01:05:26,420 --> 01:05:30,230
You should encourage people to do mathematics via competitions.

589
01:05:30,770 --> 01:05:35,390
And clearly some people really like that, but some are completely put off by it.

590
01:05:35,870 --> 01:05:37,960
I'm on the side of being put off by it.

591
01:05:37,970 --> 01:05:48,320
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.

592
01:05:51,000 --> 01:05:55,590
Make no mistake. So this question is also on the teaching theme.

593
01:05:56,220 --> 01:05:59,880
You mentioned the importance of having a great maths teacher when you're younger.

594
01:06:00,210 --> 01:06:04,470
Do you have any thoughts on how we can encourage more mathematicians to go into teaching?

595
01:06:05,630 --> 01:06:10,190
Yes. Pay them more. Yeah.

596
01:06:12,450 --> 01:06:18,150
Who is your math teacher when when you were at school? Was there one that sort of stood out that inspired you?

597
01:06:18,360 --> 01:06:25,050
Well, there were two, so there was one in junior school, Mrs. Briggs, who was a wonderful teacher.

598
01:06:25,890 --> 01:06:32,310
And there was one in my high school who had actually done a Ph.D. in number theory and gave me some books and so on.

599
01:06:32,370 --> 01:06:37,590
He was a great inspiration. Did you speak to them after you sold Fermat's Last Theorem?

600
01:06:38,710 --> 01:06:42,610
I did meet my high school teacher. I think I met both.

601
01:06:42,610 --> 01:06:46,090
Yes. Yes, afterwards. That must be a nice moment for them.

602
01:06:46,480 --> 01:06:50,620
Okay. The alarms going off saying we have to finish. But I did promise one more question.

603
01:06:50,620 --> 01:06:53,620
Which was up there. Yes. If we get microphone to this gentleman.

604
01:06:54,610 --> 01:06:59,370
Oh, okay. I'm sorry. Okay.

605
01:06:59,370 --> 01:07:05,170
Look, I'm just fascinated by that. The idea of being stuck on a problem for a long,

606
01:07:05,440 --> 01:07:10,450
long period of time and then having an insight in terms of bringing the freshness

607
01:07:11,380 --> 01:07:15,220
in terms of things that help you gain that fresh approach to the same problem.

608
01:07:15,790 --> 01:07:23,650
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?

609
01:07:25,790 --> 01:07:33,499
So. I think what the way it seems to work is you work very, very intensively on on these math problems so that,

610
01:07:33,500 --> 01:07:40,370
you know, everything that's been done, you you try out everything and your conscious mind kind of runs out of ideas.

611
01:07:41,090 --> 01:07:47,060
And then it's often said in mathematics, it's the three B's that help bus, bath and pad,

612
01:07:47,570 --> 01:07:54,020
that when your mind is relaxed, somehow your subconscious takes over and pieces together.

613
01:07:54,080 --> 01:08:06,050
So I can't say anything useful, except at some point you let your mind relax and something gets put together in your mind.

614
01:08:07,970 --> 01:08:12,580
For no prescription. Sorry. Now, technically, we should get off.

615
01:08:12,590 --> 01:08:17,870
But the thing is, I'm in charge, so I think I don't want to rob you of your chance to ask Andrew's questions.

616
01:08:17,910 --> 01:08:21,000
If we can get a microphone over here. It's good.

617
01:08:21,680 --> 01:08:25,819
Okay. Oh, was it okay? Great. It's all solved and no one need.

618
01:08:25,820 --> 01:08:32,630
Tell me off then. Okay. In which case, then all that remains is to say an enormous thank you to Andrew once.

619
01:08:33,350 --> 01:08:35,210
And thank you all for coming very much. Thank you.