Good evening, ladies and gentlemen. 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 Mathematical Institute,
which has kindly chosen the museum as the first known 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 Sir 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, and which had been formulated some 350 years earlier.
After his lecture, Sir Andrew will be joined by mathematician and broadcaster Dr. Hannah Fry of University College London,
who's a good friend of the Science Museum Group.
She presented Britain's greatest invention from our national collection centre near Swindon earlier 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 a 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 too 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 Marcus de Soto, also of the Mathematical Institute,
who's one of the Science Museum's advisors Illuminating India exhibition,
has as its centrepiece a leaf from the back Charlie manuscript kind lent to us by the Bodleian Library,
which shows a very early use of the symbol of a circle four 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 take us into the rest of the evening. It's now my pleasure to introduce Professor Sir Martin Bridson, who's Whitehead,
professor of pure mathematics and head of Oxford's Mathematical Institute.
Thank you. Thank you, Mary, for that very nice introduction.
Well, generally, I would 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 about six lectures spread throughout the year and it's proved to be fantastically successful.
It was originally the vision that grew out of conversations but was driven by Professor Alan Gabrielli,
who's here somewhere this evening and made a reality by Darren Lombard, who's made he and his team made this evening reality.
And it's been quite a revelation to us. And as Mary alluded 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.
Public lectures sell out incredibly quickly, and people have a real appetite, not just for some showmanship with with mathematical elements,
but really for delving into what mathematics is and what it's doing for society today.
And I'd also like to extend a particular welcome to our alumni and friends who are here this evening.
And 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.
And that doesn't just mean a flow of information or 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 Mathematics in the first place. So a particular welcome to all alumni and friends.
So 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 shan't delay long from that.
And then after Andrew has given his lecture, he will do a question and answer session for about 30 minutes with Hannah Fry,
and then you'll have a chance to ask some questions at the end of that.
I'm Hannah.
Many of you probably know is a distinguished author and broadcaster populariser of mathematics, but she's also an active mathematician herself.
She's the lecturer in the Mathematics of Cities.
Sounds fascinating. At the Centre for Spatial Analysis at UCL.
And 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.
So mathematics is everywhere. So the mathematics of love.
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,
unexpected about 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 of the one we're having this evening where people are genuinely curious to 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 the sort of relentless and often frustrating struggle with the great problems of mathematics looking for truth and beauty,
trying to serve mankind? What's that really like?
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.
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.
And I could start listing Andrew's achievements and honours at this point, but that would take up far too much of the evening.
So I won't. Let me just say, I. I got to know Andrew in Princeton in the early nineties when we were both,
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 Andrew struggling with this huge problem and 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.
And he is, quite rightly by far the most celebrated mathematician of modern times.
We're 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 as you observed, I am not going to talk about Fermat's Last Theorem.
That, of course, is the equation that Verma 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 solve equations in two variables.
So equations in two variables. So here I've given a typical example.
The equation y squared is x, q plus x 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 x plus b, they can solve equations like that.
But I'm afraid it's very embarrassing to say this.
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.
There is no solution for any general class of equations of degree higher than than two.
So the Greeks dealt with the case where the exponent was two.
But in the last 2000 years we've, we've made some progress, but we haven't got there.
And the second half of the 20th 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 a farmer and what he did, 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 and his
own work and his own books were the following equations which I've put here.
So y squared is x cubed minus two, and y squared is x cubed minus four.
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.
Y is plus or minus five. X is three 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 prior level.
Very, very simple. Pretty GCSE and yet it's incredibly ingenious.
Well, this equation, solving these kinds of equations and 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 family showed, 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 bequeathed to 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 that solutions in real numbers, that's the kind of graph you would normally plot if you're doing high school mathematics.
What Fermi observed was that if you take any two solutions to this, so you take the solution t is for nine.
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.
And similarly, Q is a solution with X is to and Y is five.
You draw the straight line through those two points and it will hit the curve, which is y squared is x, q plus 17.
This is this curved one one more time. At a point are.
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 two, minus three are both rational numbers, that is not chance.
That always happens. That was the observation of FMR.
And interestingly. We know it was an observation of failure because Newton thanked him for this observation much later.
So this process we call taking a chord.
So you take a chord and it beats the curve again.
One of the tricks of mathematicians is they say, well, what happens if you actually make P and Q the same?
You just take a tangent here, then it counts as if it.
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.
So Pharma 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 cord 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, Pete.
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.
But sometimes you don't and you get infinitely many. Okay.
Why do you get rational solutions for the third point?
While this is it's very elementary. Now I'll just explain why.
So if you take y squared is x Q plus x plus P, so you think of A and B as fixed rational numbers.
And then when we do algebra, we, we write a line, a straight line as is X plus C.
So again, aman 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 X plus C squared is x Q plus x plus B.
And if you know that two of the solutions of that cubic equation of rational, then it's very easy to see.
The third is rational because the sum 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 author infinitely many.
And if there are, how do we find them? Well.
This question sort of laid low for a couple of hundred years.
And then in 1901, quackery, great apologist, 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 model 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 Fahmi 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 it 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 15th century.
It is by now. But for this one, you do something more peculiar.
And that is you look first at modern arithmetic.
So let me remind you what my prismatic is. So you take a prime.
So seven as a prime number, then mod seven arithmetic, you add 2 to 6.
That makes eight, but you always subtract off a multiple of seven.
So you get back into the range 0 to 6. So two plus six is the same as one mod seven.
The difference is divisible by seven. Two times six, which should be 12.
I don't know how it comes to say 117. I'm sorry.
Two times six is 12, which should be five, not seven. Oh, it does.
Okay. Sorry. Yes, it doesn't say the same on my notes.
Okay. Two times six. This five months have a very good hour in the equations.
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.
And four times 416 take off two multiples of seven, you get two months seven.
So you can talk about equations, mod seven and the first step in trying to understand solving in rationals and integers.
Strangely enough. Is to think about the equations in modern arithmetic.
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.
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 axes, all different wise, and you count up how many of them give you inequality in modern arithmetic?
But then you do something which is much more distressing even to a mathematician.
You take three over and three that's fine. Multiply it by five over n five.
That's the number of solutions.
Modify arithmetic, then type seven over n seven and you multiply those together over all primes, another infinitely many primes.
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 zero.
If the equation y squared x x Q plus XP plus p has infinitely many solutions.
And should go to something nonzero if it has only finite remaining 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. 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 could we multiply together infinitely many things. This is just hopeless.
So people tested boats and spread it.
And I tested this on a computer in the early days of the use of computers in the late fifties.
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 apologise, but I hope I can explain some of it.
So you make a function of S where you take this product over primes P and a product just means you multiply them together.
So it's just like we did before of some strange creature here, one minus, and that's a number.
P is the prime P to the minus. S plus pizza. One minus two S to the minus one.
AP is the number one plus P minus the SNP that was counting solutions mod p, which we did before.
Okay. Now if you compute this function at the value one just formally so this doesn't actually mean anything
except I'm just putting one in for s here you got one minus ap inverse plus three to the minus one.
All the minus ones bit complicated. You simplify it.
You get back to this product of 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 and 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 have used the mod prismatic. We're now going to use complex numbers as well use s as complex numbers.
And. This function alias actually makes sense if we use complex numbers.
Okay. This may seem like a strange. Result.
But it's actually. Absolutely fundamental to the subject.
It actually was the key result that I had to prove to solve Fermat's Last Theorem.
This is what lay behind it was showing this.
This is the. The part of the tiny Amazon or a conjecture that proves Fermat's Last Theorem.
Okay. Now, now that we know.
That this alias makes sense. We can now make of actual conjecture,
which has actually defined that it has infinitely many rational solutions if and only if this function takes the value zero at one.
So this was, um.
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.
So you get $1,000,000 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 in my thesis and the 1990s when it was finally proved.
This theorem says, While One Direction is known that if this function is non zero at one, then it the elliptic curve.
The equation only has financially many rational solutions. We don't know how to go in the other direction.
Okay. 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 Fermat's Last Theorem.
It's got its roots in famous work, and it's one of the millennium problems.
This can be interpreted in a special case.
In a way that describes actually an even older problem.
This problem is a thousand years old. And the problem is.
If you take a right angle triangle with rational length sides doesn't exist.
Such a triangle whose area is an.
For an some given number, like 1 to 5 and so on.
Can you solve that while in fact you can solve some of them and you can't.
Others, for example, there is no such triangle with Area one.
You cannot find a right angle triangle whose sides have rational numbers and whose area
is one that's actually equivalent to the case and equals three of Fermat's Last Theorem.
So it was solved by Femi himself. 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.
E the equation y squared is x cubed minus n squared.
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.
And One Direction. I've just given you the proof. You can just write it down.
So you take a is X squared, minus N squared over Y and so on.
And you find that actually this right angled is all the lengths of a right angle
triangle and the area is n 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 -157 squared x.
You take that elliptic curve and you. You can't do it 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 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 boats from the dock injection.
This problem of solving these equations, as I said, is part of a greater a greater problem that number theorists,
mathematicians of worried about solving equations and 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 was 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 Gordon Townsend process?
To give you all the point.
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.
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.
Two Elliptic Curve. So when I was a student.
Everybody thought. That you could have arbitrarily high rank that is you could get.
Write 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 was 21 and so on.
So everyone thought, okay, 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 elliptic curves.
And then someone, some group of people decided to start applying probability 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 a small, finite number of things you can test the probability of.
But what do you do when you're testing the probability? Whether infinitely many objects.
It's not so clear. And the kind of property 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 cemeteries they have. So you don't consider all the ways A dies 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 Bachelor and I 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 that's only finite. Many with rank more than 22.
That is where you need more than 22 generators. I have my mind is open on this.
And I don't know. But I want to tell you something that is just from last year that's got people excited.
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.
Thank you very much. Thank you very much.
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.
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.
I. For people who don't know about them, could you tell us a little bit more?
Because, of course, as I said, I am 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, uh, 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, in 2003.
But there are six left. $6 million on a $6 million problem?
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 it's about a function that was introduced by Raymond.
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 all that.
I mean, these these problems span sort of the breadth of of mathematical research.
Are there other 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 addicted to number theory from the time I was ten years old.
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.
Then there were some areas of maths that you felt slightly weaker in than 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 them apart from I have this really irritating habit of writing in Latin.
And so it was required to learn Latin to get into Oxford.
It wasn't at the standard that I could read Fair Master, not effective.
How is your Latin now? Minimal. Yes, indeed.
In terms of the way that that mathematicians describe their subject,
it always strikes me that if you have only ever been acquainted with maths at school hearing,
mathematicians 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 subject in.
Well, I think there's two things.
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.
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 this and it was wrong and so on.
So that had a particular romantic story and a very personal one for me.
But why is why do we talk about beauty and mathematics and elegance?
I mean, as. It's hard to explain in any terms what beauty and elegance are in paintings or in music and so on.
But I think perhaps it's.
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
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 beautiful that that we feel that's mathematicians.
So I think 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 garden.
Yes. So why do people get put off by, if that's the experience that the professional mathematicians have when,
you know, doing their subject, why do people get pissed off?
Well, I think the biggest handicap for mathematics.
My impression is. That's. And certainly it was my experience in the U.S. that.
It's if you're young, you really need someone who cares about mathematics and likes it to teach you and your first steps.
And unfortunately. It's quite rare to at least it was there to have maths teachers before you reached the age
of ten 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 there 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 that enjoyment.
And after the age of ten or 11, it's often too late. Yeah.
Does the public perception of mathematics kind of contribute to that, too?
That the way that people are?
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 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.
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 learn how to adapt to that struggle.
But, you know, 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 aptitude to become a mathematician then?
Well. Okay. So that's being a professional mathematician where you do research or just being competent and say being competent.
Okay. So being competent, i. I.
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 a particular kind of personality that will struggle with things, will focus, won't give up, and so on.
So when you go to these seminars of colleagues and I mean, I know when I go to seminars of colleagues,
I often come away having 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 or a 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.
But with the file, he 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, the time heals these things and sleep heals these things.
And you replacement therapy, you know, you do something else that takes away the pain of the day.
Yeah, of course. But.
Well, is that are your views on sort of discouragement and feeling discouraged at college by the fact that you eventually were successful,
of course, with them therapy. 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 sort of coloured by your experience? I don't know, but I'm sure it must be.
Obviously, if you know that it worked out in the end, then you accept it.
You accept the. The setbacks. Yes.
Obviously, I've had a positive resolution. So I'll be more more benign about these periods of of being stuck.
But. Even in smaller problems where I haven't solved them.
I just accept it's part of being a mathematician as being stuck in weather.
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.
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,
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.
They've been very, very good top of the world, literally.
And in mathematics, competitions are an education and the homework is whatever.
And they make 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 quite famous as writing textbooks.
But this really happens. We're so frustrated at not being able to get the answers straight 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 little more used to this being stuck.
But I guess when it comes to Fermat's Last Theorem particular, I mean, you were stuck for a very long time.
Well, perhaps that's slightly unfair. It took a long time to get towards that to the end of the proof,
with ever many points during that period where it sort of felt like a chore, where where being stuck, you know, was discouraging.
I'm. No, I think I think I always felt I'd gained insights into the problem.
But, yes, I was stuck for very long periods.
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.
The whole thing. But I was getting somewhere.
And, no, I didn't. I didn't feel like giving up.
I didn't feel I 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 of all, I didn't literally have enough time.
You know, our lifetimes, though, are limited and.
And also, though not until late on I was,
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.
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, though, that proving it was possible?
Yes. So I tried it as a teenager and young person.
But then I realised when I became a professional mathematician that actually those methods,
which were basically 19th century methods, had been exhausted and everyone that had tried everything basically.
So I did put it away for a while. But what happened was in 86, get out free.
I made a connection with these actually with these elliptic curves.
With this result I, I mentioned about this L series Ali of asks 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 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.
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.
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.
How did you manage to keep your work secret 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,
but I strong it out over a few years and people said, Ah, he's gone off the boil and things like that.
And 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 imagined 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 it, but you know, you realise on the moment there's a problem, there's a gap that goes on.
Do you ever look back at the proof now? Not really, no.
I mean, I have the outline of 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.
Sometimes you forget what you've done and you're really impressed by this smart young man who runs it.
You did it, but you wonder how you did.
But I don't spend any time really doing that now.
I mean, this was solving Fermat's Last Theorem happened quite early on, really, in your 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. I could solve the Riemann hypothesis, but you know, that would be nice.
That would be nice. Yeah, but it's. It's not quite my speciality anyway, so.
Okay.
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.
For 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 on high school, while you're an undergraduate.
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.
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 settled and you've got your job and everything gets thrown into it.
Well, that's that's the point of tenure 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, uh, 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 to engage more people in mathematics.
How do you think mathematics should be viewed? How do you want the world to view a 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.
And I'd like for people to be able to.
To see it. I mean, to experience it if they can, but just to see what it is we find so appealing about it.
I mean, in some way, it's the third. It's it's something that's immutable.
I mean, people talk about other universes and everything.
I just can't imagine any other kind of mathematics that 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, and 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 queuing times for cars or whatever it is.
I mean, you know, for securing your your Internet communications, your credit card, whatever it is that underlies everything in the world.
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,
as I say, I think the most permanent thing there is. Wonderful.
Great point. I think. Okay, so there are, I think some people wandering around with microphones.
If you put your hands up, you want a 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 mike up to you in the middle there have two and then three have it.
So I first wanted to ask you whether you have read Coulson's first number theory book,
but then you said the books are the worst 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 this because it's known as arithmetic in translation, I presume.
So I haven't to once say the question again.
So the question is whether I've read Gauss's famous book on number theory called Disquisitions mathematically.
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.
No, 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. Um, second questioner.
Oh, you've got one. Very fantastic. And you said that you thought that the 19th century methods had been, uh, used, looked at.
So there was. There was nothing left that. Does that mean that you think that Fermat didn't actually have a proof?
Well, the question is whether I think Fama 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 studied 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 and it seems unnecessary to ask it a second time.
So think of a different question. Okay. Off the top of my head, your progress on Fermat's Last Theorem.
There were ups and there were downs. If you were to graph against time, what function would best?
Some of that graph. You wanted the question after cosine.
I'm sorry. Right. I like that. Yeah. Okay.
So I think there are perhaps if you start put a flagpole in the beginning, a flagpole at the end,
there'll be three flagpoles in between and each one's higher than the one before.
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.
But you went back over the same ground, tried the same tricks several times, right.
While you were working? Yes. So I find.
The 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.
Now, evolution works by making mistakes, and in some sense, mathematics, I feel, is bit the same.
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 this side. Okay. So we've got one there. Perfect. And then we'll go to just long and then three out on that.
I'm Andrea. Thank you so much. I just wanted to ask if you think maths is addictive.
Well, certainly number theory, I think the answer is obviously completely yes.
I think number theory in particular that even among professional mathematicians, there are quite a few who 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 often migrate to number theory, but they very rarely migrate away from it.
Christine. Yes, perfect. Thank you. A stranger I've got I'm a secondary school maths teacher.
So my 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 staff from about the way we teach square roots,
that whether or not the the square to the positive number is always a positive number and only a positive and a negative one,
it is the root of an equation. Or is the square root of that number always the positive square, a negative root?
Well, I think you you have to decide.
I mean, you can choose. You can choose.
It's a matter of terminology, really. So it's up to you to decide which is the standard terminology.
It's not going to solve your dispute today. I'm not going to solve this dispute now and not affect.
Oh, sorry. I'm curious to know whether the resolution of Fermat's last conjecture has 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 a much smaller length of time now because it's used in.
And security and cryptography and so on. A great deal.
So. The utility in those fields.
What I think is already it's already there.
Really. Think we've got time for one more round.
Okay, so if we go one, you there two and then anyone else three does the same.
Whereas one can't remember that we get just done here.
Oh. Okay. We'll see each month. You go first. Yeah. Hi.
I'm also a secondary school maths teacher and I was interested what you're saying about, um,
sort of spending a long time on a problem, not being successful 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?
Oh, 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.
Make no mistake. So this question is also on the teaching theme.
You mentioned the importance of having a great maths teacher when you're younger.
Do you have any thoughts on how we can encourage more mathematicians to go into teaching?
Yes. Pay them more. Yeah.
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 had actually done a Ph.D. in number theory and gave me some books and so on.
He was a great inspiration. Did you speak to them after you sold Fermat's Last Theorem?
I did meet my high school teacher. I think I met both.
Yes. Yes, afterwards. That must be a nice moment for them.
Okay. The alarms going off saying we have to finish. But I did promise one more question.
Which was up there. Yes. If we get microphone to this gentleman.
Oh, okay. I'm sorry. Okay.
Look, I'm 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.
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 pad,
that when your mind is relaxed, somehow your subconscious takes over and pieces together.
So I can't say anything useful, except at some point you let your mind relax 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 ask Andrew's questions.
If we can get a microphone over here. It's good.
Okay. Oh, was it okay? Great. It's all solved and no one need.
Tell me off then. Okay. In which case, then all that remains is to say an enormous thank you to Andrew once.
And thank you all for coming very much. Thank you.