Good evening, everyone. Welcome to the Science Museum.
I'm Roger Highfield, the science director, and it's a thrill to be here tonight to introduce the Oxford Mathematics public lectures.
They're being webcast from the IMAX of the museum. Hello, world!
Now, mathematics lies at the heart of this museum and our mathematics gallery.
Zaha Hadid crystallised abstract mathematical thinking into beautiful physical forms.
It's central to the cryptography in our top secret exhibition, to the A.I. in our driverless exhibition.
If you go to our Science City gallery, which is brand new,
you'll see Newton's Principia and the story of the most powerful way that we have to understand the universe,
which is, of course, a scientific method.
Mathematics also underpins the scanners, the crystallography, the epidemiology and more in our massive medicine galleries, which opened last week.
Now, in a lecture given 60 years ago, Eugene Wagner described the unreasonable effectiveness of mathematics.
Mathematics is indeed a universal language. It's a means to understand everything that we do.
It's a tool for in for increasing our thinking power.
Tonight,
we're going to celebrate its awesome importance with our guest of Honour Fields medallist Timothy Gower's and with the help of Hannah Fry of UCL,
who is giving the right Christmas lectures. She is the author of Hello World and also a Science Museum Group trustee.
But first of all, I'd like to introduce the Oxford Mathematical Institute's director of external relations.
Please welcome Professor Alan Gorier Lee. Thank you.
Many thanks, Roger.
You know, it's an important event when you have a series of a speaker introducing each other thoughts when maybe have a few more too.
So what we've been trying to accomplish over the last six years now with the Oxford Mathematics lecture,
but also with extended programme is to bring the best of mathematics to the public to a lecture series first in Oxford.
But then in London and now to other parts of the country. And I'm I am particularly grateful to both the Science Museum and also to an affray.
We've been great friends and partners of Oxford Mathematics and believe in what we're trying to accomplish,
and I hope we'll have many more such collaboration, but also to help us in our goal.
I have to give special thanks to the X Markets sponsor of the Oxford Mathematics Public Lecture.
Series X equity markets are leading quantitative driven electronic market maker with offices in London,
Singapore and New York, and I'm glad to see that many of you came to the event today today.
It's a great honour for me to introduce Professor Tim Grover,
the Rose Board Chair of Mathematics at Cambridge University and fellow of Trinity College.
Since you will hear more about Professor Gower's life, mathematics and opinion,
I've been given the impossible task to introduce him in a few words for inspiration and naturally turn to short description of historical figure.
I've been recently at Yale University and you'll find right at the centre of the
universe the beautiful graveyards and you'll find the grave of the great lost Gonzaga,
one of the greatest physicists of all time, and simply on his gravestone is described by Nobel laureate,
etc. So it could be appropriate to describe professor go goers as field medallists, etc. and be done with it.
But it would be a gross mischaracterisation of his other accomplishment and influence,
not only on mathematics directly but on the way mathematician work and organised is also been a passionate,
passionate about explaining mathematics to the public,
and as the unique record of having both one of the shortest introduction of mathematics with Oxford University
Press and an encyclopaedic one may be the last the longest introduction with Princeton University Press.
If you follow him on social media, as I do, you will also discover True, True Humanist,
passionate about mathematics, but also caring about all social and political aspects of today's world.
So for my second attempt and knowing that Professor Gowers is of particular interest in music,
I thought about another short posthumous description, which seems to be a good fit.
This one is about the 15th century English composer John Dunstable, who was described as mathematician, musician and whatnot.
Tonight, we will first hear about the essence of mathematics, which is about generalisation and abstraction.
Then, Professor Goes, would be in conversation with unafraid.
And my hope, of course, is that at this point, we learn all about Professor Gauss ETC's and whatnots.
So without further ado, please let me welcome Professor Gower's tonight for story.
Thank you very much. I just start by saying that this is the largest slide I've ever had the pleasure of presenting.
I also want to say that I don't think I can quiet in half an hour. So what I have?
Live up to the promise of the title of this talk, but I can do something towards it because really,
to say exactly why I believe that we'll never run out of questions is it's quite a long and complicated task.
But this is just one. I suppose that the emphasis is on one reason because there are plenty of reasons.
But I suppose the fundamental reason is that mathematics has a sort of hydrologic quality that when you answer one question,
it somehow begets 10 other questions.
And one of the things one of the sort of begetting mechanisms is generalisation, and that's what I'm focussing on this evening.
So before I get underway, there are two types of generalisation that one could be talking about.
One is generalising ideas of mathematics or definitions, and the other is generalising statements and theorems and lemons and things like that.
So I'm mainly going to be focussing on generalising statements, but I will say a little bit about generalising concepts because I think it would be.
A shame, not a little bit about that, because that is also very, very important.
So let's just dive right in.
The first two of these concepts are things that maybe I don't know how much mathematical experience people have on average in this room,
but if you've done maths at school, then I would expect that at some stage you've come across something like X two,
the three over two and you may remember. If you came to understand that if you didn't,
I'm about to explain it the sort of feeling that how connects to the three ever to make any
sense because you can't take a number one and a half times and sort of multiply those together,
that just doesn't make any sense. So how do we make sense of this concept?
Similarly, what about each of us, Z have said, is complex so raisings a complex power that's not even madder than raising two?
So what is the power one and a half? I mean, what would it mean to take one plus I lots of e and multiply those together?
That's even more sort of nonsensical.
And just to sort of get even worse, what could it possibly mean for a shape to have a dimension that wasn't an integer,
and we'll come back to that in a minute?
So let's just quickly go through those and then I'll move on to generalising statements to the main thing I want to talk about.
So the way that we decide and in each case, it's slightly different the process that we use to generalise the concept.
So if you want to generalise powers to non integer powers, one thing we do is just focus on this rule here.
That extra m plus N equals extra m times X to the end.
Although many schoolchildren will will say that it's actually having plus x to the end because they always believe that in mathematicians terms,
all functions are linear, but they're not. So there's one has the stress that X percentage x them times x to the end and not extend plus x then.
And here's the reason illustrated with an example. So take X to the five.
That's just x times exercise x times x times x.
And if I just split that half of three lots of x times two lots of X, I see that it's x three times x two to two.
Once you've seen that example, that's pretty obvious at X to the M Plus and there's always x sam times x today.
And because it's just embarrassing, X is is the same as axis times and X is where I don't really mean and X is I mean, a lot of X multiplied together.
Now, the point is that the way we then use that is we say, well, that's a rule,
and there isn't really an objective meaning to extend the three ever to.
It's up to us to choose the most sensible meaning we can. And the way to choose the most sensible one is to choose the one that preserves this rule.
So let's just assume that we've got that rule that would tell us that X Cubed would have to be X two to
three over two times x two to three over two because three of a two plus three over two equals three.
So if we want that rule to be true when and then about equal to three over two, we need this equation to hold.
But that tells us that x two to three over two has to be the square root of x cubed.
Or you're feeling very alert, you'll object that excuse might be negative or there are two square roots and so on.
But the convention is will assume that the number on the X is positive and we'll take the positive square root.
That's just convenience rather than some objective reality, but it is very convenient.
OK, so that's giving us a very good answer to what takes two to three over two hours
and then you can build on that and work out what extra any fraction should be.
And then you can use other arguments to get to infer from fractions to irrational numbers and you can carry on and on.
Let's move on. But that that it takes us so far, but it doesn't take us to something like this what would eat at a root two plus three IP?
And again, if you've done at A-level a.m. Mass, you'll know how we how we sort this out.
If you haven't, I'm going to ask you to take something on trust.
So the trick here is to think of it as the square root of two plus three.
I know it has raised me to the square root of two plus three.
But to reformulate this and the way we reformulated this to use the following formula which you need
to have done some calculus of sort of reasonably advanced kind to to be able to justify this formula.
So if you haven't seen this, this is the thing I'm asking you to take on trust. It is the case that for all real numbers X,
each of the X is one plus x plus x squared plus x x two factorial plus executed over three factorial and so on.
And this dot dot dot means you. The sum gets closer and closer to sum number,
and that is the number that we take as the definition of X own story takes the definition of the sum of the series.
And that turns out always to equal each the X. When X is real, we know how to make sense of energy x when X is real.
But the great thing about the definition on the right is that now it's much easier to
make it to make sense for complex numbers because all we need on the right is addition.
Dividing by integers. Multiplication and a limiting process and all of those addition multiplication.
Dividing by an integer and taking limits, all of those make very good sense for complex numbers that generalise straightforwardly.
So the left hand side does not generalise straightforwardly, but the right hand side generalise is very straightforward.
So we take the right hand side as the definition when we have a complex number.
OK, so now let's move on to fractional dimension,
so this this collapsed chair is supposed to represent the sort of existential despair that you might feel when
trying to conceive of a shape that has sort of two and a half degrees of freedom or something like that.
So we know there's a one dimensional shape is one where you have sort of one degree of freedom in two dimensions.
You have sort of the length and breadth of something and then the three dimensions.
You add height for what, two and a half dimensions, but it's not at all clear.
It seems as though the very notion of a degree of freedom has to be a whole number, and indeed it does.
So we have to do the same trick of finding some other way of thinking about dimension that will allow us to generalise it.
So let's see what we do. So the idea is to focus in two and three dimensions on area and volume.
But since the area is very much a two dimensional concept and volume is very much a three dimensional concept,
and I want to have something that isn't an integer. I don't want to think I don't want to call this area and volume.
I want to call the amount of stuff so.
Another familiar fact is that if you take a two dimensional shape and you expand it by a factor of two in every direction,
you get a new shape that has four times. The amount of stuff as the original shape.
So you can see it very clearly in one of these squares four times and to the big square.
And what's for what's the significance of for it is that it is one times to two times to I two squared,
if you right that it would be too little too on top. And that too is telling us that we're a two dimensional shape.
Similarly, in three dimensions, if I double every single direction of a cube, I get a new,
bigger cube into which I can fit eight copies of the original cube, and eight is two cubed.
If I instead expand by a factor of three in every direction, I get 27.
I got a shape into which I fit 27 cubes and 27 is three cubed.
And again, it's the fact that I'm saying cube, which is to the power three.
It's that three of one that comes after the words to the power that is telling me what the dimension is in that case.
So now let me show you a shape or I won't actually show you the shape, I will show you a process that eventually leads to the shape.
The process is this you start with a line segment that goes from this point to this point and you divide it
into three equal parts and you replace the middle part by the other two sides of an equilateral triangle.
And then you take each one of the sides of this new shape so that that's not shown, but it would be one two three four like that.
Here I take each of the four parts and do the same process. I divide this segment here into three.
Replace the middle bit by the other two sides of an equilateral triangle. And here I've done the same and done the same and done the same.
Now I've taken each of the 16 segments that make up this shape and replace those by four little bits that make that kind of exacting shape.
And each one of those I replace and a bit I'm not showing you is that each one of these
little segments are replaced by one size and then each one of those segments are replaced.
I I'll go on infinitely long, which is why I can't really search for one slide and there will be a limiting shape.
And that limiting slope shape is called the snowflake, and I think it's fairly clear from the picture.
That the cost snowflake is made out of four copies of itself shrunk down,
so if I look at this shape here because it's produced by basically exactly the same process as the entire Cork snowflake,
it is just a little cork snowflake. And that's another little cork snowflake. And that's another little cork snowflake.
And that's another little cork snowflake. So this shape here, it's a fractal shape.
It's made out of four copies of itself, but those four copies are a bit smaller. Let's focus on it the other way around.
So let's look at this part here and say if I were to.
Expand that shape here by a factor of three in every direction.
What would I get? Well, the distance from here to here would expand by a factor of three, so I would actually get this shape here.
So if I expand this bit here by a factor of three, I get the same shape.
So the amount of stuff goes up by four. Expanded it by a factor of three to get this shape here.
But this shape is made out of four copies of this shape. So just let me say that once more, I've got a small shape.
I expand it by a factor of three and I get four copies of the original shape.
So where the cube, when I expanded it by a factor of three, I got 27 copies of the original Q Yeah, I got four.
Twenty seven was three cubed. So what do I need to ask, I need to ask for is three to the what?
Fortunately, we thought a little bit about raising a non-intrusive house,
and therefore it turns out to be three raised to the power, the logarithm of force, the base three.
It's a log that's a number between one and two. So this is a shape.
It doesn't really matter what exactly what the dimension is if you don't like logs, but.
It is definitely between three and four, because there's between one and two, the dimension,
because it's more than three to the one smaller than three to the two three to the minus three three to the two is nine,
four is between three and nine. So the dimension of this shape?
If you use that concept to dimension, it's natural to say it's but it's between one and two.
You might say, well, that's a little bit, you know, we can talk about areas of all sorts of shapes that have to be squares,
whereas this is a very special shape of the self similarity property.
But it turns out that we can build on these ideas and make sense of dimension of lots of other shapes.
And I've now realised that I've been using slightly more time than I intended, which is always the way I was to get onto the main topic.
So I wanted to talk about a beautiful result called vanderveer and Stern and talk about generalisations of that.
One of them is the reason I chose this is that it's had a lot of connexions with my own research over the years.
So that divergence term we start by doing the following we take a large number.
I've taken fifty here, not all that large, a large number, and we assign a colour to all the numbers up to that number.
So well, I've decided either red, black or blue to all the numbers up to 50.
And now. So no, terribly obvious reason, but what mathematicians we like to do, these things are going to hunt for arithmetic progressions.
And I want my maths progressions to consist of numbers that have the same colour.
So what's an arithmetic progression at something like four, seven, 10,
13 or something where you have some numbers to get from each number to the next number,
you have a jump, which in that case was three, and the jump is the same each time. So another one would be, say, 16, 22, 28, 34, 40, 46.
Something with the jump was six in that case. So if we look around, can we see any arithmetic progressions?
I sort of, as you can see, two, four six. This is a black one, but it stops.
So can we find any of lines? Four. Well, here's one.
I've underlined so we go from two to 13 to 24, 2005,
so the step there was 11 o'clock monochromatic because it's just got one car that sets and thirty five it stops there.
I couldn't get to 46 because that was blue, not black. So anybody feel like finding me a blue one of legs for.
Thank you. Three, seven, 11, 15. And that's also has for and it also can't go any further, and I I've looked reasonably hard.
I don't think that's one of five. That's why everything is blue.
Slightly more challenging and red one, I think I wanted to give you very long for this because I need to press on.
But you have to have a chance. And three to one, here it is.
Twenty three, thirty one, thirty nine and forty seven set at eight each time.
What has found a Vaart theorem say this sort of silly game of finding arithmetic progressions.
Everything, everything's the same colour. It says this is not a formal statement to the serum, obviously, but it's it says that.
Basically, however, you colour the numbers up to and so says, if I take an end, that's large enough,
I have some palette of colours that say I got 25 colours and you want to fight.
You want better be an athletic progression of length 79 then as long as NW is large enough.
Then, however, I colour the numbers from one to end with 25 colours.
That will definitely be an aromatic progression of like 79, where all the numbers in that charismatic progression have the same colour.
There's just no way of avoiding that happening that when as long as an is large enough saying it's quite an important qualification,
but the point is that it it does exist, some with that property.
Right. So now let's get generalising.
I hope you understood that, so you colour the integers from one to end.
You can't stop that being an aromatic progression of wheat with only one colour.
So the first generation I want to talk about is called a density version,
which starts with the following question you might sort of the one thing that mathematicians do.
I don't just sort of say, Oh, great, because if firm, now let's go home, they ask themselves, as I said earlier on more questions.
So one of the questions you might ask here is, well, we know there must be one of the colours at least must contain long artificial progressions.
But can we somehow identify which colour that is? What is the best we can say that there must be some colour.
And it turns out that isn't a very, very surprisingly satisfactory answer to this question, which is a result called semi-trailers theorem.
Again, this is not a formal statement. But let me try to say what this means.
So when we have a colour, one of the colours that we use,
when we're colouring the numbers from one to and we define its density to be the number of times you use that colour divided by N.
So if you used, say your colour in numbers from one to 100 and you use the colour red 30 times, we'd say the density of red was nought point three.
So what summary this term says is if you tell me a density like one percent and you tell me the length of harassment,
a progression like a million, then it will be an N.
With the property that any colour that uses one percent of the numbers from one to N must contain an asthmatic progression of like four million.
So he actually is not important as a colour, just as any bunch of integers from one to N,
as long as there's at least you've got at least one one percent of all the integers from one to N and N was massively large,
uh will have to contain an arithmetic progression of like four million.
So Sam Register immediately implies Van 2007, because if you colour the numbers from one to end with,
say, 100 colours, this one of those colours must be used at least one percent of the time.
That's a famous principle called pigeonhole principle in mathematics, or if you've got, you know,
or similar, if you had four colours, at least one of the colours must be used 25 percent of the time.
So by summary, to serve that colour itself will contain an artificial progression.
Right. Let's move on now, so I'm going to generalise in a completely different way.
This is a firm again not stated, formally called the Hayles do theorem.
Let me move on to a picture and see if I can explain roughly what the Hell's to it, Theorem says.
So when I was young, I used to have a game called for pegs.
And this is a picture of the box or a box that was selling on eBay of the same game.
So this is three dimensional noughts and crosses, except it's not quite because it's four by four by four instead of three by three by three.
So lions have length for in this game.
Turns out, if you have three by three by three, it's a very easy win for the first player, but four by four by four is.
A rather good game.
I think it's if you play optimally, it's a draw, but it's pretty hard to play optimally and people often win if they're not super expensive, the game.
So a line here will be something like here's a simple example of a line and a slightly more complicated example,
but the exact point to that point at that point and that point where you get
more complicated going to be one that's diagonal in every possible respect.
So say that point, that point, that point and that point.
Um? And what the [INAUDIBLE] do it, Theorem says, is however long the lines are so here that four and however many colours you have,
say 100, as long as the dimension is high enough, there must be a line that consists of points of only one colour.
Let's consider what an even higher dimensional noughts and crosses might be like this one we can at least visualise.
This is an example of four dimensional noughts and crosses.
But because I can't do a slide in four dimensions or even to a slide in three dimensions of a fundamental thing,
I'm sort of projected onto two dimensions. What we have to do is find some way of representing it.
So the way I've represented it, I think you can sort of see I've just as a three dimensional noughts and crosses
board consists of a bunch of two-dimensional ones placed sort of next to each other.
I don't have to one on top of the other.
We could just place them next to each other and it would be just the same game, but it's a little bit harder to visualise what a line is.
So here I've taken a two dimensional one, and then I put those three together that makes a three dimensional board.
And then those three together make another three dimensional types. You prefer to visualise it vertically.
Actually, that looks sort of like a three dimensional board next to another one, next to another one, and that makes a four dimensional ball.
And then we can hear it. Here are some examples of lines. Let's just do the most diagonal one possible.
We go that point there, that point there, that point there.
But another one might be say that point. That point, that point, you get the general idea at that point, at that point, at that point does.
So going back to the house, do it to him again.
It says if you if you have a sufficiently high dimensional board, then how wide it is.
And have many colours you you have you cannot avoid lions and have a consequence for the game,
says, even if you have a game where you play it with sort of eight players, let's say.
If the dimension is large enough, it can't end in a draw because at least one of it,
once it once the borders filled, you've coloured the points with eight different colours.
And one of those colours has a line. Somebody must have made a line first.
So the game couldn't have ended in a draw as there was a paper, the paper of Hale's.
It talks about multidimensional tic tac toe, tic tac toe being the American for noughts and crosses.
Right, let's move on to yet another completely different generalisation this time.
What we're doing is let's think of how we might represent a general arithmetic progression
of flags for you might say you pick a point a number A and you pick a difference d.
And then the ultimate progression consists of the numbers a a plus d, a A-plus, 2D, an lost 3D.
So the expressions do 2D and 3D are very simple examples of polynomials.
Polynomial functions of D, but they're actually polynomials of degree one or linear polynomials of D,
so d 2D 3D, they're very simple expressions here. I just replace those by some rather more complicated ones.
D squared, d cubed and d-plus d squared those to each of the four.
I want the polynomial version of events, Theorem says,
is that I could choose any bunch of polynomials like that as long as they don't have constant terms.
That's a technical reason, but the theory is not true if you have constant terms.
Um? And you can find so, however, as long as I've picked a large enough integer and I cut the numbers from one to that integer,
I'll be able to find patterns like this. Let me just illustrate it, but I think it'll make it clearer.
So let's go back to this is exactly the same colouring of integers I had before,
and I promise you that this wasn't sort of fake trying to have a little search.
And eventually I found an example of a pair of numbers.
And so I said all of a velocity squared plus d cubed and a diversity of capacity to the force belonged to the set.
And here it is. I had a equals seven and D equals two.
So 11 is seven plus two squared, 15 is seven plus two cubed and twenty nine is seven plus two plus two squared plus two to the fourth.
Well, check that to the force of 16 to IS for two is to add them together at 22.
Add that to seven. You got 29, so I wasn't cheating.
Right. So as I stress, I this is just an example, but the system itself says that whatever I'd chosen, that looks a bit like this.
I could have chosen sort of d squared plus d d to the fifth twenty five d to the 17+, D to the 13 and a few more far that goes as well.
And it would still have been true. As long as I colour enough integers, I will get a pattern of Typekit desired kind.
Right. So let's just recap, here are our methods that we've got for generalising.
We go from a colouring theorem to the density theorem.
So that means that instead of colouring, we just say any old set that's reasonably dense must contain the configuration we're looking for,
right from integers to high dimensional grid sold noughts and crosses boards. I went from arithmetic progressions to more general polynomial patterns.
Those were the three ways that we came up. And so there is completely different.
They're so different than what we might call orthogonal to each other and using right orthogonal,
slightly fanciful way, but one that mathematicians often do because they are orthogonal.
We can start asking, Well, what about if we try and combine some of these generalisation methods?
Could we actually? Get even more generalisations.
So there are eight possible generalisations if you've got three methods,
if you include what mathematicians one called the trivial generalisation where you don't generalise.
So if you insist on at least some generalisation, that goes down to seven because there are seven ways of choosing at least some of these.
So I'm going to think about what happens if I combine one and two, if I combine one and three, and if I combine two and three.
So combining one and three is fairly straightforward.
The Hayles Drouet Theorem said if you colour a sufficiently high dimensional noughts and crosses border, you must get a line in one colour.
So the density of ocean says if I just take one percent of or some fixed percentage of a sufficiently high.
Dimensional noughts and crosses board.
There must be a line in that, so I fill up, let's say, one percent of the points in a very, very high dimensional noughts and crosses board.
I can't avoid making a line somewhere. That would be what the density theorem says.
The colouring Fitzgerald would say, I have to colour every single point and then one of the colours will contain a line.
The density version just says any sufficiently dense colour will have to contain a line.
But that turned out to be a much harder result than the hills to it itself.
This is due to two mathematicians called Hiro Furstenberg and Yitzhak Katz Nelson.
So let's try one on three. So now I'm thinking about going from colouring to density and from arithmetic progressions to polynomial patterns.
Well, then again, it's fairly straightforward to see how the generalisation ought to work.
We ought to say instead of colouring,
we'll just take a sufficiently dense set of numbers from one to n and hope to find a polynomial pattern inside that dense set.
And that turns out to be a true result as well.
The the theorem that you can do it with colouring was due to two mathematicians called Vitale, Berthelsen and Sasha A.
Now there's an interesting story that turned out that before they did the colouring version,
they had a sort of machinery which had developed building on the work of actually Hillel Furstenberg.
They knew that once they had the colouring version, they would be able to generalise the density version.
Not it's not obvious how you can, but then it turned out that they knew how to do that.
So that was true. And the last thing I want to talk about, although sorry,
the third of these are combining two methods of generalisation is something that's rather less
obvious now here on how it's to be a little bit of a challenge to say what's going on here.
So you think that this is a two dimensional grid that I've coloured black and red?
But you will also see I put numbers one or two in there, something a bit strange is going on.
So that's not supposed to be a two dimensional noughts and crosses board.
This is supposed to represent one point that lives inside a 64 dimensional noughts and crosses forward.
So if you think about it, just a straightforward, regular and awesome process forward, we might have a coordinate system that went north north.
Not one, not two one one one one one one two two, not two one two two in 64 dimensions.
So what are the what are the points of that board? They are pairs. Of numbers on each number in the pairs is either not one or two.
Yeah, I just got 64 of those 64 numbers, each one of which is a nought one or two.
That simple, that's just a point and sixty four dimensional space.
So why have I counted some coordinates red in some black? Well, just cause I felt like it, I have to take it on trust at this point here.
Remember, this is just one point. This is the coordinates. I happen to arrange the coordinates in an eight by grid.
This point is green. OK, so this is.
I'm just declaring it victory, and it's not some mathematical reality that you've missed.
I'm declaring this point victory. All right.
So I want to tell you what a lion looks like in this 64 dimensional space, and it's a very special sort of night.
Uh. Here it goes. So why the reason I highlighted those points in red is because they're about to change.
Maybe you see they all went there, all notes, and they're all ones and they're all tools.
So if you've got a point where there's a whole bunch of noughts in there coordinates and then another one where there's a whole bunch of ones in those
same coordinates and another point with a whole bunch of twos and those same coordinates that's going to form a line and sixty four dimensional space.
This is just a natural generalisation of what happens in two dimensions and three dimensions.
If you think about it for a while. But there's something else about this line that makes it particularly special,
which is that the points that do the changing lie in, they have further coordinates in a special set.
That set is the set one to four seven so that either and the but both the row in the column, it's it's here.
I got the first first row, second row, false row, seventh row, first row, second row, four through seven, sorry first column,
second column, fourth column, seventh column and each point that's coloured red lines in a row and a column from one to four seven.
And then those are the points that changed. So it's not just any old bunch of points of change.
This is the fact that this set here is called the set of all points,
where the row and the column lions at one two four seven is what you might call the Cartesian Square.
So what square there is important if you look at in particular, it's a focus on a number of points that are doing the changing.
There'll be four times four makes 16, which is a square number.
It turns out from this. So what of what is this?
This is the statement that however, I colour all the possible three to the 64 points of which this is just one with some small number of colours.
I will not be able to avoid getting one of these special lines.
Special line where the coordinates are vary are arranged in a nice pattern like that.
All of one colour. Um, actually, I lied to, so sixty four won't be enough, it'll be n squared for some very large n.
But again, I want to illustrate it better with just one slide.
Um. Now.
If you didn't completely follow that because I think it is challenging,
it was challenging for me when I first came across this term to work out what on Earth was going on.
Please believe me that this is a generalisation, and from it you can quite straightforward to deduce, for example,
the statement that from this sort of case here that if you cut of integers from one to N and is very large,
you can find an metric progression of like three.
But a common difference is a perfect square turns out to be a fairly easy exercise to get from this statement to that statement.
And this is just one special case which analyses the polynomial thing.
Now. Getting to the tentative version is always straightforward, you just replace is green or is some particular colour by belongs to some dense.
So here what I'm saying is if you choose one percent of all the possible points that you can get here,
you must find a line of this special form that lives inside that search.
So that's the density version of that's generalising all three different directions.
That's a density version of the polynomial hails Jewitt theorem.
But there's one difference between this result and all the other results that I've talked about,
which is this is not a result, this is an open problem.
And I wanted to get to that because that's something that I am working on right now with one of my research students.
I wanted to sort of show that just by this process of generalisation, you can get.
To the frontiers of mathematics, I will stop there. Thank you very much.
So we're going to have about 25, 30 minutes of questions and then for over two years,
if you have questions that you want to ask again and please, do you think of them as we go along?
And I think for me, what I wanted to kick off with was just ask you about those kind of problems.
What is it about them that intrigues you, I guess? How did you choose them as your your area of study?
In a sense, I didn't actually, because when I started out, I was doing a completely different area of mathematics,
so but I had a friend who worked on some great experiment related results, and I always found that interesting and at some point I.
Thought I potentially saw a way of coming up with a new proof of summary system, which quite a long way down the line.
I eventually did. And when that happened, the whole sort of focus of my research changed into something called additive combinatorics,
which concerns problems somewhat similar to the ones that I've just been talking about over and.
But maybe that's not exactly the question you're asking. So maybe you're asking these look like sort of amusing puzzles, but what?
Why? Why concern oneself is a huge effort to solve any of those puzzles, why go to that effort just for one of these puzzles?
And my answer to that is that they're not mere puzzles.
So if you want to solve one of them, so one of these results that I've just been talking about, it doesn't have that.
Many typically don't have that many direct applications even to other parts of mathematics, let alone outside mathematics.
But the techniques that you're forced to develop in order to solve one of these problems will very
often have much wider applicability and implications for other parts of mathematics very often.
So one of the great mathematicians in combinatorics, which is a sort of wider area that I belong to Paul Adams.
Was a master at asking. Problems that had this kind of quality of seeming like amusing puzzles, but they just encapsulated some difficulty.
He didn't just throw out any old question, and when you started thinking about them,
you realised that there was some difficulty that if you could solve that, you would really be in a much better position than you were before.
For a lot of other things, and not just the sort of headline problem that he asked, that's got that flavour a bit, I think.
Is that motivating motivating factor for you.
Then the fact that the the I guess the things that you're discovering along the way will be useful or useful as even enter into your your motivation.
Well. Can useful can mean useful to engineers or it can mean useful to other pure mathematicians.
I would be thrilled if something that I did was useful to engineers, but it's not my primary goal.
And if something is useful to other mathematicians and I feel as though it's helping to develop the subject.
And that's maybe more important to me or more immediately important, I say immediate importance, because if you do, then develop the whole discipline.
It's just inevitable that the applications outside mathematics follow.
It's just that it may not be my precise theorem that I still feel I'm contributing
to a big endeavour and then bits of that endeavour that almost randomly.
Chosen somehow turn out to be very helpful to people outside mathematics.
How early on did you know that you wanted to be a mathematician? I'm not very early, I was always one of my favourite subjects at school,
but it wasn't my I wouldn't say it was always by far and away my favourite subject it was.
And the other thing that makes me say not that early was that.
Until at least when I was an undergraduate and maybe even later than that, I didn't really.
Or maybe towards the end of my undergraduate time at.
Perhaps not even the to that it made up, of course. Part three of Cambridge is part three course.
Did I have the slightest conception of what research in mathematics would be like?
I think probably many people, if you are not a mathematician, you're tempted to ask, maybe have even asked if you've ever met a mathematician?
So just what could research in mathematics actually be like?
And if you're unkind, you say, is it multiplying larger and had a few times?
So then you have to explain that no maths is not just being sort of a bad pocket calculator, but more to it than that.
And I hope maybe I've conveyed something of the sort of world that mathematicians inhabit, or at least some corner of mathematics.
So. And then another another sort of thing that makes you maybe not want to say I want
to be a mathematician is that when you're starting something very intimidating
about the very notion of an open problem in mathematics because you're surrounded
by people who have a reputation for being incredibly clever and so on,
you say, Well, there are all those people out that are incredibly clever research mathematicians, and this is an open problem.
How am I going to come in and solve that problem? There is an answer to that.
And the answer is not. You have brains bursting out of your head. It's that mathematics.
Just as I've been saying, we don't run out of interesting problems that we get more and more problems in.
The more you think about it, the more questions you ask, the more problems get generated.
So actually, there are quite a lot of problems around that have not been thought about by all the world's experts for the last 20 years.
And so if you don't try to solve the remote hypothesis for your Ph.D., then there's a chance of making some progress.
And then you say the idea is you try and work on problems that are maybe not so central and then increase your level of ambition gradually.
When did you work that out? When can you remember when you stopped being intimidated?
I suppose it was when I first. Made any progress at all?
Well, then maybe that's a slight. So the first thing I did was just a tiny little tweak to someone's existing
argument that improves the answer to prove the bounds that came out of the proof.
I gave a talk on that rather sort of informal setting, and I was a mathematician by won't name who at some point sort of walked out of the door,
I think, to go smoke a cigarette and I came back for that sort of gave an indication of his level of interest.
But I later on found a much more something with the same problem that got not just an improved result,
but the best possible result that you could get by a much more complicated argument.
And so at that point, no one was going to know, and I sort of felt this.
I've definitely shown that I could do research at that point.
That's probably in the middle of my first of my second year.
And that was the thing that certain genetics, I guess something I knew, I thought something.
So talk to me then about that process of research. So you were saying there about, you know, being a mathematician on the day to day?
Tell us a little bit about your process when you are approaching a new problem. How do you do it?
How do you how do you tackle a challenge? Part of the answer, actually,
is choosing the problem in the first place to make sure that it just feels like the kind of thing that
might conceivably be amenable to the sorts of things that I have in my own mathematical toolbox.
I can very often just misjudge that completely, and the problem is much harder and not something that I can do.
So that's another thing to bear in mind that I think if you have the right level of ambition by not doing research,
you should be ready for most problems. Very much most problems of nine out of 10 problems that you try just not to get all
that far because you're looking for that one in 10 when something kind of gives.
But how do you find that? The the thing that I think is most central is the process that I alluded to at the beginning of the talk,
which is you don't try and take route one to the solution.
So if that's something that quite a lot of people take a while to learn, because when you'll set exercises at school and university,
they're carefully tailored so that think you use a standard method and you just use that method and you get to the solution?
Or perhaps there's one little trick, and once you've spotted the trick, it's plain sailing from that moment on.
But a real life open research problem is not like that at all.
So if you try and use that sort of idea of just try and hit it with standard methods.
Or just look for that one little idea after which it'll be easy.
You won't get very far. Very occasionally there are problems that surprisingly some little trick works, but that's very much the exception.
So somebody have to ask yourself when you're doing any sort of research is.
When you have an idea about how to tackle it, the first question you ought to ask yourself is, well, if that idea worked?
Why is it that this is still an open problem?
And if you've got a reasonable answer that some reasonable story to tell that this idea is a little bit left field in some way,
or it involves mixing using ideas from a totally different area of mathematics that
would not be familiar to the experts in this area of mathematics or something like that.
Some story to tell than probably your approach will fail,
although it can be valuable to see why the approach fails and then try to think something more.
Sophisticated. I've lost track of what your original question was, not when I was coming to the answer.
What's your process when you're tackling any problem? Oh yes.
The first test was you don't take the direct route, but you ask yourself other questions.
So one of the questions that you can ask if you say you're trying to prove some statement is to see whether you can prove a more general statement.
So that's a generalisation. And the reason that can work is that sometimes when you generalise the statement,
it actually paradoxically, although are proving something stronger, it's easier to do.
Why is it easier? It's because. She's trying to prove something that's stronger and more general.
You don't have as much. Room for manoeuvre, it's a bit like being if you're in a worse position in chess,
it can be easier to decide what move to make because you're more or less forced. The only way of avoiding checkmate is to do such and such.
If you make your problems harder, it can make them easier.
So that's one method, but another method is exact opposite, which is look at a very special case of what you are doing.
Save with some said you could look at progressions of like three that turned out to be a very fruitful thing to do.
And build up from a special case until you hope that you get to the point where you sort of spot a pattern and what you're
doing in the special cases and then come generalised to get back to the thing you were originally trying to answer,
sometimes in other methods. Another question you can ask as well.
I don't see how to solve this, but let me invent a different problem in a slightly different context.
That appears to resemble this problem and involve quite a lot of the same difficulties and
see whether I can just solve this somewhat similar problem and get some insights from that.
And what someone's always looking for is a new problem that should be easier than the problem you're trying to solve.
That should help you to solve the problem you're trying to solve. If you can solve that one.
And that's a process that can sort of split up and split up. So if I have, this is a problem I'm trying to solve.
This is starting from nothing. Say I invent this new problem that might be helpful for this one.
It might be easier, but I can't see how this one helps this file, and I can't see how to answer this one.
So I'll find another one in between. And another one in between here, and I will try and get a path from zero to full solution of the problem.
So this process of sort of splitting up a kind of top down approach is very important and involves a lot of that.
Very much not what one does is sort of say, I wonder what the first line of my eventual proof is going to be a lot of.
Absolutely not like that. But I mean, that process inevitably involves a lot of dead ends along the way.
Do you think it's important to how do you shield yourself against against being discouraged when you meet dead ends?
I think it's quite hard to do when you're just starting out, because you really,
really want to have got that first theorem first solution to an open problem.
Once you have that ideally maybe two or three times and then you sort of know that it can be done.
And you also know that it won't happen immediately. You have to be patient and you have to be prepared to fail several times before you succeed.
So you then sort of get used to it as being part of the job. This is part of the process of proving theorems.
It's not necessarily the most difficult part, either.
I mean, another difficult thing is when you're trying to do something and then somebody else does it basata.
And or even worse, you work hard on a problem and discover that it was done 10 years ago.
But yes, has that happened to you? Not in a sort of major way, so I haven't sort of spent six months on something and it's been my big project.
Perhaps it's one exception, actually. That was something that I. Worked on, but I was very silly in that particular case to think that it.
I was just covering all these things. I just covered various everything thinking I was.
Discovering it for myself, and then I discovered there's a whole field and all the concepts of coming up with a well known,
but even that was that wasn't a major disappointment to me really, because I was nowhere near.
The very nice student I was trying to solve, and the solution to that there was an incredibly clever and I don't think I would have found it myself.
And if you were to rediscover things, that's actually very beneficial because you understand those things in some
deep way that you don't quite get if you're just passively read a textbook.
So again, not even even from this of disappointments, one does eventually gain even that little bit crushing for a while.
Has there been an occasion where you've been picked to the post in trying to be something?
Well, they've been several occasions where. People have proved things that I would very much like to have proved, um, but.
Not so many occasions where I've really been working hard on, that's the thing I was absolutely working on at that moment and then someone did.
It is more common than I've thought really hard about something and worked pretty hard on it and then sort of thought,
Oh, maybe I'll try something else. It's still one of my favourite problems,
but you have to move around because otherwise you're going to get sort of stuck not solving a problem for five years.
That's Andrew Wiles says that's not a good example for this doesn't always work out.
No, it's not a good model. I should say it's a very good example of a good model to follow.
And. So even then,
if someone solves a problem that I have worked on hard in the past and really light and sort of thought I might well want to return to that can be it.
It also this time of disappointment. But that's again, something that one has to just take on the chin and keep going.
I am curious now which theorems would you have liked involved? Well, the one that I was talking about, first of all.
Well, I sort of developed the basics and it's sort of beginnings of the service.
I was I thought I would have a try at the P versus NP problem.
This is a long time ago and I sort of underestimated how much work had gone into that problem.
And that's a restriction of the problem to do with a concept called monitor and circuit complexity.
So if you sort of restrict what a computer can do, you know, trying to show that it then will not be able to solve certain problems efficiently.
And so I formulated that problem and thought about it quite a lot.
And then it turned out that simple Sasha Raspberry I've had years before.
Proved exactly what I was trying to prove and won prises for it.
And but that would be, you know, that would be absolutely great.
Sir, to have my name, if I could sort of choose one. But of course you I mean, you're not short of prises yourself.
I'm not complaining at all. Do you remember when you found out about the Fields medal?
Did it come as a surprise? I'm. Yes and no.
So. I it was not a surprise that I was being considered for it because I got all sorts of mysterious messages,
you know, saying, could you please send me a CV by yesterday and. But I can't tell you why and that's what I.
And it was the right time of the cycle for the decision to be.
For the committed to making its deliberations, and so I. Didn't want to.
So. I suppose I sort of felt presumptuous to assume that's what it was, but there wasn't much else that could be so anyway, um.
And then.
When it actually when I did find out, it was quite surprising because I was summoned to the office of my then head of department with someone else.
And so that couldn't really find what it was about, but it was and we both got fields, medals.
And at the same time, yes. Did it?
Did you? Does that mean that you were in some sense sharing the glory?
Well, no. I mean that they gave awards, of course, each time.
But I mean, say, within Cambridge or something, I suppose.
But I think that in a way, I think it that wasn't the problem.
I think they made more of a fuss because it was more sort of unusual to have two in the same institution at the same time.
So I think it actually works to my benefit, really. How much do those those prises mean to you?
What's made a massive difference my life in a way, because.
In a way that it shouldn't have, I would say, if the world really adjusts place because it's not the case that there are each time,
you know, four people who tower above everyone else.
It's more like, you know,
the committee that makes the decision has a difficult decision and just eventually has to sort of settle for four four people.
So that's one thing that's there's not some sort of difference in kind between someone who gets one and
someone who just misses out or indeed someone who doesn't even just miss out but does something amazing.
A few years later or something like that, I know that the longer since it happened,
the more remarkable examples of more remarkable things I see other mathematicians doing,
and the more I sort of see my actually rather small place in the vast body of vast corpus of mathematics.
But I think what it gives me is what has given me is just sort of unfair leg up in life,
really, that I just get I got lots of invitations, interesting invitations to things.
People sort of like the introduction here saying, you know, we got a field medallist giving a talk and so on.
And it's sort of it is quite nice. But.
But at the same time. In the end, I don't think it's.
I think I sort of felt after getting that I had to try to some extent to sort
of pretend I hadn't got it so as not to sort of relax too much and just not.
Another thing that could have been a mistake, I think, would have been to to say, Well, now I've got that.
I was, how can you go up from there? Well, I'll have to solve solve one of the clean millennium problems or something like that
or solve some really massive problem that's bigger than anything I've done before.
And I think if you take that attitude, the chances of success are very small and you just have to say, no, I'll just pretend it never happened.
Just keep on working on the things that interest you and stop the good things.
Happy medium things. Do you think it kind of liberated you in a way, though?
In some ways, yes. So I know that. So that freed me up to do one or two things that I think I might have thought twice about otherwise.
So one of them was editing the prison components mathematics,
which was mentioned earlier on which I would say took up roughly half my working time for about five years,
which was a sacrifice that I'm not sure I would have felt I could possibly make.
If I hadn't somehow just made any sort of reputation I needed to make.
And it was a project that I believed in. But it's not not a research project and also I didn't have.
Worry too much about people saying he's. Sort of gone soft, he's doing popularisation now, not mathematics or something like that.
And the other thing which is in the last last 10 years, I've spent quite a lot of time, although certainly not all of my time,
but quite a lot of time thinking about automatic theorem,
proving that is trying to get computers to find proofs of theorems, which is something that really interests me.
But again, it sort of doesn't really count as mathematics. So it's something that.
I think I wouldn't have felt.
I could afford the time to spend on if I was continually sort of thinking about how my CV was doing and those sorts of things.
That hasn't liberated me totally. After a while, I thought I better get back to mass just to sort of show that I really I can still prove it.
So I had about three years where I was not doing that much mass at all.
Doing a bit, but less spending less time en masse, and I was on automatic,
so I'm pretty sure that's changed now and I've for the last five years or so, it's been concentrating more on mats, but it's automatic.
They're improving. So my interest on the side. Do you think those prices are generally a good thing for the mosque community?
Um. I think they're mixed. I do think that's one good thing about the Fields Medal is that because it's happens at a young age and because it doesn't
come with a massive amount of money in the way that the Nobel prise does have a really quite small amount of money.
And so that means that the sort of pain of not getting it, I think, is much reduced because for two reasons.
One is they don't have this sort of I missed out on half a million pounds or something, which I think would be quite bad.
And the other is that even after your it's too late because you're about to turn 40 years old,
you certainly haven't stopped being able to do mass and is still plenty of chance to to to prove amazing theorems and
get an amazing reputation and maybe win other prises that are aimed at older people if you're interested in prises.
So I think there may be some degree of unhealthiness about it, but uh.
But I think in many people's cases, it gets some benefit as well as people sort of got a sort of dream, maybe.
I think it probably in my case.
When I sort of began to think maybe I would be a chance, it did sort of probably make me work quite a lot harder than I would have done for a while.
So my mother had some beneficial effect in that direction.
You've also been a really big advocate of sort of collaborative mathematics and thinking of the projects that you've that you've done recently.
Do you think there's ever a contradiction between that sort of working together on a particular theorem or
proof and the sort of thirst that might be required for a prise or sort of an individual the hunt for glory?
If we're talking about huge collaboration, then of which you're just one very small part of, which was the thing that I.
Set off at one stage, then I think not many people would be satisfied with just participating,
even if they participated very fruitfully in huge collaborations, and for that reason, it's not that method of doing things hasn't taken over.
That's one of the reasons hasn't taken over how we do mathematics. But mathematics is now very, very collaborative.
And I'd say the majority of papers are two or three courses.
And that's for the simple reason that two or three people can often work a lot more efficiently than twice as efficient.
There were three times as much as one person. For obvious reasons you can specialise.
One person might be good at one aspect of finding proofs and another person good as another one person might throw out lots of ideas.
Another person might be good at hitting them down again. Somebody might have expertise in one area, somebody might have expertise in another area,
and I think people just find it's easier to come up with to be a sort of say it's two
or two papers be easier to write fifty two or two papers and 25 one of the papers.
And another thing is that I think if you are one of two authors, you probably get.
Much more than half the credit. For being a sole author of a paper, I think that's most people, most mathematicians, I think, would agree with you.
I think you probably get 75 percent of the criticism unless you've unless there's some very
strong suspicion that you weren't really the person who had the ideas or something like that.
But that's another good thing about mathematics compared with other subjects. We have also strictly in alphabetical order,
rather than having complicated ordering that you have in some sciences where the first person did this and the last named author did that and so on,
so forth, we avoid all that in order. So exactly to sort of avoid that kind of talk about who really did this in.
And the result of that, I think, is that it is strongly encourages people to work in small collaborations quite like that.
This is that you're sort of presenting a very rational kind of calculated argument towards small collaborators.
And this I think as I regularly read your blog, I don't know how many people here also do,
but this is something have sort of known for taking a sort of rational approach to decisions that might not necessarily seem mathematical.
And if I'm worried that one can sort of kid oneself for put sort of veneer of rationality just to sort of justify one's prejudices.
But maths does give you the tools for recognising that when that happens, just so they're not always used to effect.
Is it true that you once decided whether or not to have a medical procedure based on your speculation of the risks?
It is because it was, uh, I have had problems.
Actually, they've just recently seem to be starting up again, which is annoying of arrhythmia in my heart and atrial fibrillation.
And if somebody does a procedure that can be done to correct that where they stick a wire up a vein,
go in your leg and the mine goes up into your heart and it sounds very unpleasant.
They sort of burn the side of your heart. But the reason for that is that it breaks an electrical connexion,
which is messing things up if you don't want to have and causing the rhythm to go off.
And it's a pretty safe operation, but when you read about the risks,
it all sounds quite frightening and in particular as one of the thousand mortality risk.
So I thought, I really sort of I really want to have this one in a thousand risk of dying then.
So I looked into it a bit more. And of course, it depends a lot on how old you are and various other things.
And also the benefits of operation depend on science, on how old you are.
And eventually, I decided to go ahead because I had the idea of looking up.
What the mortality risk was of just being alive for two months, age of 50.
It turned out that an operation concentrated about a month's risk into one operation.
So my risk of sort of falling under a bus or whatever it was over a month would be about one in 10000.
The idea was then I thought, Well, I'm not terribly scared of the next month, so I shouldn't be all that scared.
That was the level of rationality somehow. And then it all went fine.
And yeah. But is this something that you find yourself doing a lot?
Sort of. I mean, I noticed on your blog you often with politics, especially,
you do this sort of very involved posts where your you seem to be almost breaking
argument down to try and persuade yourself of something one way or the other.
Actually, I would say that they're I'm fairly sure what my view is in advance,
but I just want to sort of check that there is some rational justification for it.
So I wrote a post about why apologies have leave voters in the audience,
but why we should remain in the EU and which had sort of game theoretic aspect to it.
I wrote something about why everyone should vote for the alternative vote in the 2011 referendum, because see,
these posts are very successful, and I wrote something recently about tactical voting in the European elections.
But going back to your question about rationality, I do sort of sometimes take things a little bit further than maybe some people would so.
For example, if I'm in the kitchen clearing up, I sort of have a mathematician to have this thought, but the thought I'm talking about is,
say there's some stuff on the table and you've got to transfer it to the side of the sink and
there's some other stuff in the sink and needs to go in the cupboard and that sort of thing.
It's a mistake to say that first on the water tables and sink stuff,
then I'll do all the sink to the cupboard stuff because that wastes the journey where you could have when you were walking back to the table,
you could have been carrying something to that sort of thing. So I do have those kinds of thoughts.
I promise I have a life. Does it drive your family mad?
I try to keep quiet about what I'm doing.
Look, on the rare occasions when I've suggested anybody else in the family that are more sort of ergonomic ways of doing things,
I've been slapped down very quickly. I'm going to come to the audience in just one second.
Just one last question.
I want to ask you, though, do you think that being a mathematician is something that that that requires an aptitude that that can't be taught?
Do you think it's a gift? My own view for which I don't really have enough evidence to say that it's definitely correct,
but it's a little conviction is that you need to be good up to a certain level, but you don't have to be a genius level.
As long as your. Pretty good at maths and very, very keen.
To make progress, but I think the keenness is, you know,
I think if you have a choice between sacrificing a couple of points of your IQ but gaining 10 percent of enthusiasm,
go for the enthusiasm because it may take you slightly longer to have some ideas, but you'll have some.
In the end, that's not solving a problem, it's a slow process. It doesn't really matter how quick you are.
There's a long term thing. It's not to say that if you are super quick and have those kind of ideas and people can't work out where they came from.
Of course, that can help to some extent. But there are plenty of people who are like that who don't make a huge success.
I think the one example I can think of somebody who is a prodigy and is by any account,
a genius who has also made a huge success that would be Terrence Town.
But if not Terrence Taylor, you just don't. There are lots of nontariff stars have done very good things as well.
So what advice if you could go back in time and speak to your 17 year old self, what advice would you give yourself?
I think I would say. I would go back to what I was talking about earlier, that it looks.
I say don't be intimidated by all this, by the cleverness that you see around.
As long as you keep at it. Learn what you have to learn, but try to bring plenty of problems,
including problems that have been solved already, but just get practise at problem solving.
So practise, practise, practise. Keep your enthusiasm and you can make room for yourself.
Yeah, exactly. OK, we'll come to the audience then. So does anyone who wants to kick us off with questions.
OK, we have. Let's go. We'll go first here. One.
Yes, please. You have. Leave it at that. Perfect. And then Emily go behind and then we'll come over here.
Thank you. Yeah. Thank you for a very engaging talk and interview afterwards.
My question is to do with how you choose the problems.
Once you've chosen your subject area, you know you're interested. How do you go about choosing the right problems to pursue?
And then once you're pursuing them, how do you choose when to stop, when they seem harder than you thought?
I think. Roughly speaking, I would.
Part of it, I can't have a very satisfactory answer,
I just look at a problem and I sort of feel and that looks interesting, and maybe I'd have a chance of solving it.
But supposing I've got beyond that stage and I actually started thinking about it and I might decide to abandon it after an hour or two,
or I might decide to continue.
So what would decide the I think it would be going back to this process of inventing all the questions if I found I could come up with.
Promising related questions that was perhaps related to things that I did know
how to solve and had a chance of being easier than my enthusiasm would continue,
and as long as that sort of as long as I had the feeling I was making progress in my understanding, I'd be tempted to carry on.
But there comes a time when solving a problem.
Sometimes when you just sort of get this feeling a bit bogged down, you've found lots of questions and keep not being able to answer them all.
You've done calculations that get really complicated, and you're not quite sure whether it's worth pursuing those calculations.
You can't quite see where they're going to go, even if you succeed. But maybe.
But then in the other direction, maybe you do some complicated calculation and then suddenly something simplifies dramatically.
Now onto something here and then you sort of will carry on and see whether you get anywhere.
So I think that's what it is. I have my process of doing research, and as long as I feel that's moving forwards, then I'll carry on.
And if I feel it's getting sort of. I'm getting to the point where it's not getting forward very fast.
Then I'll be better off working on something else as you may be picking up.
I'm quite I'm quite calculating about the process of what to work on with any one time,
and I think you have to be to be successful because that matters because there's a lot of twin risks of giving
things up too easily and not solving anything or not giving things up easily enough and not solving anything.
And he's got to find a sweet spot in between the two. There's another question just behind you to Africa that you might find perfect.
I was just wondering if you think any of the millennium problems remaining or any problem in general are truly unsolvable or Iberville?
I don't really know. I thought of the millennium problems.
I think the one that people talk about as possibly being on unsolvable is the people's envy problem.
My own feeling about that is that it's not because not for a good reason.
But there's a way of formulating it that makes it look very combinatorial.
Very sort of like that. Doesn't mention computers at all. Doesn't sound like a problem to do with logic.
So it doesn't feel like the kind of problem that would be insoluble.
But that's not a good reason, because there are much more sophisticated reasons for thinking that it might be that by people
who are fully aware of what I just said about it having this combinatorial reformulation.
So. As for the other ones, I really have no idea at all I.
They say the Raymond hypothesis, I'd be very surprised if that were undecided.
But it always comes as a surprise when a problem that's not sort of set up to be undesirable is on the side of will.
So Bradley surprises do happen. The question just here, it will go, and it will continue.
I think it's a bitter thought that the decision will come to you in a second.
Yeah, thank you very much. It's very it's an honour to listen to you.
My first question is what is the most widely accepted mathematical proposition that could possibly turn out to be wrong?
And second is just like a quick question. If given a chance to reason with the mathematician, who would you like to be and why?
The answer the first is I really don't know at all.
I could say I'm sort of generally maybe I would mention someone I know called Kevin Buzzard, who's quite worried that there might be.
Significant parts of number theory, which is an area which rather unlike my areas, is rather a lot big hierarchy of this.
This theorem uses this term, which uses this term understanding and this term which uses this and.
And so that if you actually were to chase the proof of this term at the top through all the different branches of the tree,
you'd have to read sort of two thousand pages of very difficult stuff,
some of which have been published and some of which is only sort of known to a few experts.
And so there's just a slight anxiety that some of the results of numbers vary.
But which ones I don't know might be wrong in quite serious ways.
And he's he takes that so seriously if set up a whole programme of machine verification of of proofs and number theory.
Who would I like to be? And sort of think like that somehow?
It's a bit like saying, what other person would I like to be? I don't really have an answer.
I have a lot of give up so much else. If I had to, my name would be someone else.
OK, I think maybe was going to say you get sort of attached to your own terms as well.
If you the part of the quid pro quo of being another mathematicians,
I'd have to say goodbye to all those things that I spent a long time on this and work time for.
I think maybe two more quick questions. The answer will give you that one last word at the back end of it.
And it's been fascinating talk. Can I just ask you a bit more about the role of computers in pure mathematics research?
And you've mentioned one or two specific problems and the role of collaboration,
but in terms of fundamental change in the way pure mathematics is done, human rights research has done, as has the computer.
What has the computer given us and what do you think it will give us in the future?
It depends what you call fundamental, but it's given us quite a few things that's given us the ability to check lots of cases,
which has helped, for example, proving the full colour theorem. It's also got a very nice role in helping to.
When you're doing research and you invent questions,
sometimes the questions will be things that were the answer can be checked on a computer because not all problems are amenable to that,
but sometimes they are. So sometimes you might think if this were true, that would be really helpful for this.
But is it true? And instead of. If it's the right sort of question, you might be able to feed it into a computer and get a computer to see,
and it might after a while say here's a complicated counterexample which would have taken you a week
to find and the computer just found it in five minutes or something so that can speed things up.
Another use of a computer that's rather nice is if a rather specific thing, but I think it could be a special case of something more general,
which is that if you're working on a problem and out comes the first few terms of some sequence.
I think you generate some integers, what you can do these days is put that into something called the online encyclopaedia of integer
sequences to see whether that sequence is something that's come up in some other context.
And very often it has. Almost always it has.
And when it has, it will sometimes just give you a formula for the sequence, which would have been extremely hard to find.
It will show you that what you're thinking about might be connected with something that you hadn't thought of and so on and so forth.
So computers can be very helpful in just speeding up the natural processes of research.
Going back to your going to the question of where things might go in the future.
I personally think that computers will eventually take over and just do everything for us.
But that's a minority view. Most people think that that humans have this sort of intuition that will never be
replicated on a computer that is absolutely essential to the process of doing mass.
I I don't believe that it will take a while, but.
But I think and before that happens,
I think computers will be able to do the sort of easier parts of mathematics and will that will be very helpful as well because then again,
if you ask a question on the computer can answer easy questions. And if that question is easy or even if it's not easy for you,
but easy for an expert and the right branch of mass on the computer might then be able to help answer it.
That could be a huge speed up. I think they will at some stage be a sort of golden age of mathematics in which computers
are incredibly helpful but don't quite spoil all the fun for they may be short lived.
Yeah, I did my bit. OK, last question, plus a quick question and then one at that.
Thank you. Thank you for speaking with us tonight.
A few years ago on your blog,
you wrote about teaching mathematics to non mathematicians and as a teacher of A-level maths is here with some A-level students.
I'm wondering if you have any advice for how we should be teaching mathematics to students who are about to go off to university,
to study mathematics or other quantitative fields. So not teaching mathematics and not mathematicians, but teaching it to mathematicians.
Well, I don't know how practical this advice is, because I know that now I feel sort of wary of offering advice to schools.
You should do this because I know that if you're actually out there in the chalk face with a big class and a very busy time table and so on,
you can't just sort of do anything that suggests it. But I do think that.
As a general, I feel very lucky that I myself school was taught by people who didn't really say it my last two years didn't.
Worry too much about the A-level syllabus, but just sort of taught maths,
and at the end of the two years you found that actually you had covered material,
but we did lots of things that weren't ready to do with the A-level syllabus,
and we were given problems that were much more challenging than you'd find on an A-level.
It was necessary to do those problems to A-level, but it was really a very, very good experience.
So it may not be a practical thing to do that in every school.
Especially now when people do for A-levels and but. I think if space can be found for.
Just the more that can be taught, we are not sort of focussing on the next level module.
Perhaps. Maybe there aren't modules anymore, but that used to be when our modules.
It was particularly bad because I was always next time, just around the corner.
But the more it's practical not to focus on exams, the better.
But. As I say, I repeat that I understand this.
This is certainly easy to do that.
I think that's a very good point for us to leave it on playing outside of what you have to do and trying to find the joy in it all.
Thank you very much, everyone, for coming, and it remains to thank you very much.