Warning everybody. It's fantastic to see lots of you here and also highlight to the people who are not in this room.
So my name's Vicky Neal. Just you have some idea who I am.
I'm a lecturer here in the maths department, so I spend some of my time teaching undergraduates.
This year is teaching the first year linear algebra course.
In fact, in this world, they're not quite so many first years as there are of you today, so they have a bit more space to spread themselves out.
I spent some of my time at Paleo College on one of the mass shooters there, so I get involved in teaching students and supporting them individually.
And then I spend the rest of my time working on public engagement with maths, which means a variety of projects.
My instructions from James today are that I have half an hour to tell you about the whole of mathematics.
So we'll see how that goes. So when I'm lecturing the linear algebra, the format is kind of I write notes on the board slowly and carefully.
They're also online notes, but all of the details on the board and we have careful definitions of theorems and detailed proofs.
And this is not going to be that kind of talk because I've got half an hour for the whole of mathematics.
So what I want to give you is a flavour of some of the ideas that come up and sort of give
you an a bit of an idea of how they link to courses that we currently offer in our programme.
And I thought I'd start with an equation. And as you know, maths is not all about equations, but equations is definitely important in mathematics.
So the equation I thought we'd start with is y squared is executed minus two, which is the nice friendly sort of equation.
And I'm particularly interested in hole number solutions to this equation.
Integer solutions integer just being total number positive or negative or zero integer solutions to this equation.
So as I'm talking, you might notice solutions, possibly,
but somehow we need to make sure that we find all of the solutions so we don't have to do a little bit more than just spot solutions.
And experience suggests the one thing that can be helpful in this kind of situation
is to think about numbers being order numbers being even that kind of thing.
So let's see what happens if y is even so if Y is even then x cubed, which is why squared plus two.
So if Y is even then y squared is even so y squared plus two is even so x cubed is
even and if executed is even then X is even because if X is odd and I keep it,
I get an odd number. So if the cube is even the original number must have been even more so.
If X is even that means that X Cubed is a multiple of eight, because if X is even,
it's two times something executed is two times something times to time, something times two times something.
So it's divisible by eight but x cubed we know is y squared plus two.
So Y squared plus two is a multiple of eight. But it isn't because y is even so y squared.
It's a multiple of four y squared plus two is not a multiple of four, let alone a multiple of eight.
So what that tells us is that it's not possible for Y to be even they can't be any solutions when Y is even.
So, this is good, right? We just ruled out infinitely many possible solutions right there,
and that just leaves us with the remaining infinitely many that didn't go as well as I'd hoped.
So this says why must be odd. And I don't know have some clever kind of trick up my sleeve for showing that why can't be odd?
Because y can be. There are solutions when war is not OK.
So we need another plan. This is how much is right, so sometimes you get a massive problem and you look at it and you sort of know how to solve it,
you kind of know what techniques, what ideas,
you know, if I apply these tools carefully and accurately and work through systematically or come up with the answer,
and that can sometimes be kind of satisfying. But for me, those are not the most interesting maths problems for me, the most interesting problems,
the ones where you look at it and go, I'm not sure I've got lots of tools at my disposal, but which ones might be relevant?
Which branches of maths? Why I need to draw on to get help me to solve that problem.
And when you're in that kind of situation, you have to be prepared for the fact that the first thing you try.
All of us that 10 things you try might not work out. So you have to be willing to kind of put pen to paper, try something.
OK, well, that didn't help. We still learn something, but OK, it didn't solve the problem for us.
So we need another idea.
And my other idea is going to have think about the equation in this rewritten form, so I wrote it in this form at the top for a reason.
I'll tell you why later, but let's think about it in this rearranged form, x cubed is y squared plus two, because now I have a plan.
My plan is to take y squared plus two factories.
It's just a difference of two squares.
And that some of you are looking slightly anxious, possibly because it's not a difference, and they're not both squares.
But I think we should be open minded because if I'd be right, this is why score at minus minus two.
A difference. And if we tell ourselves minus two is a square, it's a difference of two squares.
And I can fight twice this just fine. It's y plus the square root of minus two times y minus the squared minus two.
And now some of you look as though you might be having a moment of existential crisis because I've just ripped the sweater to minus two on the boat.
Please don't have an existential crisis. I promise it's all fine. What do I mean by the square root of minus two?
Well, different question. What do I mean by the square root of two?
What I mean by the square root of two, is this a positive number and if I square which I get to?
I think that's all I know about the square root of two.
In fact, anything else I may think I know, like it's one point for whatever I did and the fact that it's positive and if I squash, I get two.
What do I mean by the square root of minus two? It's an object that when I square, which I get minus two.
So what we can do here is instead of just working with a nice, warm, friendly, cuddly integers,
go into this brave new world where we have numbers of the phone integer plus an integer times the square to minus two,
well, square minus two is this slightly mysterious thing. But any time we see it square, we'll just replace it by minus two.
Now, my guess is that some of you have met complex numbers and some of you haven't,
I know it depends which what do you do topics at school or college?
So if you haven't met complex numbers, please don't worry, it doesn't matter for this.
Talk over the float. So good. If you have that complex numbers and you're wondering why I've written the square to minus two rather than two times I.
The answer is it wouldn't be wrong to do that, but this is the convention in this branch of maths that we write numbers in this form.
So we're looking at numbers of the form integer plus in-situ Times Square to minus two.
It's really not clear that that is a good idea for solving this problem because I've taken a nice friendly
equation looking for a nice friendly solutions and turned it into something looking a little bit more alarming.
But in the absence of any other ideas, we may as well go with it.
And when you think about it for a bit, you discover that y plus the squared to minus two y minus squared minus two have highest common factor one.
I'm going to record that using this brilliant phrase. It turns out that which means I'm not going to tell you the details.
It turns out that y plus the squared minus two and y minus the square root of minus
two have highest common factor one proving that takes five minutes of algebra.
I don't have time for five minutes of algebra. In the course of that proof, though, we use the fact the Y is odd.
So this thing up here, which felt like it, it felt like, Oh, that didn't work.
Actually, we learnt something useful about the problem that helps us later on.
So when you when you're working on a problem, you try something it doesn't work out. Don't get cross in the recycling bin.
Hang on to. It might be useful later on. So what's going on here?
We've got two numbers with high school factor one the multiplied to give a cube.
If that happens in the ordinary integers, if I've got cubicles,
product of two numbers with high school and fat to one, each of those numbers individually must be a cube.
So we can use the same kind of idea here to say that y plus or minus root minus two must be cubes.
And it doesn't matter which we focus on, let's just look at one of them. So say Y.
Plus, the square root of minus two is a cube,
and a cube in this world means a cube of a number of the form a plus B minus two where A and B ordinary integers.
And that we can just multiply this out and you can do that better than I can.
Algebra in public is not my favourite activity, but let's give it a go.
So I think this is a cubed plus three squared B Times Square two minus two plus three
a b squared squared minus two squared plus B keeps Times Square to minus two cubed.
It's really not clear this is making the problem better, is it?
But we can tidy up because remember, the deal is any time I see the square root of minus two squared, I can simplify.
So let's do a bit of tidying up. This is a cubed plus three a squared B root minus two, so I get a minus two here.
So I think this is minus six a b squared and it's a minus two lurking in there.
So I get minus Typekit square to minus two.
And now I'm going to tidy this up so that it's written in the form integer plus two times the square root minus two.
So this is a cubed minus six a b squared plus three a square be minus two b cubed times the square root of minus two.
So now I have a number of the form integer plus integer times group minus two being equal to another number of that form,
and the coefficients have to match up. So it must be the case that y is equal to this bit.
And this is the coefficient of minus two, which here is one. So in particular, we get the one is three a square be minus two b cubed.
And now for the first time, I have a glimmer of hope that we might be making some progress because I just noticed that we can factor is this.
So this is b times three squared minus two b squared.
And the reason I'm pleased about that is that I've now written one as a product of two ordinary integers,
and it's really hard to get one by multiplying together two whole numbers. They have to be one times one to minus one times minus one.
So we discover the B has to be plus or minus one. We don't want to know about B.
We want to know about X and Y, but we'll come to that. We can work out a because three.
Well, three. A squared minus two b squared must also be plus or minus one.
And when you solve it, turns out that that means that a must be plus or minus one.
And now we can go back here because we know that y is so this part. So we discover that Y is a cube two minus six a b squared.
And now I've got four cases to substitute in, but I'm a mathematician, so I'm kind of lazy.
So rather than doing four cases,
I notice that this is a times a square minus six b squared because squared minus six squared doesn't matter what case where it works out the same way.
So we get plus or minus five. And if we go back to the original equation that tells us that X is three.
And so we discover that the solutions are three, five and three minus five.
Although nice numbers are small enough that I'm sure that some of you spotted those.
And hopefully, if you spotted this one, you might have spotted this one as well,
because the original equation doesn't care whether it's positive or negative. So we expect them kind of come in pairs like this.
The important thing about what we've done here is that it shows that those are the only solutions otherwise.
Well, maybe there's some solution with hundred and fifty digits we haven't thought of yet.
This shows that these are the only solutions. Good.
I have two confessions to make. I'm just wondering in which order to confess.
OK, first confession, this is 30 of undergraduate material. Did I mention that?
So if you didn't follow the details, don't worry, I didn't give you all the details.
And this is stuff that comes up in a three year course called Outbreak Number Theory.
So I don't worry about if you didn't follow the details. The second thing is that I slightly cheated.
But let me tell you where I cheated, because it's quite subtle cheating, and it's quite interesting cheating.
The cheating was that. So to tell you about the cheating, I want to talk about famous laughs there.
If you had a famous lobster, some people have had a famous lost there and let me write it down because it's
hard to guess how it's spelled from the way I said Fermat's Last Theorem.
So back in the 17th century, Foma was thinking about this equation X to the N plus, y to the N is set to the N and defenceless.
The ancient Greeks knew about this equation when and is to.
This is the familiar equation when any familiar equation for the side to a right angle triangle.
And we know that their whole numbers solutions,
people who write maths textbooks know the three four five is a right triangle of five 12 13 three square plus fourth grade is five squared and so on.
And the Greeks can find all of the solutions this question one, and it's two, and it's a beautiful problem.
I recommend it. Fama said. Well, what happens if and is at least three?
Can we somehow generalise these beautiful ideas and think about what happens when an is at least three?
And if any three, for example, you get some very interesting solutions because one cubed plus minus one cubed.
Is there a cube? That's not very interesting. So we're interested really in solutions where X, Y and Z are positive whole numbers,
and some are claims that there were no solutions to this equation in positive integers when a. at least three.
And he claimed to be able to prove that it didn't provide any evidence of that.
I don't really know why go to Theorem because he didn't provide any evidence that he could prove it, but that we go in the 19th century.
Mathematicians have this beautiful plan for proving farmers lost their here is a very high level summary of that strategy.
OK, ready. So you take X and plus y to the end, you throw in a complex,
read through each of you to see whatever that is and use it to factories x plus watch the end as a product of linear factors.
This is the same kind of idea of throwing in square ish minus two to get a difference of two squares.
So then you've got this product of any effects as being equal to Z to the end,
and you mumble a bit about the linear factors having highest common factor one and argue that
individually that means they must each be intense power and then you get a contradiction. Just like that.
And that was great until somebody pointed out why it doesn't work. And the reason it doesn't work is really subtle.
And it's to do with this property of unique prime factorisation in the integers.
If I pick a hole no bigger than one, I can write it as a product of subprimes. So if we pick like 12, we can write it as two times three times three.
So product to supply numbers. Importantly, there's only one way of doing that.
I mean, maybe some of you would like to is two times three times two or three times two times two.
It doesn't matter which order you put the factors in, but you have to use the same pool of prime factors.
We call that the uniqueness of prime factorisation, and that's not just true for 12.
That's true for any positive whole number. And I know that's true because it's a theorem.
We can prove it. It turns out that in some of these settings where we're trying to generalise the introduced,
where we throw in a complex and through to unity or square root of that or something,
we have something that behaves a lot like the integers we can add. We can multiply, we call it a ring.
But we don't necessarily have to have unique prime factorisation. And if you don't have a unique prime factorisation, then the argument here,
here's a productive end things with high school, in fact one that's equal to an end to power.
Therefore, they're all powers that doesn't work. You need unique prime factorisation in that.
And it turns out sometimes when you throw in a complex and through to unity, you don't have unique factorisation.
So this priest actually doesn't work. So it didn't prove farmers lost their home in the 19th century.
And I guess at that point, mathematicians might say, God, it's really hard to give up mathematicians, very resilient people.
And in fact, 100 150 years of modern algebraic number theory has arisen from trying to understand what's going on here.
How come the integers do have this property,
which I think almost hadn't really been appreciated until they discovered it failed in these other settings?
So how can the integers do? How come some of these other settings don't?
How can we tell whether the setting does or doesn't have this unique prime factorisation property?
What can we do about this if it doesn't? So our second year you do the rings.
The modules start to learn about some of this because we're talking about rings here, start to explore unique prime factorisation,
and that then feeds on into the third year algebra number theory course, which extends those ideas even further.
So in fact, it turns out that if you throw in the square root of minus two, you do have unique prime factorisation.
So if this does work, but there's a little bit of work to do to prove that sort of.
So I'm not really cheating. I'm just not telling you about some of the details.
So just to finish the story of farmers loss there, and for those of you who are not familiar with it, it has now been proved.
It was proved in the mid-1990s by Andrew Wiles and building on work of predecessors,
which is why, of course, we're in the Andrew Wallace building today. So he's one of my colleagues got office upstairs.
So, ah, OK. So we know about integer solutions to this equation.
What about real solutions to this equation so we could draw some graphs? Let's try sketching graphs.
This is a fun thing to do. Let's warm up. Let's try sketching a graph of Y is x cubed minus two, so Y equals x cubed looks something like this.
So Y cusecs cubed minus two looks something like this where this is minus two.
And I'm not fantastic to join Cube X, but it's just a sketch. So let's find something like that.
So what happens? If instead, we try doing y squared equals x cubed minus two.
Have you tried sketching off of y squared equals some function of X? You should try it.
Really cool. Things happen. So y squared is a negative, so anywhere over here were executed.
Minus two is negative. We're not going to get anything over here when x k minus two is positive.
We go to get two values because this equation doesn't mind whether it is positive or negative.
So we expect symmetry about the horizontal axis,
and this graph is going to kind of stretch depending on whether x q minus two is less than one or more than one,
whether it's gets bigger or smaller when you square and it turns out what you end up with is something looks a bit like this.
It is kind of intriguing, and this is an example of an elliptic curve, which is a confusing name because they're not ellipses,
so elliptic curve is something of the form y squared equals a cubic in X.
And actually, there's a nice link back here because when Andrew was proofs, farmers lost their ellipticals were a key part of that.
So if you haven't seen the BBC documentary about this, you should totally go.
And Google BBC iPlayer farmers lost their a one hour documentary.
It's been over 20 years old now, but it's fantastic. You should watch it if you have other key so you can have other shapes be elliptical.
So let me show you some of the kinds of things that can happen. So one thing that might happen is that my elliptic curve goes up and it turns around,
but it doesn't get back to the horizontal axis and then it turns round again, so I can have a cubic like that.
And the corresponding elliptic curve looks something a bit like this.
That was supposed to be symmetric, and it's actually not a bad effort, so it's just having a moment to be proud of it.
My pick might cut the horizontal axis three times, and in that case, the elliptic curve kind of falls apart into pieces.
And you end up with something maybe looking a bit like this. It's kind of intriguing.
So this one starts to link up with an area of maths called algebraic geometry, and we have a three year course called algebraic curves.
So algebraic geometry built in this idea that Descartes is famous for Cartesian coordinates other people as well.
But you can take an algebraic equation and turn it into a graph as we don't.
They don't pay attention to a problem in geometry, or you can take from geometry and turn it into algebra.
Using coordinates. So algebraic geometry kind of takes this idea of, well, how can I use tools from algebra to answer problems in geometry?
How can I use tools from geometry to answer problems in algebra? What are the parallels between these areas?
That's a really fruitful avenue for exploration. OK, so we thought about instant solutions.
We thought about real solutions. Let's think about rational solutions. Why not?
So a rational number means whole number divided by not zero whole numbers, by fractions, if you like.
And this is really exciting because if you take two rational points on elliptic curve,
meaning both coordinates are rational, you can add them and get another one.
So if you have a nicely behaved elliptic curve, you could take a rational point p.
So both its coordinates are rational, they're both fractions. I can take a rational point.
Q and then I add them are the way that I add them is that I draw a straight line through them.
Which by a miracle of mathematics, intersects the curve just once more.
And that's not the answer. We then reflect in the horizontal axis.
And that's the answer. We call that plus key. And it might feel a bit odd to call that addition.
In what sense, is this addition we're sort of doing something quite geometric working with this curve.
But it turns out that it behaves in the same way as edition of normal real numbers in various useful ways.
And this is an example of what happens in university, this process of abstraction, which comes up in algebra all the time.
This is super important.
So you notice the same phenomenon happening in seemingly disparate parts of mathematics and you go, Well, what do they have in common?
What is the underlying structure that brings those together? Because if you can then isolate that structure,
a proof theorems about that structure that applies to all the examples you've noticed and all the examples you haven't encountered yet.
So in fact, the rings that I mentioned earlier on, it's an example of that process.
The structure of choice here is the group. So you find this going on with addition of real numbers.
You find it going on with symmetries of an equilateral triangle. You find it going on with multiplying vertebral matrices.
There's lots of these kind of examples of groups. So what properties does a group must have?
I'm not going to list them all. One important property is that there has to be a point that behaves like zero.
We call it an identity. It doesn't change it. So here we need to point zero so that when you do p plus that point zero, you get back p again.
And the point zero turns out to be the point. It's infinity, which I can't do on the whiteboard because it's infinity.
So to make sense to the point it's infinity in a precise way uses ideas from projective geometry.
That's the second optional course. So you make sense of this point.
It's infinity, and it turns out the looking elliptical is using projective geometry. It's really nice that also ties in with this algebraic geometry.
This might seem quite abstract, actually. It turns out that rational points on elliptic curves,
as well as having intrinsic properties of mathematical interest, are also useful for cryptography.
So very roughly speaking, the idea is that if you pick a point,
a rational point on a nicely behaved elliptic curve, you can add it to itself lots of times, fairly easily.
So if I give you a point p, you can work out seven p fairly quickly, especially with your computer.
If I tell you that this is seven p, it's quite hard for you to work out what the original point p was.
And that asymmetry is what makes this kind of tool useful for elliptic curve cryptography for its the type of private key public key cryptography.
So people studying cryptography are thinking about rational points of elliptic curves.
They're also thinking about what's going to happen post quantum computers because quantum computers are going
to be able to solve some of these problems too quickly for our current cryptographic techniques to be robust.
So I post quantum cryptography is a hot topic. OK, so we thought about instant solutions.
We thought about real solutions. We thought about rational solutions. We should think about complex solutions, right?
Why not? So, hey, we're interested in where X and Y can be complex numbers and I guess quite hard to draw on
the board because X is in some sense two dimensional and Y is in some sense two dimensional.
Now, I don't have enough room to draw the elliptic curve, but it turns out to be a. A doughnut is a sort of surface with a hole in it.
How can we possibly know that it's a tourist? It's doughnut to shape with a hole in the middle of it?
Well, I haven't even bothered to draw on the white-board.
Well, we can use ideas from algebraic topology to help with geometry of surfaces to make sense of what's going on here.
So algebraic algebraic topology gives us powerful ways to distinguish even in many dimensions between something that looks like a beach ball
with no holes and something that looks like a doughnut with one hole or something that looks like a pretzel with more holes and so on.
So we can use these kind of ideas to discover that it's a tourist,
and one powerful tool for tackling illicit cards in the complex world uses the via Strauss P function, which has its own P written in a special fund.
My P normally looks like this. This is the P for the viceroy's p function, and the viceroy's p function is defined by W Infinite series,
and I used to feel very scared because adding together infinitely many things is a pretty risky
business and putting it together infinitely many things are two directions is kind of terrifying,
but in first year analysis, we make sense of these things.
What does it mean to add together infinitely many things? So maybe you've come across an example where you can add together infinitely?
Many things get a sensible answer, like one plus a half plus a quarter plus an eighth plus a sixteenth consultants an example of a geometric series.
You add infinitely many things. You get a sensible, finite answer. We say the series converges.
But if you're careless, you add together infinitely many things.
It doesn't make sense if you do one plus a half plus a third plus a quarter, plus a fifth in fifth and so on.
That's the harmonic series it doesn't convert. You don't get a finite answer diverges.
So in first your analysis, we're looking at these ideas. Well, what does it mean to say that a sequence converges or a series converges?
What tools do we have for looking at the series and going up? It converges, or no, it doesn't.
Analysis also includes things like what we do, differentiation. We sort of want to take a limit.
What precisely does that mean? So getting the logical, rigorous foundations of that precise, which is a really satisfying thing to do.
What is the link between differentiation? How do we formalise that?
Once you're done first, your analysis, then you can do second year complex analysis, which is like complex numbers meets calculus.
But it's much more beautiful than that makes it sound. It's full of theorems that are really surprising theories.
They shouldn't be true, but somehow all because of miracles of complex analysis and complex analysis.
These Doubly Infinite series and you've got these tools for making sense of it, so you can make sense of the Vice Jospeh function,
and you can show that it satisfies the differential equation that starts to have the form of an elliptical serve.
So you can tackle questions about ellipticals cards by relating back to the Vice Jospeh function.
Via this differential equation, I could keep talking about elliptic curves,
but I think that I have to stop because you have like another chalk or something.
I hope that's given you a bit of a flavour of some of the topics that come up in Pure Massey University, and I hope you enjoy the rest of your day.
Thank you very much.