1 00:00:11,380 --> 00:00:17,920 Warning everybody. It's fantastic to see lots of you here and also highlight to the people who are not in this room. 2 00:00:17,920 --> 00:00:20,500 So my name's Vicky Neal. Just you have some idea who I am. 3 00:00:20,500 --> 00:00:24,790 I'm a lecturer here in the maths department, so I spend some of my time teaching undergraduates. 4 00:00:24,790 --> 00:00:26,740 This year is teaching the first year linear algebra course. 5 00:00:26,740 --> 00:00:33,670 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. 6 00:00:33,670 --> 00:00:39,970 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. 7 00:00:39,970 --> 00:00:45,490 And then I spend the rest of my time working on public engagement with maths, which means a variety of projects. 8 00:00:45,490 --> 00:00:51,160 My instructions from James today are that I have half an hour to tell you about the whole of mathematics. 9 00:00:51,160 --> 00:00:59,830 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. 10 00:00:59,830 --> 00:01:06,190 They're also online notes, but all of the details on the board and we have careful definitions of theorems and detailed proofs. 11 00:01:06,190 --> 00:01:10,870 And this is not going to be that kind of talk because I've got half an hour for the whole of mathematics. 12 00:01:10,870 --> 00:01:16,540 So what I want to give you is a flavour of some of the ideas that come up and sort of give 13 00:01:16,540 --> 00:01:20,770 you an a bit of an idea of how they link to courses that we currently offer in our programme. 14 00:01:20,770 --> 00:01:28,480 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. 15 00:01:28,480 --> 00:01:39,160 So the equation I thought we'd start with is y squared is executed minus two, which is the nice friendly sort of equation. 16 00:01:39,160 --> 00:01:42,670 And I'm particularly interested in hole number solutions to this equation. 17 00:01:42,670 --> 00:01:49,480 Integer solutions integer just being total number positive or negative or zero integer solutions to this equation. 18 00:01:49,480 --> 00:01:54,100 So as I'm talking, you might notice solutions, possibly, 19 00:01:54,100 --> 00:02:00,040 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. 20 00:02:00,040 --> 00:02:03,790 And experience suggests the one thing that can be helpful in this kind of situation 21 00:02:03,790 --> 00:02:07,780 is to think about numbers being order numbers being even that kind of thing. 22 00:02:07,780 --> 00:02:19,630 So let's see what happens if y is even so if Y is even then x cubed, which is why squared plus two. 23 00:02:19,630 --> 00:02:25,840 So if Y is even then y squared is even so y squared plus two is even so x cubed is 24 00:02:25,840 --> 00:02:36,880 even and if executed is even then X is even because if X is odd and I keep it, 25 00:02:36,880 --> 00:02:41,380 I get an odd number. So if the cube is even the original number must have been even more so. 26 00:02:41,380 --> 00:02:52,600 If X is even that means that X Cubed is a multiple of eight, because if X is even, 27 00:02:52,600 --> 00:02:56,380 it's two times something executed is two times something times to time, something times two times something. 28 00:02:56,380 --> 00:03:01,900 So it's divisible by eight but x cubed we know is y squared plus two. 29 00:03:01,900 --> 00:03:12,430 So Y squared plus two is a multiple of eight. But it isn't because y is even so y squared. 30 00:03:12,430 --> 00:03:22,390 It's a multiple of four y squared plus two is not a multiple of four, let alone a multiple of eight. 31 00:03:22,390 --> 00:03:29,260 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. 32 00:03:29,260 --> 00:03:33,670 So, this is good, right? We just ruled out infinitely many possible solutions right there, 33 00:03:33,670 --> 00:03:43,980 and that just leaves us with the remaining infinitely many that didn't go as well as I'd hoped. 34 00:03:43,980 --> 00:03:54,000 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? 35 00:03:54,000 --> 00:03:59,310 Because y can be. There are solutions when war is not OK. 36 00:03:59,310 --> 00:04:05,830 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, 37 00:04:05,830 --> 00:04:07,740 you kind of know what techniques, what ideas, 38 00:04:07,740 --> 00:04:13,800 you know, if I apply these tools carefully and accurately and work through systematically or come up with the answer, 39 00:04:13,800 --> 00:04:19,620 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, 40 00:04:19,620 --> 00:04:25,470 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? 41 00:04:25,470 --> 00:04:30,360 Which branches of maths? Why I need to draw on to get help me to solve that problem. 42 00:04:30,360 --> 00:04:34,530 And when you're in that kind of situation, you have to be prepared for the fact that the first thing you try. 43 00:04:34,530 --> 00:04:41,180 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. 44 00:04:41,180 --> 00:04:46,720 OK, well, that didn't help. We still learn something, but OK, it didn't solve the problem for us. 45 00:04:46,720 --> 00:04:48,640 So we need another idea. 46 00:04:48,640 --> 00:04:55,450 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. 47 00:04:55,450 --> 00:05:05,110 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. 48 00:05:05,110 --> 00:05:08,890 My plan is to take y squared plus two factories. 49 00:05:08,890 --> 00:05:12,230 It's just a difference of two squares. 50 00:05:12,230 --> 00:05:17,600 And that some of you are looking slightly anxious, possibly because it's not a difference, and they're not both squares. 51 00:05:17,600 --> 00:05:22,610 But I think we should be open minded because if I'd be right, this is why score at minus minus two. 52 00:05:22,610 --> 00:05:27,620 A difference. And if we tell ourselves minus two is a square, it's a difference of two squares. 53 00:05:27,620 --> 00:05:36,660 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. 54 00:05:36,660 --> 00:05:42,570 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. 55 00:05:42,570 --> 00:05:48,120 Please don't have an existential crisis. I promise it's all fine. What do I mean by the square root of minus two? 56 00:05:48,120 --> 00:05:51,860 Well, different question. What do I mean by the square root of two? 57 00:05:51,860 --> 00:05:57,850 What I mean by the square root of two, is this a positive number and if I square which I get to? 58 00:05:57,850 --> 00:06:00,550 I think that's all I know about the square root of two. 59 00:06:00,550 --> 00:06:07,660 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. 60 00:06:07,660 --> 00:06:12,550 What do I mean by the square root of minus two? It's an object that when I square, which I get minus two. 61 00:06:12,550 --> 00:06:17,530 So what we can do here is instead of just working with a nice, warm, friendly, cuddly integers, 62 00:06:17,530 --> 00:06:23,110 go into this brave new world where we have numbers of the phone integer plus an integer times the square to minus two, 63 00:06:23,110 --> 00:06:29,840 well, square minus two is this slightly mysterious thing. But any time we see it square, we'll just replace it by minus two. 64 00:06:29,840 --> 00:06:32,960 Now, my guess is that some of you have met complex numbers and some of you haven't, 65 00:06:32,960 --> 00:06:35,960 I know it depends which what do you do topics at school or college? 66 00:06:35,960 --> 00:06:39,410 So if you haven't met complex numbers, please don't worry, it doesn't matter for this. 67 00:06:39,410 --> 00:06:46,580 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. 68 00:06:46,580 --> 00:06:52,190 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. 69 00:06:52,190 --> 00:06:56,090 So we're looking at numbers of the form integer plus in-situ Times Square to minus two. 70 00:06:56,090 --> 00:07:00,680 It's really not clear that that is a good idea for solving this problem because I've taken a nice friendly 71 00:07:00,680 --> 00:07:05,060 equation looking for a nice friendly solutions and turned it into something looking a little bit more alarming. 72 00:07:05,060 --> 00:07:09,210 But in the absence of any other ideas, we may as well go with it. 73 00:07:09,210 --> 00:07:19,590 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. 74 00:07:19,590 --> 00:07:25,230 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. 75 00:07:25,230 --> 00:07:33,120 It turns out that y plus the squared minus two and y minus the square root of minus 76 00:07:33,120 --> 00:07:38,550 two have highest common factor one proving that takes five minutes of algebra. 77 00:07:38,550 --> 00:07:45,300 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. 78 00:07:45,300 --> 00:07:48,270 So this thing up here, which felt like it, it felt like, Oh, that didn't work. 79 00:07:48,270 --> 00:07:51,840 Actually, we learnt something useful about the problem that helps us later on. 80 00:07:51,840 --> 00:07:58,470 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. 81 00:07:58,470 --> 00:08:02,550 Hang on to. It might be useful later on. So what's going on here? 82 00:08:02,550 --> 00:08:08,460 We've got two numbers with high school factor one the multiplied to give a cube. 83 00:08:08,460 --> 00:08:11,790 If that happens in the ordinary integers, if I've got cubicles, 84 00:08:11,790 --> 00:08:17,450 product of two numbers with high school and fat to one, each of those numbers individually must be a cube. 85 00:08:17,450 --> 00:08:25,790 So we can use the same kind of idea here to say that y plus or minus root minus two must be cubes. 86 00:08:25,790 --> 00:08:30,440 And it doesn't matter which we focus on, let's just look at one of them. So say Y. 87 00:08:30,440 --> 00:08:32,420 Plus, the square root of minus two is a cube, 88 00:08:32,420 --> 00:08:39,860 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. 89 00:08:39,860 --> 00:08:43,640 And that we can just multiply this out and you can do that better than I can. 90 00:08:43,640 --> 00:08:47,180 Algebra in public is not my favourite activity, but let's give it a go. 91 00:08:47,180 --> 00:08:54,830 So I think this is a cubed plus three squared B Times Square two minus two plus three 92 00:08:54,830 --> 00:09:03,500 a b squared squared minus two squared plus B keeps Times Square to minus two cubed. 93 00:09:03,500 --> 00:09:07,790 It's really not clear this is making the problem better, is it? 94 00:09:07,790 --> 00:09:14,240 But we can tidy up because remember, the deal is any time I see the square root of minus two squared, I can simplify. 95 00:09:14,240 --> 00:09:22,850 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. 96 00:09:22,850 --> 00:09:28,110 So I think this is minus six a b squared and it's a minus two lurking in there. 97 00:09:28,110 --> 00:09:32,390 So I get minus Typekit square to minus two. 98 00:09:32,390 --> 00:09:38,300 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. 99 00:09:38,300 --> 00:09:49,220 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. 100 00:09:49,220 --> 00:09:55,250 So now I have a number of the form integer plus integer times group minus two being equal to another number of that form, 101 00:09:55,250 --> 00:10:00,380 and the coefficients have to match up. So it must be the case that y is equal to this bit. 102 00:10:00,380 --> 00:10:10,610 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. 103 00:10:10,610 --> 00:10:18,770 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. 104 00:10:18,770 --> 00:10:24,020 So this is b times three squared minus two b squared. 105 00:10:24,020 --> 00:10:30,380 And the reason I'm pleased about that is that I've now written one as a product of two ordinary integers, 106 00:10:30,380 --> 00:10:37,040 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. 107 00:10:37,040 --> 00:10:42,440 So we discover the B has to be plus or minus one. We don't want to know about B. 108 00:10:42,440 --> 00:10:48,980 We want to know about X and Y, but we'll come to that. We can work out a because three. 109 00:10:48,980 --> 00:10:53,540 Well, three. A squared minus two b squared must also be plus or minus one. 110 00:10:53,540 --> 00:10:58,250 And when you solve it, turns out that that means that a must be plus or minus one. 111 00:10:58,250 --> 00:11:08,030 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. 112 00:11:08,030 --> 00:11:12,710 And now I've got four cases to substitute in, but I'm a mathematician, so I'm kind of lazy. 113 00:11:12,710 --> 00:11:14,870 So rather than doing four cases, 114 00:11:14,870 --> 00:11:24,020 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. 115 00:11:24,020 --> 00:11:30,110 So we get plus or minus five. And if we go back to the original equation that tells us that X is three. 116 00:11:30,110 --> 00:11:42,260 And so we discover that the solutions are three, five and three minus five. 117 00:11:42,260 --> 00:11:46,070 Although nice numbers are small enough that I'm sure that some of you spotted those. 118 00:11:46,070 --> 00:11:49,640 And hopefully, if you spotted this one, you might have spotted this one as well, 119 00:11:49,640 --> 00:11:55,460 because the original equation doesn't care whether it's positive or negative. So we expect them kind of come in pairs like this. 120 00:11:55,460 --> 00:12:01,520 The important thing about what we've done here is that it shows that those are the only solutions otherwise. 121 00:12:01,520 --> 00:12:05,060 Well, maybe there's some solution with hundred and fifty digits we haven't thought of yet. 122 00:12:05,060 --> 00:12:11,540 This shows that these are the only solutions. Good. 123 00:12:11,540 --> 00:12:17,000 I have two confessions to make. I'm just wondering in which order to confess. 124 00:12:17,000 --> 00:12:21,620 OK, first confession, this is 30 of undergraduate material. Did I mention that? 125 00:12:21,620 --> 00:12:24,920 So if you didn't follow the details, don't worry, I didn't give you all the details. 126 00:12:24,920 --> 00:12:28,760 And this is stuff that comes up in a three year course called Outbreak Number Theory. 127 00:12:28,760 --> 00:12:34,640 So I don't worry about if you didn't follow the details. The second thing is that I slightly cheated. 128 00:12:34,640 --> 00:12:40,700 But let me tell you where I cheated, because it's quite subtle cheating, and it's quite interesting cheating. 129 00:12:40,700 --> 00:12:51,700 The cheating was that. So to tell you about the cheating, I want to talk about famous laughs there. 130 00:12:51,700 --> 00:12:55,840 If you had a famous lobster, some people have had a famous lost there and let me write it down because it's 131 00:12:55,840 --> 00:13:02,860 hard to guess how it's spelled from the way I said Fermat's Last Theorem. 132 00:13:02,860 --> 00:13:12,850 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. 133 00:13:12,850 --> 00:13:16,000 The ancient Greeks knew about this equation when and is to. 134 00:13:16,000 --> 00:13:20,440 This is the familiar equation when any familiar equation for the side to a right angle triangle. 135 00:13:20,440 --> 00:13:22,480 And we know that their whole numbers solutions, 136 00:13:22,480 --> 00:13:30,280 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. 137 00:13:30,280 --> 00:13:34,510 And the Greeks can find all of the solutions this question one, and it's two, and it's a beautiful problem. 138 00:13:34,510 --> 00:13:39,370 I recommend it. Fama said. Well, what happens if and is at least three? 139 00:13:39,370 --> 00:13:44,710 Can we somehow generalise these beautiful ideas and think about what happens when an is at least three? 140 00:13:44,710 --> 00:13:51,370 And if any three, for example, you get some very interesting solutions because one cubed plus minus one cubed. 141 00:13:51,370 --> 00:13:59,860 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, 142 00:13:59,860 --> 00:14:06,610 and some are claims that there were no solutions to this equation in positive integers when a. at least three. 143 00:14:06,610 --> 00:14:10,240 And he claimed to be able to prove that it didn't provide any evidence of that. 144 00:14:10,240 --> 00:14:16,900 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. 145 00:14:16,900 --> 00:14:23,710 Mathematicians have this beautiful plan for proving farmers lost their here is a very high level summary of that strategy. 146 00:14:23,710 --> 00:14:28,000 OK, ready. So you take X and plus y to the end, you throw in a complex, 147 00:14:28,000 --> 00:14:34,030 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. 148 00:14:34,030 --> 00:14:38,440 This is the same kind of idea of throwing in square ish minus two to get a difference of two squares. 149 00:14:38,440 --> 00:14:41,950 So then you've got this product of any effects as being equal to Z to the end, 150 00:14:41,950 --> 00:14:45,910 and you mumble a bit about the linear factors having highest common factor one and argue that 151 00:14:45,910 --> 00:14:53,560 individually that means they must each be intense power and then you get a contradiction. Just like that. 152 00:14:53,560 --> 00:14:59,200 And that was great until somebody pointed out why it doesn't work. And the reason it doesn't work is really subtle. 153 00:14:59,200 --> 00:15:03,820 And it's to do with this property of unique prime factorisation in the integers. 154 00:15:03,820 --> 00:15:10,450 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. 155 00:15:10,450 --> 00:15:15,700 So product to supply numbers. Importantly, there's only one way of doing that. 156 00:15:15,700 --> 00:15:19,870 I mean, maybe some of you would like to is two times three times two or three times two times two. 157 00:15:19,870 --> 00:15:24,460 It doesn't matter which order you put the factors in, but you have to use the same pool of prime factors. 158 00:15:24,460 --> 00:15:28,480 We call that the uniqueness of prime factorisation, and that's not just true for 12. 159 00:15:28,480 --> 00:15:32,710 That's true for any positive whole number. And I know that's true because it's a theorem. 160 00:15:32,710 --> 00:15:38,170 We can prove it. It turns out that in some of these settings where we're trying to generalise the introduced, 161 00:15:38,170 --> 00:15:42,100 where we throw in a complex and through to unity or square root of that or something, 162 00:15:42,100 --> 00:15:47,230 we have something that behaves a lot like the integers we can add. We can multiply, we call it a ring. 163 00:15:47,230 --> 00:15:54,930 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, 164 00:15:54,930 --> 00:15:58,270 here's a productive end things with high school, in fact one that's equal to an end to power. 165 00:15:58,270 --> 00:16:02,830 Therefore, they're all powers that doesn't work. You need unique prime factorisation in that. 166 00:16:02,830 --> 00:16:07,000 And it turns out sometimes when you throw in a complex and through to unity, you don't have unique factorisation. 167 00:16:07,000 --> 00:16:11,500 So this priest actually doesn't work. So it didn't prove farmers lost their home in the 19th century. 168 00:16:11,500 --> 00:16:17,590 And I guess at that point, mathematicians might say, God, it's really hard to give up mathematicians, very resilient people. 169 00:16:17,590 --> 00:16:24,490 And in fact, 100 150 years of modern algebraic number theory has arisen from trying to understand what's going on here. 170 00:16:24,490 --> 00:16:27,070 How come the integers do have this property, 171 00:16:27,070 --> 00:16:31,210 which I think almost hadn't really been appreciated until they discovered it failed in these other settings? 172 00:16:31,210 --> 00:16:34,990 So how can the integers do? How come some of these other settings don't? 173 00:16:34,990 --> 00:16:39,430 How can we tell whether the setting does or doesn't have this unique prime factorisation property? 174 00:16:39,430 --> 00:16:43,870 What can we do about this if it doesn't? So our second year you do the rings. 175 00:16:43,870 --> 00:16:49,840 The modules start to learn about some of this because we're talking about rings here, start to explore unique prime factorisation, 176 00:16:49,840 --> 00:16:55,390 and that then feeds on into the third year algebra number theory course, which extends those ideas even further. 177 00:16:55,390 --> 00:17:01,420 So in fact, it turns out that if you throw in the square root of minus two, you do have unique prime factorisation. 178 00:17:01,420 --> 00:17:06,290 So if this does work, but there's a little bit of work to do to prove that sort of. 179 00:17:06,290 --> 00:17:11,860 So I'm not really cheating. I'm just not telling you about some of the details. 180 00:17:11,860 --> 00:17:16,810 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. 181 00:17:16,810 --> 00:17:22,780 It was proved in the mid-1990s by Andrew Wiles and building on work of predecessors, 182 00:17:22,780 --> 00:17:28,840 which is why, of course, we're in the Andrew Wallace building today. So he's one of my colleagues got office upstairs. 183 00:17:28,840 --> 00:17:35,620 So, ah, OK. So we know about integer solutions to this equation. 184 00:17:35,620 --> 00:17:40,030 What about real solutions to this equation so we could draw some graphs? Let's try sketching graphs. 185 00:17:40,030 --> 00:17:54,230 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. 186 00:17:54,230 --> 00:17:59,030 So Y cusecs cubed minus two looks something like this where this is minus two. 187 00:17:59,030 --> 00:18:03,640 And I'm not fantastic to join Cube X, but it's just a sketch. So let's find something like that. 188 00:18:03,640 --> 00:18:12,880 So what happens? If instead, we try doing y squared equals x cubed minus two. 189 00:18:12,880 --> 00:18:17,080 Have you tried sketching off of y squared equals some function of X? You should try it. 190 00:18:17,080 --> 00:18:24,580 Really cool. Things happen. So y squared is a negative, so anywhere over here were executed. 191 00:18:24,580 --> 00:18:29,590 Minus two is negative. We're not going to get anything over here when x k minus two is positive. 192 00:18:29,590 --> 00:18:33,610 We go to get two values because this equation doesn't mind whether it is positive or negative. 193 00:18:33,610 --> 00:18:36,310 So we expect symmetry about the horizontal axis, 194 00:18:36,310 --> 00:18:41,050 and this graph is going to kind of stretch depending on whether x q minus two is less than one or more than one, 195 00:18:41,050 --> 00:18:49,310 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. 196 00:18:49,310 --> 00:18:58,910 It is kind of intriguing, and this is an example of an elliptic curve, which is a confusing name because they're not ellipses, 197 00:18:58,910 --> 00:19:04,610 so elliptic curve is something of the form y squared equals a cubic in X. 198 00:19:04,610 --> 00:19:11,900 And actually, there's a nice link back here because when Andrew was proofs, farmers lost their ellipticals were a key part of that. 199 00:19:11,900 --> 00:19:15,230 So if you haven't seen the BBC documentary about this, you should totally go. 200 00:19:15,230 --> 00:19:19,160 And Google BBC iPlayer farmers lost their a one hour documentary. 201 00:19:19,160 --> 00:19:26,420 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. 202 00:19:26,420 --> 00:19:37,470 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, 203 00:19:37,470 --> 00:19:42,680 but it doesn't get back to the horizontal axis and then it turns round again, so I can have a cubic like that. 204 00:19:42,680 --> 00:19:53,430 And the corresponding elliptic curve looks something a bit like this. 205 00:19:53,430 --> 00:20:03,380 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. 206 00:20:03,380 --> 00:20:11,950 My pick might cut the horizontal axis three times, and in that case, the elliptic curve kind of falls apart into pieces. 207 00:20:11,950 --> 00:20:17,080 And you end up with something maybe looking a bit like this. It's kind of intriguing. 208 00:20:17,080 --> 00:20:23,740 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. 209 00:20:23,740 --> 00:20:29,720 So algebraic geometry built in this idea that Descartes is famous for Cartesian coordinates other people as well. 210 00:20:29,720 --> 00:20:34,590 But you can take an algebraic equation and turn it into a graph as we don't. 211 00:20:34,590 --> 00:20:39,940 They don't pay attention to a problem in geometry, or you can take from geometry and turn it into algebra. 212 00:20:39,940 --> 00:20:46,660 Using coordinates. So algebraic geometry kind of takes this idea of, well, how can I use tools from algebra to answer problems in geometry? 213 00:20:46,660 --> 00:20:51,070 How can I use tools from geometry to answer problems in algebra? What are the parallels between these areas? 214 00:20:51,070 --> 00:20:56,650 That's a really fruitful avenue for exploration. OK, so we thought about instant solutions. 215 00:20:56,650 --> 00:21:00,310 We thought about real solutions. Let's think about rational solutions. Why not? 216 00:21:00,310 --> 00:21:06,490 So a rational number means whole number divided by not zero whole numbers, by fractions, if you like. 217 00:21:06,490 --> 00:21:10,480 And this is really exciting because if you take two rational points on elliptic curve, 218 00:21:10,480 --> 00:21:14,110 meaning both coordinates are rational, you can add them and get another one. 219 00:21:14,110 --> 00:21:19,240 So if you have a nicely behaved elliptic curve, you could take a rational point p. 220 00:21:19,240 --> 00:21:23,330 So both its coordinates are rational, they're both fractions. I can take a rational point. 221 00:21:23,330 --> 00:21:33,070 Q and then I add them are the way that I add them is that I draw a straight line through them. 222 00:21:33,070 --> 00:21:37,840 Which by a miracle of mathematics, intersects the curve just once more. 223 00:21:37,840 --> 00:21:42,250 And that's not the answer. We then reflect in the horizontal axis. 224 00:21:42,250 --> 00:21:48,860 And that's the answer. We call that plus key. And it might feel a bit odd to call that addition. 225 00:21:48,860 --> 00:21:53,300 In what sense, is this addition we're sort of doing something quite geometric working with this curve. 226 00:21:53,300 --> 00:21:59,960 But it turns out that it behaves in the same way as edition of normal real numbers in various useful ways. 227 00:21:59,960 --> 00:22:05,990 And this is an example of what happens in university, this process of abstraction, which comes up in algebra all the time. 228 00:22:05,990 --> 00:22:07,730 This is super important. 229 00:22:07,730 --> 00:22:15,560 So you notice the same phenomenon happening in seemingly disparate parts of mathematics and you go, Well, what do they have in common? 230 00:22:15,560 --> 00:22:21,980 What is the underlying structure that brings those together? Because if you can then isolate that structure, 231 00:22:21,980 --> 00:22:28,070 a proof theorems about that structure that applies to all the examples you've noticed and all the examples you haven't encountered yet. 232 00:22:28,070 --> 00:22:32,000 So in fact, the rings that I mentioned earlier on, it's an example of that process. 233 00:22:32,000 --> 00:22:37,040 The structure of choice here is the group. So you find this going on with addition of real numbers. 234 00:22:37,040 --> 00:22:44,270 You find it going on with symmetries of an equilateral triangle. You find it going on with multiplying vertebral matrices. 235 00:22:44,270 --> 00:22:48,530 There's lots of these kind of examples of groups. So what properties does a group must have? 236 00:22:48,530 --> 00:22:53,420 I'm not going to list them all. One important property is that there has to be a point that behaves like zero. 237 00:22:53,420 --> 00:23:02,150 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. 238 00:23:02,150 --> 00:23:08,300 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. 239 00:23:08,300 --> 00:23:12,620 So to make sense to the point it's infinity in a precise way uses ideas from projective geometry. 240 00:23:12,620 --> 00:23:16,250 That's the second optional course. So you make sense of this point. 241 00:23:16,250 --> 00:23:22,910 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. 242 00:23:22,910 --> 00:23:27,080 This might seem quite abstract, actually. It turns out that rational points on elliptic curves, 243 00:23:27,080 --> 00:23:32,810 as well as having intrinsic properties of mathematical interest, are also useful for cryptography. 244 00:23:32,810 --> 00:23:36,710 So very roughly speaking, the idea is that if you pick a point, 245 00:23:36,710 --> 00:23:42,200 a rational point on a nicely behaved elliptic curve, you can add it to itself lots of times, fairly easily. 246 00:23:42,200 --> 00:23:46,730 So if I give you a point p, you can work out seven p fairly quickly, especially with your computer. 247 00:23:46,730 --> 00:23:52,130 If I tell you that this is seven p, it's quite hard for you to work out what the original point p was. 248 00:23:52,130 --> 00:24:01,550 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. 249 00:24:01,550 --> 00:24:06,140 So people studying cryptography are thinking about rational points of elliptic curves. 250 00:24:06,140 --> 00:24:10,500 They're also thinking about what's going to happen post quantum computers because quantum computers are going 251 00:24:10,500 --> 00:24:15,170 to be able to solve some of these problems too quickly for our current cryptographic techniques to be robust. 252 00:24:15,170 --> 00:24:20,790 So I post quantum cryptography is a hot topic. OK, so we thought about instant solutions. 253 00:24:20,790 --> 00:24:26,370 We thought about real solutions. We thought about rational solutions. We should think about complex solutions, right? 254 00:24:26,370 --> 00:24:32,700 Why not? So, hey, we're interested in where X and Y can be complex numbers and I guess quite hard to draw on 255 00:24:32,700 --> 00:24:36,510 the board because X is in some sense two dimensional and Y is in some sense two dimensional. 256 00:24:36,510 --> 00:24:43,530 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. 257 00:24:43,530 --> 00:24:47,740 How can we possibly know that it's a tourist? It's doughnut to shape with a hole in the middle of it? 258 00:24:47,740 --> 00:24:49,800 Well, I haven't even bothered to draw on the white-board. 259 00:24:49,800 --> 00:24:55,680 Well, we can use ideas from algebraic topology to help with geometry of surfaces to make sense of what's going on here. 260 00:24:55,680 --> 00:25:03,690 So algebraic algebraic topology gives us powerful ways to distinguish even in many dimensions between something that looks like a beach ball 261 00:25:03,690 --> 00:25:09,000 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. 262 00:25:09,000 --> 00:25:12,540 So we can use these kind of ideas to discover that it's a tourist, 263 00:25:12,540 --> 00:25:21,480 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. 264 00:25:21,480 --> 00:25:28,710 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, 265 00:25:28,710 --> 00:25:33,330 and I used to feel very scared because adding together infinitely many things is a pretty risky 266 00:25:33,330 --> 00:25:37,950 business and putting it together infinitely many things are two directions is kind of terrifying, 267 00:25:37,950 --> 00:25:41,190 but in first year analysis, we make sense of these things. 268 00:25:41,190 --> 00:25:46,620 What does it mean to add together infinitely many things? So maybe you've come across an example where you can add together infinitely? 269 00:25:46,620 --> 00:25:53,580 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. 270 00:25:53,580 --> 00:25:58,290 You add infinitely many things. You get a sensible, finite answer. We say the series converges. 271 00:25:58,290 --> 00:26:00,660 But if you're careless, you add together infinitely many things. 272 00:26:00,660 --> 00:26:06,240 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. 273 00:26:06,240 --> 00:26:11,010 That's the harmonic series it doesn't convert. You don't get a finite answer diverges. 274 00:26:11,010 --> 00:26:17,220 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? 275 00:26:17,220 --> 00:26:21,600 What tools do we have for looking at the series and going up? It converges, or no, it doesn't. 276 00:26:21,600 --> 00:26:26,440 Analysis also includes things like what we do, differentiation. We sort of want to take a limit. 277 00:26:26,440 --> 00:26:32,700 What precisely does that mean? So getting the logical, rigorous foundations of that precise, which is a really satisfying thing to do. 278 00:26:32,700 --> 00:26:36,140 What is the link between differentiation? How do we formalise that? 279 00:26:36,140 --> 00:26:42,870 Once you're done first, your analysis, then you can do second year complex analysis, which is like complex numbers meets calculus. 280 00:26:42,870 --> 00:26:47,550 But it's much more beautiful than that makes it sound. It's full of theorems that are really surprising theories. 281 00:26:47,550 --> 00:26:52,750 They shouldn't be true, but somehow all because of miracles of complex analysis and complex analysis. 282 00:26:52,750 --> 00:26:58,230 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, 283 00:26:58,230 --> 00:27:03,180 and you can show that it satisfies the differential equation that starts to have the form of an elliptical serve. 284 00:27:03,180 --> 00:27:07,710 So you can tackle questions about ellipticals cards by relating back to the Vice Jospeh function. 285 00:27:07,710 --> 00:27:11,670 Via this differential equation, I could keep talking about elliptic curves, 286 00:27:11,670 --> 00:27:15,330 but I think that I have to stop because you have like another chalk or something. 287 00:27:15,330 --> 00:27:20,760 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. 288 00:27:20,760 --> 00:27:41,178 Thank you very much.