1 00:00:18,130 --> 00:00:21,700 Good evening, everyone. Welcome to the Science Museum. 2 00:00:21,700 --> 00:00:31,210 I'm Roger Highfield, the science director, and it's a thrill to be here tonight to introduce the Oxford Mathematics public lectures. 3 00:00:31,210 --> 00:00:36,280 They're being webcast from the IMAX of the museum. Hello, world! 4 00:00:36,280 --> 00:00:40,630 Now, mathematics lies at the heart of this museum and our mathematics gallery. 5 00:00:40,630 --> 00:00:48,160 Zaha Hadid crystallised abstract mathematical thinking into beautiful physical forms. 6 00:00:48,160 --> 00:00:55,300 It's central to the cryptography in our top secret exhibition, to the A.I. in our driverless exhibition. 7 00:00:55,300 --> 00:00:58,930 If you go to our Science City gallery, which is brand new, 8 00:00:58,930 --> 00:01:05,560 you'll see Newton's Principia and the story of the most powerful way that we have to understand the universe, 9 00:01:05,560 --> 00:01:07,780 which is, of course, a scientific method. 10 00:01:07,780 --> 00:01:18,760 Mathematics also underpins the scanners, the crystallography, the epidemiology and more in our massive medicine galleries, which opened last week. 11 00:01:18,760 --> 00:01:28,330 Now, in a lecture given 60 years ago, Eugene Wagner described the unreasonable effectiveness of mathematics. 12 00:01:28,330 --> 00:01:33,640 Mathematics is indeed a universal language. It's a means to understand everything that we do. 13 00:01:33,640 --> 00:01:38,020 It's a tool for in for increasing our thinking power. 14 00:01:38,020 --> 00:01:38,500 Tonight, 15 00:01:38,500 --> 00:01:47,410 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, 16 00:01:47,410 --> 00:01:55,390 who is giving the right Christmas lectures. She is the author of Hello World and also a Science Museum Group trustee. 17 00:01:55,390 --> 00:02:01,840 But first of all, I'd like to introduce the Oxford Mathematical Institute's director of external relations. 18 00:02:01,840 --> 00:02:10,460 Please welcome Professor Alan Gorier Lee. Thank you. 19 00:02:10,460 --> 00:02:12,050 Many thanks, Roger. 20 00:02:12,050 --> 00:02:21,200 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. 21 00:02:21,200 --> 00:02:26,450 So what we've been trying to accomplish over the last six years now with the Oxford Mathematics lecture, 22 00:02:26,450 --> 00:02:34,250 but also with extended programme is to bring the best of mathematics to the public to a lecture series first in Oxford. 23 00:02:34,250 --> 00:02:42,710 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. 24 00:02:42,710 --> 00:02:48,680 We've been great friends and partners of Oxford Mathematics and believe in what we're trying to accomplish, 25 00:02:48,680 --> 00:02:53,630 and I hope we'll have many more such collaboration, but also to help us in our goal. 26 00:02:53,630 --> 00:02:59,150 I have to give special thanks to the X Markets sponsor of the Oxford Mathematics Public Lecture. 27 00:02:59,150 --> 00:03:05,900 Series X equity markets are leading quantitative driven electronic market maker with offices in London, 28 00:03:05,900 --> 00:03:13,100 Singapore and New York, and I'm glad to see that many of you came to the event today today. 29 00:03:13,100 --> 00:03:15,920 It's a great honour for me to introduce Professor Tim Grover, 30 00:03:15,920 --> 00:03:20,840 the Rose Board Chair of Mathematics at Cambridge University and fellow of Trinity College. 31 00:03:20,840 --> 00:03:25,940 Since you will hear more about Professor Gower's life, mathematics and opinion, 32 00:03:25,940 --> 00:03:35,450 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. 33 00:03:35,450 --> 00:03:39,590 I've been recently at Yale University and you'll find right at the centre of the 34 00:03:39,590 --> 00:03:45,890 universe the beautiful graveyards and you'll find the grave of the great lost Gonzaga, 35 00:03:45,890 --> 00:03:52,970 one of the greatest physicists of all time, and simply on his gravestone is described by Nobel laureate, 36 00:03:52,970 --> 00:04:02,390 etc. So it could be appropriate to describe professor go goers as field medallists, etc. and be done with it. 37 00:04:02,390 --> 00:04:07,040 But it would be a gross mischaracterisation of his other accomplishment and influence, 38 00:04:07,040 --> 00:04:13,820 not only on mathematics directly but on the way mathematician work and organised is also been a passionate, 39 00:04:13,820 --> 00:04:16,730 passionate about explaining mathematics to the public, 40 00:04:16,730 --> 00:04:22,670 and as the unique record of having both one of the shortest introduction of mathematics with Oxford University 41 00:04:22,670 --> 00:04:30,050 Press and an encyclopaedic one may be the last the longest introduction with Princeton University Press. 42 00:04:30,050 --> 00:04:35,480 If you follow him on social media, as I do, you will also discover True, True Humanist, 43 00:04:35,480 --> 00:04:42,140 passionate about mathematics, but also caring about all social and political aspects of today's world. 44 00:04:42,140 --> 00:04:47,840 So for my second attempt and knowing that Professor Gowers is of particular interest in music, 45 00:04:47,840 --> 00:04:52,280 I thought about another short posthumous description, which seems to be a good fit. 46 00:04:52,280 --> 00:05:03,200 This one is about the 15th century English composer John Dunstable, who was described as mathematician, musician and whatnot. 47 00:05:03,200 --> 00:05:09,320 Tonight, we will first hear about the essence of mathematics, which is about generalisation and abstraction. 48 00:05:09,320 --> 00:05:12,680 Then, Professor Goes, would be in conversation with unafraid. 49 00:05:12,680 --> 00:05:19,310 And my hope, of course, is that at this point, we learn all about Professor Gauss ETC's and whatnots. 50 00:05:19,310 --> 00:05:33,000 So without further ado, please let me welcome Professor Gower's tonight for story. 51 00:05:33,000 --> 00:05:46,870 Thank you very much. I just start by saying that this is the largest slide I've ever had the pleasure of presenting. 52 00:05:46,870 --> 00:05:51,790 I also want to say that I don't think I can quiet in half an hour. So what I have? 53 00:05:51,790 --> 00:05:58,780 Live up to the promise of the title of this talk, but I can do something towards it because really, 54 00:05:58,780 --> 00:06:05,140 to say exactly why I believe that we'll never run out of questions is it's quite a long and complicated task. 55 00:06:05,140 --> 00:06:11,290 But this is just one. I suppose that the emphasis is on one reason because there are plenty of reasons. 56 00:06:11,290 --> 00:06:17,350 But I suppose the fundamental reason is that mathematics has a sort of hydrologic quality that when you answer one question, 57 00:06:17,350 --> 00:06:20,020 it somehow begets 10 other questions. 58 00:06:20,020 --> 00:06:28,030 And one of the things one of the sort of begetting mechanisms is generalisation, and that's what I'm focussing on this evening. 59 00:06:28,030 --> 00:06:34,240 So before I get underway, there are two types of generalisation that one could be talking about. 60 00:06:34,240 --> 00:06:42,880 One is generalising ideas of mathematics or definitions, and the other is generalising statements and theorems and lemons and things like that. 61 00:06:42,880 --> 00:06:52,500 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. 62 00:06:52,500 --> 00:06:57,780 A shame, not a little bit about that, because that is also very, very important. 63 00:06:57,780 --> 00:07:03,800 So let's just dive right in. 64 00:07:03,800 --> 00:07:10,430 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, 65 00:07:10,430 --> 00:07:16,890 but if you've done maths at school, then I would expect that at some stage you've come across something like X two, 66 00:07:16,890 --> 00:07:24,390 the three over two and you may remember. If you came to understand that if you didn't, 67 00:07:24,390 --> 00:07:29,360 I'm about to explain it the sort of feeling that how connects to the three ever to make any 68 00:07:29,360 --> 00:07:35,420 sense because you can't take a number one and a half times and sort of multiply those together, 69 00:07:35,420 --> 00:07:40,460 that just doesn't make any sense. So how do we make sense of this concept? 70 00:07:40,460 --> 00:07:48,380 Similarly, what about each of us, Z have said, is complex so raisings a complex power that's not even madder than raising two? 71 00:07:48,380 --> 00:07:55,490 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? 72 00:07:55,490 --> 00:07:59,240 That's even more sort of nonsensical. 73 00:07:59,240 --> 00:08:07,380 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, 74 00:08:07,380 --> 00:08:11,270 and we'll come back to that in a minute? 75 00:08:11,270 --> 00:08:18,830 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. 76 00:08:18,830 --> 00:08:25,230 So the way that we decide and in each case, it's slightly different the process that we use to generalise the concept. 77 00:08:25,230 --> 00:08:32,270 So if you want to generalise powers to non integer powers, one thing we do is just focus on this rule here. 78 00:08:32,270 --> 00:08:36,830 That extra m plus N equals extra m times X to the end. 79 00:08:36,830 --> 00:08:44,090 Although many schoolchildren will will say that it's actually having plus x to the end because they always believe that in mathematicians terms, 80 00:08:44,090 --> 00:08:52,610 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. 81 00:08:52,610 --> 00:08:57,320 And here's the reason illustrated with an example. So take X to the five. 82 00:08:57,320 --> 00:09:00,500 That's just x times exercise x times x times x. 83 00:09:00,500 --> 00:09:06,240 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. 84 00:09:06,240 --> 00:09:12,380 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. 85 00:09:12,380 --> 00:09:22,680 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. 86 00:09:22,680 --> 00:09:26,460 Now, the point is that the way we then use that is we say, well, that's a rule, 87 00:09:26,460 --> 00:09:31,200 and there isn't really an objective meaning to extend the three ever to. 88 00:09:31,200 --> 00:09:38,940 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. 89 00:09:38,940 --> 00:09:43,500 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 90 00:09:43,500 --> 00:09:49,440 three over two times x two to three over two because three of a two plus three over two equals three. 91 00:09:49,440 --> 00:09:55,770 So if we want that rule to be true when and then about equal to three over two, we need this equation to hold. 92 00:09:55,770 --> 00:10:01,620 But that tells us that x two to three over two has to be the square root of x cubed. 93 00:10:01,620 --> 00:10:08,150 Or you're feeling very alert, you'll object that excuse might be negative or there are two square roots and so on. 94 00:10:08,150 --> 00:10:14,690 But the convention is will assume that the number on the X is positive and we'll take the positive square root. 95 00:10:14,690 --> 00:10:20,510 That's just convenience rather than some objective reality, but it is very convenient. 96 00:10:20,510 --> 00:10:23,960 OK, so that's giving us a very good answer to what takes two to three over two hours 97 00:10:23,960 --> 00:10:28,100 and then you can build on that and work out what extra any fraction should be. 98 00:10:28,100 --> 00:10:35,810 And then you can use other arguments to get to infer from fractions to irrational numbers and you can carry on and on. 99 00:10:35,810 --> 00:10:44,240 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? 100 00:10:44,240 --> 00:10:50,240 And again, if you've done at A-level a.m. Mass, you'll know how we how we sort this out. 101 00:10:50,240 --> 00:10:54,740 If you haven't, I'm going to ask you to take something on trust. 102 00:10:54,740 --> 00:10:59,230 So the trick here is to think of it as the square root of two plus three. 103 00:10:59,230 --> 00:11:05,150 I know it has raised me to the square root of two plus three. 104 00:11:05,150 --> 00:11:11,780 But to reformulate this and the way we reformulated this to use the following formula which you need 105 00:11:11,780 --> 00:11:17,900 to have done some calculus of sort of reasonably advanced kind to to be able to justify this formula. 106 00:11:17,900 --> 00:11:24,410 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, 107 00:11:24,410 --> 00:11:30,710 each of the X is one plus x plus x squared plus x x two factorial plus executed over three factorial and so on. 108 00:11:30,710 --> 00:11:35,870 And this dot dot dot means you. The sum gets closer and closer to sum number, 109 00:11:35,870 --> 00:11:42,770 and that is the number that we take as the definition of X own story takes the definition of the sum of the series. 110 00:11:42,770 --> 00:11:51,800 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. 111 00:11:51,800 --> 00:11:55,100 But the great thing about the definition on the right is that now it's much easier to 112 00:11:55,100 --> 00:12:02,870 make it to make sense for complex numbers because all we need on the right is addition. 113 00:12:02,870 --> 00:12:11,180 Dividing by integers. Multiplication and a limiting process and all of those addition multiplication. 114 00:12:11,180 --> 00:12:18,890 Dividing by an integer and taking limits, all of those make very good sense for complex numbers that generalise straightforwardly. 115 00:12:18,890 --> 00:12:23,210 So the left hand side does not generalise straightforwardly, but the right hand side generalise is very straightforward. 116 00:12:23,210 --> 00:12:29,640 So we take the right hand side as the definition when we have a complex number. 117 00:12:29,640 --> 00:12:34,230 OK, so now let's move on to fractional dimension, 118 00:12:34,230 --> 00:12:42,090 so this this collapsed chair is supposed to represent the sort of existential despair that you might feel when 119 00:12:42,090 --> 00:12:48,810 trying to conceive of a shape that has sort of two and a half degrees of freedom or something like that. 120 00:12:48,810 --> 00:12:53,490 So we know there's a one dimensional shape is one where you have sort of one degree of freedom in two dimensions. 121 00:12:53,490 --> 00:12:56,820 You have sort of the length and breadth of something and then the three dimensions. 122 00:12:56,820 --> 00:13:02,730 You add height for what, two and a half dimensions, but it's not at all clear. 123 00:13:02,730 --> 00:13:07,980 It seems as though the very notion of a degree of freedom has to be a whole number, and indeed it does. 124 00:13:07,980 --> 00:13:14,250 So we have to do the same trick of finding some other way of thinking about dimension that will allow us to generalise it. 125 00:13:14,250 --> 00:13:21,990 So let's see what we do. So the idea is to focus in two and three dimensions on area and volume. 126 00:13:21,990 --> 00:13:27,210 But since the area is very much a two dimensional concept and volume is very much a three dimensional concept, 127 00:13:27,210 --> 00:13:33,240 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. 128 00:13:33,240 --> 00:13:38,050 I want to call the amount of stuff so. 129 00:13:38,050 --> 00:13:45,640 Another familiar fact is that if you take a two dimensional shape and you expand it by a factor of two in every direction, 130 00:13:45,640 --> 00:13:52,570 you get a new shape that has four times. The amount of stuff as the original shape. 131 00:13:52,570 --> 00:13:59,880 So you can see it very clearly in one of these squares four times and to the big square. 132 00:13:59,880 --> 00:14:08,920 And what's for what's the significance of for it is that it is one times to two times to I two squared, 133 00:14:08,920 --> 00:14:14,940 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. 134 00:14:14,940 --> 00:14:21,030 Similarly, in three dimensions, if I double every single direction of a cube, I get a new, 135 00:14:21,030 --> 00:14:26,850 bigger cube into which I can fit eight copies of the original cube, and eight is two cubed. 136 00:14:26,850 --> 00:14:32,340 If I instead expand by a factor of three in every direction, I get 27. 137 00:14:32,340 --> 00:14:36,240 I got a shape into which I fit 27 cubes and 27 is three cubed. 138 00:14:36,240 --> 00:14:39,570 And again, it's the fact that I'm saying cube, which is to the power three. 139 00:14:39,570 --> 00:14:49,440 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. 140 00:14:49,440 --> 00:14:58,620 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. 141 00:14:58,620 --> 00:15:05,520 The process is this you start with a line segment that goes from this point to this point and you divide it 142 00:15:05,520 --> 00:15:12,540 into three equal parts and you replace the middle part by the other two sides of an equilateral triangle. 143 00:15:12,540 --> 00:15:19,590 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. 144 00:15:19,590 --> 00:15:26,460 Here I take each of the four parts and do the same process. I divide this segment here into three. 145 00:15:26,460 --> 00:15:32,490 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. 146 00:15:32,490 --> 00:15:46,710 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. 147 00:15:46,710 --> 00:15:52,260 And each one of those I replace and a bit I'm not showing you is that each one of these 148 00:15:52,260 --> 00:15:57,170 little segments are replaced by one size and then each one of those segments are replaced. 149 00:15:57,170 --> 00:16:07,470 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. 150 00:16:07,470 --> 00:16:18,580 And that limiting slope shape is called the snowflake, and I think it's fairly clear from the picture. 151 00:16:18,580 --> 00:16:25,150 That the cost snowflake is made out of four copies of itself shrunk down, 152 00:16:25,150 --> 00:16:32,230 so if I look at this shape here because it's produced by basically exactly the same process as the entire Cork snowflake, 153 00:16:32,230 --> 00:16:37,570 it is just a little cork snowflake. And that's another little cork snowflake. And that's another little cork snowflake. 154 00:16:37,570 --> 00:16:42,760 And that's another little cork snowflake. So this shape here, it's a fractal shape. 155 00:16:42,760 --> 00:16:48,430 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. 156 00:16:48,430 --> 00:16:53,290 So let's look at this part here and say if I were to. 157 00:16:53,290 --> 00:16:59,320 Expand that shape here by a factor of three in every direction. 158 00:16:59,320 --> 00:17:05,950 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. 159 00:17:05,950 --> 00:17:14,740 So if I expand this bit here by a factor of three, I get the same shape. 160 00:17:14,740 --> 00:17:21,370 So the amount of stuff goes up by four. Expanded it by a factor of three to get this shape here. 161 00:17:21,370 --> 00:17:29,070 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. 162 00:17:29,070 --> 00:17:34,520 I expand it by a factor of three and I get four copies of the original shape. 163 00:17:34,520 --> 00:17:40,490 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. 164 00:17:40,490 --> 00:17:46,840 Twenty seven was three cubed. So what do I need to ask, I need to ask for is three to the what? 165 00:17:46,840 --> 00:17:50,570 Fortunately, we thought a little bit about raising a non-intrusive house, 166 00:17:50,570 --> 00:17:57,610 and therefore it turns out to be three raised to the power, the logarithm of force, the base three. 167 00:17:57,610 --> 00:18:04,810 It's a log that's a number between one and two. So this is a shape. 168 00:18:04,810 --> 00:18:11,240 It doesn't really matter what exactly what the dimension is if you don't like logs, but. 169 00:18:11,240 --> 00:18:14,870 It is definitely between three and four, because there's between one and two, the dimension, 170 00:18:14,870 --> 00:18:21,650 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, 171 00:18:21,650 --> 00:18:25,650 four is between three and nine. So the dimension of this shape? 172 00:18:25,650 --> 00:18:32,730 If you use that concept to dimension, it's natural to say it's but it's between one and two. 173 00:18:32,730 --> 00:18:39,630 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, 174 00:18:39,630 --> 00:18:43,620 whereas this is a very special shape of the self similarity property. 175 00:18:43,620 --> 00:18:49,500 But it turns out that we can build on these ideas and make sense of dimension of lots of other shapes. 176 00:18:49,500 --> 00:18:57,180 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. 177 00:18:57,180 --> 00:19:05,760 So I wanted to talk about a beautiful result called vanderveer and Stern and talk about generalisations of that. 178 00:19:05,760 --> 00:19:12,540 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. 179 00:19:12,540 --> 00:19:16,950 So that divergence term we start by doing the following we take a large number. 180 00:19:16,950 --> 00:19:23,820 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. 181 00:19:23,820 --> 00:19:31,760 So well, I've decided either red, black or blue to all the numbers up to 50. 182 00:19:31,760 --> 00:19:44,050 And now. So no, terribly obvious reason, but what mathematicians we like to do, these things are going to hunt for arithmetic progressions. 183 00:19:44,050 --> 00:19:47,590 And I want my maths progressions to consist of numbers that have the same colour. 184 00:19:47,590 --> 00:19:51,130 So what's an arithmetic progression at something like four, seven, 10, 185 00:19:51,130 --> 00:19:55,210 13 or something where you have some numbers to get from each number to the next number, 186 00:19:55,210 --> 00:20:04,150 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. 187 00:20:04,150 --> 00:20:12,500 Something with the jump was six in that case. So if we look around, can we see any arithmetic progressions? 188 00:20:12,500 --> 00:20:17,080 I sort of, as you can see, two, four six. This is a black one, but it stops. 189 00:20:17,080 --> 00:20:22,860 So can we find any of lines? Four. Well, here's one. 190 00:20:22,860 --> 00:20:29,040 I've underlined so we go from two to 13 to 24, 2005, 191 00:20:29,040 --> 00:20:37,650 so the step there was 11 o'clock monochromatic because it's just got one car that sets and thirty five it stops there. 192 00:20:37,650 --> 00:20:49,740 I couldn't get to 46 because that was blue, not black. So anybody feel like finding me a blue one of legs for. 193 00:20:49,740 --> 00:20:59,760 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. 194 00:20:59,760 --> 00:21:04,030 I don't think that's one of five. That's why everything is blue. 195 00:21:04,030 --> 00:21:13,950 Slightly more challenging and red one, I think I wanted to give you very long for this because I need to press on. 196 00:21:13,950 --> 00:21:22,110 But you have to have a chance. And three to one, here it is. 197 00:21:22,110 --> 00:21:30,060 Twenty three, thirty one, thirty nine and forty seven set at eight each time. 198 00:21:30,060 --> 00:21:34,680 What has found a Vaart theorem say this sort of silly game of finding arithmetic progressions. 199 00:21:34,680 --> 00:21:45,320 Everything, everything's the same colour. It says this is not a formal statement to the serum, obviously, but it's it says that. 200 00:21:45,320 --> 00:21:52,000 Basically, however, you colour the numbers up to and so says, if I take an end, that's large enough, 201 00:21:52,000 --> 00:21:56,300 I have some palette of colours that say I got 25 colours and you want to fight. 202 00:21:56,300 --> 00:22:03,020 You want better be an athletic progression of length 79 then as long as NW is large enough. 203 00:22:03,020 --> 00:22:07,820 Then, however, I colour the numbers from one to end with 25 colours. 204 00:22:07,820 --> 00:22:14,660 That will definitely be an aromatic progression of like 79, where all the numbers in that charismatic progression have the same colour. 205 00:22:14,660 --> 00:22:20,840 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, 206 00:22:20,840 --> 00:22:28,380 but the point is that it it does exist, some with that property. 207 00:22:28,380 --> 00:22:33,150 Right. So now let's get generalising. 208 00:22:33,150 --> 00:22:36,510 I hope you understood that, so you colour the integers from one to end. 209 00:22:36,510 --> 00:22:43,440 You can't stop that being an aromatic progression of wheat with only one colour. 210 00:22:43,440 --> 00:22:47,340 So the first generation I want to talk about is called a density version, 211 00:22:47,340 --> 00:22:51,990 which starts with the following question you might sort of the one thing that mathematicians do. 212 00:22:51,990 --> 00:23:00,930 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. 213 00:23:00,930 --> 00:23:10,230 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. 214 00:23:10,230 --> 00:23:16,950 But can we somehow identify which colour that is? What is the best we can say that there must be some colour. 215 00:23:16,950 --> 00:23:26,820 And it turns out that isn't a very, very surprisingly satisfactory answer to this question, which is a result called semi-trailers theorem. 216 00:23:26,820 --> 00:23:31,290 Again, this is not a formal statement. But let me try to say what this means. 217 00:23:31,290 --> 00:23:36,540 So when we have a colour, one of the colours that we use, 218 00:23:36,540 --> 00:23:44,080 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. 219 00:23:44,080 --> 00:23:54,300 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. 220 00:23:54,300 --> 00:24:02,040 So what summary this term says is if you tell me a density like one percent and you tell me the length of harassment, 221 00:24:02,040 --> 00:24:06,740 a progression like a million, then it will be an N. 222 00:24:06,740 --> 00:24:17,120 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. 223 00:24:17,120 --> 00:24:21,350 So he actually is not important as a colour, just as any bunch of integers from one to N, 224 00:24:21,350 --> 00:24:28,400 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, 225 00:24:28,400 --> 00:24:34,760 uh will have to contain an arithmetic progression of like four million. 226 00:24:34,760 --> 00:24:40,050 So Sam Register immediately implies Van 2007, because if you colour the numbers from one to end with, 227 00:24:40,050 --> 00:24:45,560 say, 100 colours, this one of those colours must be used at least one percent of the time. 228 00:24:45,560 --> 00:24:51,640 That's a famous principle called pigeonhole principle in mathematics, or if you've got, you know, 229 00:24:51,640 --> 00:24:56,840 or similar, if you had four colours, at least one of the colours must be used 25 percent of the time. 230 00:24:56,840 --> 00:25:04,810 So by summary, to serve that colour itself will contain an artificial progression. 231 00:25:04,810 --> 00:25:13,200 Right. Let's move on now, so I'm going to generalise in a completely different way. 232 00:25:13,200 --> 00:25:19,800 This is a firm again not stated, formally called the Hayles do theorem. 233 00:25:19,800 --> 00:25:27,120 Let me move on to a picture and see if I can explain roughly what the Hell's to it, Theorem says. 234 00:25:27,120 --> 00:25:31,920 So when I was young, I used to have a game called for pegs. 235 00:25:31,920 --> 00:25:38,970 And this is a picture of the box or a box that was selling on eBay of the same game. 236 00:25:38,970 --> 00:25:46,350 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. 237 00:25:46,350 --> 00:25:48,890 So lions have length for in this game. 238 00:25:48,890 --> 00:25:55,180 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. 239 00:25:55,180 --> 00:25:56,510 A rather good game. 240 00:25:56,510 --> 00:26:08,810 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. 241 00:26:08,810 --> 00:26:14,930 So a line here will be something like here's a simple example of a line and a slightly more complicated example, 242 00:26:14,930 --> 00:26:18,560 but the exact point to that point at that point and that point where you get 243 00:26:18,560 --> 00:26:22,010 more complicated going to be one that's diagonal in every possible respect. 244 00:26:22,010 --> 00:26:27,230 So say that point, that point, that point and that point. 245 00:26:27,230 --> 00:26:39,790 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, 246 00:26:39,790 --> 00:26:48,040 say 100, as long as the dimension is high enough, there must be a line that consists of points of only one colour. 247 00:26:48,040 --> 00:26:53,440 Let's consider what an even higher dimensional noughts and crosses might be like this one we can at least visualise. 248 00:26:53,440 --> 00:26:57,400 This is an example of four dimensional noughts and crosses. 249 00:26:57,400 --> 00:27:03,910 But because I can't do a slide in four dimensions or even to a slide in three dimensions of a fundamental thing, 250 00:27:03,910 --> 00:27:08,350 I'm sort of projected onto two dimensions. What we have to do is find some way of representing it. 251 00:27:08,350 --> 00:27:15,220 So the way I've represented it, I think you can sort of see I've just as a three dimensional noughts and crosses 252 00:27:15,220 --> 00:27:19,260 board consists of a bunch of two-dimensional ones placed sort of next to each other. 253 00:27:19,260 --> 00:27:20,500 I don't have to one on top of the other. 254 00:27:20,500 --> 00:27:26,290 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. 255 00:27:26,290 --> 00:27:32,320 So here I've taken a two dimensional one, and then I put those three together that makes a three dimensional board. 256 00:27:32,320 --> 00:27:35,740 And then those three together make another three dimensional types. You prefer to visualise it vertically. 257 00:27:35,740 --> 00:27:42,190 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. 258 00:27:42,190 --> 00:27:46,210 And then we can hear it. Here are some examples of lines. Let's just do the most diagonal one possible. 259 00:27:46,210 --> 00:27:51,550 We go that point there, that point there, that point there. 260 00:27:51,550 --> 00:28:01,550 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. 261 00:28:01,550 --> 00:28:05,690 So going back to the house, do it to him again. 262 00:28:05,690 --> 00:28:12,100 It says if you if you have a sufficiently high dimensional board, then how wide it is. 263 00:28:12,100 --> 00:28:18,760 And have many colours you you have you cannot avoid lions and have a consequence for the game, 264 00:28:18,760 --> 00:28:24,930 says, even if you have a game where you play it with sort of eight players, let's say. 265 00:28:24,930 --> 00:28:29,490 If the dimension is large enough, it can't end in a draw because at least one of it, 266 00:28:29,490 --> 00:28:33,340 once it once the borders filled, you've coloured the points with eight different colours. 267 00:28:33,340 --> 00:28:36,900 And one of those colours has a line. Somebody must have made a line first. 268 00:28:36,900 --> 00:28:43,070 So the game couldn't have ended in a draw as there was a paper, the paper of Hale's. 269 00:28:43,070 --> 00:28:51,800 It talks about multidimensional tic tac toe, tic tac toe being the American for noughts and crosses. 270 00:28:51,800 --> 00:28:59,810 Right, let's move on to yet another completely different generalisation this time. 271 00:28:59,810 --> 00:29:04,550 What we're doing is let's think of how we might represent a general arithmetic progression 272 00:29:04,550 --> 00:29:10,010 of flags for you might say you pick a point a number A and you pick a difference d. 273 00:29:10,010 --> 00:29:18,660 And then the ultimate progression consists of the numbers a a plus d, a A-plus, 2D, an lost 3D. 274 00:29:18,660 --> 00:29:25,350 So the expressions do 2D and 3D are very simple examples of polynomials. 275 00:29:25,350 --> 00:29:31,140 Polynomial functions of D, but they're actually polynomials of degree one or linear polynomials of D, 276 00:29:31,140 --> 00:29:36,510 so d 2D 3D, they're very simple expressions here. I just replace those by some rather more complicated ones. 277 00:29:36,510 --> 00:29:41,040 D squared, d cubed and d-plus d squared those to each of the four. 278 00:29:41,040 --> 00:29:46,410 I want the polynomial version of events, Theorem says, 279 00:29:46,410 --> 00:29:51,840 is that I could choose any bunch of polynomials like that as long as they don't have constant terms. 280 00:29:51,840 --> 00:29:57,120 That's a technical reason, but the theory is not true if you have constant terms. 281 00:29:57,120 --> 00:30:05,510 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, 282 00:30:05,510 --> 00:30:11,270 I'll be able to find patterns like this. Let me just illustrate it, but I think it'll make it clearer. 283 00:30:11,270 --> 00:30:15,770 So let's go back to this is exactly the same colouring of integers I had before, 284 00:30:15,770 --> 00:30:20,030 and I promise you that this wasn't sort of fake trying to have a little search. 285 00:30:20,030 --> 00:30:23,660 And eventually I found an example of a pair of numbers. 286 00:30:23,660 --> 00:30:32,120 And so I said all of a velocity squared plus d cubed and a diversity of capacity to the force belonged to the set. 287 00:30:32,120 --> 00:30:37,730 And here it is. I had a equals seven and D equals two. 288 00:30:37,730 --> 00:30:49,190 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. 289 00:30:49,190 --> 00:30:54,770 Well, check that to the force of 16 to IS for two is to add them together at 22. 290 00:30:54,770 --> 00:31:02,060 Add that to seven. You got 29, so I wasn't cheating. 291 00:31:02,060 --> 00:31:11,630 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. 292 00:31:11,630 --> 00:31:22,260 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. 293 00:31:22,260 --> 00:31:33,080 And it would still have been true. As long as I colour enough integers, I will get a pattern of Typekit desired kind. 294 00:31:33,080 --> 00:31:38,450 Right. So let's just recap, here are our methods that we've got for generalising. 295 00:31:38,450 --> 00:31:41,000 We go from a colouring theorem to the density theorem. 296 00:31:41,000 --> 00:31:47,750 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, 297 00:31:47,750 --> 00:31:56,670 right from integers to high dimensional grid sold noughts and crosses boards. I went from arithmetic progressions to more general polynomial patterns. 298 00:31:56,670 --> 00:32:01,670 Those were the three ways that we came up. And so there is completely different. 299 00:32:01,670 --> 00:32:06,980 They're so different than what we might call orthogonal to each other and using right orthogonal, 300 00:32:06,980 --> 00:32:12,410 slightly fanciful way, but one that mathematicians often do because they are orthogonal. 301 00:32:12,410 --> 00:32:16,400 We can start asking, Well, what about if we try and combine some of these generalisation methods? 302 00:32:16,400 --> 00:32:21,590 Could we actually? Get even more generalisations. 303 00:32:21,590 --> 00:32:25,250 So there are eight possible generalisations if you've got three methods, 304 00:32:25,250 --> 00:32:29,660 if you include what mathematicians one called the trivial generalisation where you don't generalise. 305 00:32:29,660 --> 00:32:38,350 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. 306 00:32:38,350 --> 00:32:45,620 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. 307 00:32:45,620 --> 00:32:49,800 So combining one and three is fairly straightforward. 308 00:32:49,800 --> 00:32:56,780 The Hayles Drouet Theorem said if you colour a sufficiently high dimensional noughts and crosses border, you must get a line in one colour. 309 00:32:56,780 --> 00:33:05,270 So the density of ocean says if I just take one percent of or some fixed percentage of a sufficiently high. 310 00:33:05,270 --> 00:33:07,250 Dimensional noughts and crosses board. 311 00:33:07,250 --> 00:33:14,750 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. 312 00:33:14,750 --> 00:33:18,920 I can't avoid making a line somewhere. That would be what the density theorem says. 313 00:33:18,920 --> 00:33:24,830 The colouring Fitzgerald would say, I have to colour every single point and then one of the colours will contain a line. 314 00:33:24,830 --> 00:33:30,980 The density version just says any sufficiently dense colour will have to contain a line. 315 00:33:30,980 --> 00:33:35,840 But that turned out to be a much harder result than the hills to it itself. 316 00:33:35,840 --> 00:33:43,430 This is due to two mathematicians called Hiro Furstenberg and Yitzhak Katz Nelson. 317 00:33:43,430 --> 00:33:55,820 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. 318 00:33:55,820 --> 00:33:59,930 Well, then again, it's fairly straightforward to see how the generalisation ought to work. 319 00:33:59,930 --> 00:34:01,490 We ought to say instead of colouring, 320 00:34:01,490 --> 00:34:07,340 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. 321 00:34:07,340 --> 00:34:10,250 And that turns out to be a true result as well. 322 00:34:10,250 --> 00:34:18,600 The the theorem that you can do it with colouring was due to two mathematicians called Vitale, Berthelsen and Sasha A. 323 00:34:18,600 --> 00:34:22,460 Now there's an interesting story that turned out that before they did the colouring version, 324 00:34:22,460 --> 00:34:27,710 they had a sort of machinery which had developed building on the work of actually Hillel Furstenberg. 325 00:34:27,710 --> 00:34:33,240 They knew that once they had the colouring version, they would be able to generalise the density version. 326 00:34:33,240 --> 00:34:39,090 Not it's not obvious how you can, but then it turned out that they knew how to do that. 327 00:34:39,090 --> 00:34:43,590 So that was true. And the last thing I want to talk about, although sorry, 328 00:34:43,590 --> 00:34:49,410 the third of these are combining two methods of generalisation is something that's rather less 329 00:34:49,410 --> 00:34:55,080 obvious now here on how it's to be a little bit of a challenge to say what's going on here. 330 00:34:55,080 --> 00:35:03,810 So you think that this is a two dimensional grid that I've coloured black and red? 331 00:35:03,810 --> 00:35:08,550 But you will also see I put numbers one or two in there, something a bit strange is going on. 332 00:35:08,550 --> 00:35:12,720 So that's not supposed to be a two dimensional noughts and crosses board. 333 00:35:12,720 --> 00:35:19,800 This is supposed to represent one point that lives inside a 64 dimensional noughts and crosses forward. 334 00:35:19,800 --> 00:35:25,320 So if you think about it, just a straightforward, regular and awesome process forward, we might have a coordinate system that went north north. 335 00:35:25,320 --> 00:35:33,090 Not one, not two one one one one one one two two, not two one two two in 64 dimensions. 336 00:35:33,090 --> 00:35:41,410 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. 337 00:35:41,410 --> 00:35:45,940 Yeah, I just got 64 of those 64 numbers, each one of which is a nought one or two. 338 00:35:45,940 --> 00:35:50,470 That simple, that's just a point and sixty four dimensional space. 339 00:35:50,470 --> 00:35:58,270 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. 340 00:35:58,270 --> 00:36:04,120 Remember, this is just one point. This is the coordinates. I happen to arrange the coordinates in an eight by grid. 341 00:36:04,120 --> 00:36:10,180 This point is green. OK, so this is. 342 00:36:10,180 --> 00:36:14,680 I'm just declaring it victory, and it's not some mathematical reality that you've missed. 343 00:36:14,680 --> 00:36:18,830 I'm declaring this point victory. All right. 344 00:36:18,830 --> 00:36:26,370 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. 345 00:36:26,370 --> 00:36:35,550 Uh. Here it goes. So why the reason I highlighted those points in red is because they're about to change. 346 00:36:35,550 --> 00:36:40,410 Maybe you see they all went there, all notes, and they're all ones and they're all tools. 347 00:36:40,410 --> 00:36:45,690 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 348 00:36:45,690 --> 00:36:52,200 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. 349 00:36:52,200 --> 00:36:56,400 This is just a natural generalisation of what happens in two dimensions and three dimensions. 350 00:36:56,400 --> 00:37:02,100 If you think about it for a while. But there's something else about this line that makes it particularly special, 351 00:37:02,100 --> 00:37:09,570 which is that the points that do the changing lie in, they have further coordinates in a special set. 352 00:37:09,570 --> 00:37:17,550 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. 353 00:37:17,550 --> 00:37:24,240 I got the first first row, second row, false row, seventh row, first row, second row, four through seven, sorry first column, 354 00:37:24,240 --> 00:37:33,930 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. 355 00:37:33,930 --> 00:37:38,230 And then those are the points that changed. So it's not just any old bunch of points of change. 356 00:37:38,230 --> 00:37:42,810 This is the fact that this set here is called the set of all points, 357 00:37:42,810 --> 00:37:50,340 where the row and the column lions at one two four seven is what you might call the Cartesian Square. 358 00:37:50,340 --> 00:37:58,380 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. 359 00:37:58,380 --> 00:38:04,050 There'll be four times four makes 16, which is a square number. 360 00:38:04,050 --> 00:38:07,290 It turns out from this. So what of what is this? 361 00:38:07,290 --> 00:38:20,670 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. 362 00:38:20,670 --> 00:38:24,450 I will not be able to avoid getting one of these special lines. 363 00:38:24,450 --> 00:38:32,200 Special line where the coordinates are vary are arranged in a nice pattern like that. 364 00:38:32,200 --> 00:38:40,700 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. 365 00:38:40,700 --> 00:38:46,180 But again, I want to illustrate it better with just one slide. 366 00:38:46,180 --> 00:38:51,680 Um. Now. 367 00:38:51,680 --> 00:38:53,990 If you didn't completely follow that because I think it is challenging, 368 00:38:53,990 --> 00:38:59,300 it was challenging for me when I first came across this term to work out what on Earth was going on. 369 00:38:59,300 --> 00:39:04,700 Please believe me that this is a generalisation, and from it you can quite straightforward to deduce, for example, 370 00:39:04,700 --> 00:39:12,980 the statement that from this sort of case here that if you cut of integers from one to N and is very large, 371 00:39:12,980 --> 00:39:15,530 you can find an metric progression of like three. 372 00:39:15,530 --> 00:39:23,910 But a common difference is a perfect square turns out to be a fairly easy exercise to get from this statement to that statement. 373 00:39:23,910 --> 00:39:29,020 And this is just one special case which analyses the polynomial thing. 374 00:39:29,020 --> 00:39:40,040 Now. Getting to the tentative version is always straightforward, you just replace is green or is some particular colour by belongs to some dense. 375 00:39:40,040 --> 00:39:44,300 So here what I'm saying is if you choose one percent of all the possible points that you can get here, 376 00:39:44,300 --> 00:39:51,170 you must find a line of this special form that lives inside that search. 377 00:39:51,170 --> 00:39:55,340 So that's the density version of that's generalising all three different directions. 378 00:39:55,340 --> 00:40:00,660 That's a density version of the polynomial hails Jewitt theorem. 379 00:40:00,660 --> 00:40:04,620 But there's one difference between this result and all the other results that I've talked about, 380 00:40:04,620 --> 00:40:09,120 which is this is not a result, this is an open problem. 381 00:40:09,120 --> 00:40:16,230 And I wanted to get to that because that's something that I am working on right now with one of my research students. 382 00:40:16,230 --> 00:40:22,270 I wanted to sort of show that just by this process of generalisation, you can get. 383 00:40:22,270 --> 00:40:34,830 To the frontiers of mathematics, I will stop there. Thank you very much. 384 00:40:34,830 --> 00:40:40,500 So we're going to have about 25, 30 minutes of questions and then for over two years, 385 00:40:40,500 --> 00:40:45,900 if you have questions that you want to ask again and please, do you think of them as we go along? 386 00:40:45,900 --> 00:40:50,310 And I think for me, what I wanted to kick off with was just ask you about those kind of problems. 387 00:40:50,310 --> 00:40:59,060 What is it about them that intrigues you, I guess? How did you choose them as your your area of study? 388 00:40:59,060 --> 00:41:04,980 In a sense, I didn't actually, because when I started out, I was doing a completely different area of mathematics, 389 00:41:04,980 --> 00:41:20,000 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. 390 00:41:20,000 --> 00:41:27,380 Thought I potentially saw a way of coming up with a new proof of summary system, which quite a long way down the line. 391 00:41:27,380 --> 00:41:38,120 I eventually did. And when that happened, the whole sort of focus of my research changed into something called additive combinatorics, 392 00:41:38,120 --> 00:41:45,920 which concerns problems somewhat similar to the ones that I've just been talking about over and. 393 00:41:45,920 --> 00:41:52,670 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? 394 00:41:52,670 --> 00:42:02,350 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? 395 00:42:02,350 --> 00:42:10,330 And my answer to that is that they're not mere puzzles. 396 00:42:10,330 --> 00:42:18,910 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. 397 00:42:18,910 --> 00:42:25,930 Many typically don't have that many direct applications even to other parts of mathematics, let alone outside mathematics. 398 00:42:25,930 --> 00:42:30,670 But the techniques that you're forced to develop in order to solve one of these problems will very 399 00:42:30,670 --> 00:42:37,990 often have much wider applicability and implications for other parts of mathematics very often. 400 00:42:37,990 --> 00:42:47,470 So one of the great mathematicians in combinatorics, which is a sort of wider area that I belong to Paul Adams. 401 00:42:47,470 --> 00:43:01,640 Was a master at asking. Problems that had this kind of quality of seeming like amusing puzzles, but they just encapsulated some difficulty. 402 00:43:01,640 --> 00:43:05,300 He didn't just throw out any old question, and when you started thinking about them, 403 00:43:05,300 --> 00:43:11,270 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. 404 00:43:11,270 --> 00:43:17,000 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. 405 00:43:17,000 --> 00:43:18,930 Is that motivating motivating factor for you. 406 00:43:18,930 --> 00:43:28,580 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. 407 00:43:28,580 --> 00:43:37,400 Well. Can useful can mean useful to engineers or it can mean useful to other pure mathematicians. 408 00:43:37,400 --> 00:43:43,670 I would be thrilled if something that I did was useful to engineers, but it's not my primary goal. 409 00:43:43,670 --> 00:43:50,180 And if something is useful to other mathematicians and I feel as though it's helping to develop the subject. 410 00:43:50,180 --> 00:43:59,060 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. 411 00:43:59,060 --> 00:44:03,380 It's just inevitable that the applications outside mathematics follow. 412 00:44:03,380 --> 00:44:09,380 It's just that it may not be my precise theorem that I still feel I'm contributing 413 00:44:09,380 --> 00:44:18,600 to a big endeavour and then bits of that endeavour that almost randomly. 414 00:44:18,600 --> 00:44:23,700 Chosen somehow turn out to be very helpful to people outside mathematics. 415 00:44:23,700 --> 00:44:33,080 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, 416 00:44:33,080 --> 00:44:43,690 but it wasn't my I wouldn't say it was always by far and away my favourite subject it was. 417 00:44:43,690 --> 00:44:48,790 And the other thing that makes me say not that early was that. 418 00:44:48,790 --> 00:44:54,700 Until at least when I was an undergraduate and maybe even later than that, I didn't really. 419 00:44:54,700 --> 00:45:00,030 Or maybe towards the end of my undergraduate time at. 420 00:45:00,030 --> 00:45:07,350 Perhaps not even the to that it made up, of course. Part three of Cambridge is part three course. 421 00:45:07,350 --> 00:45:11,050 Did I have the slightest conception of what research in mathematics would be like? 422 00:45:11,050 --> 00:45:17,590 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? 423 00:45:17,590 --> 00:45:21,180 So just what could research in mathematics actually be like? 424 00:45:21,180 --> 00:45:28,860 And if you're unkind, you say, is it multiplying larger and had a few times? 425 00:45:28,860 --> 00:45:39,720 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. 426 00:45:39,720 --> 00:45:50,980 And I hope maybe I've conveyed something of the sort of world that mathematicians inhabit, or at least some corner of mathematics. 427 00:45:50,980 --> 00:45:59,510 So. And then another another sort of thing that makes you maybe not want to say I want 428 00:45:59,510 --> 00:46:03,770 to be a mathematician is that when you're starting something very intimidating 429 00:46:03,770 --> 00:46:09,410 about the very notion of an open problem in mathematics because you're surrounded 430 00:46:09,410 --> 00:46:13,400 by people who have a reputation for being incredibly clever and so on, 431 00:46:13,400 --> 00:46:18,920 you say, Well, there are all those people out that are incredibly clever research mathematicians, and this is an open problem. 432 00:46:18,920 --> 00:46:24,620 How am I going to come in and solve that problem? There is an answer to that. 433 00:46:24,620 --> 00:46:29,450 And the answer is not. You have brains bursting out of your head. It's that mathematics. 434 00:46:29,450 --> 00:46:33,620 Just as I've been saying, we don't run out of interesting problems that we get more and more problems in. 435 00:46:33,620 --> 00:46:37,770 The more you think about it, the more questions you ask, the more problems get generated. 436 00:46:37,770 --> 00:46:45,440 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. 437 00:46:45,440 --> 00:46:54,650 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. 438 00:46:54,650 --> 00:47:03,170 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. 439 00:47:03,170 --> 00:47:09,070 When did you work that out? When can you remember when you stopped being intimidated? 440 00:47:09,070 --> 00:47:13,940 I suppose it was when I first. Made any progress at all? 441 00:47:13,940 --> 00:47:19,850 Well, then maybe that's a slight. So the first thing I did was just a tiny little tweak to someone's existing 442 00:47:19,850 --> 00:47:25,460 argument that improves the answer to prove the bounds that came out of the proof. 443 00:47:25,460 --> 00:47:37,730 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, 444 00:47:37,730 --> 00:47:44,360 I think, to go smoke a cigarette and I came back for that sort of gave an indication of his level of interest. 445 00:47:44,360 --> 00:47:51,170 But I later on found a much more something with the same problem that got not just an improved result, 446 00:47:51,170 --> 00:47:54,950 but the best possible result that you could get by a much more complicated argument. 447 00:47:54,950 --> 00:47:59,600 And so at that point, no one was going to know, and I sort of felt this. 448 00:47:59,600 --> 00:48:03,320 I've definitely shown that I could do research at that point. 449 00:48:03,320 --> 00:48:06,590 That's probably in the middle of my first of my second year. 450 00:48:06,590 --> 00:48:12,080 And that was the thing that certain genetics, I guess something I knew, I thought something. 451 00:48:12,080 --> 00:48:17,330 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? 452 00:48:17,330 --> 00:48:21,920 Tell us a little bit about your process when you are approaching a new problem. How do you do it? 453 00:48:21,920 --> 00:48:28,520 How do you how do you tackle a challenge? Part of the answer, actually, 454 00:48:28,520 --> 00:48:35,180 is choosing the problem in the first place to make sure that it just feels like the kind of thing that 455 00:48:35,180 --> 00:48:43,420 might conceivably be amenable to the sorts of things that I have in my own mathematical toolbox. 456 00:48:43,420 --> 00:48:49,330 I can very often just misjudge that completely, and the problem is much harder and not something that I can do. 457 00:48:49,330 --> 00:48:55,480 So that's another thing to bear in mind that I think if you have the right level of ambition by not doing research, 458 00:48:55,480 --> 00:49:04,360 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 459 00:49:04,360 --> 00:49:09,730 that far because you're looking for that one in 10 when something kind of gives. 460 00:49:09,730 --> 00:49:20,360 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, 461 00:49:20,360 --> 00:49:24,590 which is you don't try and take route one to the solution. 462 00:49:24,590 --> 00:49:33,080 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, 463 00:49:33,080 --> 00:49:40,010 they're carefully tailored so that think you use a standard method and you just use that method and you get to the solution? 464 00:49:40,010 --> 00:49:47,750 Or perhaps there's one little trick, and once you've spotted the trick, it's plain sailing from that moment on. 465 00:49:47,750 --> 00:49:51,740 But a real life open research problem is not like that at all. 466 00:49:51,740 --> 00:49:56,600 So if you try and use that sort of idea of just try and hit it with standard methods. 467 00:49:56,600 --> 00:50:00,560 Or just look for that one little idea after which it'll be easy. 468 00:50:00,560 --> 00:50:08,510 You won't get very far. Very occasionally there are problems that surprisingly some little trick works, but that's very much the exception. 469 00:50:08,510 --> 00:50:14,910 So somebody have to ask yourself when you're doing any sort of research is. 470 00:50:14,910 --> 00:50:20,460 When you have an idea about how to tackle it, the first question you ought to ask yourself is, well, if that idea worked? 471 00:50:20,460 --> 00:50:23,740 Why is it that this is still an open problem? 472 00:50:23,740 --> 00:50:29,650 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, 473 00:50:29,650 --> 00:50:34,150 or it involves mixing using ideas from a totally different area of mathematics that 474 00:50:34,150 --> 00:50:38,140 would not be familiar to the experts in this area of mathematics or something like that. 475 00:50:38,140 --> 00:50:41,470 Some story to tell than probably your approach will fail, 476 00:50:41,470 --> 00:50:47,240 although it can be valuable to see why the approach fails and then try to think something more. 477 00:50:47,240 --> 00:50:54,070 Sophisticated. I've lost track of what your original question was, not when I was coming to the answer. 478 00:50:54,070 --> 00:50:57,550 What's your process when you're tackling any problem? Oh yes. 479 00:50:57,550 --> 00:51:05,590 The first test was you don't take the direct route, but you ask yourself other questions. 480 00:51:05,590 --> 00:51:13,840 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. 481 00:51:13,840 --> 00:51:21,670 So that's a generalisation. And the reason that can work is that sometimes when you generalise the statement, 482 00:51:21,670 --> 00:51:26,440 it actually paradoxically, although are proving something stronger, it's easier to do. 483 00:51:26,440 --> 00:51:32,970 Why is it easier? It's because. She's trying to prove something that's stronger and more general. 484 00:51:32,970 --> 00:51:39,290 You don't have as much. Room for manoeuvre, it's a bit like being if you're in a worse position in chess, 485 00:51:39,290 --> 00:51:45,530 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. 486 00:51:45,530 --> 00:51:49,340 If you make your problems harder, it can make them easier. 487 00:51:49,340 --> 00:51:56,150 So that's one method, but another method is exact opposite, which is look at a very special case of what you are doing. 488 00:51:56,150 --> 00:52:03,450 Save with some said you could look at progressions of like three that turned out to be a very fruitful thing to do. 489 00:52:03,450 --> 00:52:08,940 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 490 00:52:08,940 --> 00:52:15,660 doing in the special cases and then come generalised to get back to the thing you were originally trying to answer, 491 00:52:15,660 --> 00:52:19,500 sometimes in other methods. Another question you can ask as well. 492 00:52:19,500 --> 00:52:25,080 I don't see how to solve this, but let me invent a different problem in a slightly different context. 493 00:52:25,080 --> 00:52:28,620 That appears to resemble this problem and involve quite a lot of the same difficulties and 494 00:52:28,620 --> 00:52:34,910 see whether I can just solve this somewhat similar problem and get some insights from that. 495 00:52:34,910 --> 00:52:42,950 And what someone's always looking for is a new problem that should be easier than the problem you're trying to solve. 496 00:52:42,950 --> 00:52:47,750 That should help you to solve the problem you're trying to solve. If you can solve that one. 497 00:52:47,750 --> 00:52:52,820 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. 498 00:52:52,820 --> 00:52:59,690 This is starting from nothing. Say I invent this new problem that might be helpful for this one. 499 00:52:59,690 --> 00:53:04,580 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. 500 00:53:04,580 --> 00:53:14,570 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. 501 00:53:14,570 --> 00:53:21,170 So this process of sort of splitting up a kind of top down approach is very important and involves a lot of that. 502 00:53:21,170 --> 00:53:27,080 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. 503 00:53:27,080 --> 00:53:34,490 Absolutely not like that. But I mean, that process inevitably involves a lot of dead ends along the way. 504 00:53:34,490 --> 00:53:44,650 Do you think it's important to how do you shield yourself against against being discouraged when you meet dead ends? 505 00:53:44,650 --> 00:53:48,760 I think it's quite hard to do when you're just starting out, because you really, 506 00:53:48,760 --> 00:53:55,490 really want to have got that first theorem first solution to an open problem. 507 00:53:55,490 --> 00:54:01,690 Once you have that ideally maybe two or three times and then you sort of know that it can be done. 508 00:54:01,690 --> 00:54:09,700 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. 509 00:54:09,700 --> 00:54:18,310 So you then sort of get used to it as being part of the job. This is part of the process of proving theorems. 510 00:54:18,310 --> 00:54:20,140 It's not necessarily the most difficult part, either. 511 00:54:20,140 --> 00:54:25,240 I mean, another difficult thing is when you're trying to do something and then somebody else does it basata. 512 00:54:25,240 --> 00:54:31,870 And or even worse, you work hard on a problem and discover that it was done 10 years ago. 513 00:54:31,870 --> 00:54:44,250 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. 514 00:54:44,250 --> 00:54:55,070 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. 515 00:54:55,070 --> 00:55:02,230 I was just covering all these things. I just covered various everything thinking I was. 516 00:55:02,230 --> 00:55:08,560 Discovering it for myself, and then I discovered there's a whole field and all the concepts of coming up with a well known, 517 00:55:08,560 --> 00:55:16,840 but even that was that wasn't a major disappointment to me really, because I was nowhere near. 518 00:55:16,840 --> 00:55:23,460 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. 519 00:55:23,460 --> 00:55:30,630 And if you were to rediscover things, that's actually very beneficial because you understand those things in some 520 00:55:30,630 --> 00:55:37,020 deep way that you don't quite get if you're just passively read a textbook. 521 00:55:37,020 --> 00:55:45,030 So again, not even even from this of disappointments, one does eventually gain even that little bit crushing for a while. 522 00:55:45,030 --> 00:55:50,130 Has there been an occasion where you've been picked to the post in trying to be something? 523 00:55:50,130 --> 00:56:00,400 Well, they've been several occasions where. People have proved things that I would very much like to have proved, um, but. 524 00:56:00,400 --> 00:56:06,910 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. 525 00:56:06,910 --> 00:56:11,770 It is more common than I've thought really hard about something and worked pretty hard on it and then sort of thought, 526 00:56:11,770 --> 00:56:15,610 Oh, maybe I'll try something else. It's still one of my favourite problems, 527 00:56:15,610 --> 00:56:23,740 but you have to move around because otherwise you're going to get sort of stuck not solving a problem for five years. 528 00:56:23,740 --> 00:56:30,030 That's Andrew Wiles says that's not a good example for this doesn't always work out. 529 00:56:30,030 --> 00:56:38,090 No, it's not a good model. I should say it's a very good example of a good model to follow. 530 00:56:38,090 --> 00:56:43,520 And. So even then, 531 00:56:43,520 --> 00:56:51,490 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. 532 00:56:51,490 --> 00:56:58,640 It also this time of disappointment. But that's again, something that one has to just take on the chin and keep going. 533 00:56:58,640 --> 00:57:08,170 I am curious now which theorems would you have liked involved? Well, the one that I was talking about, first of all. 534 00:57:08,170 --> 00:57:13,030 Well, I sort of developed the basics and it's sort of beginnings of the service. 535 00:57:13,030 --> 00:57:16,720 I was I thought I would have a try at the P versus NP problem. 536 00:57:16,720 --> 00:57:25,300 This is a long time ago and I sort of underestimated how much work had gone into that problem. 537 00:57:25,300 --> 00:57:32,410 And that's a restriction of the problem to do with a concept called monitor and circuit complexity. 538 00:57:32,410 --> 00:57:39,370 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. 539 00:57:39,370 --> 00:57:43,690 And so I formulated that problem and thought about it quite a lot. 540 00:57:43,690 --> 00:57:52,090 And then it turned out that simple Sasha Raspberry I've had years before. 541 00:57:52,090 --> 00:57:57,650 Proved exactly what I was trying to prove and won prises for it. 542 00:57:57,650 --> 00:58:03,450 And but that would be, you know, that would be absolutely great. 543 00:58:03,450 --> 00:58:09,540 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. 544 00:58:09,540 --> 00:58:16,140 I'm not complaining at all. Do you remember when you found out about the Fields medal? 545 00:58:16,140 --> 00:58:22,450 Did it come as a surprise? I'm. Yes and no. 546 00:58:22,450 --> 00:58:29,800 So. I it was not a surprise that I was being considered for it because I got all sorts of mysterious messages, 547 00:58:29,800 --> 00:58:35,770 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. 548 00:58:35,770 --> 00:58:43,910 And it was the right time of the cycle for the decision to be. 549 00:58:43,910 --> 00:58:49,820 For the committed to making its deliberations, and so I. Didn't want to. 550 00:58:49,820 --> 00:59:03,440 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. 551 00:59:03,440 --> 00:59:06,710 And then. 552 00:59:06,710 --> 00:59:18,190 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. 553 00:59:18,190 --> 00:59:23,920 And so that couldn't really find what it was about, but it was and we both got fields, medals. 554 00:59:23,920 --> 00:59:29,890 And at the same time, yes. Did it? 555 00:59:29,890 --> 00:59:34,420 Did you? Does that mean that you were in some sense sharing the glory? 556 00:59:34,420 --> 00:59:39,400 Well, no. I mean that they gave awards, of course, each time. 557 00:59:39,400 --> 00:59:43,150 But I mean, say, within Cambridge or something, I suppose. 558 00:59:43,150 --> 00:59:46,330 But I think that in a way, I think it that wasn't the problem. 559 00:59:46,330 --> 00:59:54,190 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. 560 00:59:54,190 --> 01:00:05,540 So I think it actually works to my benefit, really. How much do those those prises mean to you? 561 01:00:05,540 --> 01:00:10,780 What's made a massive difference my life in a way, because. 562 01:00:10,780 --> 01:00:18,060 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, 563 01:00:18,060 --> 01:00:23,130 you know, four people who tower above everyone else. 564 01:00:23,130 --> 01:00:24,510 It's more like, you know, 565 01:00:24,510 --> 01:00:30,300 the committee that makes the decision has a difficult decision and just eventually has to sort of settle for four four people. 566 01:00:30,300 --> 01:00:35,820 So that's one thing that's there's not some sort of difference in kind between someone who gets one and 567 01:00:35,820 --> 01:00:43,170 someone who just misses out or indeed someone who doesn't even just miss out but does something amazing. 568 01:00:43,170 --> 01:00:47,850 A few years later or something like that, I know that the longer since it happened, 569 01:00:47,850 --> 01:00:53,610 the more remarkable examples of more remarkable things I see other mathematicians doing, 570 01:00:53,610 --> 01:01:04,340 and the more I sort of see my actually rather small place in the vast body of vast corpus of mathematics. 571 01:01:04,340 --> 01:01:12,200 But I think what it gives me is what has given me is just sort of unfair leg up in life, 572 01:01:12,200 --> 01:01:16,760 really, that I just get I got lots of invitations, interesting invitations to things. 573 01:01:16,760 --> 01:01:23,900 People sort of like the introduction here saying, you know, we got a field medallist giving a talk and so on. 574 01:01:23,900 --> 01:01:33,340 And it's sort of it is quite nice. But. 575 01:01:33,340 --> 01:01:41,580 But at the same time. In the end, I don't think it's. 576 01:01:41,580 --> 01:01:46,500 I think I sort of felt after getting that I had to try to some extent to sort 577 01:01:46,500 --> 01:01:52,740 of pretend I hadn't got it so as not to sort of relax too much and just not. 578 01:01:52,740 --> 01:01:58,680 Another thing that could have been a mistake, I think, would have been to to say, Well, now I've got that. 579 01:01:58,680 --> 01:02:04,830 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 580 01:02:04,830 --> 01:02:08,250 or solve some really massive problem that's bigger than anything I've done before. 581 01:02:08,250 --> 01:02:15,360 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. 582 01:02:15,360 --> 01:02:19,590 Just keep on working on the things that interest you and stop the good things. 583 01:02:19,590 --> 01:02:24,720 Happy medium things. Do you think it kind of liberated you in a way, though? 584 01:02:24,720 --> 01:02:33,630 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. 585 01:02:33,630 --> 01:02:36,210 So one of them was editing the prison components mathematics, 586 01:02:36,210 --> 01:02:44,250 which was mentioned earlier on which I would say took up roughly half my working time for about five years, 587 01:02:44,250 --> 01:02:50,880 which was a sacrifice that I'm not sure I would have felt I could possibly make. 588 01:02:50,880 --> 01:02:55,470 If I hadn't somehow just made any sort of reputation I needed to make. 589 01:02:55,470 --> 01:03:02,760 And it was a project that I believed in. But it's not not a research project and also I didn't have. 590 01:03:02,760 --> 01:03:12,020 Worry too much about people saying he's. Sort of gone soft, he's doing popularisation now, not mathematics or something like that. 591 01:03:12,020 --> 01:03:20,300 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, 592 01:03:20,300 --> 01:03:23,270 but quite a lot of time thinking about automatic theorem, 593 01:03:23,270 --> 01:03:29,570 proving that is trying to get computers to find proofs of theorems, which is something that really interests me. 594 01:03:29,570 --> 01:03:36,070 But again, it sort of doesn't really count as mathematics. So it's something that. 595 01:03:36,070 --> 01:03:38,520 I think I wouldn't have felt. 596 01:03:38,520 --> 01:03:49,530 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. 597 01:03:49,530 --> 01:03:58,440 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. 598 01:03:58,440 --> 01:04:05,650 So I had about three years where I was not doing that much mass at all. 599 01:04:05,650 --> 01:04:09,700 Doing a bit, but less spending less time en masse, and I was on automatic, 600 01:04:09,700 --> 01:04:16,780 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. 601 01:04:16,780 --> 01:04:23,800 They're improving. So my interest on the side. Do you think those prices are generally a good thing for the mosque community? 602 01:04:23,800 --> 01:04:37,140 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 603 01:04:37,140 --> 01:04:45,630 come with a massive amount of money in the way that the Nobel prise does have a really quite small amount of money. 604 01:04:45,630 --> 01:04:54,030 And so that means that the sort of pain of not getting it, I think, is much reduced because for two reasons. 605 01:04:54,030 --> 01:05:00,120 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. 606 01:05:00,120 --> 01:05:07,770 And the other is that even after your it's too late because you're about to turn 40 years old, 607 01:05:07,770 --> 01:05:15,780 you certainly haven't stopped being able to do mass and is still plenty of chance to to to prove amazing theorems and 608 01:05:15,780 --> 01:05:24,930 get an amazing reputation and maybe win other prises that are aimed at older people if you're interested in prises. 609 01:05:24,930 --> 01:05:31,740 So I think there may be some degree of unhealthiness about it, but uh. 610 01:05:31,740 --> 01:05:39,780 But I think in many people's cases, it gets some benefit as well as people sort of got a sort of dream, maybe. 611 01:05:39,780 --> 01:05:43,990 I think it probably in my case. 612 01:05:43,990 --> 01:05:55,770 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. 613 01:05:55,770 --> 01:05:59,350 So my mother had some beneficial effect in that direction. 614 01:05:59,350 --> 01:06:07,120 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. 615 01:06:07,120 --> 01:06:14,920 Do you think there's ever a contradiction between that sort of working together on a particular theorem or 616 01:06:14,920 --> 01:06:24,790 proof and the sort of thirst that might be required for a prise or sort of an individual the hunt for glory? 617 01:06:24,790 --> 01:06:33,420 If we're talking about huge collaboration, then of which you're just one very small part of, which was the thing that I. 618 01:06:33,420 --> 01:06:39,210 Set off at one stage, then I think not many people would be satisfied with just participating, 619 01:06:39,210 --> 01:06:47,740 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. 620 01:06:47,740 --> 01:06:53,610 That's one of the reasons hasn't taken over how we do mathematics. But mathematics is now very, very collaborative. 621 01:06:53,610 --> 01:06:58,230 And I'd say the majority of papers are two or three courses. 622 01:06:58,230 --> 01:07:06,250 And that's for the simple reason that two or three people can often work a lot more efficiently than twice as efficient. 623 01:07:06,250 --> 01:07:10,080 There were three times as much as one person. For obvious reasons you can specialise. 624 01:07:10,080 --> 01:07:15,840 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. 625 01:07:15,840 --> 01:07:23,430 Another person might be good at hitting them down again. Somebody might have expertise in one area, somebody might have expertise in another area, 626 01:07:23,430 --> 01:07:29,490 and I think people just find it's easier to come up with to be a sort of say it's two 627 01:07:29,490 --> 01:07:34,690 or two papers be easier to write fifty two or two papers and 25 one of the papers. 628 01:07:34,690 --> 01:07:42,040 And another thing is that I think if you are one of two authors, you probably get. 629 01:07:42,040 --> 01:07:51,890 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. 630 01:07:51,890 --> 01:07:58,160 I think you probably get 75 percent of the criticism unless you've unless there's some very 631 01:07:58,160 --> 01:08:02,720 strong suspicion that you weren't really the person who had the ideas or something like that. 632 01:08:02,720 --> 01:08:09,530 But that's another good thing about mathematics compared with other subjects. We have also strictly in alphabetical order, 633 01:08:09,530 --> 01:08:17,120 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, 634 01:08:17,120 --> 01:08:27,140 so forth, we avoid all that in order. So exactly to sort of avoid that kind of talk about who really did this in. 635 01:08:27,140 --> 01:08:35,060 And the result of that, I think, is that it is strongly encourages people to work in small collaborations quite like that. 636 01:08:35,060 --> 01:08:41,780 This is that you're sort of presenting a very rational kind of calculated argument towards small collaborators. 637 01:08:41,780 --> 01:08:47,600 And this I think as I regularly read your blog, I don't know how many people here also do, 638 01:08:47,600 --> 01:08:54,380 but this is something have sort of known for taking a sort of rational approach to decisions that might not necessarily seem mathematical. 639 01:08:54,380 --> 01:09:03,410 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. 640 01:09:03,410 --> 01:09:11,900 But maths does give you the tools for recognising that when that happens, just so they're not always used to effect. 641 01:09:11,900 --> 01:09:19,100 Is it true that you once decided whether or not to have a medical procedure based on your speculation of the risks? 642 01:09:19,100 --> 01:09:23,570 It is because it was, uh, I have had problems. 643 01:09:23,570 --> 01:09:31,700 Actually, they've just recently seem to be starting up again, which is annoying of arrhythmia in my heart and atrial fibrillation. 644 01:09:31,700 --> 01:09:38,600 And if somebody does a procedure that can be done to correct that where they stick a wire up a vein, 645 01:09:38,600 --> 01:09:42,990 go in your leg and the mine goes up into your heart and it sounds very unpleasant. 646 01:09:42,990 --> 01:09:50,930 They sort of burn the side of your heart. But the reason for that is that it breaks an electrical connexion, 647 01:09:50,930 --> 01:09:58,100 which is messing things up if you don't want to have and causing the rhythm to go off. 648 01:09:58,100 --> 01:10:03,920 And it's a pretty safe operation, but when you read about the risks, 649 01:10:03,920 --> 01:10:09,530 it all sounds quite frightening and in particular as one of the thousand mortality risk. 650 01:10:09,530 --> 01:10:17,270 So I thought, I really sort of I really want to have this one in a thousand risk of dying then. 651 01:10:17,270 --> 01:10:21,260 So I looked into it a bit more. And of course, it depends a lot on how old you are and various other things. 652 01:10:21,260 --> 01:10:26,120 And also the benefits of operation depend on science, on how old you are. 653 01:10:26,120 --> 01:10:32,510 And eventually, I decided to go ahead because I had the idea of looking up. 654 01:10:32,510 --> 01:10:39,020 What the mortality risk was of just being alive for two months, age of 50. 655 01:10:39,020 --> 01:10:45,530 It turned out that an operation concentrated about a month's risk into one operation. 656 01:10:45,530 --> 01:10:53,810 So my risk of sort of falling under a bus or whatever it was over a month would be about one in 10000. 657 01:10:53,810 --> 01:11:03,650 The idea was then I thought, Well, I'm not terribly scared of the next month, so I shouldn't be all that scared. 658 01:11:03,650 --> 01:11:07,940 That was the level of rationality somehow. And then it all went fine. 659 01:11:07,940 --> 01:11:12,710 And yeah. But is this something that you find yourself doing a lot? 660 01:11:12,710 --> 01:11:18,470 Sort of. I mean, I noticed on your blog you often with politics, especially, 661 01:11:18,470 --> 01:11:23,630 you do this sort of very involved posts where your you seem to be almost breaking 662 01:11:23,630 --> 01:11:28,360 argument down to try and persuade yourself of something one way or the other. 663 01:11:28,360 --> 01:11:33,370 Actually, I would say that they're I'm fairly sure what my view is in advance, 664 01:11:33,370 --> 01:11:37,870 but I just want to sort of check that there is some rational justification for it. 665 01:11:37,870 --> 01:11:42,340 So I wrote a post about why apologies have leave voters in the audience, 666 01:11:42,340 --> 01:11:50,890 but why we should remain in the EU and which had sort of game theoretic aspect to it. 667 01:11:50,890 --> 01:11:57,430 I wrote something about why everyone should vote for the alternative vote in the 2011 referendum, because see, 668 01:11:57,430 --> 01:12:11,800 these posts are very successful, and I wrote something recently about tactical voting in the European elections. 669 01:12:11,800 --> 01:12:22,280 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. 670 01:12:22,280 --> 01:12:34,730 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, 671 01:12:34,730 --> 01:12:39,350 say there's some stuff on the table and you've got to transfer it to the side of the sink and 672 01:12:39,350 --> 01:12:42,710 there's some other stuff in the sink and needs to go in the cupboard and that sort of thing. 673 01:12:42,710 --> 01:12:45,680 It's a mistake to say that first on the water tables and sink stuff, 674 01:12:45,680 --> 01:12:51,920 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, 675 01:12:51,920 --> 01:12:57,890 you could have been carrying something to that sort of thing. So I do have those kinds of thoughts. 676 01:12:57,890 --> 01:13:02,570 I promise I have a life. Does it drive your family mad? 677 01:13:02,570 --> 01:13:05,380 I try to keep quiet about what I'm doing. 678 01:13:05,380 --> 01:13:14,480 Look, on the rare occasions when I've suggested anybody else in the family that are more sort of ergonomic ways of doing things, 679 01:13:14,480 --> 01:13:21,080 I've been slapped down very quickly. I'm going to come to the audience in just one second. 680 01:13:21,080 --> 01:13:22,310 Just one last question. 681 01:13:22,310 --> 01:13:30,380 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? 682 01:13:30,380 --> 01:13:38,290 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, 683 01:13:38,290 --> 01:13:49,300 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. 684 01:13:49,300 --> 01:13:55,620 As long as your. Pretty good at maths and very, very keen. 685 01:13:55,620 --> 01:13:59,610 To make progress, but I think the keenness is, you know, 686 01:13:59,610 --> 01:14:08,040 I think if you have a choice between sacrificing a couple of points of your IQ but gaining 10 percent of enthusiasm, 687 01:14:08,040 --> 01:14:14,410 go for the enthusiasm because it may take you slightly longer to have some ideas, but you'll have some. 688 01:14:14,410 --> 01:14:19,420 In the end, that's not solving a problem, it's a slow process. It doesn't really matter how quick you are. 689 01:14:19,420 --> 01:14:30,730 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. 690 01:14:30,730 --> 01:14:36,880 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. 691 01:14:36,880 --> 01:14:42,330 I think the one example I can think of somebody who is a prodigy and is by any account, 692 01:14:42,330 --> 01:14:46,870 a genius who has also made a huge success that would be Terrence Town. 693 01:14:46,870 --> 01:14:54,040 But if not Terrence Taylor, you just don't. There are lots of nontariff stars have done very good things as well. 694 01:14:54,040 --> 01:15:01,630 So what advice if you could go back in time and speak to your 17 year old self, what advice would you give yourself? 695 01:15:01,630 --> 01:15:09,060 I think I would say. I would go back to what I was talking about earlier, that it looks. 696 01:15:09,060 --> 01:15:15,630 I say don't be intimidated by all this, by the cleverness that you see around. 697 01:15:15,630 --> 01:15:23,840 As long as you keep at it. Learn what you have to learn, but try to bring plenty of problems, 698 01:15:23,840 --> 01:15:28,040 including problems that have been solved already, but just get practise at problem solving. 699 01:15:28,040 --> 01:15:35,840 So practise, practise, practise. Keep your enthusiasm and you can make room for yourself. 700 01:15:35,840 --> 01:15:42,720 Yeah, exactly. OK, we'll come to the audience then. So does anyone who wants to kick us off with questions. 701 01:15:42,720 --> 01:15:47,160 OK, we have. Let's go. We'll go first here. One. 702 01:15:47,160 --> 01:15:51,720 Yes, please. You have. Leave it at that. Perfect. And then Emily go behind and then we'll come over here. 703 01:15:51,720 --> 01:15:57,300 Thank you. Yeah. Thank you for a very engaging talk and interview afterwards. 704 01:15:57,300 --> 01:16:00,860 My question is to do with how you choose the problems. 705 01:16:00,860 --> 01:16:07,710 Once you've chosen your subject area, you know you're interested. How do you go about choosing the right problems to pursue? 706 01:16:07,710 --> 01:16:16,040 And then once you're pursuing them, how do you choose when to stop, when they seem harder than you thought? 707 01:16:16,040 --> 01:16:22,330 I think. Roughly speaking, I would. 708 01:16:22,330 --> 01:16:24,280 Part of it, I can't have a very satisfactory answer, 709 01:16:24,280 --> 01:16:30,400 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. 710 01:16:30,400 --> 01:16:38,170 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, 711 01:16:38,170 --> 01:16:39,460 or I might decide to continue. 712 01:16:39,460 --> 01:16:50,830 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. 713 01:16:50,830 --> 01:16:56,080 Promising related questions that was perhaps related to things that I did know 714 01:16:56,080 --> 01:17:01,900 how to solve and had a chance of being easier than my enthusiasm would continue, 715 01:17:01,900 --> 01:17:10,150 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. 716 01:17:10,150 --> 01:17:14,590 But there comes a time when solving a problem. 717 01:17:14,590 --> 01:17:23,740 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. 718 01:17:23,740 --> 01:17:31,330 You've done calculations that get really complicated, and you're not quite sure whether it's worth pursuing those calculations. 719 01:17:31,330 --> 01:17:37,370 You can't quite see where they're going to go, even if you succeed. But maybe. 720 01:17:37,370 --> 01:17:43,010 But then in the other direction, maybe you do some complicated calculation and then suddenly something simplifies dramatically. 721 01:17:43,010 --> 01:17:51,280 Now onto something here and then you sort of will carry on and see whether you get anywhere. 722 01:17:51,280 --> 01:18:00,160 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. 723 01:18:00,160 --> 01:18:06,020 And if I feel it's getting sort of. I'm getting to the point where it's not getting forward very fast. 724 01:18:06,020 --> 01:18:09,020 Then I'll be better off working on something else as you may be picking up. 725 01:18:09,020 --> 01:18:14,660 I'm quite I'm quite calculating about the process of what to work on with any one time, 726 01:18:14,660 --> 01:18:22,040 and I think you have to be to be successful because that matters because there's a lot of twin risks of giving 727 01:18:22,040 --> 01:18:27,050 things up too easily and not solving anything or not giving things up easily enough and not solving anything. 728 01:18:27,050 --> 01:18:38,190 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. 729 01:18:38,190 --> 01:18:50,160 I was just wondering if you think any of the millennium problems remaining or any problem in general are truly unsolvable or Iberville? 730 01:18:50,160 --> 01:18:54,300 I don't really know. I thought of the millennium problems. 731 01:18:54,300 --> 01:19:02,460 I think the one that people talk about as possibly being on unsolvable is the people's envy problem. 732 01:19:02,460 --> 01:19:06,720 My own feeling about that is that it's not because not for a good reason. 733 01:19:06,720 --> 01:19:12,300 But there's a way of formulating it that makes it look very combinatorial. 734 01:19:12,300 --> 01:19:17,010 Very sort of like that. Doesn't mention computers at all. Doesn't sound like a problem to do with logic. 735 01:19:17,010 --> 01:19:22,320 So it doesn't feel like the kind of problem that would be insoluble. 736 01:19:22,320 --> 01:19:29,550 But that's not a good reason, because there are much more sophisticated reasons for thinking that it might be that by people 737 01:19:29,550 --> 01:19:34,650 who are fully aware of what I just said about it having this combinatorial reformulation. 738 01:19:34,650 --> 01:19:43,370 So. As for the other ones, I really have no idea at all I. 739 01:19:43,370 --> 01:19:48,800 They say the Raymond hypothesis, I'd be very surprised if that were undecided. 740 01:19:48,800 --> 01:19:55,090 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. 741 01:19:55,090 --> 01:20:06,200 So Bradley surprises do happen. The question just here, it will go, and it will continue. 742 01:20:06,200 --> 01:20:09,970 I think it's a bitter thought that the decision will come to you in a second. 743 01:20:09,970 --> 01:20:16,010 Yeah, thank you very much. It's very it's an honour to listen to you. 744 01:20:16,010 --> 01:20:24,530 My first question is what is the most widely accepted mathematical proposition that could possibly turn out to be wrong? 745 01:20:24,530 --> 01:20:34,100 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? 746 01:20:34,100 --> 01:20:39,110 The answer the first is I really don't know at all. 747 01:20:39,110 --> 01:20:48,750 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. 748 01:20:48,750 --> 01:20:57,640 Significant parts of number theory, which is an area which rather unlike my areas, is rather a lot big hierarchy of this. 749 01:20:57,640 --> 01:21:02,760 This theorem uses this term, which uses this term understanding and this term which uses this and. 750 01:21:02,760 --> 01:21:10,650 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, 751 01:21:10,650 --> 01:21:13,770 you'd have to read sort of two thousand pages of very difficult stuff, 752 01:21:13,770 --> 01:21:17,100 some of which have been published and some of which is only sort of known to a few experts. 753 01:21:17,100 --> 01:21:22,590 And so there's just a slight anxiety that some of the results of numbers vary. 754 01:21:22,590 --> 01:21:28,440 But which ones I don't know might be wrong in quite serious ways. 755 01:21:28,440 --> 01:21:37,160 And he's he takes that so seriously if set up a whole programme of machine verification of of proofs and number theory. 756 01:21:37,160 --> 01:21:42,980 Who would I like to be? And sort of think like that somehow? 757 01:21:42,980 --> 01:21:51,390 It's a bit like saying, what other person would I like to be? I don't really have an answer. 758 01:21:51,390 --> 01:21:56,660 I have a lot of give up so much else. If I had to, my name would be someone else. 759 01:21:56,660 --> 01:22:00,710 OK, I think maybe was going to say you get sort of attached to your own terms as well. 760 01:22:00,710 --> 01:22:03,830 If you the part of the quid pro quo of being another mathematicians, 761 01:22:03,830 --> 01:22:11,870 I'd have to say goodbye to all those things that I spent a long time on this and work time for. 762 01:22:11,870 --> 01:22:16,980 I think maybe two more quick questions. The answer will give you that one last word at the back end of it. 763 01:22:16,980 --> 01:22:24,650 And it's been fascinating talk. Can I just ask you a bit more about the role of computers in pure mathematics research? 764 01:22:24,650 --> 01:22:30,290 And you've mentioned one or two specific problems and the role of collaboration, 765 01:22:30,290 --> 01:22:37,970 but in terms of fundamental change in the way pure mathematics is done, human rights research has done, as has the computer. 766 01:22:37,970 --> 01:22:43,190 What has the computer given us and what do you think it will give us in the future? 767 01:22:43,190 --> 01:22:50,090 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, 768 01:22:50,090 --> 01:22:59,630 which has helped, for example, proving the full colour theorem. It's also got a very nice role in helping to. 769 01:22:59,630 --> 01:23:02,360 When you're doing research and you invent questions, 770 01:23:02,360 --> 01:23:09,380 sometimes the questions will be things that were the answer can be checked on a computer because not all problems are amenable to that, 771 01:23:09,380 --> 01:23:15,230 but sometimes they are. So sometimes you might think if this were true, that would be really helpful for this. 772 01:23:15,230 --> 01:23:23,370 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, 773 01:23:23,370 --> 01:23:29,060 and it might after a while say here's a complicated counterexample which would have taken you a week 774 01:23:29,060 --> 01:23:35,030 to find and the computer just found it in five minutes or something so that can speed things up. 775 01:23:35,030 --> 01:23:42,320 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, 776 01:23:42,320 --> 01:23:49,130 which is that if you're working on a problem and out comes the first few terms of some sequence. 777 01:23:49,130 --> 01:23:55,820 I think you generate some integers, what you can do these days is put that into something called the online encyclopaedia of integer 778 01:23:55,820 --> 01:24:00,500 sequences to see whether that sequence is something that's come up in some other context. 779 01:24:00,500 --> 01:24:05,000 And very often it has. Almost always it has. 780 01:24:05,000 --> 01:24:12,080 And when it has, it will sometimes just give you a formula for the sequence, which would have been extremely hard to find. 781 01:24:12,080 --> 01:24:17,030 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. 782 01:24:17,030 --> 01:24:23,150 So computers can be very helpful in just speeding up the natural processes of research. 783 01:24:23,150 --> 01:24:29,900 Going back to your going to the question of where things might go in the future. 784 01:24:29,900 --> 01:24:34,970 I personally think that computers will eventually take over and just do everything for us. 785 01:24:34,970 --> 01:24:43,820 But that's a minority view. Most people think that that humans have this sort of intuition that will never be 786 01:24:43,820 --> 01:24:49,180 replicated on a computer that is absolutely essential to the process of doing mass. 787 01:24:49,180 --> 01:24:55,390 I I don't believe that it will take a while, but. 788 01:24:55,390 --> 01:24:58,480 But I think and before that happens, 789 01:24:58,480 --> 01:25:04,030 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, 790 01:25:04,030 --> 01:25:09,700 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, 791 01:25:09,700 --> 01:25:17,410 but easy for an expert and the right branch of mass on the computer might then be able to help answer it. 792 01:25:17,410 --> 01:25:23,590 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 793 01:25:23,590 --> 01:25:30,610 are incredibly helpful but don't quite spoil all the fun for they may be short lived. 794 01:25:30,610 --> 01:25:35,590 Yeah, I did my bit. OK, last question, plus a quick question and then one at that. 795 01:25:35,590 --> 01:25:39,730 Thank you. Thank you for speaking with us tonight. 796 01:25:39,730 --> 01:25:41,560 A few years ago on your blog, 797 01:25:41,560 --> 01:25:49,000 you wrote about teaching mathematics to non mathematicians and as a teacher of A-level maths is here with some A-level students. 798 01:25:49,000 --> 01:25:54,670 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, 799 01:25:54,670 --> 01:26:07,370 to study mathematics or other quantitative fields. So not teaching mathematics and not mathematicians, but teaching it to mathematicians. 800 01:26:07,370 --> 01:26:17,040 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. 801 01:26:17,040 --> 01:26:26,360 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, 802 01:26:26,360 --> 01:26:32,720 you can't just sort of do anything that suggests it. But I do think that. 803 01:26:32,720 --> 01:26:44,770 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. 804 01:26:44,770 --> 01:26:49,480 Worry too much about the A-level syllabus, but just sort of taught maths, 805 01:26:49,480 --> 01:26:52,480 and at the end of the two years you found that actually you had covered material, 806 01:26:52,480 --> 01:26:56,050 but we did lots of things that weren't ready to do with the A-level syllabus, 807 01:26:56,050 --> 01:27:01,390 and we were given problems that were much more challenging than you'd find on an A-level. 808 01:27:01,390 --> 01:27:07,150 It was necessary to do those problems to A-level, but it was really a very, very good experience. 809 01:27:07,150 --> 01:27:14,270 So it may not be a practical thing to do that in every school. 810 01:27:14,270 --> 01:27:25,190 Especially now when people do for A-levels and but. I think if space can be found for. 811 01:27:25,190 --> 01:27:30,450 Just the more that can be taught, we are not sort of focussing on the next level module. 812 01:27:30,450 --> 01:27:38,790 Perhaps. Maybe there aren't modules anymore, but that used to be when our modules. 813 01:27:38,790 --> 01:27:43,590 It was particularly bad because I was always next time, just around the corner. 814 01:27:43,590 --> 01:27:49,860 But the more it's practical not to focus on exams, the better. 815 01:27:49,860 --> 01:27:54,590 But. As I say, I repeat that I understand this. 816 01:27:54,590 --> 01:27:57,540 This is certainly easy to do that. 817 01:27:57,540 --> 01:28:05,580 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. 818 01:28:05,580 --> 01:28:31,286 Thank you very much, everyone, for coming, and it remains to thank you very much.