1 00:00:00,870 --> 00:00:03,650 George Zimmerman's trial. 2 00:00:20,480 --> 00:00:29,750 Thank you very much for for the introduction, in particular for the invitation both to Alan and to the Mathematical Department. 3 00:00:30,470 --> 00:00:35,330 It's a great honour and a privilege to be in this room and to speak in this building. 4 00:00:39,230 --> 00:00:45,520 If you look at this slide. It's a familiar site. 5 00:00:45,520 --> 00:00:51,280 It's Pebble Beach, and many of the shapes on the beach look familiar. 6 00:00:51,370 --> 00:00:58,270 They are just Pebble. So it is a pebble. It would recognise Pebbles. 7 00:00:58,690 --> 00:01:07,690 You would recognise this beach. However, if I would ask you to name some of these pebbles, that would be a challenge. 8 00:01:08,810 --> 00:01:11,830 Uh, we don't have names for shapes. 9 00:01:12,160 --> 00:01:17,710 We have a name, so very few shapes. And for the majority, you can see quite a number here. 10 00:01:17,800 --> 00:01:22,150 A couple of turn the tables on that medium. Uh, Pebble Beach. 11 00:01:22,150 --> 00:01:27,800 We don't have names. And this might be related to the fact that, um. 12 00:01:28,420 --> 00:01:35,380 Linguistic skills and shape recognition skills are in different cerebellar hemispheres. 13 00:01:36,350 --> 00:01:42,360 Uh. There were comments being made on this subject. 14 00:01:43,500 --> 00:01:52,170 One is due to allow Polyclinic Guy who was a famous mathematician and he said that God created the integers all as is the work of man. 15 00:01:53,730 --> 00:02:01,320 Meaning that? Well, in today's language I would say you would like to your bank account. 16 00:02:01,320 --> 00:02:06,060 You would like to be identified by an integer number and maybe not by a drawing. 17 00:02:06,570 --> 00:02:08,460 It's a safe thing. This is the thing. 18 00:02:08,730 --> 00:02:17,430 The integers are something where we feel home, where we feel comfortable with, and the shapes are maybe a little bit more elusive. 19 00:02:18,300 --> 00:02:27,990 There is another famous citation from Galileo, which is well-known that the Book of Nature is written in the language of mathematics. 20 00:02:29,010 --> 00:02:33,510 However, the second part of that citation is probably less known. 21 00:02:34,260 --> 00:02:40,410 It says that the symbols in the book are triangles, circles and other geometrical figures. 22 00:02:40,980 --> 00:02:47,250 So Galileo says that we need to understand shapes in order to talk about natural phenomena. 23 00:02:48,790 --> 00:02:53,120 However, you need to understand that we need to talk about them and the first. 24 00:02:53,440 --> 00:02:59,290 If you want to give names to the shapes, the first thing is to assign integer numbers to this shape somehow. 25 00:02:59,860 --> 00:03:06,880 So a goal is in the both sense, to understand that you are shapes by using natural numbers. 26 00:03:06,910 --> 00:03:11,260 To what extent can we explain shapes by the means of natural numbers? 27 00:03:12,220 --> 00:03:17,800 There has been a lot of thinking about Pablo's, so this is not the first attempt to describe Pablo's. 28 00:03:18,490 --> 00:03:24,130 Aristotle heard at length about Pablo's and he actually constructed a model. 29 00:03:25,370 --> 00:03:31,960 The evolution of Pablo shapes in, if I would say that he predicted that pebbles are spherical. 30 00:03:32,320 --> 00:03:37,660 Well, as you saw in the picture, not every Pablo spherical. The Pablo in my hand is not spherical, certainly. 31 00:03:38,530 --> 00:03:45,880 But there were other speculations, David Hilbert, about his very famous book, 32 00:03:47,260 --> 00:03:53,290 Geometrical Imagination, where he made a comment that coastal paths tend to be ellipsoid though. 33 00:03:54,520 --> 00:04:00,850 And in fact, if you go to the mathematical collection of göttingen, which is a very famous mathematical collection, 34 00:04:00,940 --> 00:04:06,400 started in the 17th century, the first piece in that collection is a coastal pebble. 35 00:04:07,540 --> 00:04:11,860 So people were wondering about pebble shapes and not much later. 36 00:04:12,400 --> 00:04:19,600 Lord Reilly, who is the son of the Nobel Prize winning Lord, really pointed out that tables are not Aleph sides. 37 00:04:20,260 --> 00:04:24,250 So pebbles don't seem to be spherical. Don't seem to be ellipsoid though. 38 00:04:24,910 --> 00:04:29,800 What are pebbles like? Do we have a name for Pablo shapes? 39 00:04:31,750 --> 00:04:35,230 What are the numbers which we can use to characterise them? 40 00:04:36,700 --> 00:04:43,930 Well, the kind of numbers I'm interested in are related to mechanical properties of a shape. 41 00:04:44,650 --> 00:04:50,350 So here is a disk and well, I will. 42 00:04:51,310 --> 00:04:53,770 Now, this is a question. How shall I put it down? 43 00:04:54,310 --> 00:05:01,990 If I put it down, it is rolling along its perimeter and there are positions where it would stand still. 44 00:05:03,220 --> 00:05:07,780 And there are other positions where I can balance it like a pencil on each step. 45 00:05:07,810 --> 00:05:12,969 These are the stable points and these are the unstable points in the mathematical language. 46 00:05:12,970 --> 00:05:18,520 We can say that we describe this shape as a distance from its centre of mass and 47 00:05:18,520 --> 00:05:22,840 the maxima corresponds to unstable points and the minimum to stable points. 48 00:05:25,990 --> 00:05:34,210 So in this way we assign some numbers to this shape and in three dimensions we can do a similar thing here is, 49 00:05:35,110 --> 00:05:40,120 well, I take this, Pablo, as an illustration. This is a kind of an ellipse side like Pablo. 50 00:05:40,120 --> 00:05:51,700 It's not an ellipsoid. It this can be also described as a distance function in two variables, because now we have two spatial coordinates, two angles. 51 00:05:52,270 --> 00:06:00,750 The minima of this function are stable points. I put it down and it will stay like that. 52 00:06:01,260 --> 00:06:04,320 There are several points. And there are the maxima. 53 00:06:04,770 --> 00:06:08,010 The tip of the ellipsoid. So there are three types. 54 00:06:08,310 --> 00:06:12,450 And the numbers of these are related by a famous theorem by Poincaré and hope. 55 00:06:12,810 --> 00:06:16,889 So in two dimensions, the number of stable and unstable points is always equal. 56 00:06:16,890 --> 00:06:21,520 And in three dimensions we have this relationship. Add these numbers. 57 00:06:21,520 --> 00:06:25,120 So to each pebble or to each shape, we can assign these numbers. 58 00:06:25,120 --> 00:06:29,140 Are they meaningful? Do they tell us something about nature? 59 00:06:30,580 --> 00:06:34,120 Well, to tell. I have to go back a little bit. 60 00:06:34,610 --> 00:06:37,989 Okay. To find the numbers. 61 00:06:37,990 --> 00:06:42,250 Next time you go onto a beach, you just. You will be tempted to do this. 62 00:06:42,260 --> 00:06:50,469 So you take a pebble in your hand there, you find the table points relatively easily and also the unstable points and 63 00:06:50,470 --> 00:06:55,240 the subtle points out there rather easily identified by a hand experiment. 64 00:06:57,590 --> 00:07:02,600 Uh, to motivate. Why? Why these things might be relevant. 65 00:07:02,840 --> 00:07:12,950 Let me briefly tell you the, the story of the gun book, which I learned over the announced and this goes back 25 years. 66 00:07:13,490 --> 00:07:24,230 I was visiting Cornell University and my friend Andy, a winner suggested a very interesting problem, which is related to what I just told you. 67 00:07:25,190 --> 00:07:38,620 So here is I think this is called a verbal, uh, it's not in a good chair because my son, who is two and a half, is using it rather frequently. 68 00:07:39,050 --> 00:07:44,660 But I hope it will work. So. So you know how these things work? 69 00:07:45,030 --> 00:07:54,570 There is a little music or so, uh, no matter how you put it down, it will come back to the same position and the. 70 00:07:56,520 --> 00:08:01,679 Have you ever taken the verbal apart? Um, maybe earlier on. 71 00:08:01,680 --> 00:08:04,920 And you don't remember because my son is doing it all the time. 72 00:08:05,280 --> 00:08:09,030 So if you do it, you realise there is a weight at the bottom of the Bible. 73 00:08:09,030 --> 00:08:15,180 That is why it works. Now the question was if we take a plane, a disc like this, 74 00:08:16,740 --> 00:08:23,280 and we are not admitted to add any weight to the disc, it is just a disc and we roll it along the perimeter. 75 00:08:23,280 --> 00:08:25,620 So it is a two dimensional plane that problem. 76 00:08:26,070 --> 00:08:34,080 Is it possible to cut out the disc so it behaves like a vehicle so it has just one stable point and one unstable point. 77 00:08:35,160 --> 00:08:40,560 Now, this question is easily answered, relatively easily answered. 78 00:08:41,040 --> 00:08:50,660 And let me briefly tell you how. Uh, so if you measure the distance, as I told before, from the centre of gravity, 79 00:08:51,770 --> 00:08:59,270 having this view like this would imply that this distance function has just one maximum and one minimum, 80 00:08:59,270 --> 00:09:04,250 because then it would be behaving just a verbal coming back always to the same position. 81 00:09:05,330 --> 00:09:10,310 Now the idea is to disprove this by contradiction. 82 00:09:10,880 --> 00:09:16,630 So if we slice this function. The length of the slice is zero here. 83 00:09:16,780 --> 00:09:23,950 When it is tangent and it is going at some point, the length of the slice will be exactly 180 degrees. 84 00:09:24,820 --> 00:09:29,950 So if we translate it back to the original body, we see that a straight cut. 85 00:09:29,950 --> 00:09:30,910 So the body. 86 00:09:31,920 --> 00:09:43,170 And we can see here that any radius inside the slice is bigger than any radius outside this slice in plain English, not in a vigorous way. 87 00:09:43,620 --> 00:09:51,240 I could say that the body has obviously a fat side with all the big red and it has a slim side with all the small red. 88 00:09:52,440 --> 00:10:02,000 Well, that seems to be a contradiction because the body, if I support it on that, the centre of gravity, it's supposed to be in equilibrium. 89 00:10:02,010 --> 00:10:07,890 It's so it's supposed to be balanced. However, this body is not balanced along that blue line. 90 00:10:08,160 --> 00:10:12,140 It will taste all that to the right hand side. So we've got a contradiction. 91 00:10:12,150 --> 00:10:17,310 There is no such body. We call it a serum because we were proud of it. 92 00:10:17,790 --> 00:10:21,990 We published it and probably nobody ever read it. 93 00:10:22,560 --> 00:10:26,880 So it disappeared a couple of years later. 94 00:10:28,560 --> 00:10:38,670 I went to a large conference in Hamburg, which is probably the largest ever mathematical conference with two 3000 people in attendance. 95 00:10:39,270 --> 00:10:45,270 And I went there primarily to see Vladimir Putin not give the plenary talk. 96 00:10:47,790 --> 00:10:56,400 He gave a very interesting talk. He talked about various disciplines optics, mechanics, geometry. 97 00:10:56,730 --> 00:11:01,500 And in each case, he showed that some number is equal or bigger than four. 98 00:11:02,880 --> 00:11:04,350 Obviously. I saw that. 99 00:11:05,800 --> 00:11:15,960 Our little theorem, which was completely gone, proved actually that we had two stable points at least, and too unstable points at least. 100 00:11:15,970 --> 00:11:20,140 So all together we had at least four points of equilibrium. 101 00:11:20,800 --> 00:11:24,940 So is this the same number four as Arnold was talking about? 102 00:11:25,090 --> 00:11:32,350 A different number four? Uh, I wanted to ask him after the talk, but that didn't happen. 103 00:11:32,350 --> 00:11:38,440 There were just too many people. So the organisers offered. 104 00:11:40,020 --> 00:11:51,180 To the audience. If you pay a fee of 30 German marks, we can be seated with a celebrity of our choice, which was half of my weekly budget then. 105 00:11:51,840 --> 00:12:00,480 But I. I volunteered to pay it. I was seated with Arnold together with 15 other people, and everybody had to offer something. 106 00:12:01,140 --> 00:12:05,250 So the lunch was not very successful from my point of view. 107 00:12:05,250 --> 00:12:12,240 At the end, he asked whether I had anything to offer. I said no, because it was a painful scene and. 108 00:12:13,650 --> 00:12:19,740 He left. Uh, many of you may know the world. 109 00:12:19,920 --> 00:12:29,850 For those of you who don't. At that point, he was a little bit older than I am now, but he was already a small planet. 110 00:12:29,850 --> 00:12:33,630 Has been named after him. Uh. Um. 111 00:12:33,660 --> 00:12:45,630 Sadly, he's not alive anymore. But we could list him as one of the significant, influential mathematical thinkers of the last 50 years. 112 00:12:46,800 --> 00:12:50,480 And, um. So I got a little bit disappointed. 113 00:12:50,600 --> 00:12:53,360 I gave my own talk, which was in the wrong session. 114 00:12:54,080 --> 00:13:01,459 And just the chairman came and myself and I left the conference, and that would have been the end of the story. 115 00:13:01,460 --> 00:13:09,110 But I noticed that Arnold was talking to someone and he wanted to get rid of the person. 116 00:13:09,110 --> 00:13:13,329 And he told him that. It's obvious what you are telling me. 117 00:13:13,330 --> 00:13:16,510 And by the way, I have an appointment with a gentleman over there. 118 00:13:17,440 --> 00:13:22,780 That was me. So he told me that I saw your name tag. 119 00:13:22,780 --> 00:13:29,590 You are from Budapest. You paid a 30 mark, so you must have a very good reason to come to that lunch. 120 00:13:30,220 --> 00:13:34,390 What is your reason? I got 20 minutes. You just tell me briefly. 121 00:13:34,870 --> 00:13:42,820 So I told him briefly. He was thinking about it and I offered him to explain the proof of our little theorem. 122 00:13:43,120 --> 00:13:46,980 He said he knew the proof. He doesn't need that. And he. 123 00:13:47,020 --> 00:13:50,160 How did you think about the three dimensional case? And I said, you are. 124 00:13:50,170 --> 00:13:53,500 Of course, we already solved the problem in three dimensions. 125 00:13:53,950 --> 00:14:00,160 So what is the solution? Well, you take this kind of body. 126 00:14:00,280 --> 00:14:03,760 This is a cylinder which you cut of, like diagonally. 127 00:14:05,050 --> 00:14:09,340 How many stable points does this have? What is your guess? 128 00:14:09,910 --> 00:14:11,770 So how many positions would it be? 129 00:14:11,770 --> 00:14:19,599 Still, if I put down on this table, we will try it afterwards, where it would stand still even after I rocket a little bit. 130 00:14:19,600 --> 00:14:23,200 So it is really a stable equilibrium point. Is that infinite amount? 131 00:14:24,220 --> 00:14:29,680 You said infinity, which is a good guess if it would be a cylinder which is not orthogonal. 132 00:14:30,250 --> 00:14:33,940 Yes, it would have infinitely many, but it is not got orthogonal. 133 00:14:33,940 --> 00:14:38,600 It's got a diagonal angle. I do. 134 00:14:38,690 --> 00:14:43,860 We are going down now. So any other any other guesses? 135 00:14:43,880 --> 00:14:47,180 Yes, sir. Well, for now, we are going up again. 136 00:14:47,780 --> 00:14:51,290 I am asking about stable, stable, stable. 137 00:14:51,740 --> 00:14:55,010 We will come back to that. To stable anything. 138 00:14:55,010 --> 00:14:58,730 Anybody else? Just one. So let me try this. 139 00:14:58,940 --> 00:15:08,000 I just put it down. And you can see it just comes back to the same position, no matter how I put it down. 140 00:15:12,080 --> 00:15:17,870 So it always comes back to the same position. It's it has one stable point of equilibrium. 141 00:15:19,010 --> 00:15:21,260 So I know what's staring at me. 142 00:15:21,260 --> 00:15:31,100 And I've always, for a brief moment, impressed with myself because I saw that one of the leading mathematical thinkers, I could surprise him. 143 00:15:31,340 --> 00:15:38,450 He was genuinely surprised. And but then I realised that his he was not admiring me. 144 00:15:38,450 --> 00:15:45,640 It was more like pity. He said, you know, you realise that, that this, this is completely wrong what you are showing me. 145 00:15:45,650 --> 00:15:48,410 And I said, no, this is good, this is a good example. 146 00:15:48,830 --> 00:15:57,110 And I was so hungry that I was tempted to offer him to go to the buffet and buy a piece of sausage and make an experiment, 147 00:15:57,110 --> 00:16:01,159 hoping that he would pay for the sausage. But it never came to that. 148 00:16:01,160 --> 00:16:06,050 He told me that. Okay, yes, it has one stable point, but how many altogether? 149 00:16:06,600 --> 00:16:11,290 Well, for it has one here at the tip. 150 00:16:11,300 --> 00:16:14,720 It's an unstable point. Another one here. 151 00:16:15,670 --> 00:16:24,159 And it has here. This is a saddle point. A saddle point is unstable in this direction in at almost any direction. 152 00:16:24,160 --> 00:16:28,840 But it is stable like this. So it has four points. 153 00:16:30,190 --> 00:16:37,000 So a young man who told me once you have something which has less than for you, send me an email, but now I have to go. 154 00:16:37,840 --> 00:16:43,209 And so once he was leaving, I asked him, And do you think that this thing exists? 155 00:16:43,210 --> 00:16:50,320 And he said, Well, I think many people will try to prove that it does not exist because there are many indications that it does not exist, 156 00:16:50,680 --> 00:16:53,919 but I think it may exist. So I went home. 157 00:16:53,920 --> 00:16:58,840 I started thinking. So this is the summary of what I told you until now. 158 00:16:59,650 --> 00:17:03,940 In the plane we have the same number of stable and unstable points always. 159 00:17:04,840 --> 00:17:10,690 So it is sufficient to characterise one object with one of these numbers because 160 00:17:10,690 --> 00:17:15,460 the other number is the same and in three dimensions we have three types. 161 00:17:15,520 --> 00:17:19,090 This object has one stable to one stable and one saddle, 162 00:17:19,510 --> 00:17:24,730 but it is sufficient to characterise it with two numbers because the third one can be computed. 163 00:17:25,390 --> 00:17:28,510 So the problem is like this in two dimensions. 164 00:17:28,510 --> 00:17:34,270 We cannot have zero because that would be the perpetual mobile and we cannot have one, which I proved. 165 00:17:34,810 --> 00:17:40,510 And for any other class we have an example because four and larger or equal three, 166 00:17:40,510 --> 00:17:44,380 we have regular polygons and for an equal two, we have an ellipse or whatever. 167 00:17:45,100 --> 00:17:51,440 In 3D, the situation is utterly different. We have a matrix to a two dimensional matrix. 168 00:17:51,460 --> 00:17:57,340 This is my famous sausage. Four and four is, for example, a tetrahedron. 169 00:17:57,340 --> 00:18:01,180 And I know it was asking about this body. Now. 170 00:18:04,070 --> 00:18:10,340 Christopher Columbus was challenged once to balance an egg on its tip. 171 00:18:10,760 --> 00:18:14,190 This is an egg from the hotel. 172 00:18:15,420 --> 00:18:20,240 And if you put it down, it won't stand on its tip. 173 00:18:20,690 --> 00:18:24,020 Uh, you can say that on. On the camera. 174 00:18:25,220 --> 00:18:29,910 So what did Columbus do? Yeah. 175 00:18:30,210 --> 00:18:39,310 Let me try that. So. 176 00:18:40,750 --> 00:18:49,150 And he won the bet or whatever. Now, in that article terms, what he did, he. 177 00:18:50,290 --> 00:18:56,200 Added a new equilibrium point so that it was used to be unstable. 178 00:18:56,710 --> 00:19:00,700 He turned it into a stable one and he added other points around it. 179 00:19:01,510 --> 00:19:12,040 And the main idea is that if you do small changes to a body like that, you can always add new point and we can formalise this a. 180 00:19:13,220 --> 00:19:18,980 We can. It's not difficult to prove that if you have iron j stable and unstable points, 181 00:19:18,980 --> 00:19:24,860 you can always increase one of them by one by a suitable small truncation. 182 00:19:25,640 --> 00:19:31,610 This is what we call the Columbus algorithm, and it means that for a sufficiently small, 183 00:19:31,610 --> 00:19:37,700 specific truncation, if you have an object here, you can have an object here and an object here. 184 00:19:38,360 --> 00:19:43,530 Now, what does it mean for our knowledge? Conjecture? What does it imply? 185 00:19:44,070 --> 00:19:47,310 We still don't know whether it exists, but if it exists. 186 00:19:48,860 --> 00:19:55,850 The existence of this object would imply that any other type exists by this algorithm. 187 00:19:57,280 --> 00:20:01,660 Okay. So this algorithm is just telling you the significance of this object. 188 00:20:02,600 --> 00:20:05,770 Uh. Let me show you this. 189 00:20:06,280 --> 00:20:10,770 So here is this is very important for this talk. 190 00:20:11,820 --> 00:20:15,310 Uh, so this is this is. This is just a cuboid rectangle. 191 00:20:15,610 --> 00:20:20,350 It's a it's a it's a block. Okay. How many stable points does it have? 192 00:20:22,360 --> 00:20:31,480 Six, right? Six stable faces. Now, here, we chopped off two small pieces at the corners. 193 00:20:32,680 --> 00:20:37,060 And if I put it down on that face here. 194 00:20:39,970 --> 00:20:44,110 It falls down. So we did not by chopping off this face. 195 00:20:44,140 --> 00:20:47,650 Nothing happened. It is not a new equilibrium. 196 00:20:48,610 --> 00:20:52,850 If I put it down on this face. It stands still. 197 00:20:53,150 --> 00:20:57,020 So we added a new equilibrium, a new stable point. 198 00:20:57,050 --> 00:21:00,350 All the previous ones are still working. 199 00:21:00,720 --> 00:21:06,860 Okay, so by chopping off small pieces, you can add that is the idea of the Columbus satellite. 200 00:21:09,180 --> 00:21:12,840 Now, but they still didn't know how it works. What is the shape? 201 00:21:12,870 --> 00:21:19,109 So we went for holidays with my wife to Greece and I convinced her that as an evening 202 00:21:19,110 --> 00:21:24,810 program we should collect pebbles and classify them according to the Poincaré help formula. 203 00:21:26,100 --> 00:21:30,330 To make this more scientific, we should probably collect 2000 pebbles. 204 00:21:31,650 --> 00:21:40,080 Some people in this room do know my wife, and those who know her will not be surprised. 205 00:21:40,500 --> 00:21:44,910 But the others may be that at 1200 I give up. 206 00:21:45,450 --> 00:21:51,450 But she insisted that we finish the project. Now this is the result. 207 00:21:51,450 --> 00:21:57,510 We did not find any shape. We didn't find any shape which had this property of Arnold. 208 00:21:58,230 --> 00:21:59,100 So we flew home. 209 00:21:59,100 --> 00:22:10,260 I got a little bit depressed because, you see, I know this was a very, very significant mathematician, but on occasions he could be wrong. 210 00:22:10,590 --> 00:22:14,260 So maybe this conjecture of his was just wrong. It could be wrong. 211 00:22:14,280 --> 00:22:22,379 There is no evidence that it was good. At home I started thinking, what could have been the reason that we didn't find this shape? 212 00:22:22,380 --> 00:22:32,070 And one is that one can assign flatness and seen this as a measure of the two convex bodies. 213 00:22:32,070 --> 00:22:35,280 There are many ways to define it, this one particular way. 214 00:22:35,640 --> 00:22:39,180 So this is from Anna's office. 215 00:22:39,180 --> 00:22:43,830 This is the sphere. It is not flat and not say nobody will dispute. 216 00:22:44,640 --> 00:22:47,820 So this has minimal flatness and minimal thinness. 217 00:22:48,240 --> 00:22:52,560 So we assign the values one and one to the sphere by this definition. 218 00:22:55,230 --> 00:22:59,100 I am sorry I didn't bring a Frisbee, but most people are familiar with that. 219 00:23:00,090 --> 00:23:07,049 It these definitions would give a very large flatness and a very small thinness value. 220 00:23:07,050 --> 00:23:11,610 And for the pencil, pastel is very thin, but it is not flat. 221 00:23:12,060 --> 00:23:20,160 And what we could prove that if this object existed, it would also have minimal flatness and thinness like the sphere, but it would not be a sphere. 222 00:23:20,670 --> 00:23:27,270 So that would explain why we didn't find it on the beach, because things next to the sphere tend to be rather sensitive. 223 00:23:28,080 --> 00:23:34,110 So the idea which I showed you in the play, in our case sometimes works in 3D, sometimes it does not. 224 00:23:34,740 --> 00:23:37,590 And starting from the case, very does not work. 225 00:23:37,590 --> 00:23:47,910 We produced an example of this surface, but it could not be manufactured because the deviation from the sphere was so small. 226 00:23:48,630 --> 00:23:56,850 Somewhat later we came up with this shape. Uh, here is the shape and I thank Alain Fire for the invitation. 227 00:23:56,850 --> 00:24:04,139 I didn't thank him particularly for lending me his personal going back for the further for this presentation. 228 00:24:04,140 --> 00:24:11,880 This is a particularly nice piece, so we will need a little bit space here, but oh, the egg. 229 00:24:13,310 --> 00:24:22,280 Yes. So here is a nice piece and I let it move a little bit. 230 00:24:29,240 --> 00:24:35,600 So for this object, it takes a little bit more time to find its way back to the stable equilibrium. 231 00:24:36,080 --> 00:24:40,730 But unlike the salami, it has only one unstable point. 232 00:24:40,740 --> 00:24:44,390 That's a major difference. It's very it's very hard to manufacture. 233 00:24:46,460 --> 00:24:51,790 So very small design our street to many, many equilibrium, likely the sphere. 234 00:24:51,800 --> 00:24:55,010 If you make a small perturbation, you get many, many equilibria. 235 00:24:55,490 --> 00:24:58,550 So we resolve the Arnold's conjecture. 236 00:24:59,030 --> 00:25:06,320 This is the first really bad work in Gone Back, and we decided to give it to Professor Arnold on the occasion of his 70th birthday. 237 00:25:06,350 --> 00:25:10,670 This is the picture of it's Peter with whom we constructed the ship. 238 00:25:12,200 --> 00:25:19,130 There are a number of pieces now around the world. We are most proud, of course, of the Oxford Compact, which is outside the room. 239 00:25:20,240 --> 00:25:28,850 Uh, now a little bit of the it was so we looked at this ship for a while and, and realised that it is similar to some turtles. 240 00:25:28,850 --> 00:25:32,450 However, those turtles were couldn't be found in Hungary. 241 00:25:32,450 --> 00:25:37,640 They were tropical turtles. So we had to go to the zoo and ask for permissions. 242 00:25:38,320 --> 00:25:46,020 Uh, it took a while, but. So this is this is a gone back to just small errata. 243 00:25:46,630 --> 00:25:57,040 And then I went to the history for further to the museum for Natural History and I was given a huge turtle shell of a lower part turtle. 244 00:25:57,610 --> 00:26:04,750 I put it down on the floor and I saw that it had 16 different stable positions and I knew this was an imperfect gun. 245 00:26:05,950 --> 00:26:11,040 Now we measured many shells and we turned over many turtles. 246 00:26:11,050 --> 00:26:16,000 I am sorry to say, but we try to do it so that with each turtle only once. 247 00:26:16,390 --> 00:26:23,830 And what resulted was an observation which probably more interesting to the scientific 248 00:26:23,830 --> 00:26:30,790 community than the gone was originally that many turtles self allied by muscle power. 249 00:26:30,790 --> 00:26:34,720 So they use their necks and, and, and arms and legs. 250 00:26:36,130 --> 00:26:44,890 So this is a very nice example. If you have aquatic turtle at home, don't try it many times, but you'll try it once you will see it. 251 00:26:45,550 --> 00:26:53,740 Even large aquatic turtles can do this. And there is another technique which has been adopted only by two or three species. 252 00:26:54,460 --> 00:26:59,830 They came so close to this gumbo like shape that they serve divide by gravity. 253 00:27:00,010 --> 00:27:08,020 So this is a main tool for them to survive because being turned over is for a turtle, a life threatening situation. 254 00:27:08,620 --> 00:27:11,110 And the medium turtles cannot survive. Right. 255 00:27:11,740 --> 00:27:20,260 Told turtles found this shape or get got so close to the shape that the biologist believed us that this is not a coincidence. 256 00:27:22,000 --> 00:27:26,260 Now back to back to the story. So this was we must go. 257 00:27:27,850 --> 00:27:31,630 And I gave this talk on the going back, Professor. 258 00:27:31,690 --> 00:27:35,410 I know it was in the audience. Of course, I was a little bit nervous. 259 00:27:36,640 --> 00:27:42,490 He listened to it and after that he said, this is a good job, but now we should do something else. 260 00:27:42,940 --> 00:27:47,140 And he outlined some problems, which I didn't understand because I was too nervous. 261 00:27:47,840 --> 00:27:55,750 Later on, we met in the elevator and he told me that he liked the talk and in particular the story. 262 00:27:55,990 --> 00:28:01,600 The pebble. The pebble stuff was very interesting, and I thought he liked that. 263 00:28:01,840 --> 00:28:09,640 Marriage actually survived this trial, but his interest was more mathematical. 264 00:28:09,670 --> 00:28:13,000 He asked, Do you realise the significance of your slide? 265 00:28:13,780 --> 00:28:18,120 Well, for me it had multiple significance, first of all, which I just mentioned. 266 00:28:18,130 --> 00:28:25,510 It's a memory to this event. But also the significance was that we didn't find any piece. 267 00:28:26,480 --> 00:28:30,860 And then he said, this is obviously not the significance of your slide. 268 00:28:31,040 --> 00:28:37,730 This is just the opposite. Uh, put that two lines in your imagination at two. 269 00:28:38,690 --> 00:28:44,719 So in this regard, the first two and the first column, because they might be disregarded for physical reasons, 270 00:28:44,720 --> 00:28:48,680 which I would just describe and look at the rest, look at the numbers. 271 00:28:49,400 --> 00:28:54,200 The book is not in this table, but it is almost there. 272 00:28:54,560 --> 00:29:00,020 It is almost there. It looks as though Pablo's would be evolving towards the gumball. 273 00:29:00,560 --> 00:29:08,240 Now, this was very strange for me, really, and I hope it is also strange for you, 274 00:29:08,240 --> 00:29:12,830 because the first part of the lecture was about why this cannot happen. 275 00:29:13,550 --> 00:29:18,340 Uh, let me just briefly describe about those white barriers. 276 00:29:18,350 --> 00:29:26,179 That is the easiest part. That is the easy part. Also, he told me that he didn't formulate as a conjecture. 277 00:29:26,180 --> 00:29:30,650 He just gave us this hint that that maybe this is happening. 278 00:29:31,100 --> 00:29:35,450 And he told me that unlike the gumball, this is serious mathematics. 279 00:29:35,450 --> 00:29:42,080 So now you have to learn stuff. Okay. Uh, so this was his hint. 280 00:29:43,310 --> 00:29:49,660 And the celebration for Pablo's occurs in two ways. 281 00:29:49,670 --> 00:29:52,999 Either it is by fiction, so the parable is sliding. 282 00:29:53,000 --> 00:29:57,980 You can hear it when you go to the beat. You can hear this sliding and rolling motion. 283 00:29:58,910 --> 00:30:04,430 If something is very flat and it is sliding, it tends to become even more flat. 284 00:30:04,940 --> 00:30:09,920 If something is very sane and it is evolving, it tends to become even more thin. 285 00:30:10,550 --> 00:30:15,379 So shortly friction accounts for those. 286 00:30:15,380 --> 00:30:22,430 Various friction would accumulate things. At either two stable or too unstable equilibrium points. 287 00:30:22,970 --> 00:30:29,600 However, it is certainly doesn't account for why shapes would move in that direction. 288 00:30:29,630 --> 00:30:34,130 That has to be as that has to be driven by collisions. 289 00:30:34,820 --> 00:30:42,320 Now, as I showed you, if you take a collision in a collision, which I will not do physically, a small piece will come off. 290 00:30:42,440 --> 00:30:43,880 If I hit it very hard. 291 00:30:44,900 --> 00:30:56,330 And what I illustrated before is that if we take of small pieces the number of equilibrium points apparent in naively we think always increases. 292 00:30:57,050 --> 00:31:00,230 How is it possible that the opposite happens on the beach? 293 00:31:01,690 --> 00:31:09,800 Well, uh, so this is the naively this is the evolution which is suggested by this model. 294 00:31:09,820 --> 00:31:16,000 So you take off a small piece and you get more equilibrium, uh, to get to take less. 295 00:31:16,090 --> 00:31:20,860 Okay. Is it possible to reduce the number of equilibrium? 296 00:31:21,490 --> 00:31:29,890 Yes. If you chop off a large piece of cake, how can the chop of a large piece with chopped of say this piece here? 297 00:31:30,850 --> 00:31:33,909 I show you a and large version of this piece. 298 00:31:33,910 --> 00:31:37,420 This is here. Just you can see better. Okay, the same thing. 299 00:31:39,040 --> 00:31:42,820 We can say that this is a rectangle, but this has been chopped off. 300 00:31:43,480 --> 00:31:47,680 Or we can say that this is a rectangle where this has been chopped off. 301 00:31:49,030 --> 00:31:54,640 And if we adopt the latter view, then how many stable points does this have? 302 00:31:56,570 --> 00:32:00,760 It's a tetrahedron. Four. So this has less. 303 00:32:00,770 --> 00:32:04,550 But in order to get there, we needed to chop off almost all of the body. 304 00:32:05,060 --> 00:32:09,590 So it's not clear what's happening now. 305 00:32:11,820 --> 00:32:17,379 So. So what we did, we made a computer simulation, we took a rectangle, 306 00:32:17,380 --> 00:32:22,720 and we were chopping off diligently, small pieces, and we counted the number of balance points. 307 00:32:23,470 --> 00:32:30,850 Here is the plot. So you can see initially it went up, then it went down, then it went up, then it went down. 308 00:32:31,330 --> 00:32:42,910 So how is that possible? Well, we said that in order to reduce the number, in order to add the new point, the truncation has to be sufficiently small. 309 00:32:43,030 --> 00:32:47,530 But I never said what is sufficiently small. This depends on the shape. 310 00:32:48,070 --> 00:32:51,850 So in the natural process, the piece is being chopped off. 311 00:32:52,300 --> 00:32:58,420 Let's say they are roughly constant. What is sufficiently small depends on the shape, and that may vary. 312 00:32:58,900 --> 00:33:04,059 And when this actual piece is below the curve, then the number of equilibrium goes up. 313 00:33:04,060 --> 00:33:07,350 Otherwise it goes down. So anything can happen. 314 00:33:07,470 --> 00:33:09,910 We we don't have any prediction for the average. 315 00:33:10,390 --> 00:33:20,710 And in particular, the Columbus algorithm in this chopping algorithm cannot be regarded as a model for the natural process because of this reason, 316 00:33:20,980 --> 00:33:26,020 because in mathematics, when you say sufficiently small, you may mean very, very small. 317 00:33:26,020 --> 00:33:36,220 And in nature, collisions create on the average similar size the collision of similar sized pieces being broken off. 318 00:33:37,900 --> 00:33:45,880 We can average also in space, so assume we have a ellipsoid, that kind of thing, and we put on a very, very, very fine mesh. 319 00:33:46,750 --> 00:33:55,210 On that mesh we will see equilibrium points, many of them, but they were gathered, concentrated in flocks. 320 00:33:56,500 --> 00:34:00,790 That can be proven. So they are always, always in smaller domains. 321 00:34:01,420 --> 00:34:09,280 Now, if we track in the number of individual points, it may differ very much if we track the number of flocks. 322 00:34:09,820 --> 00:34:14,530 And indeed, if you take the number of flocks, we see that that is decreasing. 323 00:34:14,950 --> 00:34:20,860 Of course, we ran it many, many times, but this is the average. So the trend is that it is a decreasing. 324 00:34:21,850 --> 00:34:29,979 So here is that we are table. And if you scan it very carefully, you see the same phenomenon on the surface of the table. 325 00:34:29,980 --> 00:34:36,190 You see many, many, many little equilibrium point, but they appear in flocks. 326 00:34:36,190 --> 00:34:41,860 So they if if you if you ignore the details, you see the good picture. 327 00:34:41,870 --> 00:34:47,200 So in order to get a sense of the phenomenon, you have to ignore the details. 328 00:34:47,210 --> 00:34:52,450 You have to ignore the small scale structure that too scarce. And you have to think on the global scale. 329 00:34:53,350 --> 00:34:57,160 And the Columbus algorithm was very much on the local scale. 330 00:34:57,670 --> 00:35:01,980 Now the model is describing average behaviour, 331 00:35:02,050 --> 00:35:10,750 so time and space average behaviour are called partial differential equations and that is what I was referring to. 332 00:35:11,020 --> 00:35:20,979 In order to find what is happening on the time and space average, you need to understand this in terms of these models of them, 333 00:35:20,980 --> 00:35:25,240 in terms of partial differential equations, they are called PD models. 334 00:35:26,530 --> 00:35:36,670 What is a partial differential equation? Well, a simple the kind of partial differential equation we are interested in is I imagine here is a curve. 335 00:35:37,360 --> 00:35:45,760 And at each point you move this curve in time and at each point of the curve you defined a speed in the normal direction. 336 00:35:46,270 --> 00:35:53,889 And then the curve shrinks and then again the speed is given, and then again the curve is shrinking. 337 00:35:53,890 --> 00:35:57,580 So this whole phenomenon is a partial differential equation. 338 00:35:58,570 --> 00:36:05,260 Things depend also on space and also on time, uh, as this curve evolves. 339 00:36:07,720 --> 00:36:13,930 First we have a certain number of equilibrium points and in the end we have in this case, we have less. 340 00:36:14,530 --> 00:36:18,970 So equilibrium points may disappear or may appear. 341 00:36:19,630 --> 00:36:27,790 If they disappear, we call it an end. The relation they went away or if new come around. 342 00:36:27,790 --> 00:36:32,800 They call it a creation. Both events can take place in this in this process. 343 00:36:33,280 --> 00:36:40,179 Now, here are our questions. What is the formula for the speed which describes the natural abrasion process? 344 00:36:40,180 --> 00:36:43,570 So what is the partial differential equation? We have to understand? 345 00:36:44,350 --> 00:36:48,900 And if we find that formula, what is the average value for Omega? 346 00:36:48,940 --> 00:36:53,380 Would it be would it be positive or would it be negative? 347 00:36:53,980 --> 00:37:01,930 So I know hint was in the direction that if you look it in a very average way at the right equation, 348 00:37:02,170 --> 00:37:07,900 then Omega will be positive in some of the average value of omega will be will be positive. 349 00:37:09,220 --> 00:37:16,610 Here is it in three dimensions. And well, I have some models here. 350 00:37:16,970 --> 00:37:25,879 So the easy way to study it, if you go to the drugstore and you buy a bottle of soap and then you wash your hands with it, 351 00:37:25,880 --> 00:37:30,260 you will see something similar happening. We did it the hard way. 352 00:37:31,400 --> 00:37:37,440 We know that is one of these equations with the computer and printed out in 3-D. 353 00:37:37,490 --> 00:37:42,080 What came out of this? So here are these successive shapes. 354 00:37:42,830 --> 00:37:47,360 We started with exactly this cuboid. 355 00:37:47,360 --> 00:37:55,970 Okay? This had six stable points. This is the next stage and. 356 00:37:59,500 --> 00:38:07,479 This still has six. Although they appear to be slightly less stable, so to say. 357 00:38:07,480 --> 00:38:12,070 But there are still six. This is the next one. 358 00:38:12,980 --> 00:38:23,140 Sometime later, this is stable. Uh, this is still the stable, but I cannot balance it here anymore. 359 00:38:23,770 --> 00:38:27,790 So these ones somehow disappeared at the long end. 360 00:38:28,690 --> 00:38:33,150 And this is the ultimate thing, which is a kind of an ellipse. 361 00:38:33,250 --> 00:38:39,730 It's not an ellipsoid. And you probably believe me that I can only balance it here. 362 00:38:39,910 --> 00:38:46,420 No, at that point, it will always fall over. So we started with six and we ended up with two. 363 00:38:46,750 --> 00:38:55,799 So at least from the experimental or intuitive point of view, it seems to make sense that if we have all of these things in an average, 364 00:38:55,800 --> 00:39:00,370 the way we get rid of equilibrium points, is it actually too? 365 00:39:00,400 --> 00:39:04,120 Can we say anything about these mathematically? Well. 366 00:39:05,170 --> 00:39:10,330 The formula, the actual equations for these operation processes were there. 367 00:39:12,850 --> 00:39:19,360 Nobody at this paper. I think this is a brilliant paper, and it was written by Fred Blore. 368 00:39:19,390 --> 00:39:27,190 He was in Liverpool. If anyone in this room knows anything about Fred Blau, I would be grateful because we tried to contact him. 369 00:39:27,580 --> 00:39:30,940 He's now retired, but we couldn't get hold of him. 370 00:39:31,240 --> 00:39:40,120 So he wrote this paper in 1977 and he laid down the general method calculations for possible aberrations. 371 00:39:40,130 --> 00:39:46,390 So he gave he gave the right formula for a V in case of Collisional aberration. 372 00:39:47,650 --> 00:39:53,170 These are questions are very, very difficult equations for a matter from the mathematical point of view. 373 00:39:53,650 --> 00:39:59,110 Very little is known about them. On the other hand, not known to blow. 374 00:39:59,770 --> 00:40:05,169 There was a huge activity close to these equations, not quite these equations. 375 00:40:05,170 --> 00:40:15,660 So these equations tell you that the speed by which the boundary is moving inward is depending on the curvature, 376 00:40:15,670 --> 00:40:18,820 so more curved things move more speedily. 377 00:40:19,450 --> 00:40:25,089 And these three dimensions, it depends on the Gaussian curvature and the mean curvature. 378 00:40:25,090 --> 00:40:31,060 And those are constant term, rather complicated, I would say nasty partial differential equation. 379 00:40:31,990 --> 00:40:42,340 However, there was some activity which was below I'm not aware of, and that was the Poincaré conjecture which attracted enormous mathematical powers. 380 00:40:42,970 --> 00:40:48,040 And ultimately the Poincaré conjecture was proved by the way to flow. 381 00:40:48,490 --> 00:40:53,560 And the e.g. flow is a generalisation of this flow. 382 00:40:54,490 --> 00:41:00,490 This flow is a special case of the flows which blower has studied. 383 00:41:01,450 --> 00:41:06,160 So for this flow, which is called the curve, short curve, shortening flow. 384 00:41:07,150 --> 00:41:15,190 The Poincaré conjecture, the proof for the Poincare conjecture is one of the major mathematical breakthroughs of recent years, 385 00:41:15,700 --> 00:41:22,749 and it attracted top people. And that was a very lucky thing because they were certainly not okay. 386 00:41:22,750 --> 00:41:28,329 Blau didn't say anything about any relation or creation of critical point. 387 00:41:28,330 --> 00:41:36,250 He didn't study it. They were not interested in it either, but they were working on these flows for extended period of time. 388 00:41:36,250 --> 00:41:43,020 They were studying the properties of these for the ever surprising great theorems being proved about these flows, 389 00:41:43,030 --> 00:41:46,360 which I won't tell you today, but it's a very interesting topic. 390 00:41:47,500 --> 00:41:54,580 One of these papers by Matt Grayson showed that in the case of this format goes equal to one. 391 00:41:55,090 --> 00:42:02,920 So if you just take the plane out curve and you shrink it by the curvature speed, the number of critical points is going down. 392 00:42:03,970 --> 00:42:07,920 Of course. This is a special case of that. 393 00:42:08,520 --> 00:42:18,480 And to generalise it. I had to put in some weakening conditions, assuming generosity of the bifurcation for the critical point. 394 00:42:19,700 --> 00:42:27,080 But taking taking these assumptions in addition to the original problem for the play in our case, 395 00:42:27,440 --> 00:42:34,010 uh, based on Grayson's suicide, uh, one could prove that Omega is equal to one. 396 00:42:34,340 --> 00:42:37,640 So always the number of critical points is always decreasing. 397 00:42:38,240 --> 00:42:45,860 In three dimensional, the situation was much worse because it is not true that it always decreases. 398 00:42:45,860 --> 00:42:49,639 Sometimes it increases. And I was stuck. 399 00:42:49,640 --> 00:42:53,360 I couldn't do with this term anything for a couple of years. 400 00:42:53,870 --> 00:43:01,000 Then I discovered that there is another equation which is much better understood than these very special geometric equations. 401 00:43:01,010 --> 00:43:02,490 That is the heat equation. 402 00:43:02,510 --> 00:43:14,900 The heat equation is describing the, uh, dissipation of heat, uh, or in a medium it was written down and first studied by Fourier. 403 00:43:15,500 --> 00:43:22,740 And later on in the eighties it turned out that the heat equation is a good model for the blurring of images. 404 00:43:23,300 --> 00:43:28,610 This is a very fashionable topic and that started an industry of studying of the 405 00:43:28,610 --> 00:43:33,320 features of the gentle metric features of surfaces evolving under the heat equation. 406 00:43:34,250 --> 00:43:42,890 Now, uh, there was a huge literature on this, and even in 1995 it was an open question whether in the heat decoration, 407 00:43:43,130 --> 00:43:49,910 which again is a much simpler equation than blow as equation, whether creation could occur or not. 408 00:43:49,910 --> 00:43:54,680 It was an open question. And and James Damon finally showed an example. 409 00:43:54,920 --> 00:44:01,579 Evacuation could occur, but people were not satisfied with that because if they looked at the screen most of the time, 410 00:44:01,580 --> 00:44:05,960 they didn't see the C and the relations, but they didn't see questions. 411 00:44:06,590 --> 00:44:12,110 So two mathematicians, uh, actually students of conducting, uh, 412 00:44:12,260 --> 00:44:19,220 said that the term which we cannot explain the replaced by a random variable with zero expected value. 413 00:44:19,670 --> 00:44:27,260 And if we do that then we get our end on process, the expected value of which is positive. 414 00:44:28,220 --> 00:44:34,160 Now the same can be done for the more complicated, fair case for the blow equations. 415 00:44:34,880 --> 00:44:41,630 So what? It adds up to that the discrete process suggested an evolution in this direction. 416 00:44:42,050 --> 00:44:49,040 But from the average mode, as we get an opposite indication, but only in a random, in a weak sense. 417 00:44:49,040 --> 00:44:54,590 So it is moving that way, but not not maybe not always, but most of the time I would say. 418 00:44:55,630 --> 00:44:58,930 Okay. Let's go back to this chess board. 419 00:44:59,350 --> 00:45:03,460 These are the stable points, number of stable points and number of unstable points. 420 00:45:04,210 --> 00:45:07,240 All that we need from all the previous mathematics. 421 00:45:07,240 --> 00:45:10,390 Is this as Jovian an object in here? 422 00:45:10,660 --> 00:45:16,270 And we are throwing a dice, but the dice is is not correct. 423 00:45:16,870 --> 00:45:25,600 It's more often it is falling to this side. The dice will determine whether the object will move up to the right down or to the left. 424 00:45:25,960 --> 00:45:30,040 And because the dice is loaded, that is what was devised before. 425 00:45:30,670 --> 00:45:36,130 It's more often than it would go up to the left, then it would go to the right and below. 426 00:45:37,070 --> 00:45:46,510 That is a very simple statement. And in each well course we have the barriers due to friction and this process. 427 00:45:46,510 --> 00:45:51,160 So we have many, many objects and we are throwing these dice many, many times. 428 00:45:51,820 --> 00:45:55,240 The simplest assumption is all these probabilities are constant. 429 00:45:56,230 --> 00:46:06,370 This is very simple, mobile data, very simple. It's called the Markov process, and it is very well understood and it gives you a distribution. 430 00:46:07,030 --> 00:46:13,419 If you start with many particles all over the place, it gives you a very definite distribution for the end result. 431 00:46:13,420 --> 00:46:15,460 It will be a stationary distribution. 432 00:46:16,420 --> 00:46:23,440 And if you compare that with the field data, which we had that on the Greek island, you'll see you get a very nice agreement. 433 00:46:24,400 --> 00:46:29,200 So this process is a very particular process. 434 00:46:29,830 --> 00:46:37,239 So you started with an initial values to take a bunch of pebbles, you take the average number of stable and unstable points. 435 00:46:37,240 --> 00:46:41,620 So it is not an integer anymore and you start evolving this process. 436 00:46:41,620 --> 00:46:48,970 So what you will see that there is a fast descent and then it starts to element in some area. 437 00:46:49,240 --> 00:47:01,840 So there are definitely two phases. The first phase is a diffusion process which has a drift, and the second phase is a recurrent process. 438 00:47:02,890 --> 00:47:07,959 So let me show you some data. The first is what I would call coastal data. 439 00:47:07,960 --> 00:47:11,410 So we collected many, many, many pebbles on many, many coasts. 440 00:47:12,160 --> 00:47:20,650 It's a good scientific activity. So I am planning to apply for a grant to extend these activities to many other coasts. 441 00:47:21,010 --> 00:47:30,850 These bodies were collected privately, so to say we collected each of those little circles means 50 or 100 pebbles. 442 00:47:31,600 --> 00:47:36,580 Some of them were less abraded, some of them were more abraded smoother. 443 00:47:37,150 --> 00:47:47,080 And if we start this very simple model from the less abraded pebbles as the initial values, the trajectory seemed to go to the right place. 444 00:47:47,320 --> 00:47:54,760 So they seemed to find the pebbles, which are more abraded. Maybe, but there is no causal relationship. 445 00:47:54,760 --> 00:48:00,610 We collected this pebbles in many different locations, so it is not true that one of them was abraded from the other one. 446 00:48:01,210 --> 00:48:08,080 Now this is Australia. Uh, this is a small, not a smaller medium river in southwestern Australia. 447 00:48:08,590 --> 00:48:13,180 And we sampled pebbles along the river around 12 places here. 448 00:48:13,180 --> 00:48:20,410 You can see some pictures from the river. And here is the here is the plot. 449 00:48:21,560 --> 00:48:28,700 Now, if you want this Markov process here, you get a trajectory which is reasonably close to the data. 450 00:48:30,590 --> 00:48:33,860 And if you run another trajectory, it is still close to the data. 451 00:48:33,860 --> 00:48:37,880 So there is a fast descent and then there is summary current process. 452 00:48:38,520 --> 00:48:46,780 Another example always go to the topic are places as you notice, this is port of call and uh, 453 00:48:47,480 --> 00:48:57,080 actually our colleagues had a grant to go there every year and the sample tables and they had a lot of data, but it was not so meaningful. 454 00:48:57,080 --> 00:49:02,060 And we propose that include the number of equilibrium points among the staff which they measure. 455 00:49:03,110 --> 00:49:06,080 So we also offer that one of us would go and have them. 456 00:49:07,130 --> 00:49:12,980 So they accepted this proposal here is that there are actually two rivers here, a tributary and a main river. 457 00:49:13,700 --> 00:49:15,529 And here is the plot. 458 00:49:15,530 --> 00:49:26,840 And again, you can see that there is a part where it is descending, the numbers descending very fast and then it seems to be oscillating. 459 00:49:27,980 --> 00:49:30,889 And if you go to a lab and run this experiment, 460 00:49:30,890 --> 00:49:38,780 you just put in a number of tables in advance and rotate it and you measure the number of equilibrium points you cannot get it wrong. 461 00:49:39,290 --> 00:49:45,050 So it is the is the process is so stable, so robust that there is no way you can get it wrong. 462 00:49:45,500 --> 00:49:51,050 You always get a similar process. So this seems to be a do a natural process. 463 00:49:51,980 --> 00:49:55,970 The mathematical statements which I made could be made probably more rigorous. 464 00:49:56,990 --> 00:50:06,920 Oh, here's the lab data measured. So as a summary, I might say that next time you go to a nice Pebble Beach, uh, think about this. 465 00:50:07,550 --> 00:50:15,050 That that pebbles maybe not every aspect of pebbles, but some aspect of pebbles. 466 00:50:15,470 --> 00:50:19,370 The story about pebbles can be told with natural numbers. 467 00:50:19,700 --> 00:50:23,480 Maybe only a small slice of the story, but it is a true story. 468 00:50:24,110 --> 00:50:38,540 It's only happening. So this is what Michael Berry summarised, is that the gamut exists in nature, but only as a dream. 469 00:50:39,740 --> 00:50:44,330 So all pebbles dipping down in their heart. They want to become a gumball. 470 00:50:44,960 --> 00:50:48,560 But. But few manage. 471 00:50:49,880 --> 00:50:52,670 Uh, thank you very much for the attention.