1 00:00:02,310 --> 00:00:11,520 Was. Okay. 2 00:00:12,110 --> 00:00:16,010 I think we're going to soon get started so everybody can get a seat. 3 00:00:16,580 --> 00:00:20,540 Excellent. I've been told that I should introduce myself. 4 00:00:20,720 --> 00:00:28,160 So my name is and I go, Really? I'm the professor of mathematical modelling and I'll be talking about mathematical modelling. 5 00:00:31,260 --> 00:00:35,270 Today is a special day. It's a special place. 6 00:00:35,280 --> 00:00:37,230 We are in the Andrew Wise building. 7 00:00:38,610 --> 00:00:47,020 It was opened a year ago and just became the focal point for mathematics but also for a lot of different activities. 8 00:00:47,060 --> 00:00:54,990 So it's wonderful to welcome you. And as I said, Andrew Wiles is also professor here and gets a lot of mail. 9 00:00:55,020 --> 00:00:57,360 Randomly addressed to Andrew Wiles at Mass. 10 00:00:58,610 --> 00:01:07,250 And it's a special day, not only because we have the pleasure of your company, but it is also a special day for Andrew Wiles. 11 00:01:07,460 --> 00:01:12,710 Here is I found a stamp from the Czech Republic celebrating Andrew Wyeth. 12 00:01:12,710 --> 00:01:18,410 And you see here it's the Fermat's Last Theorem. And it says here Andrew was 1995. 13 00:01:19,340 --> 00:01:32,149 So that that was about 19 years ago. But what is maybe more interesting is that it was that day in September 1994 that Andrew Wiles 14 00:01:32,150 --> 00:01:38,180 made the final discovery that allowed him to finally resolve the problem and prove the theorem. 15 00:01:38,720 --> 00:01:42,740 And we know exactly when. Because he remembers. And that day. 16 00:01:44,310 --> 00:01:49,440 That day was September 1994. So exactly two days. 17 00:01:49,500 --> 00:01:58,800 20 years. The 20 year anniversary of the discovery which led him to his fame, the proof of the theorem, the creation of the building, and so on. 18 00:01:58,810 --> 00:02:02,820 So it's all related to chain of events. Okay. 19 00:02:03,300 --> 00:02:09,750 But we're not here to talk about interesting mathematics like Andrew is doing in number theory. 20 00:02:10,020 --> 00:02:19,740 What I really want to talk today about is how mathematics is related to science, society, engineering, all of the other areas of knowledge. 21 00:02:20,850 --> 00:02:27,600 So my talk will be divided into roughly three parts and I want to have time for discussion. 22 00:02:28,950 --> 00:02:34,740 First, I will share general thoughts about the act of modelling what we what we actually do. 23 00:02:34,830 --> 00:02:39,890 Very, very, very general thought and remark. I'm not going to go all philosophical on you. 24 00:02:39,900 --> 00:02:45,740 I just want to be very pragmatic. Then I will see what his ideas apply to the climate. 25 00:02:45,770 --> 00:02:50,890 Take the climate modelling as an example of how to implement some of these ideas, 26 00:02:50,900 --> 00:02:56,000 and maybe we learn also something about how people actually do the modelling of climate in the process. 27 00:02:56,480 --> 00:03:04,250 And then I want to tell you about problems that I've been working over the last couple years or so will be something to save the planet, 28 00:03:04,250 --> 00:03:07,970 something to save mankind, and just something for the beauty of it. 29 00:03:08,750 --> 00:03:11,390 And that I reserve, you'll see exactly what I'm talking about. 30 00:03:12,110 --> 00:03:20,990 So let's start with some general remarks, and I'm going to start with some remarks from Lord Kelvin here, also seen in a stamp. 31 00:03:21,620 --> 00:03:31,010 Hughes A stamp from West Africa. And Lord Kelvin was very interested in applying ideas of mathematics and general physics to the world. 32 00:03:31,040 --> 00:03:41,030 Here we see the Great Eastern. He was actually on the boat that laid the first oceanic cable between between Europe and in America for the Telegraph. 33 00:03:42,640 --> 00:03:48,910 And Lord Calvin's talking about science, about modelling, says the following thing. 34 00:03:49,360 --> 00:03:55,420 I often say that when you can measure what you are speaking about and express it in numbers, you know something about it. 35 00:03:55,840 --> 00:04:00,550 But when you cannot express it in numbers, your knowledge is of meagre and unsatisfactory kind. 36 00:04:00,820 --> 00:04:07,510 It may be the beginning of knowledge, but you have scarcely, in your thought, advanced to the stage of science, whatever the matter may be. 37 00:04:07,750 --> 00:04:14,770 So he said, If you really want to make progress at the scientific level, you have to translate the idea into into mathematics, 38 00:04:14,770 --> 00:04:19,300 into number, into quantities for which you can make a very specific statement. 39 00:04:20,590 --> 00:04:24,620 You also said in science there is only physics. 40 00:04:24,640 --> 00:04:29,140 All the rest is stamp collecting. So not always ideas. 41 00:04:29,380 --> 00:04:33,190 You also said that at a time that vectors were completely useless and things like that. 42 00:04:33,210 --> 00:04:37,150 But can we take what we can log? Having certainly a great, great mind. 43 00:04:37,570 --> 00:04:41,570 So what does it mean by all, all inside that there is only physic? 44 00:04:41,590 --> 00:04:46,810 What's what's the idea of physics? It was very, in a broad sense, was really natural philosophy in 19th century. 45 00:04:48,040 --> 00:04:55,090 So the way physics approach a problem is in the physics paradigm is the following where you start with some data, 46 00:04:55,540 --> 00:05:03,160 then you identify some patterns, and based on the first principle, you would go about that patterns and write a model for it. 47 00:05:03,490 --> 00:05:08,490 And the simplest and classical example for that is the evolution of celestial mechanics, 48 00:05:09,130 --> 00:05:15,300 where we start with Tycho Brahe in the 16th century and who catalogue numbers, 49 00:05:15,310 --> 00:05:21,430 look at planets and look at the position in the in the sky very extensive table. 50 00:05:21,910 --> 00:05:25,660 So that was really data gathering, what we call big data and ideas. 51 00:05:26,170 --> 00:05:36,610 Then after that came Kepler. He also West Africa stamp, who says, I'm going to look at this data and what is going to do what we do today, 52 00:05:36,610 --> 00:05:41,500 which is called statistics, analyse the data, trying to extract loads directly from the data. 53 00:05:42,100 --> 00:05:47,440 And that's how Kepler came with Kepler's law like planets go wrong in an ellipse and so on. 54 00:05:47,950 --> 00:05:55,720 Okay, very deep and profound. But then came Newton and we wrote a mathematical based on physical, simple physical laws. 55 00:05:56,480 --> 00:05:59,740 Right. So you had to find the laws himself, of course, because he did both. 56 00:06:00,010 --> 00:06:05,770 He says, if I apply simple physical laws, essentially F equals them with the right force, gravitational force. 57 00:06:06,070 --> 00:06:11,020 I can explain all of it. I can explain Kepler's law. I cannot print all of it and much more. 58 00:06:11,470 --> 00:06:18,970 So summarising into a few equation mathematical model, all of the knowledge that is required to do celestial mechanics. 59 00:06:20,140 --> 00:06:26,050 Here is a stamp that I found from the Benner, which is actually the same country as the down me. 60 00:06:26,410 --> 00:06:30,970 But it was the Republic Populaire Jupiter that was during its Marxist-Leninist time. 61 00:06:33,610 --> 00:06:39,460 So they also evolved from Kepler naturally in the 20 years to, uh, to, to Newton in this time. 62 00:06:39,760 --> 00:06:44,050 So, but when you actually do modelling, you do it as a profession. 63 00:06:44,260 --> 00:06:48,700 Turns out that the first, the physics paradigm, that's why we put in textbook. 64 00:06:48,700 --> 00:06:53,540 But the reality is quite different. What do we do when we do start with this data? 65 00:06:53,560 --> 00:06:58,680 Some observation. We have a friends who's a biologist or geophysicist or material scientist. 66 00:06:58,750 --> 00:07:01,990 Look at this nice experiment. Say, Oh, that's wonderful. 67 00:07:02,290 --> 00:07:06,000 And so you go back in with your student, you say, okay, well, let's write a model. 68 00:07:06,040 --> 00:07:13,449 We very happy with the model. Then you go back to the scientist and you say, Okay, here is a model, can you tell me? 69 00:07:13,450 --> 00:07:17,980 And it shows you more data. And invariably, the model is wrong. 70 00:07:18,340 --> 00:07:21,190 The simple model never happens. It's right. 71 00:07:21,760 --> 00:07:28,840 Now you have to go and explain to your poor first year student that is making great progress because his first model is already wrong. 72 00:07:31,180 --> 00:07:35,229 But the real problem is I show you wrong. Model is fine. You are you really understand something. 73 00:07:35,230 --> 00:07:41,440 Often it may seem strange, but often the model wrong model tell you more about the nature of the problem than a correct one. 74 00:07:41,620 --> 00:07:45,190 But the real question is why is the model wrong? What did we miss? 75 00:07:45,580 --> 00:07:51,190 What are the important physics or important chemistry? All part an aspect that is not in the model. 76 00:07:52,090 --> 00:07:57,190 And so you create a new model and the model invariably also has more variables, more parameter. 77 00:07:57,190 --> 00:08:03,549 You start throwing things. Maybe there is an effect here that we neglected, maybe there is another scalar came in the problem and so on. 78 00:08:03,550 --> 00:08:10,270 So you go there and then you go back and discuss and try to validate with the model with new experiment. 79 00:08:10,780 --> 00:08:12,909 Then you see, is the model self-consistent? 80 00:08:12,910 --> 00:08:19,270 Does it provide me any insight or does it correspond to other data or the system model experiment in the literature? 81 00:08:19,480 --> 00:08:25,180 Do I understand? Does does that make sense essentially? And if it does, you have you can do two things. 82 00:08:25,300 --> 00:08:29,830 You can do a simpler model, throwing away the stuff that may not be important. 83 00:08:30,130 --> 00:08:34,390 Or you can say, no, I really want to predict things and you would make a much more complex model. 84 00:08:34,900 --> 00:08:41,110 So you can you can go both way. You can go do a little bit of math or little bit of engineering or computational science. 85 00:08:41,110 --> 00:08:43,450 Let's put that on a big computer and see what happens. 86 00:08:44,210 --> 00:08:50,540 And then you go back, you try to validate, make more experiments, and then you refine the models based on that. 87 00:08:50,930 --> 00:08:55,430 And then maybe eventually you may have some prediction and predictive value for the model. 88 00:08:56,060 --> 00:09:01,310 And then when you do that, you go back here and then you start this loop again and again and again. 89 00:09:01,310 --> 00:09:08,450 And when it comes to the climate, this loop has been done hundreds or maybe thousands of times trying to refine based on it and move on. 90 00:09:10,070 --> 00:09:13,830 But the question is, we have a motherlode. Do we know or do we know how far we should go? 91 00:09:13,850 --> 00:09:17,060 What's what's what is a mother quality? 92 00:09:17,810 --> 00:09:21,110 And there are two important aspects of quality and complexity. 93 00:09:21,500 --> 00:09:24,860 So quality can be points, instance, a predictive value. 94 00:09:24,980 --> 00:09:30,590 And we'll show you example of what I mean by that. Complexity is really the number of things you've added in your system. 95 00:09:30,590 --> 00:09:33,800 The number of parameter, the number of variable are because your system. 96 00:09:34,370 --> 00:09:39,089 It turns out that. If the brother becomes very complex, 97 00:09:39,090 --> 00:09:46,320 usually you losing in quality is not because you take into account all possible effects that you model is actually better. 98 00:09:46,770 --> 00:09:50,010 There is a point where you add things and it gets worse. 99 00:09:50,130 --> 00:09:54,090 And why is that? There are many different reason. One of them is computational time. 100 00:09:54,090 --> 00:09:57,899 Of course we know that computational time the model becomes complex. 101 00:09:57,900 --> 00:10:03,300 Then the computation time required to get an output out of it becomes increasingly large. 102 00:10:05,270 --> 00:10:09,880 Somewhere. Maybe we'd been here in the sixties. This line is always moving up. 103 00:10:09,890 --> 00:10:13,130 We know that computers are getting faster and all that. 104 00:10:13,670 --> 00:10:22,340 But the other problem, and that's what I call insight, is that as you model become more complex, you can do more things, but you understand less. 105 00:10:23,000 --> 00:10:28,220 Okay, you know something? You get a number, you get the temperature in year two 2100. 106 00:10:28,400 --> 00:10:33,170 But have you understood something about how the system or the climate works? 107 00:10:33,440 --> 00:10:39,080 It's usually which is a little system that you really gain understanding about the system, the system. 108 00:10:39,320 --> 00:10:42,350 It was the larger system that you can do prediction. 109 00:10:42,560 --> 00:10:47,959 So what you really want is to have model and study them in that whole range from different sites. 110 00:10:47,960 --> 00:10:50,390 Small, medium, large, extra large. 111 00:10:50,600 --> 00:10:58,909 But you want to try to avoid the American way, the extra, extra large right here by because the quality of the model decreases. 112 00:10:58,910 --> 00:11:03,799 You take an infinite long time to compute anything and you don't understand what you're doing. 113 00:11:03,800 --> 00:11:07,340 So, so you really want to, to work in that range. 114 00:11:08,360 --> 00:11:11,930 So let's see how it works on, on more practical example. 115 00:11:12,320 --> 00:11:16,130 So now I want to apply this basic concept on the, on climate modelling. 116 00:11:18,440 --> 00:11:21,950 So let's start with very the simplest possible model. 117 00:11:22,610 --> 00:11:26,660 And that's what in physics we call a back of the envelope equations models. 118 00:11:28,340 --> 00:11:30,020 So that's for the younger audience. 119 00:11:30,020 --> 00:11:37,520 At some point in time there were stamps and they were put on a little piece, piece of paper, which I call envelopes to send to somebody else. 120 00:11:38,990 --> 00:11:44,959 So the envelope and the stamps, as you saw, are very important for us because they contain information when you receive it. 121 00:11:44,960 --> 00:11:51,980 You can do computation on the back of the envelope. And so the basic computation when it comes to the climate is the following. 122 00:11:52,250 --> 00:11:54,860 Let's just look at how much energy is coming from the sun. 123 00:11:54,890 --> 00:12:00,570 There is no really very few other source of energy maybe from inside us, but it's a fraction of a percent. 124 00:12:00,890 --> 00:12:04,010 Doesn't matter. So let's look at the energy coming in. 125 00:12:04,010 --> 00:12:07,100 The flux of energy, 342 watt per square metre. 126 00:12:07,700 --> 00:12:12,139 And we know that the albedo, the reflection of the planet is about 0.3. 127 00:12:12,140 --> 00:12:16,459 So that's because we see it from outside. That means photons are coming out. 128 00:12:16,460 --> 00:12:21,010 So you lose energy, right? And that's about a third is reflected solar radiation. 129 00:12:21,020 --> 00:12:26,030 There is nothing you can do. It's gone. And so the back of the envelope computation say, well, 130 00:12:26,060 --> 00:12:32,090 obviously the difference because energy's concern is that of the US reflector 131 00:12:32,090 --> 00:12:37,280 235 what two square metres is there is there is no discussion about this. 132 00:12:37,280 --> 00:12:41,080 This is basic, basic physics. Okay. 133 00:12:41,090 --> 00:12:43,970 So that that's very good. Now you can go one step more, he said. 134 00:12:44,420 --> 00:12:51,979 We know from physics that a black body like a planet is well approximated by what we call a black body, 135 00:12:51,980 --> 00:12:56,510 whose flux of temperature is proportional to the temperature, to the powerful. 136 00:12:57,020 --> 00:13:00,500 And so if you look at that, if you if you play with these numbers, 137 00:13:00,920 --> 00:13:08,750 you'll get to the following prediction is that the temperature of the Earth should be -18 degrees on average, 138 00:13:09,110 --> 00:13:12,560 where it turns out the temperature of the earth is about 15 degrees. 139 00:13:12,800 --> 00:13:17,480 So the error is only 33 degrees. Okay, that's great for us. 140 00:13:17,480 --> 00:13:26,450 Wrong models. Whatever we learn is that there is some energy that is trapped inside and that as a whole it doesn't work. 141 00:13:26,450 --> 00:13:32,210 As a black body. There is something and that something is the atmosphere and it's the greenhouse effect. 142 00:13:32,660 --> 00:13:37,729 So some way in the modelling scheme, in my mother diagrams, we really don't hear, 143 00:13:37,730 --> 00:13:46,460 we've learned a lot is that we can only understand the temperature on earth if we take into account the atmosphere, not just the physics of energy. 144 00:13:47,480 --> 00:13:50,810 So we understand a lot and the prediction is obviously completely wrong. 145 00:13:52,650 --> 00:13:55,850 So so we can start a little bit more complicated. 146 00:13:55,860 --> 00:13:57,570 We can say a one dimensional model. 147 00:13:57,870 --> 00:14:05,100 So what happens is the energy comes here, bounce back and forth between different layers of the atmosphere that have different chemical composition. 148 00:14:05,930 --> 00:14:09,020 And so you can do that balance, which is called radiative forcing. 149 00:14:09,020 --> 00:14:16,100 How much CO2, for instance, the most important one with refracts energy coming at certain wavelengths to other wavelengths and so on. 150 00:14:16,580 --> 00:14:23,090 And you can do that computation. And for instance, you would get that from that simple computation. 151 00:14:23,330 --> 00:14:34,190 You know, that if you double the green, the CO2 content in the atmosphere, then you would automatically on average get a temperature of 21.1 Kelvin. 152 00:14:34,640 --> 00:14:40,970 There is no discussion. Nobody discusses that. It's very clear that this is not controversial at all. 153 00:14:41,840 --> 00:14:47,390 They said, but if you do the computation based on that, you'll get the wrong result as temperature in different zone. 154 00:14:47,960 --> 00:14:53,550 And even the average temperature is because there is another effect that we have not taken into account. 155 00:14:53,570 --> 00:15:02,540 So that model also fails. There is another effect that say the latitude effect that the poles are colder and that the equator is much warmer. 156 00:15:02,540 --> 00:15:05,540 So the radiative forcing taking place there is quite different. 157 00:15:06,020 --> 00:15:12,590 So you said, okay, now I can just do a layer here. I really have to take into account the transfer between these different zone. 158 00:15:13,370 --> 00:15:16,700 So you would go about. So we are here. We are about here. 159 00:15:16,850 --> 00:15:24,350 We have a prediction of very poor. But now we know that greenhouse gas is very important for anything related to the atmosphere. 160 00:15:24,530 --> 00:15:32,240 And you can make very good prediction based on that. Now you say, okay, now I'm going to consider a different zone. 161 00:15:32,540 --> 00:15:36,920 And by convection, warm air goes up and colder air goes down. 162 00:15:37,100 --> 00:15:43,670 I can start computing the motion of the mass of air with different temperature in different zone. 163 00:15:44,120 --> 00:15:49,279 And what I would use in this case is very simple equation, actually, the set of one, two, 164 00:15:49,280 --> 00:15:55,849 three, four, five, five equation that comes from 19th century basic fluid mechanics, 165 00:15:55,850 --> 00:16:00,770 and this one is from thermodynamics that just essentially it's a reformulation 166 00:16:00,770 --> 00:16:07,129 that's f equals me telling you the velocity of air in different zone based on the 167 00:16:07,130 --> 00:16:12,190 forcing F Landsat F data and the rotation of the things are really the core 168 00:16:12,200 --> 00:16:16,280 primitive equation when it comes to atmospheric science that the basic equation, 169 00:16:16,640 --> 00:16:19,640 nothing you can't you cannot escape physics at that level. 170 00:16:20,120 --> 00:16:26,179 Just say things are moving from one side to the other one. The global masses conserve and the thermodynamics. 171 00:16:26,180 --> 00:16:32,120 It's a statement about conservation of energy. So these are five equation that describe that and you can run them on different zone. 172 00:16:32,660 --> 00:16:39,020 We say, okay, now if I can run them on different zone, I can probably run them on a fine of grid. 173 00:16:39,440 --> 00:16:45,290 Now, instead of looking at three zone, I can say now I want to go a little bit further. 174 00:16:45,530 --> 00:16:52,760 I'm going to run the same equation of transfer of linear momentum, transfer of mass, going from one to the other one, 175 00:16:52,940 --> 00:17:00,890 but on a grid that is that spanned the entire earth and that like a typical grid would be a few degrees by a few degrees. 176 00:17:01,370 --> 00:17:06,620 So I make a little box and I say, what is the what are much masses going from one to the other one? 177 00:17:06,720 --> 00:17:08,660 Much energy is going to the other one and so on. 178 00:17:08,960 --> 00:17:16,670 And I can put that on a computer where I say, What if I can make a grid like that as long as I'm going? 179 00:17:17,000 --> 00:17:20,690 These are called general circulation modalities are the typical models use. 180 00:17:21,080 --> 00:17:29,660 If I if I can go like that, well, I can also look at the level in the atmosphere and I can have up to 30 typically level of atmosphere. 181 00:17:30,050 --> 00:17:35,120 So I make box and I just say each box exchange energy and mass with the other one and so on. 182 00:17:35,780 --> 00:17:43,640 And that's the that's essentially all there is to this climate models, but it's how we do it that is important. 183 00:17:44,180 --> 00:17:51,740 So an interesting computations is to see how much how many computation do we have to do to get any answers. 184 00:17:52,490 --> 00:17:58,899 So let's compute together. So I tell you, I told you, we have 2.5 to 2.5 degrees. 185 00:17:58,900 --> 00:18:02,950 That's about 10,000 cells, 10,000 bucks in each box. 186 00:18:02,950 --> 00:18:07,810 I define the velocity going north, the velocity going as the west. 187 00:18:08,110 --> 00:18:14,889 Right. And the temperature there and the humidity, typically, you can reduce that for four, five, six, seven variables. 188 00:18:14,890 --> 00:18:21,520 Seven variable is kind of standard to what you need to get all the information about how temperature move from one to the other, 189 00:18:21,520 --> 00:18:28,090 what are the mass that moves and so on. So 10,000 says that just for the mapping, mapping the earth. 190 00:18:28,510 --> 00:18:37,720 Now I want to map the atmosphere also, and I see about 30 layers in the vertical direction, and so that's about 300,000 red boxes. 191 00:18:38,050 --> 00:18:44,420 So now I have 300,000. That's good. But I have at least seven unknown velocity of the night. 192 00:18:44,470 --> 00:18:52,300 My mass in south, north, east, west, I have the temperature, I have humidity and a couple more. 193 00:18:53,170 --> 00:18:57,430 Okay. So that gives me 2.1 million variables. 194 00:18:58,060 --> 00:19:01,480 Okay. Now that's starting to add up and multiply up really. 195 00:19:01,480 --> 00:19:07,780 In case now if I assume 20 calculation for each variable, that means why not? 196 00:19:07,780 --> 00:19:13,670 If I want to know the velocity at one point I have to do two multiplication, then one addition and blah blah blah. 197 00:19:13,690 --> 00:19:16,660 Do a certain number of operation 20 is actually very low. 198 00:19:17,020 --> 00:19:25,300 If I have to do that for each variable, how many times my little CPU, how many operation it has to do every time it wants to compute? 199 00:19:25,780 --> 00:19:30,220 That's about 20 computation. That means 42 million calculation per time step. 200 00:19:30,760 --> 00:19:34,540 So if I just want to know how things evolve, the temperature supposed to take, 201 00:19:34,600 --> 00:19:38,350 let's think of the temperature evolve from one time step to the next time steps. 202 00:19:38,530 --> 00:19:43,149 I would have to do 40 million computation. It's very easy now do 42 million, right? 203 00:19:43,150 --> 00:19:48,880 We do, you know, giga flux, mega flops and all that mega flops, million of operation per second. 204 00:19:48,890 --> 00:19:57,129 So it's very doable. But then of course, the time step is only 30 minutes or there are plenty of way to estimate time step and all that. 205 00:19:57,130 --> 00:20:05,980 That's that's the typical time step. But what we want for climate is we really want to know the climate up to 20, 100, 100 years in the future. 206 00:20:06,430 --> 00:20:16,900 Okay. So you need you need 2 billion calculations per day to move the system, the mother for one day, and then you need 100 years of simulation. 207 00:20:17,320 --> 00:20:24,610 So you multiply that by blah blah and you get about 73 trillion computation that you CPU's have to do. 208 00:20:24,790 --> 00:20:30,790 That's quite a lot. It's quite doable, but it's, it's getting, it's getting very expensive. 209 00:20:31,150 --> 00:20:35,170 And of course as long as you there, you want to add other effects, 210 00:20:35,380 --> 00:20:41,020 ocean and ice or the interacts or the effect of aerosol, the chemistry, the biomass, the hydrology. 211 00:20:41,260 --> 00:20:47,030 So by the end, what you want is to call our earth simulator that take all these aspect, all these books, 212 00:20:47,080 --> 00:20:53,410 all the chemistry, the geochemistry of the ocean, the interaction with the ice and so on, and I putting all together. 213 00:20:53,860 --> 00:20:55,629 And that's why you need a supercomputer. 214 00:20:55,630 --> 00:21:03,250 All the people who do climate science tell us they really need very big computers to do that and contributing to global warming, 215 00:21:03,250 --> 00:21:09,250 of course, because this computers, you know, it takes a little a little town just to power them up. 216 00:21:09,250 --> 00:21:13,750 That's why they are in remote places and all that. I don't know. Nobody has done the computational much. 217 00:21:13,750 --> 00:21:21,070 The computation of climate change is actually contributing to the effect, and I'm sure it's negligible, but it's pretty big already. 218 00:21:21,910 --> 00:21:28,510 Anyway, you do need that, but so that you can do it and people do that and they've very refined ways of doing essentially. 219 00:21:28,510 --> 00:21:33,400 Tens of thousands of people have been doing that for years and years and trying to refine these models. 220 00:21:33,730 --> 00:21:36,910 But the real question now is where are we? 221 00:21:36,910 --> 00:21:41,440 Are we really here the best possible outcome in terms of model quality, 222 00:21:41,710 --> 00:21:48,250 or are we going down here where we've added so many effect that we've added so many error in the numerical computation, 223 00:21:48,460 --> 00:21:51,250 that we not really know what's really going on. 224 00:21:51,580 --> 00:22:00,790 And a lot of the parts of these very large, complex system is really to try to estimate these how much are we sure about the solution? 225 00:22:01,180 --> 00:22:06,610 So do we know that the solution that we get is actually a good solution, a valid solution? 226 00:22:09,230 --> 00:22:17,080 So what you can do. So you want to do model validation and it's always easier to predict the past in the future. 227 00:22:18,340 --> 00:22:27,249 So you can start by doing that. So here is a computation by one of these big, big models, and it looks back in the past, it said, 228 00:22:27,250 --> 00:22:34,390 Let's start a 1900 and let's take out this false and little function f that I show you a default forcing coming from outside, 229 00:22:34,660 --> 00:22:40,200 particularly volcanoes, turns out to be quite important in putting energy in in the atmosphere. 230 00:22:40,420 --> 00:22:47,770 So here you have different volcano, eruption, Pinatubo, etc. Santamaria from 1900 up to 2000. 231 00:22:47,980 --> 00:22:52,000 The black line is the recorded temperature evolution around that time. 232 00:22:52,810 --> 00:22:58,240 And the red. The red line is an average of a lot of different simulation. 233 00:22:58,240 --> 00:23:00,690 And you see it is weekly and we're going to talk about the wiggle. 234 00:23:01,120 --> 00:23:07,179 But what you see and it's also take into account what we know about the increase of greenhouse gas over this time, 235 00:23:07,180 --> 00:23:11,890 the measure of all the greenhouse gas. So you see that you get a very good job. 236 00:23:12,990 --> 00:23:16,650 At measuring and getting the the general trend. 237 00:23:16,770 --> 00:23:21,840 Of course, there are some years where the red line is not at all like the black line. 238 00:23:22,230 --> 00:23:26,400 That's why people say, you see, there is no climate change. Your prediction is wrong. 239 00:23:26,670 --> 00:23:30,870 But if you see the trend, there is no doubt it's it's very good. 240 00:23:30,900 --> 00:23:36,520 It's very, very good on many different measures. You can do the same computation now. 241 00:23:36,520 --> 00:23:40,930 You can do a thought experiment and say our greenhouse gas really that important? 242 00:23:41,020 --> 00:23:46,390 What's what if we keep them at the level of 1900? Maybe it's really the volcano that creates all that mess. 243 00:23:47,450 --> 00:23:53,690 Okay. So you start again to the same computation with the volcanoes, but with no greenhouse gas increase. 244 00:23:53,960 --> 00:23:59,240 And this is what you get. You get no increase of temperature at all from the 1900s. 245 00:24:00,590 --> 00:24:11,840 So, you know. You know, with very, very likely almost certainty that the greenhouse gases have been responsible for the evolution of these. 246 00:24:13,010 --> 00:24:22,489 Okay. So when you look at a model, it's important notion of either predictability or uncertainty. 247 00:24:22,490 --> 00:24:25,910 So it's important to find a way. What do we really know? 248 00:24:25,910 --> 00:24:30,530 What comes into the model that messes up our computation if we want to do a projection? 249 00:24:31,190 --> 00:24:36,769 So therefore, typical source of uncertainty in climate projection, there is what's called scenario. 250 00:24:36,770 --> 00:24:41,989 Uncertainty is said. Well, we really don't know if the government are finally going to come to their senses and 251 00:24:41,990 --> 00:24:46,700 do something about the problem so we can have we don't know what's going to happen. 252 00:24:46,700 --> 00:24:48,020 We can run to scenario. 253 00:24:48,350 --> 00:24:56,149 People are going to realise that we should stop using fossil fuel and do a lot of different things or people are just not going to care. 254 00:24:56,150 --> 00:24:59,610 We can do is to and anything in between we can run. Okay. 255 00:25:00,730 --> 00:25:02,140 There is modern uncertainty. 256 00:25:02,440 --> 00:25:09,520 There's still plenty of people working on the physics of this problem, all due to the ice, interact with the atmosphere, with the water and all that. 257 00:25:09,700 --> 00:25:13,540 There's a lot of physics we don't quite know yet, and there's a lot of work to be done. 258 00:25:13,750 --> 00:25:17,830 So. And which are the effects important? So what? What is the best model? 259 00:25:18,640 --> 00:25:23,130 Then there is a lot of parameter uncertainty. You don't know the data perfectly. 260 00:25:23,140 --> 00:25:29,140 You know, you don't know the parameter. You know, the temperature maybe in a box, but that comes with a certain error and all that. 261 00:25:29,140 --> 00:25:30,370 So you have a lot of things. 262 00:25:31,510 --> 00:25:40,030 And finally, more important and more interesting for us, you have internal variability, and that's intrinsic uncertainty in the climate system. 263 00:25:41,050 --> 00:25:47,440 And that means in the model itself, whatever you do, even if you have perfect knowledge of all this, you'd still be in trouble. 264 00:25:47,920 --> 00:25:52,380 And why is that? And that's really a mathematical question. And why is that? 265 00:25:52,410 --> 00:25:56,530 It really goes back to 1963 with the notion of chaos. 266 00:25:57,520 --> 00:26:01,670 And here is a stamp from. I think he's getting there. 267 00:26:03,110 --> 00:26:10,970 They seem to very much like physics and mathematics and things like that that celebrated the death of at Lawrence in 2008. 268 00:26:11,480 --> 00:26:16,790 At Lawrence, a very strong influence. And in 1963 wrote a paper, a very influential paper. 269 00:26:17,520 --> 00:26:23,900 He says it started actually with a convection problem in simplified, simplified, simplified all the way down to three equation. 270 00:26:24,560 --> 00:26:28,640 And that became a problem that had no relevance for the problem he started with. 271 00:26:28,940 --> 00:26:33,769 But he noticed there was something very strange in the system when he studied in more detail, 272 00:26:33,770 --> 00:26:39,440 more mathematically transforming to a purely mathematical problem. The equation a very simple three equation. 273 00:26:39,650 --> 00:26:44,600 The Z TD is a d t is what we call ordinary differential equation. 274 00:26:44,600 --> 00:26:51,560 Just about the simplest, the simplest type, except they have nonlinearity instead of X and Y and Z being by themselves. 275 00:26:51,770 --> 00:26:58,820 There are two places where they make a product nonlinear or what we call nonlinear and non in narrative, a very drastic effect on the solution. 276 00:27:00,080 --> 00:27:06,040 And when you run that on this own computer at the time, you can do it on your smart watch and things like that. 277 00:27:06,050 --> 00:27:15,380 It's is very it's very simple system. You notice something very remarkable that is fully understood at the time that was also quite remarkable. 278 00:27:15,710 --> 00:27:23,360 So let me tell you what what what we going to do this equation that if I give you the value at a time to equal to zero, 279 00:27:23,360 --> 00:27:31,669 think of it temperature or something like that. You can run them and know the value at a time later and there is no no is there is no problem. 280 00:27:31,670 --> 00:27:35,780 There's something that we know for sure. There is one equation that we call deterministic. 281 00:27:35,780 --> 00:27:44,989 It's Turing fully known in the future. So if you know here at times equals zero, I can go all the way to this position at 90. 282 00:27:44,990 --> 00:27:49,850 At six is three equation. I can represent that in three dimension X, Y and Z. 283 00:27:50,710 --> 00:27:54,400 So I can run the system. I can just let it go and see where it goes. 284 00:27:55,150 --> 00:27:58,590 And this is where it goes. When you let it run. 285 00:28:00,290 --> 00:28:03,650 The access atoning for the added visual appeal. 286 00:28:06,830 --> 00:28:13,219 And now what? Let it go. And the system, it doesn't go away and it doesn't go to zero itself. 287 00:28:13,220 --> 00:28:15,260 It does something different keeps turning around. 288 00:28:15,800 --> 00:28:24,950 What I haven't fully told you is that I actually started to solution two points and there are slightly, slightly, slightly different initial data. 289 00:28:25,520 --> 00:28:29,600 So it's like if you want to run a weather projection, 290 00:28:29,780 --> 00:28:39,020 I would take the temperature today at 22 degrees and I would run another scenario with 22.0001 and let them evolve, right? 291 00:28:39,230 --> 00:28:46,250 So started with two actually 2.111 green and one blue and one yellow. 292 00:28:47,210 --> 00:28:50,980 And this is what happens if you let it run. Very soon. 293 00:28:50,990 --> 00:28:56,790 The point I'm going to separate. And the separate. 294 00:28:58,460 --> 00:29:03,650 I can start as close as I want. Eventually they will end up a completely different position. 295 00:29:05,200 --> 00:29:13,149 And that was truly remarkable discovery for the for the weather system at Lawrence was a meteorologist who was interested in the weather, 296 00:29:13,150 --> 00:29:16,840 not so much in the climate, but in the weather for sure. So. 297 00:29:19,590 --> 00:29:23,670 If I just look at one of the variable X as a function of time, for instance, 298 00:29:23,790 --> 00:29:27,510 and I start with the two points exactly the same one, I wouldn't see any difference, 299 00:29:28,020 --> 00:29:35,909 but eventually a time 20 or something in arbitrary units you'll see the two diverging and you'll see right here there'd be a different position. 300 00:29:35,910 --> 00:29:45,480 So if this was temperature or rain or anything like that, if I run the model, I don't know the difference between is it really 24 or 24.000001? 301 00:29:45,960 --> 00:29:53,010 Well, what if I choose? Even if I had perfect knowledge, a little bit of difference would tell me that. 302 00:29:53,340 --> 00:29:56,460 Why two weeks later it would either rain or we sunny. 303 00:29:56,610 --> 00:30:00,540 I have no knowledge. It's impossible to know. It's mathematically impossible. 304 00:30:01,080 --> 00:30:06,360 Okay. I would need perfect knowledge and perfect ability to do computation to do that. 305 00:30:06,360 --> 00:30:11,160 But every single measurement is an initial data which come with a little bit of difference and error. 306 00:30:11,180 --> 00:30:18,510 There is nothing we can do about that. So it is two weeks window is what stop us from doing any good weather prediction past two weeks. 307 00:30:18,960 --> 00:30:22,050 Maybe we'll be able to do a few, maybe have another week. 308 00:30:22,320 --> 00:30:26,790 But we know for a fact mathematically we'll never be able to predict the weather for years time. 309 00:30:27,720 --> 00:30:31,770 We know that numbers the numbers are very clear. And in his paper, 310 00:30:32,100 --> 00:30:40,140 Lawrence says to stay differing by impossible imperceptible amounts may eventually evolve into two into two considerably different state. 311 00:30:40,470 --> 00:30:48,120 In view of the inevitable inaccuracy and incompleteness of weather observation, precise, very long range forecasting would seem to be non-existent. 312 00:30:48,840 --> 00:30:51,149 It was very clear it completely understood the problem. 313 00:30:51,150 --> 00:30:59,549 A few years later, you wrote a famous another famous paper saying that the the flap of a butterfly could trigger a tornado many years later. 314 00:30:59,550 --> 00:31:02,400 The difference and that was the amplification of this effect. 315 00:31:02,760 --> 00:31:11,070 And, you know, it's important when when they start making bad Hollywood movies with that with puppy face actor. 316 00:31:12,180 --> 00:31:15,540 So it really came in to that knowledge not only as a culture, 317 00:31:15,750 --> 00:31:22,530 but really completely changed the way of thinking of everybody doing climate and whether that was in 63. 318 00:31:22,650 --> 00:31:29,250 And people understood at that time that that it was important, but they still used the same model to try to say, 319 00:31:29,250 --> 00:31:33,180 okay, now we know we can of precisely can we can we use that? 320 00:31:33,180 --> 00:31:39,559 Can we use that knowledge? So the idea is to do what's called and somebody averaging. 321 00:31:39,560 --> 00:31:44,600 I said, I don't know where I'm going to start, so I'm going to take a bunch of people close by together. 322 00:31:45,140 --> 00:31:51,620 I've made it 24 degrees today. I'm going to try all the temperature between 23 and 25 and run my mother with that. 323 00:31:52,160 --> 00:31:57,920 And if I do that under Laurence Attractor, it's called chaotic attractor. 324 00:31:58,250 --> 00:32:01,340 If I do that, I can have different scenario depending on where I am. 325 00:32:01,550 --> 00:32:07,100 I can start with a bunch of data here and I look at the evolution and two weeks later I'm here. 326 00:32:07,340 --> 00:32:11,420 And basically if I look at the centre of here, it's still the centre of that blob. 327 00:32:12,050 --> 00:32:15,620 So I know that I will have a good knowledge. It's very high predictability. 328 00:32:15,620 --> 00:32:19,040 I know that the trajectories have not gone too far away. 329 00:32:19,880 --> 00:32:28,040 I can have medium predictability. If I start doing things, I could have very low predictability if I start here and all the solution diverge. 330 00:32:28,700 --> 00:32:32,240 So this is a way to quantify uncertainty. 331 00:32:32,360 --> 00:32:33,979 So you don't run a single model. 332 00:32:33,980 --> 00:32:39,920 You would run a model with very different initial, very close, but different initial data and look of farther diverge. 333 00:32:40,160 --> 00:32:46,010 I would say, well, I since I don't know my initial one, I cannot my prediction stop at a certain time. 334 00:32:46,190 --> 00:32:51,350 So you can quantify how much you know about the model. It's this idea of ensemble average that's so important. 335 00:32:51,590 --> 00:33:00,260 And this is from a paper prime, Tim Palmer from also from Oxford, who has done a lot of work on t on this type of problem, 336 00:33:00,260 --> 00:33:05,900 trying to find ways to use this idea to a benefit rather than being an inference. 337 00:33:10,010 --> 00:33:19,250 Okay. So what's the final picture at times of climate? This is the result of the Intergovernmental Panel on Climate Change. 338 00:33:19,870 --> 00:33:25,940 John It was July 2014 is the fifth assessment report of that panel, the IPCC. 339 00:33:26,300 --> 00:33:31,680 And they do exactly what I told you the start here, 2010 or 2014. 340 00:33:31,910 --> 00:33:36,830 And they run the model with different scenario, which is the model where people are responsible. 341 00:33:36,830 --> 00:33:40,250 And this is the model where people as they are. 342 00:33:41,560 --> 00:33:48,250 I would say it's a different, different scenario. And you see the scenario leads to very different, as you've heard in the news. 343 00:33:48,280 --> 00:33:52,090 Whatever we do, even if we start behaving, there'll still be some increase. 344 00:33:52,480 --> 00:33:56,620 But otherwise, inevitably we do that with very lightly. 345 00:33:56,800 --> 00:34:04,720 And this breadth and average are all coming forward. Is this idea of an average all the way to 2100, so you can say one four degrees. 346 00:34:04,720 --> 00:34:07,750 That's okay. I mean, I'll water my loan a few more times. 347 00:34:07,750 --> 00:34:12,130 Right. But if you actually you can since you've done all that computation on that fine grid, 348 00:34:12,340 --> 00:34:17,260 you have information about the climate change on average in all these different positions, these different place. 349 00:34:17,830 --> 00:34:22,120 So you can go back to your data and look at the evolution of the temperature. 350 00:34:22,270 --> 00:34:27,460 This the two scenario and you see that this is where we'd be if we still have increase everywhere. 351 00:34:27,760 --> 00:34:37,370 And that's where. What we are facing and what we are facing are temperature increases up to 11 degrees close to the pole, you know, so. 352 00:34:37,610 --> 00:34:41,810 So it is true. It doesn't increase that much. But that's because the pole don't change that much. 353 00:34:42,350 --> 00:34:48,320 Oh, the the ocean the oceans don't change that much. But the rest of the land is dramatic increase. 354 00:34:49,640 --> 00:34:53,240 And the theory has become quite reliable. 355 00:34:53,270 --> 00:34:57,499 I mean, there's been a lot of evolution, very quick evolution over the last five, 356 00:34:57,500 --> 00:35:01,910 ten years or so in terms of understanding what are we actually saying? 357 00:35:02,060 --> 00:35:10,310 Are we are we sure about what we're saying? And so on that community that there's very little doubt now that that's what we are facing? 358 00:35:11,960 --> 00:35:19,750 Okay. So I want to to stop about the climate here and move to other problems of application of modelling. 359 00:35:20,270 --> 00:35:28,010 I said, Well, since we have only sun, maybe we can try to find a way to be more responsible with it and try to do something about climate change. 360 00:35:28,430 --> 00:35:34,310 And I will tell you a little bit about modelling that I've done over the last few years on photovoltaics. 361 00:35:35,540 --> 00:35:41,870 So whatever we we do, we know that photovoltaics using the sun is going to be part of our energy portfolio. 362 00:35:42,410 --> 00:35:49,610 And today, today, the silicon technology, the silicon panels that we see everywhere, 363 00:35:49,610 --> 00:35:57,170 it's really the dominating technology, but it's a very restricted in many ways, not as flexible as we'd like. 364 00:35:57,440 --> 00:36:04,870 And so people have been looking forward to a different type of photovoltaic, and that search has not been doing too well. 365 00:36:04,880 --> 00:36:14,020 I mean, a lot of people in a lot of different departments find alternative, but nothing really came to to challenge the silicon technology. 366 00:36:14,030 --> 00:36:22,549 But that was until two years ago when the group of and we Snaith here in physics discovered that certain type of 367 00:36:22,550 --> 00:36:29,330 material that use in some of these photovoltaic cells they use just by themselves has remarkable properties. 368 00:36:31,280 --> 00:36:41,090 And so we started actually four or five years ago with thanks to the Oxford Martin School, which fund this type of collaboration with in Oxford, 369 00:36:41,300 --> 00:36:47,570 to look at the mathematics, the modelling of this problem with Henry Snow trying to get some insight into this process. 370 00:36:48,850 --> 00:36:56,440 So let me give you the general picture. We measured a quantity of photovoltaic cells in terms of efficiency. 371 00:36:56,710 --> 00:37:00,610 It says, how much energy can I get back from from the sun? 372 00:37:01,090 --> 00:37:06,490 Okay. And so it's a percentage of energy I can extract out of of out of light. 373 00:37:07,810 --> 00:37:11,190 And here we just going to be interested in the silicon here. 374 00:37:11,500 --> 00:37:14,920 Gallium arsenic is is very good, but it's quite expensive. 375 00:37:15,340 --> 00:37:23,350 And here is the evolution of the silicon technology from the seventies, about 12, 15% all the way to 25%. 376 00:37:23,560 --> 00:37:30,550 And you can buy now the silicon reliable silicon 20% from Chinese manufacturer. 377 00:37:30,730 --> 00:37:35,140 And they've done a great deal to actually reduce dramatically the price of silicon. 378 00:37:35,440 --> 00:37:41,260 So silicon will be with us and will be used. But as I show you, we want to have other technology for the purpose. 379 00:37:42,190 --> 00:37:53,560 Now, forget about all these. I just want to point out here, this is where we started in 2012 and we started around 15% and Resnais's group. 380 00:37:53,830 --> 00:38:00,700 And now actually last months, people have shown that they can have 20% in two years efficiency. 381 00:38:00,880 --> 00:38:07,810 It's very rapid growth, but it's not so much that it's so now it's we know that t cells are potentially very efficient, 382 00:38:08,140 --> 00:38:11,610 but it's all the made to silicon. So you have to grow a crystal of that. 383 00:38:11,660 --> 00:38:17,410 It's very rigid. But the perovskite, you can actually just lay them down by different technique of vapour, 384 00:38:17,410 --> 00:38:20,800 deposition, spin coating, which are very cheap way of doing it. 385 00:38:21,740 --> 00:38:28,220 Okay. So this is one of the modelling problem that, that we worked on with, with, with the Henry. 386 00:38:28,520 --> 00:38:36,890 It says, okay, I know that I can lead on my, my perovskite miracle compound that works so well, 387 00:38:37,130 --> 00:38:44,450 but when I do it by vapour deposition instead of being all flat, it has a lot of holes and you can see where it has a lot of hole. 388 00:38:44,450 --> 00:38:50,749 You're going to lose a lot of coverage and efficiency and things like that. So when you do that, you start making a lot of holes. 389 00:38:50,750 --> 00:38:57,050 So the problem is the problem of material is the problem of the wetting is just like when you have a bunch of water on a table, 390 00:38:57,320 --> 00:39:00,410 the little bubble start making bigger bubbles and things like that. 391 00:39:00,920 --> 00:39:03,140 So we look at that problem from that perspective. 392 00:39:03,980 --> 00:39:11,670 So that it could be good because if it doesn't fully cover, then you can say, Well, I could use these and I could make windows, for instance. 393 00:39:12,050 --> 00:39:15,410 Right. Because the light would still go through and I would still get some energy out of it. 394 00:39:15,770 --> 00:39:20,429 That's a good idea. But it looks like that. Really. That's okay. 395 00:39:20,430 --> 00:39:24,650 But really, who wants to live in a brand building? Right. So say where. 396 00:39:24,870 --> 00:39:27,930 This is probably not going to be very, very popular on the market. 397 00:39:28,230 --> 00:39:33,059 So we went back and look at the probably in more details and we created a little model. 398 00:39:33,060 --> 00:39:34,950 I just give you the basic idea of the model. 399 00:39:35,130 --> 00:39:41,670 What we want is when we lay down things, it's essentially it's called annealing, but it's really called cooking. 400 00:39:41,670 --> 00:39:45,390 If you want to make brownies and things like that, to put it in the oven and you heat it up, 401 00:39:45,630 --> 00:39:52,380 you put a layer and then you heat it up in another 100 degrees for 60 minutes, just like a cookie recipe. 402 00:39:52,800 --> 00:39:57,090 And what happens? The solvent evaporates and forms the structure that you want at the end. 403 00:39:57,690 --> 00:40:05,009 And the only thing that you can really control is the the temperature and the thickness of the film that you start. 404 00:40:05,010 --> 00:40:07,229 And, you know, if you make brownies or things like that, 405 00:40:07,230 --> 00:40:13,200 it might be very dry or very moist where you have the same type of problem you can you can do the same optimisation for brownies. 406 00:40:13,470 --> 00:40:17,070 So what we did is say, okay, or layer, it's really a full layer. 407 00:40:17,070 --> 00:40:24,420 It has holes and each of these holes, the total, a certain diameter, a certain diameter and certain lengths. 408 00:40:24,630 --> 00:40:29,940 And we can write a full energy fatty system for depending on the different radius. 409 00:40:30,270 --> 00:40:35,459 And each radius we know from fundamental physical principle can change a size. 410 00:40:35,460 --> 00:40:41,430 The whole can change the size and interact with each other. And if you do that, I spare you the detail. 411 00:40:41,430 --> 00:40:46,320 But the problem is not too hard. I mean, it does require putting things together, but it's not too hard. 412 00:40:46,740 --> 00:40:50,520 And when you do that. Sorry, let me go back here. 413 00:40:51,920 --> 00:40:57,440 When you do that, you get exactly what you want. You get the coverage as a function of the temperature. 414 00:40:58,950 --> 00:41:01,050 And as a function of the film thickness. 415 00:41:01,500 --> 00:41:09,150 So you know how much thickness you need and or what the temperature you should set your oven in order to get a certain level of coverage. 416 00:41:09,600 --> 00:41:16,530 And he had the here the the boxes, actually, the experimental data and the curves are what we predict. 417 00:41:17,430 --> 00:41:20,850 I said that's very interesting because you can really optimise your coverage. 418 00:41:21,090 --> 00:41:26,940 But then we talk with with with Henry in Henry's group and he says, but look, we can do much better than that. 419 00:41:27,090 --> 00:41:29,760 Sure, we can optimise the coverage, but no, 420 00:41:29,760 --> 00:41:37,659 we can go back to the problem and we can choose the right conditions so that we the coverage is what we want to hold. 421 00:41:37,660 --> 00:41:43,620 The distribution is exactly what we want. So that as it goes, as the light goes through it as a given column, 422 00:41:44,490 --> 00:41:47,940 because the light is going to refract in different ways for different wavelength and so on. 423 00:41:48,330 --> 00:41:51,510 And so this is what he actually made with this group. 424 00:41:51,990 --> 00:41:58,980 Okay. So now you have a technology that allows you to have a piece of photovoltaic cells are given colour. 425 00:42:00,270 --> 00:42:05,460 And that you can put on windows and things like that and get maybe not 25% because there is still a lot of light. 426 00:42:05,610 --> 00:42:09,270 But you get 5%, maybe 10%. So. Okay. 427 00:42:09,750 --> 00:42:17,069 So you can have buildings like that where you can have beautiful blue buildings and you still collect the light as you go along. 428 00:42:17,070 --> 00:42:20,090 So it creates a new market, a new way of using solar energy. 429 00:42:21,300 --> 00:42:26,400 So we patterned that that I don't know what's going to happen with that, but this is the kind of thing that you have to do. 430 00:42:27,420 --> 00:42:30,540 And we applied the same type of idea to different context. 431 00:42:32,540 --> 00:42:36,560 I think it's a good example because not only to some predictive values, 432 00:42:36,890 --> 00:42:43,219 but really the mathematical models was part of the creative process in understanding 433 00:42:43,220 --> 00:42:46,220 the material and coming up with alternative of what we can do with it. 434 00:42:49,490 --> 00:42:56,270 Okay. So part three B, the second example, the first one was aiming at saving the planet. 435 00:42:56,600 --> 00:43:03,410 Now we try to save mankind with mathematics. Let me tell you a little bit about the brain as it was advertised in the title. 436 00:43:04,910 --> 00:43:08,950 So there's a lot of wonderful mathematical problem with the brain. 437 00:43:08,960 --> 00:43:15,170 I'm going to concentrate on one that's not the kind that you hear with this huge, big initiative, 438 00:43:15,170 --> 00:43:19,100 the Human Brain Project and things like that, which is try to understand the connection. 439 00:43:19,430 --> 00:43:23,270 I'm much more interested to see the brain as a real physical entity. 440 00:43:24,490 --> 00:43:31,960 And so my motivation and initial motivation was trying to understand trauma and brain injury or traumatic brain injury. 441 00:43:32,530 --> 00:43:40,170 It can result of, uh, unhealthy sports events or car accidents or stroke or things like that. 442 00:43:42,870 --> 00:43:52,469 What happens in this case. Invariably what happens is that part of the brain becomes depleted of oxygen and 443 00:43:52,470 --> 00:43:57,570 then you have an imbalance of ions and you see that the tissue starts swelling. 444 00:43:58,440 --> 00:44:01,830 It is swelling can be very dangerous, as I'll show you in a second. 445 00:44:02,190 --> 00:44:07,950 And sometimes you do have to open the skull and do a craniotomy to relieve that part of the swelling. 446 00:44:08,370 --> 00:44:14,879 So we really wanted I find that very interesting. But because once you know that if you know the geometry and the change of geometry, 447 00:44:14,880 --> 00:44:19,410 then you can try to understand what happened all the way to the axon as the axon a stretch. 448 00:44:19,920 --> 00:44:22,890 Okay. What is the damage that's done to the axon and so on? 449 00:44:23,190 --> 00:44:30,120 All of these are very important, not only for for accident, but also in a lot of different diseases, epilepsy and things like that. 450 00:44:30,420 --> 00:44:40,530 So when we we've been thinking about this type of project for a couple years and together with with Ottawa and Jerusalem in engineering, 451 00:44:40,740 --> 00:44:46,260 we decided to create an informal group we called the International Brain Mechanics and Trauma Lab. 452 00:44:46,860 --> 00:44:54,780 We started that in 2013. Try to piggyback different speciality, different expertise from different field to address this problem, 453 00:44:54,780 --> 00:45:02,579 because we quickly realised that you need to go. You need to have the, the medical doctors, the biologists, the physiologists, the engineers, 454 00:45:02,580 --> 00:45:06,840 the mathematician and all that, all together, because these are very, very complex problems. 455 00:45:07,140 --> 00:45:15,630 So I'll just tell you about one of the problems that we're working on now, it is has grown to an international network of scientists in the world. 456 00:45:15,660 --> 00:45:19,750 Try. Trying to answer some of these problems. 457 00:45:20,320 --> 00:45:25,240 I just want to look at the simple problem. There's no simple problem when it comes to the brain. 458 00:45:25,480 --> 00:45:33,550 But the problem of swelling. And here is a swelling of brain of a rat that was induce a stroke in the in the right hand side. 459 00:45:33,880 --> 00:45:40,240 And we wanted to understand what's what actually happened. It's very one describing the physiology textbook and so on. 460 00:45:40,450 --> 00:45:44,400 But again, just like Kelvin says, there is no numbers attached to it. 461 00:45:44,410 --> 00:45:51,130 There may be measurement, but you you there was no way really to try to understand what happens, what is the sequence that happens? 462 00:45:51,580 --> 00:45:58,180 So what we did is we combine both with mechanics, solid mechanics, because you have a soft tissue and the electrochemistry, 463 00:45:58,180 --> 00:46:01,600 which is the balance of ions, interaction of ions and electrolytes. 464 00:46:02,170 --> 00:46:06,790 To try to understand that and we created a model with a quadriplegic theory that 465 00:46:06,790 --> 00:46:10,689 means a four different phase interacting with each other to try to understand that. 466 00:46:10,690 --> 00:46:17,620 And it's really the Ph.D. thesis of Georgina Lange that she she she finished yesterday. 467 00:46:17,620 --> 00:46:23,410 I think she signed the thesis, and that's also with Dominic Vella and Sarah Waters. 468 00:46:24,430 --> 00:46:32,680 So the first part was to understand the swelling. And so we built this model and fitted a number of parameters, an independent experiment. 469 00:46:33,010 --> 00:46:38,710 And then we look we went back to some data that people do and very, very controlled system. 470 00:46:38,890 --> 00:46:42,240 And we saw that we could we could we could have a very good match of them. 471 00:46:42,250 --> 00:46:51,640 I'm not going to go into detail, but based on that, we really understood the different effect of different component of of the swelling process. 472 00:46:52,120 --> 00:46:56,050 That was we the first step was to create a model that would just explain the swelling. 473 00:46:56,350 --> 00:47:02,280 And when you do that, you can go to the next step, says, okay, now I can induce a damage in my mother. 474 00:47:02,320 --> 00:47:09,310 I would induce damage on poor little rat or anything, but at least on my mother and I can see if I can predict this type of event. 475 00:47:09,910 --> 00:47:12,760 Okay. That's that was the second step. Try to do the damage. 476 00:47:13,180 --> 00:47:18,430 And the third step was to say, okay, now all this the damage propagates because that's what we really started. 477 00:47:19,060 --> 00:47:21,130 And the damage propagate the following way. 478 00:47:21,610 --> 00:47:28,570 If we start with the brain and all these red dots are capillaries, if you have a lack of oxygen, at some point you restrict the capillaries. 479 00:47:28,870 --> 00:47:33,849 What you'll have is that the tissue is going to swell and where it swell is going to compress its neighbour and 480 00:47:33,850 --> 00:47:40,180 it compress enough its neighbour that does not produce the don't give you oxygen anymore in the in the tissue. 481 00:47:40,480 --> 00:47:44,380 So they're going to die again and they're going to start swelling again. Okay. 482 00:47:44,470 --> 00:47:48,160 So to swell again and compute and so the damage propagate like that. 483 00:47:48,490 --> 00:47:55,660 This is all swelling propagate in the brain. When start, start, started, push each other so it prevent the other guys next to you to have oxygen. 484 00:47:55,810 --> 00:48:00,280 They die or they're very unhappy. And so they start sweating again and so on and so on. 485 00:48:00,580 --> 00:48:05,050 So we wanted to know what is damage propagate? And we built a model based on that. 486 00:48:05,170 --> 00:48:12,810 And we look at two scenario where the skull is not open and we look at depending on parameter, the critical stress at which the, 487 00:48:12,950 --> 00:48:20,080 the capillaries would be squeezed and as a function of the initial size of the damage, whether or not it would naturally propagate to the system. 488 00:48:20,470 --> 00:48:26,350 And we see that in this case, if the damage is if damage is large enough, it would always propagate. 489 00:48:26,650 --> 00:48:29,650 If the critical stress is large enough. 490 00:48:30,520 --> 00:48:34,509 However, if you do open it as a simple geometry, I show you that for picture. 491 00:48:34,510 --> 00:48:40,990 But the geometry is much simpler. If you open this, you relieve part of the stress and then the damage stop at a certain point. 492 00:48:41,560 --> 00:48:43,150 So this is a conceptual model. 493 00:48:43,150 --> 00:48:51,040 It has no direct predictive value, but it shows that we understood all the steps, all the critical elements that came in, the problem and the process. 494 00:48:51,040 --> 00:48:54,159 We took away some that were not important, that were thought to be important. 495 00:48:54,160 --> 00:48:58,059 Important. So now we can go back and implement that in the correct geometry. 496 00:48:58,060 --> 00:49:06,300 And that's the work that we going to do in the next in the coming years. We also look at problem of brain folding. 497 00:49:06,310 --> 00:49:15,000 That's another story I want to go on to to my last application, trying to keep on time, which is seizures. 498 00:49:16,210 --> 00:49:24,340 So obviously, I'm not going to try to save mankind of planets working on seashells, but they're there and they're quite nice. 499 00:49:25,570 --> 00:49:27,940 So the motivation is really for the beauty of it. 500 00:49:27,940 --> 00:49:35,320 As I, as I said, I can't really justify on any other way I could if I have to write a proposal for an agency, 501 00:49:37,150 --> 00:49:46,420 because we've become really good at that. But between you and I, I just do it because I think it's the problem is beautiful and interesting by itself. 502 00:49:46,600 --> 00:49:50,060 It's really curiosity driven and it's often teased out. 503 00:49:50,090 --> 00:49:55,989 We can have we can talk about that. But it's often my problem that I curiosity driven that the bigger sleep comes. 504 00:49:55,990 --> 00:49:59,560 So I have no problem doing that. And what is the problem. 505 00:49:59,590 --> 00:50:05,350 So it is his work with Derek Moulton, who is in mathematics and the biologists from in France, which is Serra. 506 00:50:05,980 --> 00:50:12,660 So what is the problem? Well. Here is a seashell I wanted to show you. 507 00:50:12,660 --> 00:50:16,830 Is the camera to come is not working. Here is a seashell and here is another one. 508 00:50:17,610 --> 00:50:20,790 And they're very beautiful shape and all that. 509 00:50:21,000 --> 00:50:24,570 But some of them have what's called morphology. 510 00:50:24,660 --> 00:50:29,700 You know, they have this pine and on top of them. And we wanted to know, can we explain? 511 00:50:29,710 --> 00:50:34,290 These people have been describing that for hundreds of years. Spine ts, rip that and all that. 512 00:50:34,420 --> 00:50:40,830 Plenty of people very important for evolutionary process because it helps you to classify, see the trees and structure and all that. 513 00:50:41,070 --> 00:50:50,310 We wanted to see what are the minimal, minimal processes that can explain the creation of this shade, that can create teeth patterns? 514 00:50:50,400 --> 00:50:52,410 What is the morphogenesis process? 515 00:50:56,440 --> 00:51:04,300 So the first part was to really try to understand the main geometric shape, the overall shape of the of the seashell. 516 00:51:04,330 --> 00:51:07,780 And we built a model based on that that tells you if, 517 00:51:08,080 --> 00:51:13,660 if I know the growth process right at the edge of the seashell here at the opening of the seashell, 518 00:51:13,930 --> 00:51:17,410 then naturally it would create one of these basic shape. 519 00:51:18,040 --> 00:51:21,790 Okay. So I'm not interested that the pattern they're just interested in the global shape. 520 00:51:21,810 --> 00:51:30,580 That was the first step. Now, after that, we say, okay, now how the o is always the the shell created and always the pattern created. 521 00:51:31,600 --> 00:51:41,170 And the basic idea is the following You have an animal most of the time before they die inside the mollusc, and it's a soft tissue. 522 00:51:41,800 --> 00:51:48,080 And the way it creates its shell is by coming out, going on the edge, depositing a new material, 523 00:51:48,080 --> 00:51:52,480 liquid accreting new material that get calcified and then retreating. 524 00:51:53,360 --> 00:51:59,299 So what you have is a soft tissue that goes on top of a heart tissue and then it goes back, 525 00:51:59,300 --> 00:52:02,720 it changes boundary and then the next day it comes out and does it again. 526 00:52:03,140 --> 00:52:07,310 Next day comes off. So what you have is you have something like that. 527 00:52:07,310 --> 00:52:13,160 You have the you have the mantle that comes out, goes on the aperture, and then lay new material. 528 00:52:14,000 --> 00:52:25,010 Now the animal itself grows as a certain piece and the accretion process is a different a different mass increase. 529 00:52:25,280 --> 00:52:31,490 There are two different process. One is for the soft part and one is for how much material I'm going to put down at each level. 530 00:52:32,330 --> 00:52:36,260 Now, if you do that, you start with one and then you go out and put a little one. 531 00:52:36,500 --> 00:52:44,420 Now, if the soft mantle grow faster than the aperture, it doesn't fit nicely on top of it. 532 00:52:44,930 --> 00:52:46,400 So it can make a nice shape. 533 00:52:47,010 --> 00:52:54,079 And just like you have an elastic or something like that or a rubber band and you push it, it cannot really put itself at the right place. 534 00:52:54,080 --> 00:52:56,030 So what's going to happen is the following. 535 00:52:57,150 --> 00:53:03,420 If you look at the edge, you're going to the mantle comes and going to try to adhere glue on itself and put a new layer. 536 00:53:03,420 --> 00:53:08,370 But it's too long for the shell edge. So it's going to do this little whoop here, it's going to buckle. 537 00:53:08,910 --> 00:53:17,880 And that's how it's going to present is going to put the new layers and comes up, it comes back in and then goes back out again. 538 00:53:18,060 --> 00:53:21,230 So if you are a little problem here, it's going to get amplified this time. 539 00:53:21,900 --> 00:53:25,920 So the equation are quite simple. Again, for this type of process. 540 00:53:25,920 --> 00:53:30,899 It's an evolutionary process where the boundary evolves with the system as time goes on. 541 00:53:30,900 --> 00:53:34,710 So it's a it's a non-standard problem when it comes to mathematical modelling, 542 00:53:35,010 --> 00:53:42,900 but it's very easy to go on and simulate and this is what you get when you vary the parameters, the two important parameters. 543 00:53:43,760 --> 00:53:48,170 Depending on the parameter that you have, you either get is very sharp spine. 544 00:53:50,520 --> 00:54:05,970 You get very worried. One. And so depending on these two parameters, the growth rate and the stiffness or stiffness you mental, 545 00:54:06,750 --> 00:54:10,830 then you can you can easily map all the shape that are around that iPhone. 546 00:54:11,040 --> 00:54:19,800 You know, either you have like a beach could be beautiful, very, very, very thin spines or or very wide spines like like this one. 547 00:54:24,360 --> 00:54:27,419 So this is this is one way to reduce a lot of knowledge. 548 00:54:27,420 --> 00:54:30,090 And we say, okay, now we have a model, but if we want to push it, 549 00:54:30,090 --> 00:54:35,760 can we apply the same idea to other patterns that are fun without without fiddling, anything? 550 00:54:35,760 --> 00:54:43,140 We go with the basic idea. So we went back and say, Well, there is another pattern that's found in seashells in Ammonites. 551 00:54:43,680 --> 00:54:47,070 No, Ammonites have been extant for about 100 million years. So obviously. 552 00:54:48,310 --> 00:54:51,190 We don't have that much data on the living animal, 553 00:54:51,370 --> 00:55:01,510 but we have a lot of data on fossil records and people for hundreds of years have been looking at these ribs on Ammonites, classified them. 554 00:55:01,540 --> 00:55:06,100 Look at evolutionary trend and all that. A lot of very nice work under very sad. 555 00:55:06,370 --> 00:55:10,450 If the only thing I know is how fast these teeth. 556 00:55:12,600 --> 00:55:17,700 These aperture opens can actually predict the form and the shape of the pattern. 557 00:55:18,390 --> 00:55:27,660 And so this is the result in blue line of the simulation value, same type of equation based on just the knowledge of the expansion rate of the inside. 558 00:55:27,960 --> 00:55:34,010 If I know the red one, I can compute the because again, the, the mental part, 559 00:55:34,020 --> 00:55:37,740 the soft tissue is going to grow faster and then it's going to be pulled back and so on. 560 00:55:37,860 --> 00:55:44,549 Essentially really was the basic simple principle and this was just accepted last week. 561 00:55:44,550 --> 00:55:57,750 I travelled this week. So what you can do is you can go back to all the biology textbook and you can map based on the parameter that they have. 562 00:55:57,750 --> 00:56:01,680 You can predict all the shape and all the features that they observe based on that. 563 00:56:01,920 --> 00:56:09,420 And you really have a tool based on mathematics that tells you can teach a very simple model what you should expect from the patterns. 564 00:56:13,830 --> 00:56:22,800 So I want to conclude with a few thought about what are the new challenges for mathematical modelling, who show you a little bit what they are. 565 00:56:23,160 --> 00:56:30,940 Turns out that a lot of complex biological system, what we have nowadays is the system we look at climate, brain or anything. 566 00:56:31,170 --> 00:56:34,950 Our multi-scale smaller scale as an infant on the larger scale. 567 00:56:35,400 --> 00:56:42,920 They are all multi physics. We have to have chemistry with physics, with ice modelling and all that. 568 00:56:42,930 --> 00:56:49,540 They involve many different process. There is always a component that stochastic and stochastic system have become more important. 569 00:56:49,560 --> 00:56:55,200 We should really look at deterministic problem with things, have natural noise and we should compute directly with that. 570 00:56:55,990 --> 00:57:00,570 Things evolve on Dynamic Network, for instance, the connection in the brain and things like that. 571 00:57:00,580 --> 00:57:06,340 So there is a lot of work now in the modelling for networks that's very important and all these problems, 572 00:57:06,340 --> 00:57:09,820 these complex problem always need integration of many disciplines, 573 00:57:10,090 --> 00:57:16,660 which makes also Oxford particularly good for that because there is a natural connection, a natural way to interact between different disciplines. 574 00:57:18,420 --> 00:57:19,590 One last slide. 575 00:57:21,190 --> 00:57:30,190 I tried to show you that mathematical modelling is a discipline, it's part of a craft, but it's also a tool for invention and creativity. 576 00:57:30,610 --> 00:57:35,320 It's needed in facing all the new challenge, climate and all that. 577 00:57:35,560 --> 00:57:39,400 There is an aspect that I didn't talk to you about is that all this modelling 578 00:57:39,400 --> 00:57:44,110 aspect also naturally creates interesting mathematical problem on their own. 579 00:57:44,110 --> 00:57:49,839 It's a natural source of interesting creative problems, even for pure mathematics. 580 00:57:49,840 --> 00:57:51,400 And I can show you examples of that. 581 00:57:51,940 --> 00:57:59,829 And if you don't believe me that mathematics is everywhere as you go out of this building, you'll see that on the glass panel over there. 582 00:57:59,830 --> 00:58:04,450 There is a a very strange form. It's a gumbo. I'm not going to tell you too much about it, 583 00:58:04,450 --> 00:58:08,500 except that we'll have a lecture later this year of the inventor of the gumbo 584 00:58:08,590 --> 00:58:12,910 and all its beautiful mathematical property and know it's related to turtles. 585 00:58:14,920 --> 00:58:20,470 Then if you go out, you'll see the Penrose stylings, of course, that I'm sure you've seen. 586 00:58:20,830 --> 00:58:24,559 And I encourage you, if you not if you have a chance to go to the lecture tomorrow, 587 00:58:24,560 --> 00:58:31,750 friend of Roger Penrose on architecture and mathematics with this tile as an important piece. 588 00:58:32,470 --> 00:58:41,680 And if you like stamps, there'll be a Christmas lecture by Robyn Wilson on which going to show is collection of mathematical stamps. 589 00:58:43,270 --> 00:58:45,700 And based on that, I thank you very much for your friendship.