Was. Okay. I think we're going to soon get started so everybody can get a seat. Excellent. I've been told that I should introduce myself. So my name is and I go, Really? I'm the professor of mathematical modelling and I'll be talking about mathematical modelling. Today is a special day. It's a special place. We are in the Andrew Wise building. It was opened a year ago and just became the focal point for mathematics but also for a lot of different activities.
So it's wonderful to welcome you. And as I said, Andrew Wiles is also professor here and gets a lot of mail. Randomly addressed to Andrew Wiles at Mass. 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. Here is I found a stamp from the Czech Republic celebrating Andrew Wyeth. And you see here it's the Fermat's Last Theorem. And it says here Andrew was 1995.
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 made the final discovery that allowed him to finally resolve the problem and prove the theorem. And we know exactly when. Because he remembers. And that day. That day was September 1994. So exactly two days. 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.
So it's all related to chain of events. Okay. But we're not here to talk about interesting mathematics like Andrew is doing in number theory. What I really want to talk today about is how mathematics is related to science, society, engineering, all of the other areas of knowledge. So my talk will be divided into roughly three parts and I want to have time for discussion. First, I will share general thoughts about the act of modelling what we what we actually do.
Very, very, very general thought and remark. I'm not going to go all philosophical on you. I just want to be very pragmatic. Then I will see what his ideas apply to the climate. Take the climate modelling as an example of how to implement some of these ideas, and maybe we learn also something about how people actually do the modelling of climate in the process.
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, something to save mankind, and just something for the beauty of it. And that I reserve, you'll see exactly what I'm talking about. 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.
Hughes A stamp from West Africa. And Lord Kelvin was very interested in applying ideas of mathematics and general physics to the world. 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. And Lord Calvin's talking about science, about modelling, says the following thing. I often say that when you can measure what you are speaking about and express it in numbers, you know something about it.
But when you cannot express it in numbers, your knowledge is of meagre and unsatisfactory kind. 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. So he said, If you really want to make progress at the scientific level, you have to translate the idea into into mathematics, into number, into quantities for which you can make a very specific statement. You also said in science there is only physics.
All the rest is stamp collecting. So not always ideas. You also said that at a time that vectors were completely useless and things like that. But can we take what we can log? Having certainly a great, great mind. So what does it mean by all, all inside that there is only physic? What's what's the idea of physics? It was very, in a broad sense, was really natural philosophy in 19th century.
So the way physics approach a problem is in the physics paradigm is the following where you start with some data, then you identify some patterns, and based on the first principle, you would go about that patterns and write a model for it. And the simplest and classical example for that is the evolution of celestial mechanics, where we start with Tycho Brahe in the 16th century and who catalogue numbers, look at planets and look at the position in the in the sky very extensive table.
So that was really data gathering, what we call big data and ideas. 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, which is called statistics, analyse the data, trying to extract loads directly from the data. And that's how Kepler came with Kepler's law like planets go wrong in an ellipse and so on.
Okay, very deep and profound. But then came Newton and we wrote a mathematical based on physical, simple physical laws. Right. So you had to find the laws himself, of course, because he did both. He says, if I apply simple physical laws, essentially F equals them with the right force, gravitational force. I can explain all of it. I can explain Kepler's law. I cannot print all of it and much more.
So summarising into a few equation mathematical model, all of the knowledge that is required to do celestial mechanics. Here is a stamp that I found from the Benner, which is actually the same country as the down me. But it was the Republic Populaire Jupiter that was during its Marxist-Leninist time. So they also evolved from Kepler naturally in the 20 years to, uh, to, to Newton in this time. So, but when you actually do modelling, you do it as a profession.
Turns out that the first, the physics paradigm, that's why we put in textbook. But the reality is quite different. What do we do when we do start with this data? Some observation. We have a friends who's a biologist or geophysicist or material scientist. Look at this nice experiment. Say, Oh, that's wonderful. And so you go back in with your student, you say, okay, well, let's write a model.
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? And it shows you more data. And invariably, the model is wrong. The simple model never happens. It's right. 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. But the real problem is I show you wrong. Model is fine. You are you really understand something.
Often it may seem strange, but often the model wrong model tell you more about the nature of the problem than a correct one. But the real question is why is the model wrong? What did we miss? What are the important physics or important chemistry? All part an aspect that is not in the model. And so you create a new model and the model invariably also has more variables, more parameter.
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. So you go there and then you go back and discuss and try to validate with the model with new experiment. Then you see, is the model self-consistent? Does it provide me any insight or does it correspond to other data or the system model experiment in the literature?
Do I understand? Does does that make sense essentially? And if it does, you have you can do two things. You can do a simpler model, throwing away the stuff that may not be important. Or you can say, no, I really want to predict things and you would make a much more complex model. 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. Let's put that on a big computer and see what happens.
And then you go back, you try to validate, make more experiments, and then you refine the models based on that. And then maybe eventually you may have some prediction and predictive value for the model. And then when you do that, you go back here and then you start this loop again and again and again. 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.
But the question is, we have a motherlode. Do we know or do we know how far we should go? What's what's what is a mother quality? And there are two important aspects of quality and complexity. So quality can be points, instance, a predictive value. 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. The number of parameter, the number of variable are because your system.
It turns out that. If the brother becomes very complex, usually you losing in quality is not because you take into account all possible effects that you model is actually better. There is a point where you add things and it gets worse. And why is that? There are many different reason. One of them is computational time. Of course we know that computational time the model becomes complex. Then the computation time required to get an output out of it becomes increasingly large.
Somewhere. Maybe we'd been here in the sixties. This line is always moving up. We know that computers are getting faster and all that. 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. Okay, you know something? You get a number, you get the temperature in year two 2100. But have you understood something about how the system or the climate works?
It's usually which is a little system that you really gain understanding about the system, the system. It was the larger system that you can do prediction. So what you really want is to have model and study them in that whole range from different sites. Small, medium, large, extra large. But you want to try to avoid the American way, the extra, extra large right here by because the quality of the model decreases.
You take an infinite long time to compute anything and you don't understand what you're doing. So, so you really want to, to work in that range. So let's see how it works on, on more practical example. So now I want to apply this basic concept on the, on climate modelling. So let's start with very the simplest possible model. And that's what in physics we call a back of the envelope equations models. So that's for the younger audience.
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. So the envelope and the stamps, as you saw, are very important for us because they contain information when you receive it. You can do computation on the back of the envelope. And so the basic computation when it comes to the climate is the following. Let's just look at how much energy is coming from the sun.
There is no really very few other source of energy maybe from inside us, but it's a fraction of a percent. Doesn't matter. So let's look at the energy coming in. The flux of energy, 342 watt per square metre. And we know that the albedo, the reflection of the planet is about 0.3. So that's because we see it from outside. That means photons are coming out. So you lose energy, right? And that's about a third is reflected solar radiation.
There is nothing you can do. It's gone. And so the back of the envelope computation say, well, obviously the difference because energy's concern is that of the US reflector 235 what two square metres is there is there is no discussion about this. This is basic, basic physics. Okay. So that that's very good. Now you can go one step more, he said.
We know from physics that a black body like a planet is well approximated by what we call a black body, whose flux of temperature is proportional to the temperature, to the powerful. And so if you look at that, if you if you play with these numbers, you'll get to the following prediction is that the temperature of the Earth should be -18 degrees on average, where it turns out the temperature of the earth is about 15 degrees. So the error is only 33 degrees. Okay, that's great for us.
Wrong models. Whatever we learn is that there is some energy that is trapped inside and that as a whole it doesn't work. As a black body. There is something and that something is the atmosphere and it's the greenhouse effect. So some way in the modelling scheme, in my mother diagrams, we really don't hear, 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.
So we understand a lot and the prediction is obviously completely wrong. So so we can start a little bit more complicated. We can say a one dimensional model. So what happens is the energy comes here, bounce back and forth between different layers of the atmosphere that have different chemical composition. And so you can do that balance, which is called radiative forcing.
How much CO2, for instance, the most important one with refracts energy coming at certain wavelengths to other wavelengths and so on. And you can do that computation. And for instance, you would get that from that simple computation. 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. There is no discussion. Nobody discusses that. It's very clear that this is not controversial at all.
They said, but if you do the computation based on that, you'll get the wrong result as temperature in different zone. And even the average temperature is because there is another effect that we have not taken into account. 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. So the radiative forcing taking place there is quite different.
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. So you would go about. So we are here. We are about here. We have a prediction of very poor. But now we know that greenhouse gas is very important for anything related to the atmosphere. And you can make very good prediction based on that. Now you say, okay, now I'm going to consider a different zone. And by convection, warm air goes up and colder air goes down.
I can start computing the motion of the mass of air with different temperature in different zone.
And what I would use in this case is very simple equation, actually, the set of one, two, three, four, five, five equation that comes from 19th century basic fluid mechanics, and this one is from thermodynamics that just essentially it's a reformulation that's f equals me telling you the velocity of air in different zone based on the forcing F Landsat F data and the rotation of the things are really the core primitive equation when it comes to atmospheric science that the basic equation,
nothing you can't you cannot escape physics at that level. Just say things are moving from one side to the other one. The global masses conserve and the thermodynamics. It's a statement about conservation of energy. So these are five equation that describe that and you can run them on different zone. We say, okay, now if I can run them on different zone, I can probably run them on a fine of grid. Now, instead of looking at three zone, I can say now I want to go a little bit further.
I'm going to run the same equation of transfer of linear momentum, transfer of mass, going from one to the other one, 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. So I make a little box and I say, what is the what are much masses going from one to the other one? Much energy is going to the other one and so on. 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?
These are called general circulation modalities are the typical models use. 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. So I make box and I just say each box exchange energy and mass with the other one and so on. And that's the that's essentially all there is to this climate models, but it's how we do it that is important.
So an interesting computations is to see how much how many computation do we have to do to get any answers. So let's compute together. So I tell you, I told you, we have 2.5 to 2.5 degrees. That's about 10,000 cells, 10,000 bucks in each box. I define the velocity going north, the velocity going as the west. Right. And the temperature there and the humidity, typically, you can reduce that for four, five, six, seven variables.
Seven variable is kind of standard to what you need to get all the information about how temperature move from one to the other, what are the mass that moves and so on. So 10,000 says that just for the mapping, mapping the earth. 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. So now I have 300,000. That's good. But I have at least seven unknown velocity of the night.
My mass in south, north, east, west, I have the temperature, I have humidity and a couple more. Okay. So that gives me 2.1 million variables. Okay. Now that's starting to add up and multiply up really. In case now if I assume 20 calculation for each variable, that means why not? If I want to know the velocity at one point I have to do two multiplication, then one addition and blah blah blah. Do a certain number of operation 20 is actually very low.
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? That's about 20 computation. That means 42 million calculation per time step. So if I just want to know how things evolve, the temperature supposed to take, let's think of the temperature evolve from one time step to the next time steps. I would have to do 40 million computation. It's very easy now do 42 million, right?
We do, you know, giga flux, mega flops and all that mega flops, million of operation per second. 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. 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.
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. So you multiply that by blah blah and you get about 73 trillion computation that you CPU's have to do. That's quite a lot. It's quite doable, but it's, it's getting, it's getting very expensive.
And of course as long as you there, you want to add other effects, ocean and ice or the interacts or the effect of aerosol, the chemistry, the biomass, the hydrology. So by the end, what you want is to call our earth simulator that take all these aspect, all these books, all the chemistry, the geochemistry of the ocean, the interaction with the ice and so on, and I putting all together. And that's why you need a supercomputer.
All the people who do climate science tell us they really need very big computers to do that and contributing to global warming, of course, because this computers, you know, it takes a little a little town just to power them up. That's why they are in remote places and all that. I don't know. Nobody has done the computational much. The computation of climate change is actually contributing to the effect, and I'm sure it's negligible, but it's pretty big already.
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. Tens of thousands of people have been doing that for years and years and trying to refine these models. But the real question now is where are we?
Are we really here the best possible outcome in terms of model quality, or are we going down here where we've added so many effect that we've added so many error in the numerical computation, that we not really know what's really going on. 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? So do we know that the solution that we get is actually a good solution, a valid solution?
So what you can do. So you want to do model validation and it's always easier to predict the past in the future. 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, 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, particularly volcanoes, turns out to be quite important in putting energy in in the atmosphere.
So here you have different volcano, eruption, Pinatubo, etc. Santamaria from 1900 up to 2000. The black line is the recorded temperature evolution around that time. And the red. The red line is an average of a lot of different simulation. And you see it is weekly and we're going to talk about the wiggle. But what you see and it's also take into account what we know about the increase of greenhouse gas over this time, the measure of all the greenhouse gas. So you see that you get a very good job.
At measuring and getting the the general trend. Of course, there are some years where the red line is not at all like the black line. That's why people say, you see, there is no climate change. Your prediction is wrong. But if you see the trend, there is no doubt it's it's very good. It's very, very good on many different measures. You can do the same computation now. You can do a thought experiment and say our greenhouse gas really that important?
What's what if we keep them at the level of 1900? Maybe it's really the volcano that creates all that mess. Okay. So you start again to the same computation with the volcanoes, but with no greenhouse gas increase. And this is what you get. You get no increase of temperature at all from the 1900s. So, you know. You know, with very, very likely almost certainty that the greenhouse gases have been responsible for the evolution of these.
Okay. So when you look at a model, it's important notion of either predictability or uncertainty. So it's important to find a way. What do we really know? What comes into the model that messes up our computation if we want to do a projection? So therefore, typical source of uncertainty in climate projection, there is what's called scenario.
Uncertainty is said. Well, we really don't know if the government are finally going to come to their senses and do something about the problem so we can have we don't know what's going to happen. We can run to scenario. 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. We can do is to and anything in between we can run. Okay. There is modern uncertainty.
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. There's a lot of physics we don't quite know yet, and there's a lot of work to be done. So. And which are the effects important? So what? What is the best model? Then there is a lot of parameter uncertainty. You don't know the data perfectly.
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. So you have a lot of things. And finally, more important and more interesting for us, you have internal variability, and that's intrinsic uncertainty in the climate system. 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.
And why is that? And that's really a mathematical question. And why is that? It really goes back to 1963 with the notion of chaos. And here is a stamp from. I think he's getting there. They seem to very much like physics and mathematics and things like that that celebrated the death of at Lawrence in 2008. At Lawrence, a very strong influence. And in 1963 wrote a paper, a very influential paper.
He says it started actually with a convection problem in simplified, simplified, simplified all the way down to three equation. And that became a problem that had no relevance for the problem he started with. But he noticed there was something very strange in the system when he studied in more detail, more mathematically transforming to a purely mathematical problem. The equation a very simple three equation. The Z TD is a d t is what we call ordinary differential equation.
Just about the simplest, the simplest type, except they have nonlinearity instead of X and Y and Z being by themselves. 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. And when you run that on this own computer at the time, you can do it on your smart watch and things like that.
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. 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, 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. There's something that we know for sure. There is one equation that we call deterministic.
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. At six is three equation. I can represent that in three dimension X, Y and Z. So I can run the system. I can just let it go and see where it goes. And this is where it goes. When you let it run. The access atoning for the added visual appeal. And now what? Let it go. And the system, it doesn't go away and it doesn't go to zero itself.
It does something different keeps turning around. 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. So it's like if you want to run a weather projection, I would take the temperature today at 22 degrees and I would run another scenario with 22.0001 and let them evolve, right? So started with two actually 2.111 green and one blue and one yellow. And this is what happens if you let it run. Very soon.
The point I'm going to separate. And the separate. I can start as close as I want. Eventually they will end up a completely different position. And that was truly remarkable discovery for the for the weather system at Lawrence was a meteorologist who was interested in the weather,
not so much in the climate, but in the weather for sure. So. If I just look at one of the variable X as a function of time, for instance, and I start with the two points exactly the same one, I wouldn't see any difference, 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.
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? Well, what if I choose? Even if I had perfect knowledge, a little bit of difference would tell me that. Why two weeks later it would either rain or we sunny. I have no knowledge. It's impossible to know. It's mathematically impossible. Okay. I would need perfect knowledge and perfect ability to do computation to do that.
But every single measurement is an initial data which come with a little bit of difference and error. 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. Maybe we'll be able to do a few, maybe have another week. But we know for a fact mathematically we'll never be able to predict the weather for years time.
We know that numbers the numbers are very clear. And in his paper, Lawrence says to stay differing by impossible imperceptible amounts may eventually evolve into two into two considerably different state. In view of the inevitable inaccuracy and incompleteness of weather observation, precise, very long range forecasting would seem to be non-existent. It was very clear it completely understood the problem.
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. The difference and that was the amplification of this effect. And, you know, it's important when when they start making bad Hollywood movies with that with puppy face actor. So it really came in to that knowledge not only as a culture, but really completely changed the way of thinking of everybody doing climate and whether that was in 63.
And people understood at that time that that it was important, but they still used the same model to try to say, okay, now we know we can of precisely can we can we use that? Can we use that knowledge? So the idea is to do what's called and somebody averaging. 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. 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.
And if I do that under Laurence Attractor, it's called chaotic attractor. If I do that, I can have different scenario depending on where I am. I can start with a bunch of data here and I look at the evolution and two weeks later I'm here. And basically if I look at the centre of here, it's still the centre of that blob. So I know that I will have a good knowledge. It's very high predictability. I know that the trajectories have not gone too far away.
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. So this is a way to quantify uncertainty. So you don't run a single model. You would run a model with very different initial, very close, but different initial data and look of farther diverge. I would say, well, I since I don't know my initial one, I cannot my prediction stop at a certain time.
So you can quantify how much you know about the model. It's this idea of ensemble average that's so important. 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, trying to find ways to use this idea to a benefit rather than being an inference. Okay. So what's the final picture at times of climate? This is the result of the Intergovernmental Panel on Climate Change.
John It was July 2014 is the fifth assessment report of that panel, the IPCC. And they do exactly what I told you the start here, 2010 or 2014. And they run the model with different scenario, which is the model where people are responsible. And this is the model where people as they are. 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. Whatever we do, even if we start behaving, there'll still be some increase.
But otherwise, inevitably we do that with very lightly. 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. That's okay. I mean, I'll water my loan a few more times. Right. But if you actually you can since you've done all that computation on that fine grid, you have information about the climate change on average in all these different positions, these different place.
So you can go back to your data and look at the evolution of the temperature. This the two scenario and you see that this is where we'd be if we still have increase everywhere. 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. So it is true. It doesn't increase that much. But that's because the pole don't change that much.
Oh, the the ocean the oceans don't change that much. But the rest of the land is dramatic increase. And the theory has become quite reliable. I mean, there's been a lot of evolution, very quick evolution over the last five, ten years or so in terms of understanding what are we actually saying? 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?
Okay. So I want to to stop about the climate here and move to other problems of application of modelling. 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. And I will tell you a little bit about modelling that I've done over the last few years on photovoltaics. So whatever we we do, we know that photovoltaics using the sun is going to be part of our energy portfolio.
And today, today, the silicon technology, the silicon panels that we see everywhere, it's really the dominating technology, but it's a very restricted in many ways, not as flexible as we'd like. And so people have been looking forward to a different type of photovoltaic, and that search has not been doing too well. I mean, a lot of people in a lot of different departments find alternative, but nothing really came to to challenge the silicon technology.
But that was until two years ago when the group of and we Snaith here in physics discovered that certain type of material that use in some of these photovoltaic cells they use just by themselves has remarkable properties. 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, to look at the mathematics, the modelling of this problem with Henry Snow trying to get some insight into this process.
So let me give you the general picture. We measured a quantity of photovoltaic cells in terms of efficiency. It says, how much energy can I get back from from the sun? Okay. And so it's a percentage of energy I can extract out of of out of light. And here we just going to be interested in the silicon here. Gallium arsenic is is very good, but it's quite expensive. And here is the evolution of the silicon technology from the seventies, about 12, 15% all the way to 25%.
And you can buy now the silicon reliable silicon 20% from Chinese manufacturer. And they've done a great deal to actually reduce dramatically the price of silicon. So silicon will be with us and will be used. But as I show you, we want to have other technology for the purpose. 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.
And now actually last months, people have shown that they can have 20% in two years efficiency. 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, but it's all the made to silicon. So you have to grow a crystal of that. It's very rigid. But the perovskite, you can actually just lay them down by different technique of vapour, deposition, spin coating, which are very cheap way of doing it.
Okay. So this is one of the modelling problem that, that we worked on with, with, with the Henry. It says, okay, I know that I can lead on my, my perovskite miracle compound that works so well, 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. 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.
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, the little bubble start making bigger bubbles and things like that. So we look at that problem from that perspective. 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. Right. Because the light would still go through and I would still get some energy out of it.
That's a good idea. But it looks like that. Really. That's okay. But really, who wants to live in a brand building? Right. So say where. This is probably not going to be very, very popular on the market. So we went back and look at the probably in more details and we created a little model. I just give you the basic idea of the model. What we want is when we lay down things, it's essentially it's called annealing, but it's really called cooking.
If you want to make brownies and things like that, to put it in the oven and you heat it up, you put a layer and then you heat it up in another 100 degrees for 60 minutes, just like a cookie recipe. And what happens? The solvent evaporates and forms the structure that you want at the end. And the only thing that you can really control is the the temperature and the thickness of the film that you start.
And, you know, if you make brownies or things like that, 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. So what we did is say, okay, or layer, it's really a full layer. It has holes and each of these holes, the total, a certain diameter, a certain diameter and certain lengths. And we can write a full energy fatty system for depending on the different radius.
And each radius we know from fundamental physical principle can change a size. The whole can change the size and interact with each other. And if you do that, I spare you the detail. But the problem is not too hard. I mean, it does require putting things together, but it's not too hard. And when you do that. Sorry, let me go back here. When you do that, you get exactly what you want. You get the coverage as a function of the temperature. And as a function of the film thickness.
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. And he had the here the the boxes, actually, the experimental data and the curves are what we predict. I said that's very interesting because you can really optimise your coverage. But then we talk with with with Henry in Henry's group and he says, but look, we can do much better than that.
Sure, we can optimise the coverage, but no, 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. The distribution is exactly what we want. So that as it goes, as the light goes through it as a given column, because the light is going to refract in different ways for different wavelength and so on. And so this is what he actually made with this group.
Okay. So now you have a technology that allows you to have a piece of photovoltaic cells are given colour. And that you can put on windows and things like that and get maybe not 25% because there is still a lot of light. But you get 5%, maybe 10%. So. Okay. So you can have buildings like that where you can have beautiful blue buildings and you still collect the light as you go along. So it creates a new market, a new way of using solar energy.
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. And we applied the same type of idea to different context. I think it's a good example because not only to some predictive values, but really the mathematical models was part of the creative process in understanding the material and coming up with alternative of what we can do with it. Okay. So part three B, the second example, the first one was aiming at saving the planet.
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. So there's a lot of wonderful mathematical problem with the brain. I'm going to concentrate on one that's not the kind that you hear with this huge, big initiative, the Human Brain Project and things like that, which is try to understand the connection. I'm much more interested to see the brain as a real physical entity.
And so my motivation and initial motivation was trying to understand trauma and brain injury or traumatic brain injury. It can result of, uh, unhealthy sports events or car accidents or stroke or things like that. What happens in this case. Invariably what happens is that part of the brain becomes depleted of oxygen and then you have an imbalance of ions and you see that the tissue starts swelling. It is swelling can be very dangerous, as I'll show you in a second.
And sometimes you do have to open the skull and do a craniotomy to relieve that part of the swelling. 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, then you can try to understand what happened all the way to the axon as the axon a stretch. Okay. What is the damage that's done to the axon and so on?
All of these are very important, not only for for accident, but also in a lot of different diseases, epilepsy and things like that. 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, we decided to create an informal group we called the International Brain Mechanics and Trauma Lab. We started that in 2013. Try to piggyback different speciality, different expertise from different field to address this problem,
because we quickly realised that you need to go. You need to have the, the medical doctors, the biologists, the physiologists, the engineers, the mathematician and all that, all together, because these are very, very complex problems. 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. Try. Trying to answer some of these problems.
I just want to look at the simple problem. There's no simple problem when it comes to the brain. 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. And we wanted to understand what's what actually happened. It's very one describing the physiology textbook and so on. But again, just like Kelvin says, there is no numbers attached to it.
There may be measurement, but you you there was no way really to try to understand what happens, what is the sequence that happens? So what we did is we combine both with mechanics, solid mechanics, because you have a soft tissue and the electrochemistry, which is the balance of ions, interaction of ions and electrolytes. To try to understand that and we created a model with a quadriplegic theory that means a four different phase interacting with each other to try to understand that.
And it's really the Ph.D. thesis of Georgina Lange that she she she finished yesterday. I think she signed the thesis, and that's also with Dominic Vella and Sarah Waters. So the first part was to understand the swelling. And so we built this model and fitted a number of parameters, an independent experiment. And then we look we went back to some data that people do and very, very controlled system. And we saw that we could we could we could have a very good match of them.
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. That was we the first step was to create a model that would just explain the swelling. And when you do that, you can go to the next step, says, okay, now I can induce a damage in my mother. 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.
Okay. That's that was the second step. Try to do the damage. And the third step was to say, okay, now all this the damage propagates because that's what we really started. And the damage propagate the following way. 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.
What you'll have is that the tissue is going to swell and where it swell is going to compress its neighbour and it compress enough its neighbour that does not produce the don't give you oxygen anymore in the in the tissue. So they're going to die again and they're going to start swelling again. Okay. So to swell again and compute and so the damage propagate like that.
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. They die or they're very unhappy. And so they start sweating again and so on and so on. So we wanted to know what is damage propagate? And we built a model based on that.
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, 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. And we see that in this case, if the damage is if damage is large enough, it would always propagate. If the critical stress is large enough. However, if you do open it as a simple geometry, I show you that for picture.
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. So this is a conceptual model. 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. We took away some that were not important, that were thought to be important. Important. So now we can go back and implement that in the correct geometry.
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. That's another story I want to go on to to my last application, trying to keep on time, which is seizures. 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. So the motivation is really for the beauty of it.
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, 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. It's really curiosity driven and it's often teased out. We can have we can talk about that. But it's often my problem that I curiosity driven that the bigger sleep comes. So I have no problem doing that. And what is the problem.
So it is his work with Derek Moulton, who is in mathematics and the biologists from in France, which is Serra. So what is the problem? Well. Here is a seashell I wanted to show you. Is the camera to come is not working. Here is a seashell and here is another one. And they're very beautiful shape and all that. But some of them have what's called morphology. You know, they have this pine and on top of them. And we wanted to know, can we explain?
These people have been describing that for hundreds of years. Spine ts, rip that and all that. Plenty of people very important for evolutionary process because it helps you to classify, see the trees and structure and all that. We wanted to see what are the minimal, minimal processes that can explain the creation of this shade, that can create teeth patterns? What is the morphogenesis process?
So the first part was to really try to understand the main geometric shape, the overall shape of the of the seashell. And we built a model based on that that tells you if, if I know the growth process right at the edge of the seashell here at the opening of the seashell, then naturally it would create one of these basic shape. Okay. So I'm not interested that the pattern they're just interested in the global shape.
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. 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. And the way it creates its shell is by coming out, going on the edge, depositing a new material, liquid accreting new material that get calcified and then retreating.
So what you have is a soft tissue that goes on top of a heart tissue and then it goes back, it changes boundary and then the next day it comes out and does it again. Next day comes off. So what you have is you have something like that. You have the you have the mantle that comes out, goes on the aperture, and then lay new material. Now the animal itself grows as a certain piece and the accretion process is a different a different mass increase.
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. Now, if you do that, you start with one and then you go out and put a little one. Now, if the soft mantle grow faster than the aperture, it doesn't fit nicely on top of it. So it can make a nice shape. 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.
So what's going to happen is the following. 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. But it's too long for the shell edge. So it's going to do this little whoop here, it's going to buckle. 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. So if you are a little problem here, it's going to get amplified this time.
So the equation are quite simple. Again, for this type of process. It's an evolutionary process where the boundary evolves with the system as time goes on. So it's a it's a non-standard problem when it comes to mathematical modelling, 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. Depending on the parameter that you have, you either get is very sharp spine.
You get very worried. One. And so depending on these two parameters, the growth rate and the stiffness or stiffness you mental, then you can you can easily map all the shape that are around that iPhone. 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. So this is this is one way to reduce a lot of knowledge.
And we say, okay, now we have a model, but if we want to push it, can we apply the same idea to other patterns that are fun without without fiddling, anything? We go with the basic idea. So we went back and say, Well, there is another pattern that's found in seashells in Ammonites. No, Ammonites have been extant for about 100 million years. So obviously.
We don't have that much data on the living animal, 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. Look at evolutionary trend and all that. A lot of very nice work under very sad. If the only thing I know is how fast these teeth. These aperture opens can actually predict the form and the shape of the pattern.
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. If I know the red one, I can compute the because again, the, the mental part, the soft tissue is going to grow faster and then it's going to be pulled back and so on. Essentially really was the basic simple principle and this was just accepted last week.
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. You can predict all the shape and all the features that they observe based on that. 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.
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. Turns out that a lot of complex biological system, what we have nowadays is the system we look at climate, brain or anything. Our multi-scale smaller scale as an infant on the larger scale. They are all multi physics. We have to have chemistry with physics, with ice modelling and all that.
They involve many different process. There is always a component that stochastic and stochastic system have become more important. We should really look at deterministic problem with things, have natural noise and we should compute directly with that. Things evolve on Dynamic Network, for instance, the connection in the brain and things like that.
So there is a lot of work now in the modelling for networks that's very important and all these problems, these complex problem always need integration of many disciplines, which makes also Oxford particularly good for that because there is a natural connection, a natural way to interact between different disciplines. One last slide. 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.
It's needed in facing all the new challenge, climate and all that. There is an aspect that I didn't talk to you about is that all this modelling aspect also naturally creates interesting mathematical problem on their own. It's a natural source of interesting creative problems, even for pure mathematics. And I can show you examples of that. 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.
There is a a very strange form. It's a gumbo. I'm not going to tell you too much about it, except that we'll have a lecture later this year of the inventor of the gumbo and all its beautiful mathematical property and know it's related to turtles. Then if you go out, you'll see the Penrose stylings, of course, that I'm sure you've seen.
And I encourage you, if you not if you have a chance to go to the lecture tomorrow, friend of Roger Penrose on architecture and mathematics with this tile as an important piece. And if you like stamps, there'll be a Christmas lecture by Robyn Wilson on which going to show is collection of mathematical stamps. And based on that, I thank you very much for your friendship.