I should also introduce Stephen Smale. He received the Fields Medal in, when was that? It was early, it was in 1966, actually when I started my studies and he received the Fields Medal

for his work in differential topology, in particular improving the Poincare conjecture for dimensions bigger or equal to five. So that was a kind of, when I started, one of the biggest results I could imagine. But at that time, he was also a hero to me because of his engagement against the Vietnam War.

That's a long time ago, but he is also known for that engagement. Later, he turned his interest also to computer science and to complexity theory and his work about tuning. And that's what he is going to talk about,

about Turing and his interest in it. Thank you. Well, thanks very much. It's a great pleasure to be back in Heidelberg

for this forum again. So in a way, I'm talking about Turing as an inspiration for me among other scientists. Well, with partly the goal of answering the question,

who am I? Because as a mathematician, I don't think that I fit the label of applied mathematician at all. And I think conversely applied mathematics doesn't see me as an applied mathematician.

On the other hand, I certainly am not a pure mathematician. So what am I? I think I am a mathematician, but I don't fit that dichotomy. And so this talk has a little bit to do with trying to see this kind of mathematics that I do,

which has been inspired especially by Turing, but also I'll talk a little bit about the influence of these other people who've inspired me, Newton and von Neumann

and the Watson-Cret pair. So I'm bringing those in too. And this has to do with this question about my work. So often I'm asked about what some people might call

models or mathmatization of some subjects. And they ask me, is it realistic? And I have a big problem with that question.

For me, it's a question more of being idealistic. So I want to contrast this kind of different philosophy of idealism versus realism. So I'm not trying to make a realistic aspect of a certain subject, but I would like to idealize a lot

when I develop that subject through my whole career essentially. I'm thinking now of mathematics as embedded in the very large body of science and even outside of science. So mathematics for me has got this wide area

of its relationships. Okay, so let's see a little bit how this relates to Turing. And there are two examples I can think of. Turing has been a great help for me in theory of computation,

but also his theory of morphogenesis. But both share the same aspect. Turing's theory of computation I think is great. It's inspiring. But is it realistic? I think most people might agree with me that it is not.

It's idealizations. It's great because it idealized just a few key things about computation. Certainly in real life computation is not unlimited. It's finite.

And everything in real life is so finite, but a Turing machine is not. And it's a little bit more pervasive in applications of mathematics and physics and within mathematics. This notion of taking a limit as time goes to infinity is extremely insightful.

But time doesn't go to infinity, at least in our lifetimes. So this is another idealization, but leads to great insights throughout math. So those are the kind of things that I feel that idealization captures.

Okay, with Turing, this idea of a theory of computation, a theory which is measured with Turing more literally is a number of bits in a computation or in a version of a computation that I've worked out with Lenore Blum, Mike Shube,

and also with Felipe Kucher. We use arithmetic operations. And so some aspects, many, some might argue most, but scientific computation where differential equations are involved, then it's the arithmetic operations that count.

For example, I'm thinking just of the problem of measuring how long it will take for Newton's method to converge to a zero of a polynomial, a single polynomial of a real variable, how to understand Newton's method

as a method of computation. For that, I think Turing does not give insights to that problem. But in other aspects of science and life, the Turing bit model is exactly the right form for that.

Okay, but the point I want to make is that Turing theory of computation idealizes a lot, and it's important for success. The whole theory, for example, of NP completeness is based on infinite input size.

Again, idealization, a big idealization, but still the theory of NP completeness gives us a lot of understanding and ideas about what can't be computable. So these idealizations are great, but I think to say it's Turing realistic, I'd say no.

Okay, but it's not just Turing. I'm thinking also of Newton, Newton's laws of physics, his differential equations of mechanics.

There, is that an idealization? I would say that's a, except for maybe the astronomical aspects, it's a gross idealization. For example, it took a hundred years after Newton to integrate in friction.

And if you're going to apply laws of motion and so on, or Newton's laws to phenomena on earth, you need to really take into account friction. And Newton did not. And that took a hundred years before it was incorporated. So Newton, what Newton did was idealize away friction

when he talked about the apple falling. Now, similar thing is going on with von Neumann. Von Neumann's idealizations and his models of quantum mechanics.

See, I think Newton, the von Neumann did something that's not appreciated so well, but his work on the foundation of quantum mechanics, he introduced the notion of Hilbert space.

The concept of Hilbert space was first really elucidated by von Neumann in his work on quantum mechanics. I think oftentimes these days that's not recognized. They will say von Neumann made the definition of Hilbert space, but that undercuts the importance of the introduction

of the concept of a Hilbert space. But that's certainly was a huge idealization of what was going on in quantum mechanics, but extraordinarily useful and not too many decades later, because then it led to the whole theory of questions of operators, bunches of operators on Hilbert space.

So that idealization again was the key thing. And one shouldn't ask, it's not so useful to ask the question, was von Neumann's foundations, was it realistic?

Okay. And the other main example I have, since I'm working in biology these days, I'm paying a lot of attention to some kind of foundations of modern biology, especially Watson and Crick. So Watson and Crick and their monumental discovery of DNA

that was, you know, people oftentimes don't appreciate what a great idealization that was. See the theory of DNA, according to Watson and Crick does not have anything to do with proteins, but the real DNA is discovered a couple of decades later

is wound around a car formed by proteins called the histone proteins. And those structures are chromatin. So they are absolutely critical to get this deeper picture, more detailed picture of what is DNA.

It's this spiral of the helix around a core of proteins which dictates so much of the physics and chemistry of DNA. That's called chromatin, that core. And again, this was a huge idealization. And, you know, moreover, this is a phenomenon

that one sees, maybe not so much with Newton, but the other examples. Watson and Crick, their work was not addressed to biologists. When I was a student, biology was divided into two parts,

botany and zoology. And they had biologists as a whole, almost all, had no knowledge of chemistry. And Watson and Crick published in, maybe the magazine, Nature, and it was addressed to chemists.

And I think it took a long time before the biologists really appreciated it or could read it. I think even the chemists took a while except for maybe Pauling. So anyway, these idealizations I think are crucial in that my own work goes around these gross idealizations.

So I'm not trying to make realistic models. And even when it comes to predictions, I think predictions are okay, but all of the examples I've mentioned did not make predictions.

And in fact, the models I gave them hardly even had laboratory experiments. These pioneers did not use laboratory experiments, their own at least, but they used data that had been accumulated before them.

And that's, I think is great, but I'm using the data from biology. There's such a vast amount of data in biology. So my perspective is to use that data, but not raw data. That data from biology has been digested

and incorporated into articles by biologists to see a lot of down to earth phenomena in biology where the data is behind it, but it's might say second order because the output of the data is in these papers, biologists, right? And that's what I depend on a lot in biology

is to use that kind of data that's been accumulated. And the same, I think goes for Newton also. He used the data accumulated by Galileo, Copernicus and Kepler beforehand, mostly.

I don't think he really was doing much experiments himself, but he used that data to put together a very beautiful unified picture of physics of mechanics. Anyway, those are my inspirations and I try to follow a little bit in those footsteps.

Okay, so maybe I should give a little bit of illustration of what I'm saying in terms of my current work in biology. So I've written, let's see if I can use these things.

I've written papers with Indica. This isn't too dark, Rajapakse.

Okay, so this joint, three joint papers we have published and we're working on one now. So the one I'm working on with Indica is the following. I want to understand the heart.

I want to understand how the heart beats. And so this is an old problem. Beating in unison is a nickname of the problem. Why does the heart beat in unison? Unison means how do all these myocytes in the heart, all these cells in the heart called myocytes,

part of the populations of cells in the heart, maybe half. They each have their own rhythm, independent. But somehow they get together and cooperate

to obtain a full beating heart. So the heartbeat is associated to the coordinated action of these myocytes, these cells in the heart. So what we want to do is to give some kind

of a deeper understanding of that phenomena through mathematics. And again, a big part of that is due to the work of Turing. Turing has an article written maybe as the last paper before he died on morphogenesis.

And he's got some differential equations in there that he used and maybe people after him to understand stripes on a zebra and so on. And so we're using development of those equations. Actually, it wasn't too many years after that paper

that I wrote a paper using the same system of Turing's equations for some other understanding. Turing's equations were also big idealization. He assumed they're all linear, which I don't think you can get too far in biology by using only linear differential equations.

And what happened, it was okay for what he was doing because he was looking at the limit of an infinite number of cells. And in this infinite limit, he got partial differential equations.

And those partial differential equations, he could see these patterns in animals. And that has become a big subject ever since. But I think most biologists did not appreciate Turing and with good reason,

because morphogenesis is something that's involving what's happening with just a few cells, one, two, three, embryogenesis. That's where the new animals are formed, starting with just a few cells.

And when you take the continuum limit of cells to get the PDE, that's not useful. And that's why I think the biologists have not paid attention to that paper. And I think that's right. So I think morphogenesis involves what happens just on the cell cycle, where one cell is turned into two cells,

one perhaps differentiated, and you begin to see a morphogenesis in those terms. Okay, so for this problem of beating in the heart, this is part of a huge area of investigation

going back 100 years to Huygens, but especially developed by Arthur Winfrey, the biology of time, his book. And it has a lot to do with synchronization. For example, when people clap for very long,

they start clapping in unison, they synchronize. And there are many, many other phenomena where you see this synchronization effect of a lot of people doing the same thing, but where it's not synchronized, then it becomes synchronized. And that's been a beautiful and big subject.

Okay, so for me, there was a background to this. There was some work I did with indica. There are three papers of indica that I can refer you to, but one paper had to do with the tissues,

tissues, tissues are like an organ of the body, could be hard, but we were thinking more like the liver and to understand how the liver functions. So we were working on the problem of the function of a tissue.

Now the tissues are made up of cells, all of the same cell type, but they're a priori, not necessarily coordinated, but what's going on in tissues and the heart too are proteins, proteins play a big role in function, the main role of that.

So cell type, that means cell types are those that are working in a single tissue. One finds the basic proteins for the function of that tissue. So what we did in that paper,

it was published a few months ago, was to see how that would work, how they would synchronize with those proteins to be able to function together in a tissue, could be the liver or anywhere. And that relied, that work relied on earlier work,

and this was the beginning of a big sequence of idealizations. The earlier work was to construct in the genome. So we have the genome in each cell, the set of genes, and those genes are all listed,

they're all known, biologists know them all, but how they work, they work together. So one has to study how these proteins work, how these proteins are working in the cell,

and that's going to be given by a system of differential equations, which goes back even before Watson-Crick, differential equations, which regulate either mRNA or proteins. It's not too, they're not too much different.

Okay, for the equation, those differential equations we constructed in a paper just a year or two ago, all these references you can find out on the internet. So what we did was to construct ordinary differential equations for the system of gene activity in a single cell.

So it's the differential equations on the values of the output of this gene network. So it's based on the notion of transcription factor, which is a central gene, which controls other genes.

So you make the nodes of a network of computer science into the genes themselves, and each node you associate the concentration for say of proteins, so it'll be a positive real number. Okay, then on that, we want to have this differential equation

on the whole set of 20,000 genes. Okay, so this goes back a long way, but what we did was to formalize that, which was very useful to have for us later to have this formal picture of a dynamical system or ordinary differential equation on this 20,000 dimensional space.

And so we started out with building a graph, a network, graph in the sense of computer science so that each node corresponded to a gene. And we would get edges, directed edges, when one transcription factor is controlling a target gene.

There are about 10, transcription factors are few, not much over a thousand transcription factors, but they can target a number of genes. And when they target a gene, they can elevate the activity and initiate that activity of that target gene.

And you can write the differential equations, which are quite old, go back a hundred years, Hill equations, which say how that works. And what we did was to put all that together in the network and from the network, then we're able to get a single system of differential equations, which tell us how the genes regulate each other,

mainly from going from the transcription factors to the target genes. So elements of that go way, way back, but scientists have never done anything systematic or something, I would say maybe bold or wrong

to write down the differential equations for the whole genome in this way. So that's what we did. And we were the first to say that this was a vast idealization. And I think to a certain extent, biologists could understand that most, but not all.

We were idealizing drastically. I mean, there were, we recognized in our model, some inputs from outside of the cell, signals coming from outside of the cell and some other things. But one thing that came out of this,

maybe the same paper or the next paper, was this notion of hard wiring of the genome. And I think this didn't shock biologists, but they did never formalize it. But we're able to formalize the notion of hard wiring. So let me explain a bit about that.

Hard wiring has to do with the original idea that every cell in every human and every organ consists of the same genes. It's universal. So this is well understood by biologists, but what we said more is true.

The system of equations is hard wired. So that essentially in every cell and every human, you have the same differential equations operating on the genes. Okay, one has to do that with a little care

because some tissues, some of these edges will be silenced. So one has to take that into account, they're silenced by chromatin essentially. So one has to be a little careful about just how extensive this goes.

But the idea is in the same tissue, same cell type now, they will all be silent together or active together. That's what makes up the characteristic of a tissue. Okay, so let's see, I guess I don't have too much time.

So what we did is to use this hard wiring hypothesis. And we did it in two ways. In our paper that was published, we made this second very drastic idealization of what those equations look like. We said in each cell,

there's going to be a single basin equilibrium. So all the equations have solutions, almost all, which will go to that equilibria. So it's a global basin. And the basin is going to have then a description in terms of the variables, which would be the proteins.

So that equilibrium describes the composition of the proteins, which fits because in the cell type, all the proteins have the same composition,

the same ratios. They're all the same. The mounts are the same. As you go over cells of the same cell type or in a single organism, like the liver. So we made that hypothesis, again, drastic idealization.

That began to disturb some biologists, not others. But I think it was consistent with known biology very much. And part of what we're doing is trying to give some suggestions which make different parts of biology consistent

and give some kind of insights into the workings of biology in a broader sense. And so this is what we were trying to do, I think, with some success. So that was a hypothesis we made. And then we proved a theorem using these equations

of Turing extended to these non-linear equations, which we needed to describe the equilibria. This guy described what's going on in each cell. So we extended, yeah. So what we did then was to get some kind of theorem with some kind of new hypothesis.

We call it monotonicity. It wasn't an old hypothesis of monotonicity. It was a new one, which we express mathematically. And with that hypothesis on the actions of these cells, then we were able to get by using the Turing equations, which took into account that fusion between the cells,

we were able to get this coordination of function. So we got the function of the tissue described now by all the cells working together. Okay, yeah.

So then more recently, this suggested to me at first that we could do the same thing to get this problem of beating in unison

for describing the heart. And actually somebody came from Hong Kong, a young woman, Lin Liu, to work with me. She was an undergraduate in Hong Kong. So in June, she came to work with me and some problem, and I suggested to her that she work on the genome of the heart,

of the heart muscles. So I learned a lot over one month of talking to her about this problem. She did a lot of research. I did too, but that gave me some picture where there are two proteins that are fundamental for the heartbeat, actin and myosin.

And so then we're able to use some computation to see that this fit together pretty well with Turing's equation, where you had the fusion given by a transport about two membranes. So you get a Laplacian defined by a matrix,

which is a, what's it called? I guess a neighborhood matrix, adjacency matrix, essentially done by transport of the proteins through membranes.

So nearby these myocytes, these cells in the heart, they'll have a membrane type diffusion affecting them. And you can put all that together to get a nice Laplacian defined. And that Laplacian has an averaging effect then

on the proteins. And one has to be careful because there are counter examples, we found counter examples, but with this viral tenacity hypothesis, we were able to get this coordination. And then that was, oh, when you had an equilibrium.

Now for the heartbeat, we have to make a little broader hypothesis on the dynamics. It's going to be now a stable periodic solution to represent the heartbeat. And each myocyte, each of these cells in the heart, they do have such a rhythm that's known.

In fact, a lot of these things were known biologically and so on, but they hadn't been put together very well mathematically, not at all. So there was some appreciation that maybe the fusion between the myocytes could give rise to the heartbeat,

but it was actually using those Turing equations that were able to, and this is our work in progress, see the things I would say, we can see the end of the tunnel and we are beginning to have a theory emerge, which will give some kind of model of the heart being in rhythm.

Thank you very much.