George Zimmerman's trial.
Thank you very much for for the introduction, in particular for the invitation both to Alan and to the Mathematical Department.
It's a great honour and a privilege to be in this room and to speak in this building.
If you look at this slide. It's a familiar site.
It's Pebble Beach, and many of the shapes on the beach look familiar.
They are just Pebble. So it is a pebble. It would recognise Pebbles.
You would recognise this beach. However, if I would ask you to name some of these pebbles, that would be a challenge.
Uh, we don't have names for shapes.
We have a name, so very few shapes. And for the majority, you can see quite a number here.
A couple of turn the tables on that medium. Uh, Pebble Beach.
We don't have names. And this might be related to the fact that, um.
Linguistic skills and shape recognition skills are in different cerebellar hemispheres.
Uh. There were comments being made on this subject.
One is due to allow Polyclinic Guy who was a famous mathematician and he said that God created the integers all as is the work of man.
Meaning that? Well, in today's language I would say you would like to your bank account.
You would like to be identified by an integer number and maybe not by a drawing.
It's a safe thing. This is the thing.
The integers are something where we feel home, where we feel comfortable with, and the shapes are maybe a little bit more elusive.
There is another famous citation from Galileo, which is well-known that the Book of Nature is written in the language of mathematics.
However, the second part of that citation is probably less known.
It says that the symbols in the book are triangles, circles and other geometrical figures.
So Galileo says that we need to understand shapes in order to talk about natural phenomena.
However, you need to understand that we need to talk about them and the first.
If you want to give names to the shapes, the first thing is to assign integer numbers to this shape somehow.
So a goal is in the both sense, to understand that you are shapes by using natural numbers.
To what extent can we explain shapes by the means of natural numbers?
There has been a lot of thinking about Pablo's, so this is not the first attempt to describe Pablo's.
Aristotle heard at length about Pablo's and he actually constructed a model.
The evolution of Pablo shapes in, if I would say that he predicted that pebbles are spherical.
Well, as you saw in the picture, not every Pablo spherical. The Pablo in my hand is not spherical, certainly.
But there were other speculations, David Hilbert, about his very famous book,
Geometrical Imagination, where he made a comment that coastal paths tend to be ellipsoid though.
And in fact, if you go to the mathematical collection of göttingen, which is a very famous mathematical collection,
started in the 17th century, the first piece in that collection is a coastal pebble.
So people were wondering about pebble shapes and not much later.
Lord Reilly, who is the son of the Nobel Prize winning Lord, really pointed out that tables are not Aleph sides.
So pebbles don't seem to be spherical. Don't seem to be ellipsoid though.
What are pebbles like? Do we have a name for Pablo shapes?
What are the numbers which we can use to characterise them?
Well, the kind of numbers I'm interested in are related to mechanical properties of a shape.
So here is a disk and well, I will.
Now, this is a question. How shall I put it down?
If I put it down, it is rolling along its perimeter and there are positions where it would stand still.
And there are other positions where I can balance it like a pencil on each step.
These are the stable points and these are the unstable points in the mathematical language.
We can say that we describe this shape as a distance from its centre of mass and
the maxima corresponds to unstable points and the minimum to stable points.
So in this way we assign some numbers to this shape and in three dimensions we can do a similar thing here is,
well, I take this, Pablo, as an illustration. This is a kind of an ellipse side like Pablo.
It's not an ellipsoid. It this can be also described as a distance function in two variables, because now we have two spatial coordinates, two angles.
The minima of this function are stable points. I put it down and it will stay like that.
There are several points. And there are the maxima.
The tip of the ellipsoid. So there are three types.
And the numbers of these are related by a famous theorem by Poincaré and hope.
So in two dimensions, the number of stable and unstable points is always equal.
And in three dimensions we have this relationship. Add these numbers.
So to each pebble or to each shape, we can assign these numbers.
Are they meaningful? Do they tell us something about nature?
Well, to tell. I have to go back a little bit.
Okay. To find the numbers.
Next time you go onto a beach, you just. You will be tempted to do this.
So you take a pebble in your hand there, you find the table points relatively easily and also the unstable points and
the subtle points out there rather easily identified by a hand experiment.
Uh, to motivate. Why? Why these things might be relevant.
Let me briefly tell you the, the story of the gun book, which I learned over the announced and this goes back 25 years.
I was visiting Cornell University and my friend Andy, a winner suggested a very interesting problem, which is related to what I just told you.
So here is I think this is called a verbal, uh, it's not in a good chair because my son, who is two and a half, is using it rather frequently.
But I hope it will work. So. So you know how these things work?
There is a little music or so, uh, no matter how you put it down, it will come back to the same position and the.
Have you ever taken the verbal apart? Um, maybe earlier on.
And you don't remember because my son is doing it all the time.
So if you do it, you realise there is a weight at the bottom of the Bible.
That is why it works. Now the question was if we take a plane, a disc like this,
and we are not admitted to add any weight to the disc, it is just a disc and we roll it along the perimeter.
So it is a two dimensional plane that problem.
Is it possible to cut out the disc so it behaves like a vehicle so it has just one stable point and one unstable point.
Now, this question is easily answered, relatively easily answered.
And let me briefly tell you how. Uh, so if you measure the distance, as I told before, from the centre of gravity,
having this view like this would imply that this distance function has just one maximum and one minimum,
because then it would be behaving just a verbal coming back always to the same position.
Now the idea is to disprove this by contradiction.
So if we slice this function. The length of the slice is zero here.
When it is tangent and it is going at some point, the length of the slice will be exactly 180 degrees.
So if we translate it back to the original body, we see that a straight cut.
So the body.
And we can see here that any radius inside the slice is bigger than any radius outside this slice in plain English, not in a vigorous way.
I could say that the body has obviously a fat side with all the big red and it has a slim side with all the small red.
Well, that seems to be a contradiction because the body, if I support it on that, the centre of gravity, it's supposed to be in equilibrium.
It's so it's supposed to be balanced. However, this body is not balanced along that blue line.
It will taste all that to the right hand side. So we've got a contradiction.
There is no such body. We call it a serum because we were proud of it.
We published it and probably nobody ever read it.
So it disappeared a couple of years later.
I went to a large conference in Hamburg, which is probably the largest ever mathematical conference with two 3000 people in attendance.
And I went there primarily to see Vladimir Putin not give the plenary talk.
He gave a very interesting talk. He talked about various disciplines optics, mechanics, geometry.
And in each case, he showed that some number is equal or bigger than four.
Obviously. I saw that.
Our little theorem, which was completely gone, proved actually that we had two stable points at least, and too unstable points at least.
So all together we had at least four points of equilibrium.
So is this the same number four as Arnold was talking about?
A different number four? Uh, I wanted to ask him after the talk, but that didn't happen.
There were just too many people. So the organisers offered.
To the audience. If you pay a fee of 30 German marks, we can be seated with a celebrity of our choice, which was half of my weekly budget then.
But I. I volunteered to pay it. I was seated with Arnold together with 15 other people, and everybody had to offer something.
So the lunch was not very successful from my point of view.
At the end, he asked whether I had anything to offer. I said no, because it was a painful scene and.
He left. Uh, many of you may know the world.
For those of you who don't. At that point, he was a little bit older than I am now, but he was already a small planet.
Has been named after him. Uh. Um.
Sadly, he's not alive anymore. But we could list him as one of the significant, influential mathematical thinkers of the last 50 years.
And, um. So I got a little bit disappointed.
I gave my own talk, which was in the wrong session.
And just the chairman came and myself and I left the conference, and that would have been the end of the story.
But I noticed that Arnold was talking to someone and he wanted to get rid of the person.
And he told him that. It's obvious what you are telling me.
And by the way, I have an appointment with a gentleman over there.
That was me. So he told me that I saw your name tag.
You are from Budapest. You paid a 30 mark, so you must have a very good reason to come to that lunch.
What is your reason? I got 20 minutes. You just tell me briefly.
So I told him briefly. He was thinking about it and I offered him to explain the proof of our little theorem.
He said he knew the proof. He doesn't need that. And he.
How did you think about the three dimensional case? And I said, you are.
Of course, we already solved the problem in three dimensions.
So what is the solution? Well, you take this kind of body.
This is a cylinder which you cut of, like diagonally.
How many stable points does this have? What is your guess?
So how many positions would it be?
Still, if I put down on this table, we will try it afterwards, where it would stand still even after I rocket a little bit.
So it is really a stable equilibrium point. Is that infinite amount?
You said infinity, which is a good guess if it would be a cylinder which is not orthogonal.
Yes, it would have infinitely many, but it is not got orthogonal.
It's got a diagonal angle. I do.
We are going down now. So any other any other guesses?
Yes, sir. Well, for now, we are going up again.
I am asking about stable, stable, stable.
We will come back to that. To stable anything.
Anybody else? Just one. So let me try this.
I just put it down. And you can see it just comes back to the same position, no matter how I put it down.
So it always comes back to the same position. It's it has one stable point of equilibrium.
So I know what's staring at me.
And I've always, for a brief moment, impressed with myself because I saw that one of the leading mathematical thinkers, I could surprise him.
He was genuinely surprised. And but then I realised that his he was not admiring me.
It was more like pity. He said, you know, you realise that, that this, this is completely wrong what you are showing me.
And I said, no, this is good, this is a good example.
And I was so hungry that I was tempted to offer him to go to the buffet and buy a piece of sausage and make an experiment,
hoping that he would pay for the sausage. But it never came to that.
He told me that. Okay, yes, it has one stable point, but how many altogether?
Well, for it has one here at the tip.
It's an unstable point. Another one here.
And it has here. This is a saddle point. A saddle point is unstable in this direction in at almost any direction.
But it is stable like this. So it has four points.
So a young man who told me once you have something which has less than for you, send me an email, but now I have to go.
And so once he was leaving, I asked him, And do you think that this thing exists?
And he said, Well, I think many people will try to prove that it does not exist because there are many indications that it does not exist,
but I think it may exist. So I went home.
I started thinking. So this is the summary of what I told you until now.
In the plane we have the same number of stable and unstable points always.
So it is sufficient to characterise one object with one of these numbers because
the other number is the same and in three dimensions we have three types.
This object has one stable to one stable and one saddle,
but it is sufficient to characterise it with two numbers because the third one can be computed.
So the problem is like this in two dimensions.
We cannot have zero because that would be the perpetual mobile and we cannot have one, which I proved.
And for any other class we have an example because four and larger or equal three,
we have regular polygons and for an equal two, we have an ellipse or whatever.
In 3D, the situation is utterly different. We have a matrix to a two dimensional matrix.
This is my famous sausage. Four and four is, for example, a tetrahedron.
And I know it was asking about this body. Now.
Christopher Columbus was challenged once to balance an egg on its tip.
This is an egg from the hotel.
And if you put it down, it won't stand on its tip.
Uh, you can say that on. On the camera.
So what did Columbus do? Yeah.
Let me try that. So.
And he won the bet or whatever. Now, in that article terms, what he did, he.
Added a new equilibrium point so that it was used to be unstable.
He turned it into a stable one and he added other points around it.
And the main idea is that if you do small changes to a body like that, you can always add new point and we can formalise this a.
We can. It's not difficult to prove that if you have iron j stable and unstable points,
you can always increase one of them by one by a suitable small truncation.
This is what we call the Columbus algorithm, and it means that for a sufficiently small,
specific truncation, if you have an object here, you can have an object here and an object here.
Now, what does it mean for our knowledge? Conjecture? What does it imply?
We still don't know whether it exists, but if it exists.
The existence of this object would imply that any other type exists by this algorithm.
Okay. So this algorithm is just telling you the significance of this object.
Uh. Let me show you this.
So here is this is very important for this talk.
Uh, so this is this is. This is just a cuboid rectangle.
It's a it's a it's a block. Okay. How many stable points does it have?
Six, right? Six stable faces. Now, here, we chopped off two small pieces at the corners.
And if I put it down on that face here.
It falls down. So we did not by chopping off this face.
Nothing happened. It is not a new equilibrium.
If I put it down on this face. It stands still.
So we added a new equilibrium, a new stable point.
All the previous ones are still working.
Okay, so by chopping off small pieces, you can add that is the idea of the Columbus satellite.
Now, but they still didn't know how it works. What is the shape?
So we went for holidays with my wife to Greece and I convinced her that as an evening
program we should collect pebbles and classify them according to the Poincaré help formula.
To make this more scientific, we should probably collect 2000 pebbles.
Some people in this room do know my wife, and those who know her will not be surprised.
But the others may be that at 1200 I give up.
But she insisted that we finish the project. Now this is the result.
We did not find any shape. We didn't find any shape which had this property of Arnold.
So we flew home.
I got a little bit depressed because, you see, I know this was a very, very significant mathematician, but on occasions he could be wrong.
So maybe this conjecture of his was just wrong. It could be wrong.
There is no evidence that it was good. At home I started thinking, what could have been the reason that we didn't find this shape?
And one is that one can assign flatness and seen this as a measure of the two convex bodies.
There are many ways to define it, this one particular way.
So this is from Anna's office.
This is the sphere. It is not flat and not say nobody will dispute.
So this has minimal flatness and minimal thinness.
So we assign the values one and one to the sphere by this definition.
I am sorry I didn't bring a Frisbee, but most people are familiar with that.
It these definitions would give a very large flatness and a very small thinness value.
And for the pencil, pastel is very thin, but it is not flat.
And what we could prove that if this object existed, it would also have minimal flatness and thinness like the sphere, but it would not be a sphere.
So that would explain why we didn't find it on the beach, because things next to the sphere tend to be rather sensitive.
So the idea which I showed you in the play, in our case sometimes works in 3D, sometimes it does not.
And starting from the case, very does not work.
We produced an example of this surface, but it could not be manufactured because the deviation from the sphere was so small.
Somewhat later we came up with this shape. Uh, here is the shape and I thank Alain Fire for the invitation.
I didn't thank him particularly for lending me his personal going back for the further for this presentation.
This is a particularly nice piece, so we will need a little bit space here, but oh, the egg.
Yes. So here is a nice piece and I let it move a little bit.
So for this object, it takes a little bit more time to find its way back to the stable equilibrium.
But unlike the salami, it has only one unstable point.
That's a major difference. It's very it's very hard to manufacture.
So very small design our street to many, many equilibrium, likely the sphere.
If you make a small perturbation, you get many, many equilibria.
So we resolve the Arnold's conjecture.
This is the first really bad work in Gone Back, and we decided to give it to Professor Arnold on the occasion of his 70th birthday.
This is the picture of it's Peter with whom we constructed the ship.
There are a number of pieces now around the world. We are most proud, of course, of the Oxford Compact, which is outside the room.
Uh, now a little bit of the it was so we looked at this ship for a while and, and realised that it is similar to some turtles.
However, those turtles were couldn't be found in Hungary.
They were tropical turtles. So we had to go to the zoo and ask for permissions.
Uh, it took a while, but. So this is this is a gone back to just small errata.
And then I went to the history for further to the museum for Natural History and I was given a huge turtle shell of a lower part turtle.
I put it down on the floor and I saw that it had 16 different stable positions and I knew this was an imperfect gun.
Now we measured many shells and we turned over many turtles.
I am sorry to say, but we try to do it so that with each turtle only once.
And what resulted was an observation which probably more interesting to the scientific
community than the gone was originally that many turtles self allied by muscle power.
So they use their necks and, and, and arms and legs.
So this is a very nice example. If you have aquatic turtle at home, don't try it many times, but you'll try it once you will see it.
Even large aquatic turtles can do this. And there is another technique which has been adopted only by two or three species.
They came so close to this gumbo like shape that they serve divide by gravity.
So this is a main tool for them to survive because being turned over is for a turtle, a life threatening situation.
And the medium turtles cannot survive. Right.
Told turtles found this shape or get got so close to the shape that the biologist believed us that this is not a coincidence.
Now back to back to the story. So this was we must go.
And I gave this talk on the going back, Professor.
I know it was in the audience. Of course, I was a little bit nervous.
He listened to it and after that he said, this is a good job, but now we should do something else.
And he outlined some problems, which I didn't understand because I was too nervous.
Later on, we met in the elevator and he told me that he liked the talk and in particular the story.
The pebble. The pebble stuff was very interesting, and I thought he liked that.
Marriage actually survived this trial, but his interest was more mathematical.
He asked, Do you realise the significance of your slide?
Well, for me it had multiple significance, first of all, which I just mentioned.
It's a memory to this event. But also the significance was that we didn't find any piece.
And then he said, this is obviously not the significance of your slide.
This is just the opposite. Uh, put that two lines in your imagination at two.
So in this regard, the first two and the first column, because they might be disregarded for physical reasons,
which I would just describe and look at the rest, look at the numbers.
The book is not in this table, but it is almost there.
It is almost there. It looks as though Pablo's would be evolving towards the gumball.
Now, this was very strange for me, really, and I hope it is also strange for you,
because the first part of the lecture was about why this cannot happen.
Uh, let me just briefly describe about those white barriers.
That is the easiest part. That is the easy part. Also, he told me that he didn't formulate as a conjecture.
He just gave us this hint that that maybe this is happening.
And he told me that unlike the gumball, this is serious mathematics.
So now you have to learn stuff. Okay. Uh, so this was his hint.
And the celebration for Pablo's occurs in two ways.
Either it is by fiction, so the parable is sliding.
You can hear it when you go to the beat. You can hear this sliding and rolling motion.
If something is very flat and it is sliding, it tends to become even more flat.
If something is very sane and it is evolving, it tends to become even more thin.
So shortly friction accounts for those.
Various friction would accumulate things. At either two stable or too unstable equilibrium points.
However, it is certainly doesn't account for why shapes would move in that direction.
That has to be as that has to be driven by collisions.
Now, as I showed you, if you take a collision in a collision, which I will not do physically, a small piece will come off.
If I hit it very hard.
And what I illustrated before is that if we take of small pieces the number of equilibrium points apparent in naively we think always increases.
How is it possible that the opposite happens on the beach?
Well, uh, so this is the naively this is the evolution which is suggested by this model.
So you take off a small piece and you get more equilibrium, uh, to get to take less.
Okay. Is it possible to reduce the number of equilibrium?
Yes. If you chop off a large piece of cake, how can the chop of a large piece with chopped of say this piece here?
I show you a and large version of this piece.
This is here. Just you can see better. Okay, the same thing.
We can say that this is a rectangle, but this has been chopped off.
Or we can say that this is a rectangle where this has been chopped off.
And if we adopt the latter view, then how many stable points does this have?
It's a tetrahedron. Four. So this has less.
But in order to get there, we needed to chop off almost all of the body.
So it's not clear what's happening now.
So. So what we did, we made a computer simulation, we took a rectangle,
and we were chopping off diligently, small pieces, and we counted the number of balance points.
Here is the plot. So you can see initially it went up, then it went down, then it went up, then it went down.
So how is that possible? Well, we said that in order to reduce the number, in order to add the new point, the truncation has to be sufficiently small.
But I never said what is sufficiently small. This depends on the shape.
So in the natural process, the piece is being chopped off.
Let's say they are roughly constant. What is sufficiently small depends on the shape, and that may vary.
And when this actual piece is below the curve, then the number of equilibrium goes up.
Otherwise it goes down. So anything can happen.
We we don't have any prediction for the average.
And in particular, the Columbus algorithm in this chopping algorithm cannot be regarded as a model for the natural process because of this reason,
because in mathematics, when you say sufficiently small, you may mean very, very small.
And in nature, collisions create on the average similar size the collision of similar sized pieces being broken off.
We can average also in space, so assume we have a ellipsoid, that kind of thing, and we put on a very, very, very fine mesh.
On that mesh we will see equilibrium points, many of them, but they were gathered, concentrated in flocks.
That can be proven. So they are always, always in smaller domains.
Now, if we track in the number of individual points, it may differ very much if we track the number of flocks.
And indeed, if you take the number of flocks, we see that that is decreasing.
Of course, we ran it many, many times, but this is the average. So the trend is that it is a decreasing.
So here is that we are table. And if you scan it very carefully, you see the same phenomenon on the surface of the table.
You see many, many, many little equilibrium point, but they appear in flocks.
So they if if you if you ignore the details, you see the good picture.
So in order to get a sense of the phenomenon, you have to ignore the details.
You have to ignore the small scale structure that too scarce. And you have to think on the global scale.
And the Columbus algorithm was very much on the local scale.
Now the model is describing average behaviour,
so time and space average behaviour are called partial differential equations and that is what I was referring to.
In order to find what is happening on the time and space average, you need to understand this in terms of these models of them,
in terms of partial differential equations, they are called PD models.
What is a partial differential equation? Well, a simple the kind of partial differential equation we are interested in is I imagine here is a curve.
And at each point you move this curve in time and at each point of the curve you defined a speed in the normal direction.
And then the curve shrinks and then again the speed is given, and then again the curve is shrinking.
So this whole phenomenon is a partial differential equation.
Things depend also on space and also on time, uh, as this curve evolves.
First we have a certain number of equilibrium points and in the end we have in this case, we have less.
So equilibrium points may disappear or may appear.
If they disappear, we call it an end. The relation they went away or if new come around.
They call it a creation. Both events can take place in this in this process.
Now, here are our questions. What is the formula for the speed which describes the natural abrasion process?
So what is the partial differential equation? We have to understand?
And if we find that formula, what is the average value for Omega?
Would it be would it be positive or would it be negative?
So I know hint was in the direction that if you look it in a very average way at the right equation,
then Omega will be positive in some of the average value of omega will be will be positive.
Here is it in three dimensions. And well, I have some models here.
So the easy way to study it, if you go to the drugstore and you buy a bottle of soap and then you wash your hands with it,
you will see something similar happening. We did it the hard way.
We know that is one of these equations with the computer and printed out in 3-D.
What came out of this? So here are these successive shapes.
We started with exactly this cuboid.
Okay? This had six stable points. This is the next stage and.
This still has six. Although they appear to be slightly less stable, so to say.
But there are still six. This is the next one.
Sometime later, this is stable. Uh, this is still the stable, but I cannot balance it here anymore.
So these ones somehow disappeared at the long end.
And this is the ultimate thing, which is a kind of an ellipse.
It's not an ellipsoid. And you probably believe me that I can only balance it here.
No, at that point, it will always fall over. So we started with six and we ended up with two.
So at least from the experimental or intuitive point of view, it seems to make sense that if we have all of these things in an average,
the way we get rid of equilibrium points, is it actually too?
Can we say anything about these mathematically? Well.
The formula, the actual equations for these operation processes were there.
Nobody at this paper. I think this is a brilliant paper, and it was written by Fred Blore.
He was in Liverpool. If anyone in this room knows anything about Fred Blau, I would be grateful because we tried to contact him.
He's now retired, but we couldn't get hold of him.
So he wrote this paper in 1977 and he laid down the general method calculations for possible aberrations.
So he gave he gave the right formula for a V in case of Collisional aberration.
These are questions are very, very difficult equations for a matter from the mathematical point of view.
Very little is known about them. On the other hand, not known to blow.
There was a huge activity close to these equations, not quite these equations.
So these equations tell you that the speed by which the boundary is moving inward is depending on the curvature,
so more curved things move more speedily.
And these three dimensions, it depends on the Gaussian curvature and the mean curvature.
And those are constant term, rather complicated, I would say nasty partial differential equation.
However, there was some activity which was below I'm not aware of, and that was the Poincaré conjecture which attracted enormous mathematical powers.
And ultimately the Poincaré conjecture was proved by the way to flow.
And the e.g. flow is a generalisation of this flow.
This flow is a special case of the flows which blower has studied.
So for this flow, which is called the curve, short curve, shortening flow.
The Poincaré conjecture, the proof for the Poincare conjecture is one of the major mathematical breakthroughs of recent years,
and it attracted top people. And that was a very lucky thing because they were certainly not okay.
Blau didn't say anything about any relation or creation of critical point.
He didn't study it. They were not interested in it either, but they were working on these flows for extended period of time.
They were studying the properties of these for the ever surprising great theorems being proved about these flows,
which I won't tell you today, but it's a very interesting topic.
One of these papers by Matt Grayson showed that in the case of this format goes equal to one.
So if you just take the plane out curve and you shrink it by the curvature speed, the number of critical points is going down.
Of course. This is a special case of that.
And to generalise it. I had to put in some weakening conditions, assuming generosity of the bifurcation for the critical point.
But taking taking these assumptions in addition to the original problem for the play in our case,
uh, based on Grayson's suicide, uh, one could prove that Omega is equal to one.
So always the number of critical points is always decreasing.
In three dimensional, the situation was much worse because it is not true that it always decreases.
Sometimes it increases. And I was stuck.
I couldn't do with this term anything for a couple of years.
Then I discovered that there is another equation which is much better understood than these very special geometric equations.
That is the heat equation.
The heat equation is describing the, uh, dissipation of heat, uh, or in a medium it was written down and first studied by Fourier.
And later on in the eighties it turned out that the heat equation is a good model for the blurring of images.
This is a very fashionable topic and that started an industry of studying of the
features of the gentle metric features of surfaces evolving under the heat equation.
Now, uh, there was a huge literature on this, and even in 1995 it was an open question whether in the heat decoration,
which again is a much simpler equation than blow as equation, whether creation could occur or not.
It was an open question. And and James Damon finally showed an example.
Evacuation could occur, but people were not satisfied with that because if they looked at the screen most of the time,
they didn't see the C and the relations, but they didn't see questions.
So two mathematicians, uh, actually students of conducting, uh,
said that the term which we cannot explain the replaced by a random variable with zero expected value.
And if we do that then we get our end on process, the expected value of which is positive.
Now the same can be done for the more complicated, fair case for the blow equations.
So what? It adds up to that the discrete process suggested an evolution in this direction.
But from the average mode, as we get an opposite indication, but only in a random, in a weak sense.
So it is moving that way, but not not maybe not always, but most of the time I would say.
Okay. Let's go back to this chess board.
These are the stable points, number of stable points and number of unstable points.
All that we need from all the previous mathematics.
Is this as Jovian an object in here?
And we are throwing a dice, but the dice is is not correct.
It's more often it is falling to this side. The dice will determine whether the object will move up to the right down or to the left.
And because the dice is loaded, that is what was devised before.
It's more often than it would go up to the left, then it would go to the right and below.
That is a very simple statement. And in each well course we have the barriers due to friction and this process.
So we have many, many objects and we are throwing these dice many, many times.
The simplest assumption is all these probabilities are constant.
This is very simple, mobile data, very simple. It's called the Markov process, and it is very well understood and it gives you a distribution.
If you start with many particles all over the place, it gives you a very definite distribution for the end result.
It will be a stationary distribution.
And if you compare that with the field data, which we had that on the Greek island, you'll see you get a very nice agreement.
So this process is a very particular process.
So you started with an initial values to take a bunch of pebbles, you take the average number of stable and unstable points.
So it is not an integer anymore and you start evolving this process.
So what you will see that there is a fast descent and then it starts to element in some area.
So there are definitely two phases. The first phase is a diffusion process which has a drift, and the second phase is a recurrent process.
So let me show you some data. The first is what I would call coastal data.
So we collected many, many, many pebbles on many, many coasts.
It's a good scientific activity. So I am planning to apply for a grant to extend these activities to many other coasts.
These bodies were collected privately, so to say we collected each of those little circles means 50 or 100 pebbles.
Some of them were less abraded, some of them were more abraded smoother.
And if we start this very simple model from the less abraded pebbles as the initial values, the trajectory seemed to go to the right place.
So they seemed to find the pebbles, which are more abraded. Maybe, but there is no causal relationship.
We collected this pebbles in many different locations, so it is not true that one of them was abraded from the other one.
Now this is Australia. Uh, this is a small, not a smaller medium river in southwestern Australia.
And we sampled pebbles along the river around 12 places here.
You can see some pictures from the river. And here is the here is the plot.
Now, if you want this Markov process here, you get a trajectory which is reasonably close to the data.
And if you run another trajectory, it is still close to the data.
So there is a fast descent and then there is summary current process.
Another example always go to the topic are places as you notice, this is port of call and uh,
actually our colleagues had a grant to go there every year and the sample tables and they had a lot of data, but it was not so meaningful.
And we propose that include the number of equilibrium points among the staff which they measure.
So we also offer that one of us would go and have them.
So they accepted this proposal here is that there are actually two rivers here, a tributary and a main river.
And here is the plot.
And again, you can see that there is a part where it is descending, the numbers descending very fast and then it seems to be oscillating.
And if you go to a lab and run this experiment,
you just put in a number of tables in advance and rotate it and you measure the number of equilibrium points you cannot get it wrong.
So it is the is the process is so stable, so robust that there is no way you can get it wrong.
You always get a similar process. So this seems to be a do a natural process.
The mathematical statements which I made could be made probably more rigorous.
Oh, here's the lab data measured. So as a summary, I might say that next time you go to a nice Pebble Beach, uh, think about this.
That that pebbles maybe not every aspect of pebbles, but some aspect of pebbles.
The story about pebbles can be told with natural numbers.
Maybe only a small slice of the story, but it is a true story.
It's only happening. So this is what Michael Berry summarised, is that the gamut exists in nature, but only as a dream.
So all pebbles dipping down in their heart. They want to become a gumball.
But. But few manage.
Uh, thank you very much for the attention.