 Thank you very much, and good afternoon to everyone. Thank you very much for coming to this basic notion seminar. So as the word suggests, I've tried to follow the spirit of these basic notion seminar presentations, which is really to focus on a fairly fundamental topic of general interest. This topic of factual dimensions is not actually my research topic. So if you start asking two technical questions, I might not know the answer. But I think it's a really fascinating topic. It has connections to my research interests. And so I will try to illustrate and try to make it interesting for everyone. Usually people say that you're welcome to ask questions. And of course, you're welcome to ask questions during the talk. But most importantly, you will have to answer questions. So I will be asking you several questions. So get ready for audience participation here. So my very first question is, what is dimension? What is dimension? I think we all will agree that a line has dimension one, a square has dimension two, a cube has dimension three. But how do we really explain the fact that these are the dimensions? Any suggestions? I mean, this is an honest question to start some discussion going for this. How would you define the fact that a square has two dimensions? Any ideas? Come on, you won't get to a PhD in mathematics if you can't say why a square is two dimensional when your family asks you, yes. OK, that is a very good suggestion, right? And it's the most intuitive. And in some sense, the one we learn at school. So in the case of a one-dimensional segment, you basically can identify any position on the segment just by the one parameter. If you have the origin, then you have one parameter. If you have a square, then you have a kind of two coordinates. And in the cube, you need three coordinates to specify a point on the cube. That's how we intuitively think of dimensions, how many coordinates you have. So this is a definition that is effective and good in many ways. It has a very nice property that it allows to easily generalize to higher dimensions, OK, even though we cannot visualize, we can just think of a higher dimensional space as a five coordinates identifying a point in five dimensional space. But it has a problem is that it does not really help us to give any meaning to integer dimensions. What is one and a half dimensions? Two and a half dimensions, right? Any dimension that is not integer, what do you need? Two and a half coordinates? It's not really clear how you can do that, OK? So the purpose, often when I give this talk, I say that there's a secondary subtext in the talk, which is that I would like to show how mathematics really, let's face it, is much more creative than physics, right? No competition. We don't want to set any competition here. But I want to illustrate a little bit how the beauty in mathematics, part of the beauty in mathematics is really the creation of new ideas and new concepts, right? And this is, I think, a beautiful example in which you can really see how a new concept can be created. So the concept here is that we have these dimensions. We want to give meaning to non-integer dimensions. We really need to develop some new ideas for that. So I will illustrate a little bit. One possibility will start somewhere to start. So let's look at these objects that are one, two, and three dimensional. And let us see what happens when we stretch these objects by a factor two, for example. So we take objects which are twice as big in the sense that we stretch. The distance between any two points is twice as much as the previous one. And you'll notice that in all three cases, the new bigger object is formed up by a certain number of copies of the original object. So here, this twice segment that's twice as big is made up of two segments. And in this case, it's made up four copies of the original. And in this case, how many copies? Eight. OK. This is just warm up. And OK. So of course, I don't need to tell you. You've all been to elementary school. And you know that two is two to the one. Four is two squared. And eight is two cubed. OK. So we'll come back to that in a second. Let's just see what happens when we do the same thing three times. So this is made of three copies. The square is made of nine copies of the original. And the cube is made of 27 copies of the original, which again, are just three squared and three cubed. So this may suggest that there is some kind of that the dimension is hidden or is somehow intrinsic in this property. You enlarge an object, and you see how many copies of the original object are contained. So we have not proved anything yet. We've just looked at these particular examples. But more or less, if we were to try to draw some conclusions, we would say that if an object has dimension d and we scale it by a factor k, we take something that is k times bigger, like two times or three times or four times bigger, then it constitutes of k to the d copies. It is made up of k to the d copies. That's exactly at least the examples we had before. So we have this formula n equals k to the d. So just to review, when dimension equals one, when k equals two, we have two copies. k equals three. We have three copies and so on for the other cases. When dimension two, we have two squared and three squared. And when dimension three is eight and 27. So this is a formula that, at least in the examples we've shown, relates the scaling factor, the dimension, and the number of copies. Here, we wrote it as a formula for n. But of course, we can rearrange the formula. And we can write it as a formula for d. So we can at least hope to use this in some way as a possible definition of the dimension, possibly. We can say, OK, look at an object, scale it by a factor k, count how many copies of the original object are contained in the scale copy, and apply in this formula, and get the dimension. Now, this works at least for the examples in two and three that we gave above. There's not a technical class. I will not prove anything rigorously. But let's take this as a kind of work in definition. And let's suppose that this works. And what I want to do is I want to use this definition to construct a geometric object that has dimensions strictly between one and two, non-integer. It's a very nice construction. And we will use this formula. And the example I want to give is called a snowflake. And it was written out by von Hock in 1904. So it's a new, very classical thing. This is the picture that it looks like. It is a curve, in some sense. But as we shall see, it is very, very wiggly curve. And that's what's going to give it the extra dimensionality. So to do the construction, I want to show you, first of all, that in fact, what looks like something that is built on a kind of a circle is actually built on a triangle. So here, you really have three sides. So I'm going to do the construction just on one side. Then the construction is identical on the three sides. So we're going to construct this object and then discuss the dimension of this object. So we start with an interval, unit interval. And then what is the first step in the construction is that we're going to chop this unit interval into three equal parts and replace the middle part. So each of these two is 1 third. This is 1 third. The middle part is 1 third. And we delete that. And we replace it with a little kind of tent formed by two segments, which have exactly the same length. So now we start with something of length one. And we have something of length 4 thirds. This is the first step. So it's very simple, but I'm going slowly because soon we're going to speed up. And then what we're going to do is we're going to do the same thing on this. So now we have this. And we're going to do the same thing on each of these four bits. So each of these four sides, we're going to chop into three equal parts and then replace the middle part with a little tent. And we get exactly this. So you can see this is the middle part. We've replaced it with a bit. So what is the length of this? So we can count. Exactly, right? So we can count. So that is the first way to measure stuff is to count. So we know that each of these lengths is 1 third. We've divided into three. So each of these bits is 1 ninth. So we just count them and we see there's 16. And so you see there's 16 ninths. But now we already know that we don't want to keep counting. So let's try to make an observation that makes us a little bit smarter. And notice that what we did, why did we get four thirds here? Because we replaced three segments with four segments all of the same length. That's why we got four thirds. And it's the same thing we did here. Everywhere where we have three segments, we replace it with four segments. So the new length is going to be four thirds of the length we started with. So in this case, this is just going to be four thirds squared, right? Because the length we started with was four thirds. So we take four thirds squared. Just to get ready to make it easier the next time around. Because of course, what we're going to do now is again the same thing. So now we have 16 small intervals. And we're going to take each one. So each one is length 1 ninth. We're going to take each one and divide it into three. So each one is going to be length 127th. We have this. And we end up with this. So this is just the same thing. So we've taken each one of these and replaced it by four. And now we can count again still. But already we can be clever and see that this is four thirds times four thirds, which was the length of this times four thirds. Of course, it's four thirds cubed, which is 64 27ths. So in particular, notice that the length is increasing, obviously, because we're increasing. So how many times can we do that? Well, we can keep doing that. And at the nth stage, we will get a very weekly curve made up of a finite number of tiny, tiny little segments. And the length will be four thirds to the n. So there will be three to the n. Each length of each segment will be one over three to the n. And there will be exactly four to the n such segments. So we are keeping track exactly of what's going on. And of course, we can continue. And we can define, which is the only kind of non-trivial technical step here, the limit. There is a topology in the space of compact sets, so-called the House of Topology. But you can define the limit geometric object. So as you continue this construction, there's a sequence of compact sets. It converges to some compact set, which is going to be our object, which is the limit. And notice that the length, and I will come back to that in making observation, notice that in this limit, whatever this object is, which is not completely clear, it will be some kind of curve to look at. It will not look very different from this, because at this stage, it's already very difficult to tell these tiny little intervals. To look at it, it will look like this. But the length, if we were to try to measure the length, it would be infinite, because it's a limit of objects for all of whose the length is very well defined, and it's getting larger and larger. So this is it. This is the object we've constructed it. So now we are going to try to understand it. In particular, with regard to the dimension. What dimension does it have? Does it have some strange dimension? Well, now we have our method. And what happens if we scale this by a factor 3 or 2 or 3 or 4? How many copies? Remember, our approach is to scale it up and see how many copies it is made out of. So the key observation is the following. Is that actually, if we look at this piece of this object, it is exactly a scaled down version of the full object. If you multiply three times, if you scale this by three, you get exactly the original object. You can see that because, well, you can kind of believe that. You need to think about it a little bit completely convince yourself. But you know that this piece is built on something of length 1 third, because that's where we started the construction, by removing the middle third here. So this length from here to here is 1 third. So when you scale by a factor 3, you end up with this horizontal length is exactly the same as the original piece. And then if you think about this horizontal, because of the way it's constructed in different stages, then each of these pieces exactly 1 third of this piece. So when you scale it up by 1 third, you will get that this height here will be exactly that height. And you can easily see that you will get, when you scale it by three, you get exactly the original object that you started with. And this is not true if you do this with an object at a finite stage of the construction. If you do this at a finite stage of the construction, then when you scale it up, the two scales correspond, the two objects are at different levels of the construction, and you'll get a different number of hats in each side. But in the limit, they're exactly the same. You have hats everywhere. In the limit, it's a very weakly object because it does not have any segments in it. Every time there was a segment, at some point, you substituted the middle part with a hat. So this is extremely zaggy object. So this whole thing, yes. Yes. No, no, that's fine. I did not go because this requires so you can define the fact that there is a topology on the space of compact subsets of the plane. And in this topology, this sequence of compact subsets of the finite stages, they're converging to some compact subset of the plane. Yes. Well, the convergence inside your square. So the set is changing, and you are converging to some subset of the square. You remain inside your square, yes. You just convert, yes. So how many copies? So let's think of this. So this big copy is three times the little piece, yes. And it is formed by a certain number of smaller copies of itself. How many copies of itself does it contain? Four, right? Here is one. Here is the second, because this one is also identical, because and here is the third, and here is the fourth, right? So you have, at the stage one of the construction, if you remember, we had four segments. And then with each of those, we just started the construction exactly the same as we did from the beginning. So the construction, when you restrict yourself to each of the four segments we did at the beginning, will give you an exact copy of the whole object from the beginning, OK? So we have here that we have an object, a geometric object, which has this remarkable property that if you scale it by a factor of three, OK, it contains four copies of itself. If you remember, an object that had one dimension would contain, when you scale it by a factor three, would contain three copies of itself. And a square, which has dimension two, would contain four copies, would contain, when you scale it by a factor three, would contain eight copies of itself, OK? But in this case, it contains four. So now each side contains four copies, each side of the triangle. So I mean each of these object scaled down by a factor three. So we can use the formula we had before to compute the dimension. And we get log four of a log three, which is exactly one comma two, six, 1.261, something, OK? So this is an object whose dimension, at least if we believe that that notion of dimension makes sense, we can generalize this formula to an object that has dimension not one and not two, but something in between. And there's many different examples. We can play around with lots of different nice cases. We can do similar stuff and get objects with many different dimensions. So here, for example, you can do the same. You start with an interval, and you cut off the middle third. And instead of just putting a little hat, you put a little house. Instead of putting a little tent, you put a little house. And then you do the same thing on each. So now you have five sides instead of four, right? And then you just repeat exactly the same. So on each piece, you do the same thing. And then on each piece, you do the same thing. And you have exactly the same argument. But here you will have five pieces, right? And so the numbers are five and three. And so you get dimension 1.464 for that. And of course, you can have many objects which have very different shapes that have the same dimension, right? Just like in one or two dimensions, the dimension is not related to the shape. For example, you can do a similar construction here, which is quite different from this one here. You just take three, and you don't actually delete anything. You just take your interval, right? And now you add. You still divide it into three equal parts. But then you just add these two segments, right? So you end up with these five parts instead of three. And then you do the same thing on each one. And then again, you just repeat the same. And this one actually has exactly the same dimension, OK, for the same reason. This one here, also the same. So here, you can actually divide into five parts. One, two, three, five, you see? And then you, sorry, four. One, two, three, four, the original interval. So each interval is one, each piece is one fourth. And then you replace it by this structure here. So you end up with eight. So you replace four by eight. So you're actually doubling the length each time. And then it's very easy here to see also that you get the dimension is log eight over log four. And a couple of simple steps of calculation give exactly 1.5. This is nice because it's got exactly one and a half dimensions, right? So this is it. So as you can see, this really opens up a whole new set of geometric figures that Euclid did not realize were possible. And we can also, so these have these, they're constructed in such a way that it's easy to apply that formula, right? I mean, they're kind of artificial. We construct them so that they're very self-similar. And it's that same similarity that we use to actually calculate the dimension. But notice that if we were nasty, we could mess with your mind and change things around, right? So if you want, you can vary the algorithm at different scale. So suppose I start with this picture here, OK? And I use this algorithm for 1,000 steps, OK? So I end up with really what looks like this picture made out of millions and millions of tiny little segments, OK? But still now I can zoom in into one of those segments. And I can say, OK, wait a second. After 1,000 steps, I want to now change the algorithm. And I'm going to start putting the original snowflake things there, right? And so then for another million steps, I construct a snowflake, OK? So now if I look at the picture like this, what I see is this picture, OK? But then if I take one of these tiny intervals and I zoom in and I see that on that interval, actually it's got the structure of the snowflake and so on. And then after 1 million iterations, I still have a finite number of segments, even though they're very small. And I can change the algorithm and I can start doing one of these other examples that I have, OK? So this would not be self-similar, because this is really at different scales. You would be seeing different things. So this is a very interesting concept. It's fine. We will see some more examples later, where at different scales you see different things, OK? And in this case, it's not clear that we can use this definition of dimension. And there is, in fact, so this is why mathematicians are so good. They are not discouraged by this, OK? So they say, OK, let's generalize even more this definition of dimension. So here enters Mr. Hausdorf, OK? The end of the 1800s, beginning of 1900s. And he developed the notion of Hausdorf dimension in 1918. And this definition applies to any set. You can use it as not use a self-similarity, OK? It's a very powerful definition. But it is based very much on that idea. And the interesting thing is that it involves a notion of measure. So there is a connection between the notion of dimension and the notion of measure, right? So let's try to illustrate a little bit. So let's go back to our original picture. And in the beginning, we focused on the dimension. But now, let's talk about the measure. How do we measure the sizes of these objects? So you all know that we have Lebesgue measure. But you all know that we have one dimensional Lebesgue measure, two-dimensional Lebesgue measure, a three-dimensional Lebesgue measure. So when we measure the segment, we use length, which is one-dimensional Lebesgue measure. When we use a cube, we use area as a measure of the size of a two-dimensional object. And as a measure of the size of a three-dimensional object, we use volume, right? And each of these notions of size goes with the dimension that it comes with. Because if you try to measure the area or the volume of a segment, yeah, OK, you can do it. But you get zero, right? If you try to measure the size of a volume with length, you will get infinity, right? If you try to say how long it is, you try to put a curve, that will get infinity, OK? So if you want something that makes sense, you need to use the right measure for all of this. And this turns out to be another way into the concept of dimension. Because in some sense, you can say, OK, well, actually, the dimension of an object is the one that makes sense for the measure that I'm using, OK? So the idea of Hausdorff was very, very clever. And he defined, so each dimension has its own measure. And he defined the measure, s-dimensional measure, right? Now, this is the only formula that I have in this talk. It's not actually as scary as it looks, OK? Because the idea is very simple. You're trying to measure f, set, set that exists, OK? And you're trying to find its s-dimensional measure, OK? So what you do is you take a delta cover of f, which means a cover basically by open sets of diameter less than delta, right? And then you take the sum of these covers to the power s. And then you take the optimal cover. So this means you try to minimize what you can do. And then you take for different deltas, OK? As delta goes to 0, this turns out to be a monotone sequence. So it converges. So this is basically always defined, OK? We're not going to detest technicalities. But let me point out one simple case. For example, suppose the set f is just a finite set of points, OK? If it's a finite set of points, then what happens here when s is equal to 0? So let's try to measure the 0-dimensional measure of this set, right? So when s is equal to 0, this is just 1, OK? For any set, ui, OK? So this sum is just measuring how many sets you need to cover f, delta cover f, OK? And if these sets are finite, then they will be all isolated points. If f is finite, it constitutes a set of isolated points. So when delta is small enough, what you will need is exactly the number of sets that you need. It would be exactly the number of points that you have. And therefore, this will be just the number of points that you have, OK? So it's quite easy to see that the 0-dimensional measure is just the number of points that you have. On the other hand, it's also fairly easy to see, if you think about it, that the house of measure of a finite set will be 0 for any other s, OK? Once you put s here, then this will go to 0 if you have a finite set. Yes, the measure of ui. Yes. Sorry, I just want to keep an eye on the time here, because I'm not sure what the diameter. Yeah, well, OK. Let's suppose we're in a Euclidean space. Let's suppose we're in a subset of a Euclidean space. So this is the metric space, the diameter. Sorry, it's not the measure. It is the diameter of the set. Yes, sorry. Thank you. We don't need any measures. It's the diameter of the set. So it's the maximum distance between any two points in the set. So the crucial my own objective here is not that we try to internalize or digest this formula. But the key observation that turns out to be really remarkable is the following. Is that, in general, there's a cutoff point. So if f is an infinite set, OK, then there will be a particular value, which we call s, which depends on the set f, which has this property, that if you try to measure with this formula for every s less than sf, then the s-dimensional measure of f will be infinite. And if you try to measure the s-dimensional measure of the set for any s bigger than sf, it will be 0. This is not that difficult to see, actually, but it is like a bit, if you think of the one-dimensional, two-dimensional, three-dimensional measure, it's like that. If you're trying to measure something that's two-dimensional and you use length to try to measure it, it will be infinite. If you use volume, the measure of this two-dimensional object will be 0. So it means that you're using a dimension that's either too small or too big. If you're trying to measure it with a dimension that is too small, the size of f will be infinite. And if you try to measure it with a dimension that's too big, the size will be 0. So there is a cut-off point. And this is the most amazing thing. So just this simple formula, automatically from the properties of the formula, it gives you this cut-off. And so this is the natural candidate for the dimension of that set. And this is, indeed, the definition of the house of dimension of the set. It's this cut-off point. Anything, if you measure it, so this is the connection between measure and dimension in this way. We use measure to establish the dimension. So this is gone one step beyond. We do not use the self-similarity or very detailed geometric properties. This is a very coarse, in some sense, but it still gives this object. So once you establish this dimension, then this is the correct dimension for you to use. And then you use this formula to measure the size of the set with this dimension. And then you may still get 0 infinity, right? You may still, just like if you use one dimensionally big measure of the whole real line, it's still infinite. But you may also get a finite. So you don't know. This is a jump between infinite and 0. And this may be either 0 or infinite or finite. So this really is the first part of my talk, which is to do with the definition, with the concept of dimension, and the way that mathematicians have been able, I think in a really fascinating way, to develop some really quite amazing ideas to generalize the concept of dimension. So I promised in the title some also something about factors in nature and mathematics. So in the second half of the talk, I want to talk a little bit about this. So why this does not just have an abstract interest. In some sense, it really has relevance both in nature and in mathematics. So one of the people who mostly advocated the relevance of these concepts of fractal dimensions in nature was Mandelbrot. And he was a physicist, French physicist. And he wrote a book in 1980, took on the Fractal Geometry of Nature, which is a very remarkable book. I claim, and in this book he says, I claim that many patterns of nature are so irregular and fragmented that compared with standard geometry, nature exhibits not simply a higher degree, but an altogether different level of complexity. So he's saying, when you look at, for example, the clouds, and you try to describe the shape of the clouds, it's not just lots of tiny little combination of triangles and squares and pentagons and stuff like that. Euclidean geometry is not the right approach. There's a different level of complexity that requires a different kind of geometry. And he basically is very easy to see many examples of objects in nature. Even here, in trees, you have a kind of self-similarity in the sense that if you break off a branch and you don't know what scale you're at, it's not easy to always to say whether this is a branch or whether this is the whole tree. There's this kind of self-similar structure. Of course, not a full asymptotic one as we have in mathematics, but you have lots of examples. You cut off a piece, and it looks like the whole piece. So besides this observation, what can you do to make an interesting remark about this? And one of the most interesting things that he talked about is that he argued that many objects have a fractal structure. And therefore, how do you measure their size? We just saw that when you measure something, the dimension is related to your measurements. I mean, you cannot just measure something without keeping the dimension. And one of his examples is he wrote an article, actually was published in 1967 in Science, called How Long Is the Cost of Britain? And he made a very important point in that. And I will try to illustrate that point. So we're going to use the coast of Italy. And suppose you want to measure some, like from the top to the bottom, like from Trieste to Messina. Well, I did the very tip. OK, so we have Google Maps these days. We can actually try some measurements. So the very first half approximation is not the coastline, it's just the distance as the crow flies, as they say. And according to Google Maps, this is 843 kilometers. But of course, that's not a very good approximation. You can do much better by trying to follow the coast at a certain scale. And according to my own tracing, you get already much, much bigger value of 1,400 kilometers. But you can try to be a little bit more precise. Look at the boundaries a little bit better. You get something larger, 1,500 kilometers. You can try to be even more precise, and you get almost 1,700 kilometers. But you know where do you stop. So now, if you continue, you look at a little piece, and you actually try to go into a bit more careful. Of course, this is always going to increase. Because I am replacing what looks like just state curves, I'm following the contours of the coast a little bit more. And I can even go a little bit more. So where do I stop? So at some point, you end up really wandering exactly where the land ends and where the sea begins. It's actually not even clear. Once you get to a sufficiently small scale, where exactly do you draw the boundary? Even if you could, if the sea was frozen, at some point you still would have to go around all the boundaries of these stones and of these rocks. And you would end up getting a very, very large value. So it's really, I think, a Mandelbrot thought, and I think with him that this is a non-trivial issue, when we say that Great Britain or Norway or Italy has so many kilometers, of course, how exactly are these coasts measured? And can we get into competition between different countries? So the length of the coastline really depends on the scale at which it is measured. So Mandelbrot's idea, as far as I can understand, and I don't know that I really understand this exactly completely, but his idea, well, that these coastlines, they have a kind of fractal structure. So although you cannot go to infinite details like you do with the mathematical construction, you can look at the way the size increases as you look at larger and larger scale. And you can kind of approximate a kind of dimension for this coastline. So his suggestion was to think of the coastline as a fractal and in some approximate physicist's way, I'm not a physicist, I'm good for something, I'm not saying there's nothing bad about this, find some way to say that we can make sense of the fact that to some approximately reasonably good approximation. And then once we've got the dimension, maybe you can use the host of measure. Is the correct measure to estimate the coast? To be honest, I'm not exactly sure whether he did it or someone did it, but according to Wikipedia, the dimension of the coast of written is 1.43. And the host of measure with this dimension is 28,000 kilometers. And there's a page on Wikipedia that has the length of the coastlines of all countries, but this one is the only one where it's not clear what scale these are measured at. There's a remark that says we don't know which scale because there's not like a standardized way of measuring the coastline. And I did not find the value of this measurement of the house of dimension for different countries. I think someone must have tried to do it or do it anyway. This is just to show you the kind of approach you can take in these kind of situations. I guess so, yes, you're right, you're right. In fact, now that I think of it, that's how it's written in Wikipedia, it's kilometer. It's not kilometer, it's not kilometer squared, but it's kilometer to the 1.43. I forgot for that, you're right. Yes. OK, so this is really some ways, some possible ways in which these factual dimensions and factual measures can come into our understanding, or our perception, or our measurement of the real world, as opposed to the even more real world of mathematical objects. And factual objects do occur naturally in many areas of mathematics. My field is dynamical systems, so I will stick to that. And I want to give some examples here. There's some very, very nice pictures. And I want to explain a little bit in what context they arise. So one of the most well-known is when you have a map of the complex plane to itself. And in particular, the simplest is just a map f of z goes to z squared. So I think several of you have seen this before. I think those of you who took my course, we mentioned this at some point. And relevant here is the unit circle in the complex plane, because as you know, when you take z squared, if you have a point whose absolute value, whose norm is bigger than 1, so if it's outside the unit square, then its image is bigger than the point. Because the distance from when you take the z squared, the modulus of the distance increases when you take squared. So this means that if you apply the f iterate many times, then the value of this point is moving away from the origin is moving to infinity, in fact. So if you take a point that's outside the unit square, and you iterate it, it will move somewhere that is further away, and then further away, and then further away, and it will kind of move towards infinity. If you take a point that is inside the unit square, then it moves further inside. And it's easy to see z is a fixed point, and in this particular case, you take a point inside, it will do some kind of spiraling in towards 0. And on the other hand, if you take a point that's on the unit square, then it will remain on the unit square. So the unit square is an invariant set. Everything inside also remains bounded, stays 0. Everything outside. I will not be cruel and ask my students what happens on the unit square, but those of you who remember, we studied there's a very interesting dynamics. It's a period doubling, 2x mod 1, very chaotic and very interesting dynamics on the unit square. But this is not the focus on my talk, but there's a lot of interesting dynamics that goes with these concepts. But for the moment, I just want to introduce in this context a definition, which is the Julia set. And the Julia set of the map F, in this case, it's equal to the unit circle, and it is the topological boundary of the set of points whose future orbits remain bounded. I'm giving this formal definition because I want to apply the same definitions in some more general situations in a moment. But you can see here that that's why it's the S1. So in this case, remain bounded forever in future time. So all the points outside the unit circle, the orbits do not remain bounded, because when you iterate, they converge to infinity. On the other hand, all the points on the unit square and inside the unit square, they all stay within the unit circle. So that they remain bounded. So the boundary is exactly the unit square. So this is my definition of the Julia set, which I will use. So now, things get interesting when you add a little term to z squared plus something, small. So imagine you take a very small complex number, and now we want to iterate this map, z squared plus c, where c is very small. So it might be very small, but it creates a lot of mess. Because suppose c is a real number, for example. Suppose c is some epsilon, real epsilon. Then what happens is if you are a point here on this side, if you're image, so you take some point z on the unit circle, for example. Then you take z squared. And then suppose z squared is on this side, and you add c, so you're pushing the point a little bit outside the circle. But now suppose that you're on this side, and then you're pushing it always to the right. If c is positive, suppose p is some small, positive, real number, then what it means is that you take z squared and you push it a little bit to the right in the positive direction. So when you're on this side, you tend to be pushing it away from the circle. But then when you're on the other side, you tend to be pushing it inside towards the circle. So for the points that are far from the unit circle, so if you take a point that is very far from the unit circle, this c will be negligible. So these points will still be just moving to infinity as before. And if you take points that are very close to zero, also when you take z squared, they will move in a lot, and the effect of this c will be negligible. The problem is the points that are close to the unit circle, because then this c can push them off to one side, towards the outside or to the inside. And what happens is the question. And it's not at all obvious what happens in this case. And what happens is really, really interesting, is that you still get a Julia set. So you still get a boundary. So you still get some points that remain bounded. You still get some points that move to infinity. And the boundary is still a kind of closed curve, but it has dimensions strictly greater than one. So what you get with c small is you get here. You still get something that looks a little bit like a circle, but this time it is a bit like a snowflake curve. So it's very, very jaggedy. And that's because of the uncertainty of this c, because you take a point very close here, and you try to determine whether it's going to be inside or outside. In other words, whether it's going to be eventually going to be one of the points that remains bounded, or eventually one of the points that goes to infinity. And it's very hard to decide. And if you change the point a little bit, like this point maybe stays, the orbit stays bounded, but then there's a point very close by, a bit to the left, a bit to the right, and under the orbit of z squared, the iterates do different things, and eventually this one stays bounded, and this one leaves. So anyway, you get a picture where the boundary has this fractal structure, which means if you take a point close to this boundary, it's very hard for you to tell if it's on the inside or the outside, because it is kind of zigzagging all the time. That's the difficulty there. So let me show you some examples of other Julia sets. If you take, you don't even need to take. This is still a little bit like you can see this almost as a perturbation of the circle. This is still also something like this. This is c is not that small either. c is 0.25 plus 0.52, the complex number, and we're iterating this map. And it turns out that when you iterate this map, you have this set of points whose orbit stays inside here all the time, so stays bounded. You have these points outside whose orbit eventually moves to infinity. And then you have the Julia set, which is the boundary between these two sets, which has a fractal, which has dimension strictly greater than 1. So this is one example. And you can see that this is a little bit like a perturbation of the circle. You start the circle, and you start perturbing it as a certain structure. There's many, many different pictures. This here is for c equals minus 0.5 plus 0.5i. And you also get something that is a little bit like a perturbation of the circle. And then as you move around your complex parameter c, for each complex parameter c, you can construct the Julia set and you get the most incredible, fascinating pictures. I'm sure many of you have seen many of them before. You can look for them. And as you can see bigger, some things can change. So you also can get some strange situations like this is called a dendrite, because it does not have any points, any kind of interior. So the Julia set, by definition, is the boundary. But here it coincides. All the points that remain bounded are actually in the Julia set, so it does not contain any boundary. And you can also, when c is getting a little bit bigger, get a cantor set. So this Julia set fragments. So what was a kind of circle with a very fractal boundary? It starts getting into very strange shape, and then it breaks apart completely. And really, quite extraordinary fact, is that every single Julia set is either connected, like the examples we say before, or totally disconnected, which means it's a cantor set. It never breaks up into two pieces. So either it is just one connected piece, or it is a cantor set. Yes? Excuse me? OK, that's a very good question. I was going to comment on that. The colors are somewhat artificial. They are made. But they do have some dynamical significance. So points that are very close to the boundary, they take a long time to leave. So points that are on the outside, they're defined by the fact that eventually, they always go to infinity. So basically, most of the time, these colors are based on that. They're the level sets. So they show that points that are within a certain band, they take a certain number of iterations. The closer you get and the more time it takes to escape, and that's how the colors are usually. Some of the people who thought these colors might use other methods to ask the computer to color things in a certain way. But that's the most usual way to do that, yes. Yes, this is of the form that we had before. It's just fz squared plus c for different values of c. In fact, this theorem makes it very natural to say, OK, if every Julia set is either connected or totally disconnected, which values of c correspond to which of the two cases? And you can draw a map of the complex plane where every point is labeled as to whether the Julia set is connected or it's totally disconnected. And you get a beautiful picture like that. This is sometimes called the Mandelbrot set. This is exactly the set of parameters c for which the Julia set is connected, and it looks like that. Looks like this means it's the black part. The outside is the Julia sets which are disconnected. It turns out that the Mandelbrot set has lots of properties that have been studied. There was a huge amount of work done, especially in the 90s, but there are still many interesting open questions on this. It is known that this Mandelbrot set is connected itself, and it is known that it is closed. So the boundary of this Mandelbrot set, the Julia set, is still connected. But as you cross the boundary, that's when the Julia set breaks up into a canto set, breaks up into pieces. As you cross the boundary. The zero is here, exactly. So this is the real quadratic family. This is the real plane. This is, I think, it's symmetric with respect to the real axis here. And zero is here. This is minus 1 fourth. And this is plus 2. This tip here. Yes, just to give you minus 2. Sorry, minus 2 and plus 1 fourth. Yes, sorry. Minus 2 and plus 1 fourth. It depends on parametrization of the family. And this, I can't remember. Anyway, yes, this is the scale of this point. So it's kind of between minus 2 and plus 1 half or something. This is the kind of region of parameter values. So for more, it's a relatively small, containing a relatively small area. And there's people, as I said, especially in the 90s, Duadi, Yokoz, many mathematicians studied in detail so many properties of his Mandelburt set and the corresponding Julia sets. There's a really fascinating theory. There is a kind of landscape. There's a geometry to the different regions of the Mandelburt set. People literally explore it because the different regions give rise to different Julia sets that have different structure, especially in the subject that do not include me. Look at the Julia set and the shape of the Julia set. And they say, oh yeah, it comes from somewhere here. There are different shapes of the Julia set. Just the one theorem that I want to mention, which is related to what we've been talking about, to the dimension, is that Shishikura in a big paper that was published in the Annals of Mathematics, he proved that the boundary is house of dimension 2, the boundary. Even though it's a kind of a curve, you expect it to have kind of house of dimension 1 point something. It actually has house of dimension 2, even though it kind of feels like a curve. So something can still have house of dimension 2. It is one something in house of dimension 2. It could potentially have positive. So house of dimension 2, then the house of measure is Lebesgue measure 2. And if something has house of dimension 2, it could potentially have positive 2 dimension Lebesgue measure. And this is an open question. It is not known, still it's one of the main outstanding open questions. It is not known if the boundary of the Mandelbrot set has positive dimension. I think people believe not. But I'm not sure that there's a very clear reason why it should have one positive or not measure. Anyway, what did I want to say? Yes, this is it. And all it's glory. And I will just finish first with some pictures. They're really beautiful pictures that you get zooming into the Mandelbrot set. And again, these colors, of course, the colors make a big difference to how much we appreciate the pictures. And as you said, and the colors can also be given some kind of dynamical meaning. But I don't know if some of these are just colored in a way just to make them more nicer to look at. I'm not sure. But you really get very nice pictures. Now the Mandelbrot set and the Julia sets, they're not completely self-similar. But they do have some self-similarity. So often, as you zoom in, you get small copies of the whole thing, or almost copies. So here, like here, you see, there's like a little something that looks a little bit like the Mandelbrot set. And when you zoom in, you get all these different copies. And you get these really. And here you get one copy. And in fact, you can find on YouTube some videos. And I prepared one just to finish a little video of a zooming. So you can draw this, you can take different pictures, and you can zoom in. So let me just see if I can draw this. This is a video of the Mandelbrot set. I'm zooming in and seeing the different structures that you get at different scales. There's a little Mandelbrot set turning up at a much smaller scales, another little one there. And then I've been told that we can make this even a bit more exciting by, I don't remember what I was supposed to press. There was something that speeded it up, anyway. Ah, there we go. We want to make us dizzy. We have the speed of light here. It is so big, it's got to have a positively vague measure. It's a Mandelbrot set. Do you like it? Yeah? Yeah. You can ask your mom and dad to take it home and watch it in the evening. OK, well, thank you very much. On facto sets. Yes, but I'm not. Maybe there's other people here who would know better than me. I think you can do a lot of stuff on facto sets. But I am not an expert. But I think there has been done some stuff. You can certainly do a lot on facto sets. And there's a huge amount of mathematical theory on objects that have factual dimensions. Yes. That's a good question. I would presume certainly yes. But I cannot say that I have immediately, my fingertips are justifiable proof of that. But I think undoubtedly yes. Yes, it's a good question. Yes, thank you. You mostly spoke about deterministic objects, constructions. I know that there are statistical notions of fractals. What's the connections with what you presented? What do you mean by statistical notions of fractals? Like objects that have fractal properties, but not like their statistical properties behave in a fractal way. And they have fractal dimensions in their, I don't know, in their statistics in some way. So can you comment? I'm not sure what you mean. What I can say, and I don't know if this is what you're referring to, is that, for example, in dynamical systems where you study the statistical properties of the dynamics, the fractal structures occur very naturally. Even in such systems as simple examples like 2x mod 1 or 10x mod 1, which we did in our course, or even in this case, like in f of z equals z squared, when I talked about the dynamics there, the dynamics is very rich. And different points have different orbits. And you can, for example, say, OK, look at all the orbits that have a certain statistical behavior. And that set of orbits will have a factor measure and a factor dimension. So there's definitely a connection between these two in many dynamical systems where you look at the set of points that have certain statistical behavior, and that's connected to the fractal structure and geometric structure of that set of points. That's the closest I can get to that. Very naive questions. All the examples that you have shown can be sort of represented on a screen or on a sheet. Is something that, to us, has to suggest something flat? Let's see. Thank you. Yes. No, that's a very good question. In fact, absolutely, you can do it in any dimension. So there's many. I did not show the pictures, but in the same way that we constructed the snowflake, there are examples. You can take the something which I call, I think, is called the Czerpinski sponge. So you take a cube. And then you remove a middle third. And then you cut it into cubes. And then you remove middle thirds from each one. And then you take what's left. And you remove all the cubes. And you're left with something that still looks like a cube, but it's all full of air. It has dimensions strictly less than three, right? But it's like a sponge. So it has no volume. So jokingly, people call it something like Swiss cheese or something. You can sell it, and you're cheating, because there's no actual cheese in it. But it's empty, but it looks like cheese. But there's many examples you can construct, just as well examples for three-dimensional objects and for arbitrary dimension formula. Yes, yes. Yeah, yeah, you can remove n minus 1-dimensional cubes in a way that basically it's, ultimately, it's a bit like the construction of the cantor set, where you just remove intervals. In the examples we gave, we made a geometric object by adding pieces. But the cantor set itself, I didn't give that example. The cantor set itself has dimension between 0 and 1, right? And you can calculate this dimension, the middle third cantor set. You just remove the middle third, and then the middle third from each one. And it has this self-similarity. And you can use that formula to calculate the dimension. So using that calculation with that algorithm is very easy in any dimension. You just remove pieces, and you get something with smaller factual dimension. Thank you very much.