Knots and dynamics
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00:00
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Transcript: English(auto-generated)
00:02
It's a great pleasure to introduce A.C.N. Jiske who will talk today on knots and dynamics. Jiske got his PhD at Lille in 1979 and immediately after that he was appointed to a position at CNRS. He also hold a position from 1979 to 1981 in Brazil at IMPA and after
00:27
that he went back to CNRS. In 1982-83 he spent a year at City University of New York and since 1987 he moved from Lille to Lyon where he also holds, he holds two positions, one
00:45
at CNRS and now at the Cornell Memorial Superior. A.C.N. Jiske is a member of French Academy of Sciences and he is a recipient of a silver medal of CNRS and also has some other honors.
01:00
A.C.N. Jiske works in the general area of dynamics or better to say dynamics and related topics. Actually some of these topics became related to dynamics due to the work of Jiske. Mathematical interests of Jiske are extremely broad. He obtained fundamental results in
01:21
geometry of foliations, group actions, actions mainly on homogeneous spaces, homomorphic vector fields and also flows, actions of group bach homeomorphisms on the circle and many other fields. A.C.N. Jiske has many students and in general he is very supportive of young mathematicians. Finally I would like to say that Jiske, as you will
01:46
see in a moment, is a fantastic speaker. A.C.N. Jiske. Thank you very much. So as you can see
02:02
from my title I want to discuss some relationship between dynamical systems, especially vector fields in low dimension and topology. The central idea that I will repeat again and again in this talk is that sometimes it is a good idea to think of a vector field
02:29
in three space as being some kind of knot whose length would be infinite. It is my ambition to give a talk that could be understood by everybody in this audience. This implies
02:50
that I spend a fair amount of time recalling some very classical motivations that are well known to the experts. But this also implies that I avoid any kind of technicality so that my
03:09
presentation of the current state of research will be very impressionistic, so to speak. However, in the final part of my talk I plan to go to one example that I want to describe
03:28
with some detail and I hope it will shed some new light on a venerable dynamical system coming from number theory. So let's go. Let me begin my journey in 1963 when the
03:45
meteorologist Edward Lorenz was studying some very, very simplified model for the motion of the atmosphere and his model produced the following differential equation, an ordinary
04:01
differential equation in three space. And when he plotted the solutions of this differential equation on his very modest computer, this is what he saw. He saw this remarkable picture and what is most amazing with this picture is that it is very robust. If you change the
04:30
coefficients of this differential equation a little bit, if you add some extra small term to the equation, you still see the same thing. The orbits try to approach some object that is
04:46
called today a strange attractor. So I will not discuss today the actual relevance of this picture to the fluid motions, but what I can say is that when you see this picture and when you
05:05
realize how stable it is, you just want to understand it. And in any case you want to put it in the mathematical zoo of remarkable objects. Well, today this strange attractor,
05:25
this Lorenz picture, has become the most paradigmatic picture in chaos theory. So that was 63. In the 80s, Biermann and Williams had a wonderful idea, a simple and
05:44
crucial idea, that if you take such a differential equation in three space and if you pick an orbit and if this orbit turns out to be a periodic orbit, after all it's a closed curve in space and a closed curve is a knot. So they had the idea that maybe one could approach
06:04
the topological analysis of this object through the analysis of the periodic orbits that it contains. So let's have a look at some of these periodic orbits. Here's the first one which is kind of a trivial from the topological point of view.
06:25
Let's have a look to a second one which also is more complicated, but from the topological point of view it's also trivial. And if you wait enough you have this red curve which turned out to be a trefoil knot. So their idea was, well, maybe we could try to understand
06:45
the nature of the knots which appear inside the Lorentz attractor. Here's a collection of several periodic orbits and when you see this picture you say, well, maybe we could approximate
07:02
the Lorentz attractor and think of it as being a limit, so to speak, of a collection of knots. So please keep this picture in the back of your mind because you will see it again, or a variation on it, later in this talk.
07:22
So Biermann and Williams started a study trying to understand which knots appear and which knots don't appear inside the Lorentz attractor. And they proved that indeed the knots that appear in the Lorentz attractor are very peculiar. For instance, they proved that they are prime,
07:49
that they are linked, they are fibered, they have positive signatures, but probably most of you don't know what these words mean. But don't worry, let me show you a collection of eight knots. And as a sample of Biermann and Williams
08:08
theorems, let me tell you that among the 250 knots with less than or equal to 10 crossings,
08:20
only eight of them appear in the Lorentz attractor, those and no other knots. If you go to more crossings, for example, 16 crossings, which is the maximum that has been tabulated, the total number of knots with 16 crossings is 1,701,936 knots.
08:49
Among them, I checked, or better my computer checked, that only 21 of them appear in the Lorentz attractor. So really they are pretty special. Let me just show you
09:05
one example. The figure 8 knot that you see on the right part of the screen is the first example of the knot that you will never meet inside the Lorentz attractor as a periodic orbit. How does one prove such a theorem? Well, the trick of Biermann and Williams
09:29
was to use some kind of combinatorial approach. They built what is called Lorentz template. Rather than giving formal definitions, let me show you a little film
09:45
about that. You start with two bands of paper in the space that you unfold as you see on the screen. At the end, you produce inside space a branched surface. This branched surface is
10:03
a long-term interval that you see in the middle of the screen. This two-dimensional object is embedded in space and carries a natural dynamical system that I'm going to show you now. As a matter of fact, this dynamical system is a semi-flow. Each orbit of it
10:31
intersects the vertical segment exactly as the orbit of the doubling map. In other words, if you parametrize the interval, the central interval,
10:45
by some number between 0 and 1, the return map is just x goes to 2x modulo z. So, if you look for periodic orbits and you want to understand their topology, everything will be coded by periodic orbits of the doubling map. So, for example, one-third
11:07
multiplied by two is two-thirds multiplied by two is four-thirds, but four-thirds is one-third modulo one. So, you are back to one-third. Let's have a look. This is one-third,
11:22
two-thirds, back to one-third. The next is one-seventh, two-seventh, four-seventh, back to one-seventh. The next is one-fifteenth, two-fifteenth, four-fifteenth, eight-fifteenth,
11:41
back to one-fifteenth. And the last one is one-fifth, two-fifths, four-fifths, three-fifths, back to one-fifth. Not easy to pronounce. Okay, so what Biermann and Williams did was a combinatorial analysis of this picture, and they understood the knots that are created
12:05
by this procedure. But the fact that this picture does correspond to the Lorenz attractor with the actual differential equation I put on the board or the screen at the beginning of my talk is a much more recent result. It has been proved by Warwick Tucker in 2002.
12:28
So, the theorem I put before that Lorenz knots are five, blah, blah, blah, actually is a recent theorem, so to speak. Anyway, this kind of idea provides a clue
12:46
to a rough classification of dynamical systems. One could use as a scale to measure the dynamical system the complexity of the knots that it contains.
13:04
So, for example, on top of this hierarchy, you could put the wonderful, amazing examples of Robert Grist, who constructed a vector field in three-space, which has so many periodic orbits that they represent all possible knots. Can you believe that? This is just
13:28
incredible. He will give a talk in this congress, I think, tomorrow. I recommend you to attend. However, some dynamical systems are pretty complex with complicated dynamics in it
13:45
and have a small number of periodic orbits. Even some have no periodic orbits, like the amazing theorem of Christina Kupferberg. It could have no periodic orbit at all. So, maybe we should change a little bit our point of view, and the idea is that one should
14:03
not only concentrate on periodic orbits, but we should also look at knottyness of non-closed orbits. This is an idea, which is a very old idea, that I would like to concentrate under the name of a principle of Schwartzman, Sullivan, and Thurston,
14:27
and many others, of course, that really it is a good idea to think of a volume-preserving or measure-preserving dynamical system as being some kind of asymptotic
14:42
cycle. Let me show you a picture to explain to you what I have in mind. Here's a dynamical system, a vector field in free space, and in light blue, you have a long piece of orbit, which is long but not closed.
15:01
So, the easiest way to make it close is to connect the two endpoints by a segment, and after doing that, you get a closed curve in space, depending on the time you spend and initial point. Now, what you would like to say is that when time is very big,
15:27
and if you pick many points x as original points, what you will create in this way is a good approximation of the vector field. So, I would like to write the symbolic equation that the flow should be thought as being a limit when t goes to infinity of the average
15:48
of one over t k of t x d mu x. So, these equations means nothing. You know, multiplying a knot by a number is a crazy thing, and integrating a function with values in the space
16:03
of knots is even more crazy. But anyway, it is a very productive idea. Think of a vector field preserving a measure as being some kind of limit of average values of knots. This is something
16:20
like an ergodic point of view, mixing dynamics and topology. Well, let me give you one historical example where this approach was very productive. This was very productive in the explanation of an invariant called helicity in fluid mechanics,
16:44
and which is related to the asymptotic linking number. Before I told you I want to be understood by everybody, and since maybe not everybody knows what is the linking number, let me, allow me to spend a few seconds recalling what is a linking number. So, suppose you have two
17:06
closed curves, oriented closed curves in space, the pink one and the blue one. They are disjoint in space, but of course their projections on the screen are not disjoint. You can see that on one point the pink is over the blue, and on some other point the
17:23
blue is over the pink. So, let us concentrate on the pink over crossings, and let us look at locally what is happening. You could have basically two possible local situations. It could be like that, or like that. So, you attach the sign plus one to the first case,
17:45
and the sign minus one to the second case. You sum these local numbers to all over crossings of the pink over the blue, and you get what is called the linking number of the two curves. The main point, of course, is that this number does not depend on the way you project
18:05
your two curves on the plane, and it is invariant if you move by some isotopy the two curves without creating intersection points. So, for example, in this case you have linking number plus one because there is only one overcrossing of the pink with a positive sign,
18:23
and here you have two overcrossings of the pink over the blue with opposite signs, so that the linking number is zero. Okay, so now that we know what linking number is between two knots, disjoint knots, and since I told you that we should think of dynamical
18:44
systems as being limits of knots, it is a very obvious idea to try to compute linking numbers between dynamical systems, and this is what Arnold did. Here's the theorem of Arnold.
19:03
His idea was to compute the linking number of the flow with itself. So here's the theorem. You take a flow, let's say in some bounded domain of R3. You assume it's preserving some probability measure mu, ergodic, and you pick now two points, x1 and x2, and two times t1 and t2, which are very big.
19:29
Now you create the two knots, k of t1, x1, k of t2, x2, that you see on the picture. You compute their linking number. You divide by t1, t2, and go to the limit,
19:46
and a version of Birko's ergodic theorem implies that this limit exists almost everywhere and is independent of the choice of x1 and x2 almost everywhere, and this produces a number which is
20:03
called helicity. Let me go to an open question which has been mentioned by Arnold. Is this number a topological invariant? So what do I mean by that? I mean the following.
20:24
Suppose you have two dynamical systems, two flows, phi t1 and phi t2, which are preserving two measures, mu1 and mu2, and suppose they are conjugated by some homeomorphism H. This homeomorphism H is supposed to be orientation preserving
20:42
and sending mu1 to mu2. The question is, does that imply that the helicities of phi1 and phi2 are the same? This is, I think, a nice question. Maybe I should tell you why I am emphasizing this question and not others. One of them is
21:09
that I would be happy if somebody could answer it because it has been preventing me from sleeping for about 10 years. I think it's enough conditions. The second reason is probably more serious is that if you have a look at any picture
21:27
of a fluid from real life or even from fluid in numerical simulation like this one, for example, of a turbulent flow past an obstacle. Well, when you see this kind of picture,
21:45
you cannot prevent yourself from a quasi-philosophical comment that real life is maybe not so smooth. And really, if we want to produce invariants that have some useful,
22:03
which could be useful for our colleagues from dynamic, from fluid dynamics, our invariants have to be topological invariants. If they are not topological invariants, forget it. So that's why I believe this is a good question.
22:23
Well, this question is open. However, like any mathematician in front of a difficult question, I'll make it simpler. So let's go to a simpler question. And a way to make it simpler is to go from dimension three to dimension two. So let me recall a very classical construction
22:47
in dynamical systems, which is called a suspension. It is a way to go from dimension two to dimension three. Suppose you have a diffeomorphism of the disc. Let me call it F.
23:02
Now you take a cylinder, product of a disc and an interval, you bend it in space, and you glue the two faces by using the diffeomorphism. In this way, you produce a solid cylinder in space, and in this cylinder, you have a vector field which is
23:23
rotating around and around, and which is such that the orbits of this vector field intersect the transversal discs exactly like the orbits of the diffeomorphism. So it is a way to produce from a diffeomorphism a vector field called the suspension
23:43
of the diffeomorphism. Now clearly, not all flows are given by the suspension of diffeomorphism because not all flows have the properties of rotating always in the same direction. However, the category of flows given by suspensions is already a rather big category
24:09
that is worth studying. And I think you understand that I want to tell you that this question of Arnold is settled in the case of flows which are obtained by suspension.
24:25
Well, the other good point with this procedure is that you go to the, instead of looking for flows, you look for diffeomorphisms, and diff of the disc respecting the area is a group, and I like groups. We can begin playing games with groups and algebra.
24:50
So let me mention a sequence of theorems in a row, and then I will comment on them. First theorem is a theorem due to Calabi. It conceals the algebraic structure
25:07
of the group of all say infinity diffeomorphisms of the disc, which are area preserving and which are the identity in the neighborhood of the boundary. And the theorem says
25:23
that there is a non-trivial homomorphism from that group to the real numbers. I will come back to this homomorphism later. The second theorem is due to Banyaga.
25:41
It says essentially that Calabi's homomorphism is the only homomorphism. More precisely, it says that if you take the kernel of Calabi's homomorphism, which is a normal subgroup of diff respecting the area, this normal subgroup has no normal subgroup.
26:05
So the kernel of Calabi is a simple group. The third theorem is a connection between Calabi and helicity of Arnold. This is a theorem of Gombodu and myself.
26:21
It says the following. If you take a diffeomorphism of the disc, if you take its suspension, it produces a vector field in 3-space. This vector field has a helicity, as defined by Arnold, and the claim is that the helicity of the suspension of f is equal to the Calabi of f.
26:45
This theorem would be just a simple observation or nice observation, which would be kind of useless if the next theorem would not be whole true. This is also a theorem of myself and Gombodu that the Calabi homomorphism is a topological invariant,
27:04
which means that if you have two diffeomorphisms of the disc, which are conjugated by a homeomorphism, area preserving, then these two diffeomorphisms of the disc have the same Calabi number. So if you put all these theorems together,
27:24
you get as a corollary the answer to Arnold's question in the special case of flows, which are given by suspension of diffeomorphisms. And since many people in dynamical systems
27:44
tend to believe that suspensions are kind of significant in the world of all dynamical systems, I think it's a good hint that the original question of Arnold might be true. Well, I cannot give you the proofs of all these theorems. I would love to do so, but
28:07
time is limited. The only thing I can do is to give you a few keywords and give you some rough ideas of why something like that could be true. And the good point is that, thanks to Albert Fati, we have a nice definition of Calabi's
28:28
homomorphism. So let me recall or tell you what is this definition of Albert Fati. Start with the diffeomorphism of the disc as before, which is area preserving
28:43
and identity close to the boundary. Now, it turns out that this group, the group of all say infinity diffeomorphisms, blah, blah, blah, is a contractible group. In particular, it is connected and simply connected. This means that you can connect
29:04
your diffeomorphism to the identity by some path, F sub T, the path starting from the identity and going to F. This path may not be unique or is not unique, but its homotopy class is unique since the group is contractible.
29:26
So the idea now is this. You pick two points in your disc and you look at the motion of these two points under the isotopy F sub T. These two points move and you look at the way the interval connecting them is moving
29:43
and you get some rotation. Let's have a look at the picture. You have a fluid motion in the disc. It's an area preserving motion in the disc. You have two points which are moving and the interval connecting them is rotating, sometimes positively, sometimes negatively.
30:02
And now the definition of Calabi's homomorphism from Fati is that you just integrate the total angle of variation of this interval over the space of pairs of points. So if you don't know what was the original definition of Calabi, don't worry,
30:23
take this as a definition. It is the average of all pairs of points of the rotation angle in some isotopy connecting the diffeomorphism to the identity. If you do that, you clearly see it's a homomorphism.
30:42
What is not clear is Banneker's theorem that this is the only one. Now, to sum up this part, let me just say that Calabi's homomorphism measures the amount of rotation contained in a diffeomorphism.
31:04
Can we go further? Well, there's an open question, which I think goes back to John Mather. It's another bright question about the nature of the group of all homeomorphisms
31:21
of the disc which are the identity of the boundary and which respect the area. The question is, is this group a simple group? Note that if you replace homeo by diff, the answer is no, because Calabi's homomorphism is a homomorphism, therefore the kernel is a normal subgroup.
31:46
One way to prove or disprove this question of Mather would be to try to extend the definition of Calabi's homomorphism from diffeores to homeores. The definition of fatigue makes it clear what you want to do.
32:03
We take points, measure the amount of rotation, integrate. Why not? The bad point is that when you integrate, the function might be not integrable, so you're kind of lost. So this is an open question. Can you do something like that? A second guess is a good candidate
32:23
for a normal subgroup which has been created by O. This is a very nice normal subgroup of the group of homeomorphisms of the disc, blah, blah, but nobody knows if this normal subgroup is trivial or not.
32:41
This is called the subgroup of homeomorphisms of the disc. O will speak about that next Saturday. I recommend that you attend his talk. He will probably discuss this kind of topic. Let's try to go further. Can we find more invariants?
33:04
We have Calabi's invariant. Can we find more? Well, we know that there is nothing more than Calabi as a homomorphism. I told you this is due to Baniaga. Now there's a nice algebraic concept
33:20
which seems to be more and more useful in dynamical systems these days, which is the concept of quasi-morphism. The definition is very simple. You have a group, gamma, and a map, chi, from gamma to R.
33:41
One says that it is a quasi-morphism if chi of gamma one, gamma two, minus chi of gamma one, minus chi of gamma two is bounded. It would be zero if it is a homomorphism if bounded is a quasi-morphism. One says that this is quasi-morphism is homogeneous if it satisfies chi of
34:04
gamma to the N is N times chi of gamma. This concept originates from a wonderful theory called bounded cohomology, which has been basically developed by Misha Gromov.
34:22
I would really love to have a few hours to discuss it, and I cannot. So fortunately, I'm lucky because there will be a talk in this congress by Nicolas Monod. I think it's next Tuesday, and I'm sure he will discuss connections and motivations for the introduction of this concept.
34:43
But as an appetizer for his talk, let me just mention two theorems that you can find as a crude idea of what the meaning of this kind of object. A simple example is this. If you have an abelian group,
35:01
or even a solvable group, or even an amenable group, then it turns out that this concept is not useful. Quasi-morphism from abelian groups to R when they are homogeneous are the same thing as homomorphisms. So this is not a good concept.
35:26
Oh, I did it wrong. Sorry, I made it wrong. However, if you take a free non-abelian group, non-abelian group, or more generally, a Gromov hyperbolic group, this concept is new and very rich, and the space of non, of homogeneous
35:46
quasi-morphisms on Gromov hyperbolic groups is infinite dimensional. So you have two kinds of opposite situations, the non-abelian free group or hyperbolic groups having a lot of quasi-morphisms, and the opposite case is the case of abelian groups,
36:03
and I should say also higher rank lattices in semi-simple groups. This is the theorem of Bioger and Monod. Some groups have no quasi-morphisms. And the theorem I wanted to discuss here is that for the group that we are interested in today,
36:20
the group of all diffeomorphisms of the disk which are the identity near the boundary and which are area preserving, it turns out that these groups are on the side of hyperbolic groups. They have a space of homogeneous quasi-morphisms of infinite dimension.
36:45
I won't prove it, I will just show you the picture. The idea is just, let's do the same thing as we did with Calabi homomorphism, but instead of taking two points, let us take more than two points. For example, here's a picture using six points.
37:04
So on the vertical disk, you take six points. And then as the motion f sub t evolves from zero to one, these six points move in the disk, and they describe in the cylinder six curves
37:28
that of course might not come back to the original points. However, what you can do is exactly what we did in the case of Arnold Helicity. You can add at the end of this cylinder a smaller, thinner cylinder, the blue thin part
37:46
on the right, in which the points come back to the original position just by a segment. If you do that, you produce a braid. It's a collection of six curves starting from six points and coming back to themselves.
38:07
So with this kind of construction, you have a way of constructing algebraic objects, braids. And if you are clever enough to find quasi-morphisms on the braid group BM,
38:23
you can use this quasi-morphism, chi, to produce numbers. And these numbers, you average them instead of averaging of the space of pairs, you average of the space of n-tuples. Well, this is a rough idea of the way you can construct many quasi-morphisms
38:42
on the group of area-preserving diffeomorphisms. So, so to speak, Calabi homomorphism was a measure of the amount of rotation contained in a diffeomorphism, but these numbers measure the amount of braiding
39:01
contained in a diffeomorphism. Of course, in the case when n is equal to two, you have a braid on two strings, and a braid of two strings, the only thing you are counting is the rotation number, and you are back to Calabi's number. Well, I should tell you what kind of quasi-morphism I use, or we use, on the braid group,
39:25
and the keyword is a signature. Signature is a traditional classical invariant of braids and knots, and turns out to be a quasi-morphism, which is very useful in this context. Let's go further. Let's try to find more quasi-morphisms.
39:44
Well, I could speak for hours on this topic, and I think this is not the point in this meeting. So, for some reason, it seems that quasi-morphisms are blossoming. In all papers these days, and it would take me an infinite time to describe all these new objects,
40:02
so I decided to choose only two recent constructions, and to stop there. So, the first theorem I want to mention is a beautiful theorem by Entoff and Polterowicz, where they study the group of area-preserving
40:23
diffeomorphisms of the sphere, and what they did is that they constructed very nice quasi-morphism of that group, chi, going from diff S2 area to the ligand numbers, which, first of all, is a quasi-morphism, and second of all, has this wonderful property that
40:43
if, by chance, a diffeomorphism of the sphere has the property that it has compact support contained in a small disk, d, small means of area less than one half of the full sphere, then the Entoff-Polterowicz number of f is equal to the Calabi number of the restriction of
41:08
f to the disk. So, it's a nice way of constructing a quasi-morphism out of the many small morphisms, homomorphisms, defined on all disks. As I told you, I would love to
41:25
go into details and speak about that for a long time, but this is really a nice function, functional on the group of diffeomorphisms of the sphere, which is involving a lot of symplectic modern topology. I would have to speak about quantum homology,
41:43
I would have to speak of flow homology, that would take me too long time. However, I have to say something that the construction of Entoff-Polterowicz do not restrict to the two-dimensional case. They have similar constructions for more symplectic
42:03
manifolds going from the sphere to the projective space of any dimension. For example, it is not known whether or not you can define such a quasi-morphism for any symplectic manifold.
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The second example I wanted to mention is a theorem by Pierre P, which is very recent, which is essentially the same statement, except that you replace the two spheres by a surface, a compact surface of higher genus. So you take any compact surface of genus G, strictly positive, it could be the Taurus or some higher genus surface,
42:50
and then the theorem says that there is a Calabi quasi-morphism on the group of all Hamiltonian diffeomorphisms of that surface. With the same property as before,
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if the diffeomorphism F has a support counting in a disk, then the P's homomorphism coincides with Calabi homomorphism. Well, instead of going further and giving you more and more quasi-morphisms and more and more questions, I think it's time to concentrate on a few
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open questions or directions that might be interesting for the future. Let me mention a few of them. The first would be a hope that maybe it's possible to come back from
43:41
dimension two to dimension three. Do not forget that we come from dimension three and we came back to dimension two because I was forced to do so. You remember that Calabi's homomorphism was kind of a weak version of helicity. Helicity of the suspension is Calabi of the diffeomorphism.
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So what would be very nice is to be able to construct invariants for flows, which in the case of suspensions would reduce to the invariants I just explained to you. What would be even nicer is that these invariants of flows would be topological invariants.
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I told you I am only interested by topological invariants. And the final general question is this construction of n-topol-terovitch shows that
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maybe we are just seeing the tip of the iceberg. Maybe we should not restrict ourselves to low dimension like two or three. We need more energy and we have to go to higher dimension. We have to deal the play the game of symplectic or Hamiltonian dynamics in any dimension.
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So really there is a need for similar constructions in higher dimension. So I told you I would prefer to stop here the presentation of the present state of affairs because that would be technical and this is not my purpose.
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Instead I want to make a break in that talk and I want to present you one example. A good point is that it is an opportunity for those of you who are lost or to get a fresh restart. I want to take one example of a well-known dynamical system
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and I want to look at it with new eyes. I want to look at it with eyes of a topologist. This dynamical system is called the modular flow on SL2R modulo SL2Z.
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But before I describe the dynamics, I will begin by describing the phase space, then its topology and finally the dynamics. So the phase space is SL2R modulo SL2Z.
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So if you like groups or lie groups, you are happy already. It's the quotient of the three-dimensional lie group of two by two matrices with determinant one by the subgroup of discrete subgroup of matrices with integral coefficients. So as a space it's a three-dimensional manifold.
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Now if you want to get a better visualization of it, the best way is to say well this is the space of lattices in R2 of area one. Let me explain what I mean. A lattice in R2 is just a subgroup,
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discrete subgroup of R2 which is isomorphic to ZT. Area one is the quotient as area one. So you have one picture on the left on the right showing one example of a lattice. Of course if you have a lattice you can transform it by a matrix. In SL2R you get another lattice of area one. And in this way you get all lattices.
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In other words the space of lattices of area one is a homogeneous space under the action of SL2R. And the stabilizer of the lattice Z2 is of course SL2Z by definition. So really SL2R
47:47
modulo SL2Z is nothing more than the space of lattices of area one in the plane. Okay so now we have the space. So what about its topology? It has been known for many years,
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probably 150 years, that this space as a topological space is homeomorphic to the complement of a trifoil knot in the three sphere. Let me explain that a little bit. There are many many ways of seeing that but let me restrict to the number theoretical point
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of view which is the easiest way to explain it. If you have a lattice lambda traditionally people from number theory associate to it two numbers G2 and G3. They are defined by Eisenstein series. So G2 is 60 times, forget the 60 it's a matter of number theorist,
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60 times the summation over all non-zero elements of the lattice of omega to the minus four seen as a complex number. And G3 is the same thing with 140,
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the summation of the minus six powers of all elements of the lattice. And it is a very old theorem 19th century that these two numbers, these two complex numbers characterize the lattice. More precisely the fact is the following. If you have a map,
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if you map for each lattice lambda you associate two complex numbers G2 and G3, it turns out that this is a map which is a bijection between the space of lattices on the
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way from the so-called discriminant locus where G2 cubed is equal to 27 G3 squared. So the space again, the space of lattices is homeomorphic to the complement in C2
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of this algebraic curve G2 cubed is equal to 27 G3 squared. Let's have a look at a picture. This represents C2. The horizontal blue axis is G2 is equal to zero. The vertical green axis
50:22
is G3 is equal to zero. And in yellow, if you can see it, is the algebraic curve G2 cubed is equal to 27 G3 squared. But be careful, C2 is R4. So this picture is actually a four-dimensional picture. The blue axis is actually a copy of C,
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or as you prefer, a copy of R2. So this is two-dimensional. And if you want to look at the space of lattices of area one, you have to impose a co-dimension one condition, which means in practice that you have to slice this picture
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by the unit sphere S3 in C2, which will intersect the yellow curve, which is not a curve, it's a surface, which will intersect the yellow curve on the knot. And this knot is a trifold
51:21
knot. So since this picture is very important to understand, I will show you a little film to show it. Here's the picture. What you see here is the three-sphere, or better to say, the stereographic projection of the three-sphere, and the sphere is rotating. In yellow,
51:42
you see a trifold knot. G2 cubed is equal to 27 G3 squared. In blue, you see one of the two axes, and in green, you see the other axes. So now we have a space. We kind of understood its topology. Now dynamical system. The dynamics is incredibly easy.
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If you have a lattice in R2, you can just transform it by diagonal matrix e to the t, zero, zero, e to the minus t. That's another lattice with the same area.
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So this formula defines a dynamical system in the space of lattices, which is SL2R, Mars-0, SL2Z, which is the complement of the trifold knot in three-sphere. This is the one dynamical system I want to look at. And I want to look at it, as I told you, using the glasses from a
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constitute knot, et cetera, et cetera. So let's have a look at periodic orbits
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of this particular dynamical system. Well, I must say that these periodic orbits have a very, very long tradition in mathematics. Let me first of all show you how to construct them. Take a two-by-two matrix, SL2 and SL2Z, A. So it's a two-by-two matrix with integral
53:26
coefficients. So by definition, this two-by-two matrix preserves the integral lattice Z2, clear enough. Now assume your matrix A is hyperbolic. Hyperbolic means the trace has
53:44
modulus at least three. Or if you prefer, assume that the matrix A is diagonalizable. Then in this case, you can make it diagonal by some change of, by some matrix P. There is a matrix P which makes it diagonal. So you have a matrix P such that P, A, P inverse is plus
54:05
or minus one the diagonal matrix. So if you define lambda to be the image of the lattice Z2 by P, of course the lattice lambda is fixed by exponential by diagonal matrix, and you have
54:24
a periodic point. So you can see on the right a little film showing this film is a periodic film. So it corresponds to a periodic curve in the space of lattices. But look carefully. A single
54:40
point is not periodic. A single point is moving along a hyperbola. But the lattice is periodic. So in other words, for each matrix A in SL2Z, which is hyperbolic, I created a closed loop in the space of lattices. Let me mention just a few cases where these
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periodic orbits appear in mathematics. They appear, of course, as conjugacy classes of hyperbolic elements in PSL2Z. They correspond to closed geodesics on the modular surface.
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They are coded by ideal classes in quadratic fields, in quadratic real fields. They correspond to the classification of indefinite quadratic integral forms in Z2.
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This was the point of view of Gauss. One can reformulate everything in terms of continuous fractions, etc. I think most of you have met in one way or another way these periodic orbits in your life. So now it's time for me to go to the question I want to address. We have for
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each matrix a periodic orbit. This periodic orbit is a closed curve. It's a closed curve in the space of lattices. The space of lattices is the complement of the trefoil knot. So question, what kind of knots do we get?
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So let me call them for a moment modular knots. What are the knots that we get by this procedure in free space? And second question, how do they link with the trefoil? For each matrix we have a knot in free space in the complement of trefoil knot.
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How can we compute the linking number of this knot with the trefoil knot as a function of the matrix? So let's have a look at some examples. This is the matrix 1 1 1 2,
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in yellow as usual the trefoil knot, in green the corresponding knot. This is 1 1 2 3, you see that it's trivial but it's not situated in the same way as the first example. This is a trefoil knot, the red situation. Here's another one. You see for each matrix you
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can ask the computer to draw these pictures. I will make a comment at the end of my talk on the making of these fields. And if you pick a random matrix, we want to understand that.
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Before I answer question 1, let me answer question 2. Before I tell you what kind of knots we get, let me tell you how they link with the trefoil knot. So here's a theorem. The linking number between the knot KA associated to a matrix A is equal to the Rademacher's
58:05
function R of A. Well, you might say, who knows what the Rademacher's function is? So I should recall what is Rademacher's function. Michael Attila wrote a wonderful paper
58:23
on Dedekind eta function. And he says in the introduction that Dedekind eta function is one of the most studied functions in mathematics. So I thought maybe somebody in
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function. So let me recall it just in case Attila was wrong. So Dedekind eta function is defined by this infinite product. Eta of tau is the product of exponential of
59:01
I pi tau divided by 12 times this infinite product that you can see on the screen. Tau is a complex number in the upper half space. Imaginary part of tau is positive. The nice point about this eta function is that its 24th power is a modular function,
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which means that if you evaluate the 24th power of eta on A tau plus B divided by C tau plus D, you get eta of tau to the 24th power multiplied by the modular factor C tau plus D
59:42
to the 12th power. And this is true for any two by two integral matrix A, B, C, D. And this is the raison d'être of this eta function. Now, as you see from the definition of eta function,
01:00:00
It is a product of non-vanishing terms. So this is a non-vanishing function, which is a holomorphic non-vanishing function on the upper half-place. And the upper half-plane is simply connected. So it does make sense to take the log of eta, because we have a non-vanishing holomorphic
01:00:21
function on a simply connected domain. So let's put a log in front of this modular equation, and you get this, 24 times blah, blah, blah, plus an integral multiple of 2i pi. And this integral multiple of 2i pi
01:00:40
depends on the matrix A, and it's called the Hadamard's function. It is a very natural function defined on PSL2Z, on SL2Z, very natural from the point of view of modular functions. So this is the original definition that one finds in the manuscripts,
01:01:01
unpublished manuscripts of Riemann. In Atiyah's paper, a wonderful paper, I just recommend this paper. In Atiyah's paper, he gives seven definitions of seven functions on SL2Z, and then at the end of the paper,
01:01:23
there is a theorem that they are all equal. And these seventh definitions are really amazingly different. Some are dealing with combinatorics. Some are dealing with number theory. Some are dealing with L functions. Some are dealing with topology, in dimension two,
01:01:41
in dimension three, even in dimension four. And all of them are equal, and of course, as you got it, equal to this Hadamard's function. So the purpose of this talk is to add an eighth definition.
01:02:01
It is not difficult to see that this Hadamard's function is kind of trivially a quasi-morphism, and this is related, of course, to my talk. Okay, so now you know what Hadamard's function is. So let me give you not a proof, but a sketch, or not even a sketch, some words about the proof
01:02:22
of the theorem that the linking number of KA and the trifold knot is indeed what I told you. So the key word is the theorem of Jacobi. Jacobi proved that there is an incredible relation between the eta function of dedicant and the discriminant.
01:02:42
So here's the theorem of Jacobi, that if you take the lattice Z plus tau Z, it is a lattice, you compute its G2, its G3, then you compute its G2 cubed minus 27 G3 squared. The number you get is essentially
01:03:01
the 24th power of the eta function. So this provides a link between the eta function of dedicant and the discriminant locus. Now, I hope you remember that G2 cubed minus 27 G3 squared is equal to zero
01:03:22
is the equation of the trifold knot. And I also hope you remember this equation that the log of the complex number is basically the same thing as its argument. So if you have a curve in the complement of trifold knot,
01:03:40
and if you want to measure its linking number with the trifold knot, the thing you have to do is to take G2 cubed minus 27 G3 squared and measure the variation of its argument. But the variation of the argument is essentially the same thing as the variation of the log of this complex number. But the log of this complex number
01:04:01
is the same as the log of the eta function by Jacobi, QED. The linking number of Ka is indeed related to the eta function and hence to the Hadamard's function. That was fast, but I promise it is not difficult.
01:04:21
Well, let's have a look to make it clearer. If, instead of looking at the set of points where this discriminant is zero, you look at the set of points where this discriminant is a positive real number. This is now, could I mention one? It is a surface. It's not a knot, it's a surface.
01:04:41
It's called a Seifert surface. Let's have a look. This is our trifold knot that I'm constructing in front of you, the Seifert surface. This is the space of points, locus of the points where the discriminant is a positive real number.
01:05:02
Now, if you move in the complement of trifold knot, the discriminant is moving and each time it becomes real, you cross the Seifert surface. So let's have now a look at the combination of this picture and our knot, KA.
01:05:24
Here you can see a knot, KA, which is developing in space and which intersects from time to time the Seifert surface. It can intersect it from the red side to the yellow one
01:05:43
or from the yellow one to the red one according to whether G2 cubed minus 27 G3 squared goes from negative imaginary part to positive imaginary part or the other way around. So each time you have an intersection between the knot and the surface,
01:06:02
you count one point, a positive point or a negative point according to the sign of the intersection. And the linking number we are discussing is just the total number of algebraic intersection between the knot and the Seifert surface.
01:06:21
Here, for example, you see two red balls and one blue ball corresponding to two plus one signs and one minus one sign. Okay, so we basically answered question two. Now the question is, what about question one? What are these knots?
01:06:44
And the answer is that we are back to the beginning of this talk. The modular knots are exactly the same as the Lorenz knots. I will not prove that today.
01:07:01
That would take some time. I will just explain to you a few keywords, two little films, and wave my hands a little bit. The proof is in two steps. Since we want to prove that modular knots are the same as Lorenz knots,
01:07:22
we have to find somewhere in our phase space Lorenz template. And to create a Lorenz template, we do it this way. First, we look at those lattices in the plane which have an hexagonal shape,
01:07:42
a regular hexagonal shape. This is a one-dimensional object because you can rotate it in any direction. So the space of lattices which are hexagonal is a circle. Then you look at the space of lattices whose fundamental domain is a rhombus,
01:08:02
a horizontal rhombus where the free parameter is the angle of the rhombus that you allow to move from 60 degree to 120 degrees. This is an interval in the space of lattices. Don't worry, you will see a picture in a moment. These two pieces, this circle and this interval
01:08:23
together create an object which is basically in the shape of a theta. And then you take this one-dimensional thing and you push it in the unstable direction of the modular flow, whatever it means.
01:08:42
Let's have a look. This is the trifle knot. I'll put it in right position to see something. And then we create this one-dimensional object. And then we push in the unstable direction to create this object which looks like the Lorenz template.
01:09:09
Now we have this template. And we want to show that modular knots are Lorenz knots. So the trick is we take our knots and you want to push them
01:09:22
to put them in the right position on the template. So we need what's called an isotopy. We want to push everything to the template. To do it, I will not explain it, have no time. But for those of you who are familiar with number theory,
01:09:42
it is nothing more than the Gauss reduction process for quadratic forms. Just put it in right position. Or for those of you who prefer the modular surface, I'm just contracting the modular surface on its spine, the geodesic connecting the two singular points. Anyway, let's have a look.
01:10:06
Oops. So here you see one knot, Ka,
01:10:21
in presence of the template as before. And then here's the deformation. And you clearly see that the knot goes to the template and after the deformation is exactly in the position as I explained to you for the Lorenz knot.
01:10:43
Here's another example with a green one, more complicated. I've chosen these matrices in such a way that the corresponding knots are kind of nice. This is a forgot trefoil knot. No, this is not trefoil knot. Here's deformation.
01:11:00
I will show you a third example where you see that really these knots can be isotoped to what they should be in the case of the Lorenz attractor. Here's a third example with a more complicated matrix.
01:11:23
And finally, I will show you all these examples together to show you that not only this is true for knots, but this is all true for links. If you take together several periodic orbits, you can simultaneously map them to the Lorenz template.
01:11:43
Look at the picture and you see the final picture is what I told you at the beginning. Keep this picture in the back of your mind. This is the same one.
01:12:04
Well, what's next? Well, what's next is, well, I'm quite happy with this theorem because we have connections between two different parts.
01:12:21
So, it would be nice to go from one part to the other part. And I did some progress recently, especially understanding the vibration, but this is too recent so that I cannot explain that to you. I wish I had more time to explain that. Well, we did some long way from meteorology
01:12:44
to numbers through dynamics and topology and some group theory. Let me finish by a quick summary and maybe a short comment. The summary is that, I think the best is to say
01:13:01
what I told you at the beginning. Well, I think it's a good idea to think of a dynamical system as being some kind of infinitely long knot. This will provide some new insight to dynamics, I think. For the comments, no, no, no. Sorry, I'm finished in a minute.
01:13:20
For the final comment, let me tell you. A few months ago, I met via internet a non-mathematician. His name is Joss Les. He's a mechanical engineer. His mathematical background is more or less, according to him, first year of the university.
01:13:43
However, he loves doing mathematics and producing nice pictures. I suggest that you go to this website or you go to the exhibition hall here on fractals where one of his images have won a competition. And what he, we started the collaboration together
01:14:03
and the films that you have seen today and many more are really the result of a joint work. I wanted to thank him for that, but not only for that. You know, I have the feeling that in the last decade, mathematics has been blowing up into a multitude of little spheres
01:14:24
which are kind of disconnected. And I think it is our duty to connect them and to work hard to understand better ourselves. But not only between mathematicians. I believe that it is the duty
01:14:42
of every mathematician, whatever its field, to work hard to get contact with non-mathematicians. I really believe it's a necessary condition for our survival. And I wanted to thank Joss for reminding me of this.
01:15:03
And in order to support this last assertion, I wanted to hide myself between two quotations from Hilbert in this congress, 160 years, 106 years ago, when he said the following. The mathematical theory is not to be considered complete
01:15:23
unless you made it so clear that you can explain it to the man you meet on the street. And he also said something which is very dear to me and which is probably the shortest summary of why I love mathematics for what is clear
01:15:41
and easily comprehended, attracts and the complicated repairs us. Thank you.