 Good morning everybody. I'm Jens Markloff and this is my friend and colleague Andrea Ström-Werkson. We've worked together for many many years, wrote many papers together and but this is a first of the first lecture in the series we're going to give together. Okay, so I hope it'll be entertaining. Now this will be a lecture course about homogeneous dynamics, which in the past 20 years has seen many very spectacular applications and in number theory and mathematical physics and in some other areas and the first lecture here will give a survey of the subject and then the plan is for the remaining nine lectures these will be on the blackboard and will start from zero from scratch because I know many people here are not in this field and want to learn the basics. Okay, so the first lecture will be a fast outline of what's there in the field and it's not necessarily going to be the topics that we discuss but it's going to be sort of the the historic highlights and some of the basic ideas. Now who is a PhD student here? Okay, very good. That's one third maybe. Who is a postdoc? Very good and who is a senior academic? Ah, Modulo, good. Yeah, you're the only one it seems. Okay, so this is really aimed for the PhD students and also for people who might not be in the area, okay? If there's something unclear, please ask immediately, raise your hand and we'll see you. We have four eyes here. We will see you. If it's a question that goes a little bit further then maybe you hold that back for the tutorials, okay? So let's get going. Applications of homogeneous dynamics from number theory to statistical mechanics. So in this lecture, we'll first give you a little introduction about what homogeneous dynamics is. A particular important and powerful tool in this field is measure rigidity and that's what we will explain in the second paragraph. Then we'll talk to three classic historic applications, the proof of the Oppenheim Conjecture by Margules, which was the first spectacular use of measure rigidity. Then an application in quantum chaos will tell you various applications including Lyndon Strauss Fields Medal winning work and then we'll come to a set of very cute problems that are extremely easy to formulate but very hard to prove and there'll be an opportunity for you to, for those who are interested to, to some little numerical experiments with these very simple number theoretic sequences and then finally we'll tell you about what's coming next in the, in the subsequent nine lectures. So what is homogeneous dynamics? Homogeneous dynamics is a shorthand for dynamics on a homogeneous space. So what is a homogeneous space? Maybe I'll take the point there. Yeah. A homogeneous space is constructed by taking a lead group G and in our lectures that will be coming we'll talk about very specific examples like the group of invertible two-by-two matrices and gamma is a subgroup of G that's called a lattice. So a lattice in a lead group is a discrete subgroup so that the fundamental domain of the action of it has finite volume and that's also something we'll, we'll explain. Okay, you don't need to understand this now if you don't know what it is and then you form this quotient space. So you form the set of all cosets, left cosets in this case and these are the points in this homogeneous space. Okay, so that's an algebraic construction and now we want to have dynamics on this homogeneous space and the way we're going to achieve this is we're going to take a subgroup of our lead group, a one-parameter subgroup that's generated by one element phi t and then we multiply the elements in our space from the right by the elements of this subgroup. Okay, and that generates a map on this space. Parameterized by t which can be either a discrete variable in this case you call this discrete time dynamics or a continuous variable in this, in this case you call it continuous time dynamics or that would be a homogeneous flow. So that's the setting. And as I say in the tutorials and later we're going to talk through very specific examples. I'm going to talk about now I'll just flash the most trivial non-trivial one. So you take as your lead group G the real numbers and the real numbers of course are to make it a group you use addition as your group law. So you can think of this as the group of translations and then as your letters you take z the integers they form a group under addition as well. And then your homogeneous space will be the quotient r modulo z and you all know what that is. Well you take a fundamental domain which would be the unit interval and you just identify opposite points or you get a circle. So r modulo z is simply a circle and that is an example of a homogeneous space. We will do agotic theory that is we will talk about measures and invariant measures on the dynamics and in this case a very natural measure is the back measure or in terms of the groups that will also be the harm measure. Again we'll explain exactly what this is in the next lecture. Now to create the dynamics here a particular simple example that fits in as an example for homogeneous dynamics is taking phi to be translation by alpha where alpha is a fixed number. And if say an irrational number or rational number whatever you want can take alpha to be zero in that case we just get the identity map so that's a little boring. But take alpha to be you know translation by pi for example. Then you get this dynamics which is just rotation by alpha of the circle. Okay so that's the simplest example that we have here. Okay so as Jens mentioned measure rigidity is a really important concept or highly attractive goal maybe in homogeneous dynamics. So measure rigidity or rigidity of invariant measures what it means it's not a precise concept but it means that the set of invariant measures for the given flow or the given map. This set is much smaller than what you would have expected from the definition in some sense much smaller and also every invariant measure has some kind of strong algebraic structure to it. So let me illustrate this in just a very very simple case. Oh now it works. So the case that Jens just mentioned the case of a circle rotation or a translation of a torus. So this is a very well known equidistribution result let's consider this translation for an irrational translation so alpha is irrational. And consider the orbit just of some point say the orbit of the point zero. This orbit consists of the point zero alpha two alpha three alpha and so on. And it's a classical result proved independently by by by Zipinski and Boll more than 100 years ago that as we follow this orbit. If alpha is irrational then this orbit tends to become more and more equidistributed on the circle according to the big measure. So in particular what this means is that if I take any subinterval of the unit circle and then as I follow the orbit longer and longer I will visit this subinterval. The number of times will be roughly proportional to the length of the interval. And another way of formulating this is that's taken a continuous test function on the circle. Then if I take the average of the function along the orbit this average tends to the volume average or in this case the Lebesgue integral of the test function F. So okay I will just discuss this simple case in a way to illustrate measure rigidity. So let me outline two proofs. Are you doing it for me? So first the proof not using measure rigidity directly just using trigonometric sums. So this would be today this is a kind of standard application of harmonic analysis. It can be seen as a really good example very down to our example of how harmonic analysis can be applied to equidistribution problems. So the idea is that take the test function the given test function F and expand it as a Fourier series. And then we just have to prove equidistribution for each harmonics individually because well for instance by Weierstrass approximation finite Fourier series are dense in the space of continuous functions with respect to supremum norm. So it suffices to consider finite Fourier sums like this. And then by linearity we may in fact assume that the given test function F is just one single harmonics just e to the 2 pi i kx. And we for just that function for each individual k if we can prove this limit result then we have proved the whole thing for an arbitrary test function F. And if k is equal to zero this result is obvious because for any n this is just equal to the constant value of the function. So then it's really obvious. But then the hard thing for a more difficult problem than this this would be the hard thing to prove that every non constant harmonics that the sum tends to zero. Often then you end up having to bound some difficult trigonometric sum. You need to prove that this for any given integer k not equal to zero that this tends to zero. Now in this case it's easy because it's a geometric sum. So we can just give an explicit formula for this sum. Here it's important that alpha is assumed to be irrational. So this number here is not one so the denominator is non zero. And the numerator is bounded in absolute value by it's less than or equal to two. And then we have this denominator. So clearly this tends to zero as n tends to infinity. So that gives the result. That's a simple proof using harmonic analysis. But another proof thanks would be as follows using measured identity in this very simple case. Then let's reformulate the problem. Let's look at the let's define new n to be the measure that you get by putting the rack mass of weight one over n at each point of the first end points of this orbit. So in other words new n is the measure which when you integrate the function against it it gives us this ergodic average. And then the task is to prove that this sequence of measures new n tends to the Lebesgue measure in the weak start apology as n tends to infinity. And so recall that this space of probability measures or this sequence of probability measures is locally compact. So any sub sequence of such measures will have a sub sequence that tends to some limit. And that's using that fact it's suffices to just discuss what can we say about a limit measure. If I have a some sub sequence of these new n's and this sub sequence tends to some limit measure new then what can we say about this probability measure new. So here the convergence is always in the weak start apology. So then what you if new is such a weak limit then it's very easy to show that new has to be invariant under the map that I'm studying the translation by alpha. And it's just to write the thing out. It's written out in fairly detailed way but the point is that if I take a test function and I compose it with a map and then I apply new n some ergodic average with large n. Then this ergodic average is very close to the ergodic average where I took the f instead of f composed with alpha. It's just some small correction terms. So it follows that new if now we are always assuming that new is a weak limit of some sub sequence of new n's. Then we have it follows that since I have both of these limits it follows that new of f must be equal to new of f composed with t alpha. And this is just the same thing as saying that new if I translate the measure new by my given map then it's equal to its I get the same measure back. So new is invariant under the translation. Okay so new has this property and then what does it give us? Well it follows that new is also invariant under the map composed with itself and in number of times. So new is actually invariant under this discrete group of translations. And now it is a well-known fact and this is actually easier than proving a distribution at least if you in some ways. So this is what's known several decades before it's due to chronicle that the point alpha 2 alpha 3 alpha and so on they are dense on the unit interval. So given any number on the torus we can find the sub sequence of n values such that n alpha tends to y on the torus. And then it follows that that also these if you think about the topology it follows that new also has to be invariant under the translation by y. Using the fact that new is invariant under t alpha raised to n i for all these i's. So actually we've proved that new is invariant under all translations. All translations and then it's a simple exercise you do when you start doing measure theory to prove that such probability measure, the Borel probability measure on the torus which has all these invariants that has to be Lebesgue measure on the torus. So we've proved that any limit of some sub sequence of these new n's has to be Lebesgue measure. And then it's a simple argument this uses reductive ad absurdum that actually it follows that the full limit must tend to new. This uses the fact that any sub sequence has a sub sub sequence that converges. So it follows that then we prove that the full sequence must converge and we again get this equidistribution result that was the second proof. Okay, so the key point here was actually a measure we did it. We proved that any Borel probability measure on the torus which is invariant under the circle rotation by an irrational quantity that has to be the Lebesgue probability measure. And this is actually called unique ergodicity that this circle rotation is uniquely ergodic. So it's Borel is irregular. In principle, is it possible that some very other absolutely other measure is not Borel etc. I always think of Borel measures. When you start asking about not Borel measures, I am completely lost. Yeah, yeah. Yeah. But sure, I mean uniform equidistribution on the on the torus or the circle or whatever that really has to do with the topology. I say for any open set or something. I want that. No, no, I continue. So again, so it's a special case of measuring the duty. Here we had only one ergodic invariant measures, but in more complicated situations, measuring the duty gives us, if we can prove it, it gives us that the set of invariant measures is much smaller than what you might have expected from the definition. And then every invariant measure has some kind of algebraic structure, as I said, and yeah. So now that was a really simple homogenous space. Here I will discuss briefly some more complicated homogenous space that will play a really important role in throughout our course. And this is also very important for many applications also that we will not talk about. So, so yes. Archetypical example, maybe that you are working on when you are working on homogenous dynamics. Let G be the special linear group of n by n matrices with determinant one, considered under multiplication. So this is an n square minus one dimensional manifold. And let gamma be the subgroup of such matrices which have integer matrix, integer entries. This is often called the modular group and consider the quotient. So the set of left gamma cosets. This again turns out to have finite volume with respect to our measures. So one can normalize so that the harm measure gives a probability measure on this space. And on this space, there are many interesting, interesting one parameter subgroups that we can study the dynamics of. Here are the archetypical examples, maybe the diagonal. So any, take any diagonal matrix or any diagonal one parameter subgroup and study its action on the quotient space acting from the right. Or take some unipotent group like this and study that one parameter group. These two flows have really very different properties and we will come to that. Okay, so the next. So one reason that this homogenous space is so important for applications is that this homogenous space. Where G is SLNR and gamma is SLNZ can be identified with the space of Euclidean lattices. So Euclidean lattices in RN. So recall that the Euclidean lattice is, it's just the subset of RN that if you fix a basis, any linear basis of RN. And you consider all the integer linear combinations of this basis, then you get the lattice. And for such a lattice we have a fundamental cell, which is a fundamental region for RN modulo the lattice. So there are two lattices floating around, one is commutative and the other is not. So RN modulo L, if that has volume one with respect to Lebesgue measure, then we say that we have covolume one. And the space of Euclidean lattices of covolume one can be parametrized by this homogenous space where G is SLNR. And okay, I won't describe the details but this is one of the reasons why it has so many applications in number theory, for example. Okay. Okay, so I think you have now an impression of what measure rigidity is about. You've seen a simple proof of a classical equidistribution theorem using measure rigidity. And what I'm going to talk now about is three spectacular applications of this idea of measure rigidity. And again, this is just to give you an idea of the field. This is nothing we will discuss in detail later on. This is just to give you a little bit of flavor of the most important things that have happened. And the first one was Margulis' proof of the Oppenheim conjecture, which was the first application of homogeneous flows in a long, long standing problem in analytic number theory. And as well quantitative versions of this, which have been proved in the last 10 years, have had a major impact in the field. And the reason is that analytic number theorists couldn't prove these things. They tried for a hundred years and they couldn't do it. Okay. Only measure rigidity was able to solve these problems. So the Oppenheim conjecture is concerned with the distribution of values of a quadratic form evaluated at integer points. So you take a quadratic form Q. Could I just borrow the other? So you take a quadratic form Q up here, let's say in n variables. And we want to study in this scenario an indefinite form, which means that the symmetric matrix that defines the form has both positive and negative eigenvalues and no zero eigenvalues. So if they're P positive and Q negative, you say the form has signature PQ. And we are going to look here at irrational matrices, which means that it's not proportional to a rational matrix. So if you take your matrix Q and there's no scalar multiple of it that makes it rational, you call it irrational. And it's formula in this projective sense, of course, because the value distribution properties will not change if you multiply just by a scalar number. Okay. And then the celebrated theorem by Margules asserts that if you take an irrational form, any irrational form that's indefinite and has at least three values. Then you get values of the form evaluated at integers that get arbitrarily close to zero. In fact, he proves more, he proves that the set of values is dense in all of R, but this is the original formulation. And let me just write it down. Okay. I'm removing zero here because obviously the form evaluated at zero is zero. So then that statement would be trivial. So what it says is that no zero integer that you can always find a non-zero integer to get arbitrarily close to zero. Now, the assumption that you have at least three variables is absolutely crucial. Here's an example of an indefinite quadratic form in two variables for which this theorem fails. And that's something we can talk about in the tutorials. It'll also become a little more apparent when I explain to you the approach of Margules' theorem. So the first step in Margules' proof of the Oppenheim conjecture was to translate the problem to a question homogenous dynamics. Observation that goes back to a celebrated paper by Cussles and Swinit and Dyer in the 1955. And that then was rediscovered by Raga Newton who formulated very influential conjectures in this connection. And Margules' contribution was that he proved topological version of these Ragonata conjectures. And here is how it goes. So let's first translate the problem into a problem of homogeneous flows. You already have an impression what homogeneous flows are. It's dynamics on a homogeneous space. So what is the homogeneous space here and what is the flow? And I'm going to explain this to you in the simple case of quadratic forms in two variables. And you immediately remember, ah, but that's the case where it didn't work. Okay, but nevertheless, let me do that because the geometry will be much simpler. Plus you also see where it fails. Why two variables are bad? Okay, so first of all, we can bring our quadratic form into a simpler form by applying a linear transformation. That's basically linear algebra. So you know that you can diagonalize matrices and can do all sorts of things. By diagonalizing this matrix here, I'm bringing it in a particular form that's sort of exactly off-diagonal. And so what this says is we can write our quadratic form q as a form q naught with a certain linear transformation applied and some multiplying factor. Now we are interested only in the value distribution, so multiplying it by anything doesn't really matter. And so now we have this standard form here and the price we paid is that we have a matrix element m in here. Now the second observation is that if you look at this form, q naught, this one here, it's invariant under a certain group of matrices. Yes? Namely this group of matrices here. You've seen this group before in one of the previous slides, this group of diagonal matrices. Can everybody check this quickly? Right? So this here multiplies the x1 coordinate by e to the t and this one multiplies the x2 coordinate by e to the minus t. And if you look at the form, they just cancel. So it's invariant under this matrix. So this is the orthogonal group with respect to this matrix q naught. So the idea now is that in order to show that the values of this form can be arbitrarily close to zero when evaluated along the lattice points. What we need to show now is that we've replaced the original lattice c2 here by a lattice c2m that we can find a lattice point, a lattice here with c2m phi t. We can multiply any phi t because it leaves g naught invariant so that q naught evaluated at this particular vector y is less than epsilon for any given epsilon. That's the idea. That's what we want to show. And then the next observation is that this precisely holds if this orbit here, so now you think of this as an orbit in the space of two-dimensional Euclidean lattices, if this orbit is dense in g mod gamma. And this is a certain and this is now a homogeneous space of the type that you've seen before. That's SL2r modulo SL2z and we'll come back to that so you don't need to understand every little detail. Observation three simply says if you know the density of a certain orbit in your homogeneous space then you're done. Why? Well, because what you can show is that if you're equidistributed in the whole space you have the density of values given automatically because whenever you put a very short vector in here you will become arbitrarily close to zero. And so if your orbit is dense it will come close to a short vector in your space of lattices. That's the observation. Now that's what Margules proved for more than three variables. But why does it go wrong here? Well, there are orbits that do not become equidistributed in the case n equal to two. So we don't get a short vector and these forms that leads to an m, the forms q that leads to an m whose orbit is not dense will be exactly counter examples to the Oppenheim conjecture for forms with two variables. So that's the idea, okay? You take a problem in number theory, you translate it into a problem of equidistribution in a homogeneous space. You prove something about orbits in that homogeneous space. In this case you only need density, need to prove density to get the density of the values of the quadratic form. Later we will want more, we want equidistribution, not just density. But that's the idea here of Margules' proof. Does that give you a little insight of what Margules did? I have to say Margules' proof is extremely complicated, it's not as simple as this. That's why I went to two dimensions, two variables. And then there are quantitative versions that I mentioned where you're not only interested in the density of the values of quadratic forms but actually that quantitative equidistribution. So you're counting the number of lattice points in a big ball of radius t so that the quadratic form that you're looking at has values in some interval a, b and you count all these and you want to understand their frequency. So you normalize this by what you think is the right normalization which is the volume of the corresponding geometrical object that is defined by these conditions. And this wonderful proof of asking Margules and Moses in 98 shows that as long as you have a form with more than four variables and p greater equal than 3, then indeed this becomes uniformly distributed. And only in 2005 they could extend this to forms of signature 2, 2 and under certain deophantine conditions on the coefficients. Very difficult papers and very difficult proofs but the conceptual idea is exactly the same as on the previous slide where you now replace the density of certain orbits by their equidistribution plus many beautiful technical subtleties that I won't go into now. Now the reason why the quantitative versions of the Oppenheim conjecture could be proved is an absolutely amazing fantastic breakthrough of Marina Ratna. I think probably one of the most important theorems in ergodic theory in the last 100 years well ergodic theory I guess is not much older than that but this is really a theorem that is at the heart of measure rigidity and that has lead led to so many applications and without that theorem I wouldn't be standing here in front of you today that's for sure. I might do something else but nothing that beautiful. So what is Ratna's theorem? Well Ratna's theorem does exactly the same thing that Andreas told you about circle translations. Andreas told you that every Borel measure, every Borel probability measure that's invariant under irrational rotations is Lebesgue measure. There's no other measure. Ratna's theorem generalizes this observation vastly to any Lea group. And you take a lettuce in that Lea group and the circle rotations are generalized to groups that are generated by so-called unipotent orbits. And I'm just going to give you one example of a unipotent orbit here. That's a unipotent orbit in the group of two by two matrices and you see it sort of got one on the determinant. This would be an example of an orbit that's not unipotent and that's a very classic example of a very unstable hyperbolic flow and we'll discuss these two things in great detail in our lectures. Okay, so you learn all about these things back then. Now, what Ratna's theorem tells us is that when you have a measure on a homogeneous space, G mod gamma, that is invariant under a group of unipotent elements and if this measure is ergodic as well, well, forget about that. Those who don't know what an ergodic measure is, forget about it. Just think of an invariant measure. Then you can actually produce a whole list of these invariant measures. It will not just be one measure, as in the case of the circle translations. It can be a whole list of measures. But these measures have a very, very strong algebraic structure. They live on nice embedded sub-varieties in your homogeneous space and they can be completely classified. So in these circumstances, you can do a proof that's very similar to the one that Andreas showed you in the case of circle rotations where you say, I want to prove some equidistribution theorem of a sequence of probability measures that are generated by this unipotent dynamics. I prove that the sequence is relatively compact, which means you can go and study limits of subsequences. And then if you can show that any limit of a subsequence is invariant under such a unipotent orbit, then you can use Ratner's theorem to look at the list that she gives you and just look at every probability measure in that list and maybe find some other criterion that rules out this measure or rules out that measure. And if you're very lucky, in the end you only got one measure left and then you're in the same situation as Andreas in the circle rotations. That's the key idea. Now, oh, okay. Okay, so there has also been many applications of homogeneous dynamics to questions in mathematical physics, in particular to quantum chaos. So quantum chaos is the study of how if a classical mechanical system is chaotic, how is this reflected in the corresponding quantum mechanical system? Or if the classical mechanical system is not chaotic, how is this reflected in the corresponding quantum mechanical system? So here's, let me explain this in a really simple example, that of a billiard. So the classical mechanical system then is just that of a particle moving with constant velocity until it hits the boundary of the billiard and then it is reflected according to the standard laws of reflection. Here's one example of a billiard, the so-called cardioid billiard. This billiard, the classical system has been proved to be strongly chaotic. It's very new year, for instance. So how is this fact reflected for the corresponding quantum mechanical system? Well, then we will be considering a quantum particle inside this torus and this particle is described by its wave function and the evolution of the wave function is given by the Schrodinger equation. If we just look at stationary waves, wave functions, then this reduces to just studying the eigenvalues of the Laplace operator with Dirichlet boundary conditions. And since the region is compact, this has a completely discrete spectrum. So we can just order the eigenvalues or the energy levels, if you want, of these stationary states and we have corresponding eigenstates or eigenfunctions. And so in particular, one is interested in the high energy limit and how is the fact that the classical system is chaotic, how is this reflected in these eigenfunctions and in the spectrum? So if we look at the eigenstates first, the eigenfunctions, then here are two examples at a fairly high energy or high eigenvalue of such eigenfunctions computed by Arne Specker. And in the left side, okay, so the physical interpretation of such an eigenfunction is that if I take its absolute value square, this gives a probability density for finding the particle in a given region. So in the left eigenfunction, it seems that this density is fairly well spread out all over the surface and that seems to reflect nicely the fact that the classical motion is chaotic. But then also high up in the spectrum, one finds these examples where the eigenfunction is highly localized. It is localized around a certain simple closed geodesic. And so it's a question, how should you... This is called scarring, by the way, scars when the eigenfunction localizes around some closed geodesic or some other shape when it's not nicely spread out all over the region. So that's one question. So we can also look at the spectrum, at the eigenvalues or energy levels. So I'd like to... I can't... Can you shift to next? So for the eigenvalues at high energies, it's been projected by Bohigas, Giannone and Schmidt in 84 that if the classical mechanics system is chaotic and some generosity, then the eigenvalues should behave just as the eigenvalues of a large random matrix from the Gaussian orthogonal ensemble. And this you can study using various local statistics. So study the sequence of eigenvalues, statistical properties of this sequence on the level of mean spacing 1. So one such measure is to just look at the level spacing distribution. So the distribution of gaps between consecutive eigenvalues. And if you study this distribution for the several first 100 eigenvalues for this cardioid biliad, it matches very nicely what you would get for a Gaussian orthogonal ensemble. On the other hand, so next slide please. Yeah. And this is in sharp contrast to what one expects for an integrable system, so a highly non-chaotic system. If the classical mechanics is integrable, then it has been conjectured by Bérit-Abourg that generically the eigenvalues will behave just as the points of a Poisson process along the real line. So this would mean for instance that the level spacing distribution will be just exponential distribution. For example, by the several first 100 eigenvalues have been computed for this circular biliad, which is of course integrable in the classical mechanics. Okay. So next, let us study this for some other biliads. Let's consider a hyperbolic biliad. So let us take a compact hyperbolic surface and play a game of biliads on that one. Now we have no boundary. So the biliad flow in the classical mechanics description is just a geodesic flow on the hyperbolic surface. And this is well known to be strongly chaotic, and the question is how is this reflected in the quantum mechanical problem? So then we are studying, again, eigenfunctions of the Laplace operator. Now this is the Laplace-Beltrami operator for the hyperbolic metric. And since the surface is compact, we can order the eigenvalues. We have a completely discrete spectrum, and we can order the eigenvalues as a sequence of numbers tending to infinity. And so then the question is, what will the eigenfunctions look like? Here's one plot of such an eigenfunction at a fairly high energy level, again by Aorish and Ull this time. Okay, so we can ask again, will all the eigenfunctions at high and eigenvalues be nicely spread out like this one? Or can we have some kind of scarring? A precise formulation of this is as follows. Consider this measure. This is again the probability measure that gives the probability for finding this quantum particle in some given region. What are the possible limits of this sequence of probability measures as the eigenvalue goes to infinity? In fact, for a hyperbolic surface, if you want to switch, it was conjectured by Rudin-Kesarnak that the only possible limit is the hyperbolic area measure. So there is no scarring, no possible scarring as lambda goes to infinity. So all these Wigner measures become nicely spread out over the surface. And this conjecture is called quantum unique ergodicity. And this is still a wide open problem. It seems to be a very difficult problem. But still, there has been some quite remarkable progress in special cases. So perhaps the most remarkable case is what Elon Lindenstrauss proved. He has actually proved that quantum unique ergodicity holds for special hyperbolic surfaces, arithmetic hyperbolic surfaces. So these are hyperbolic surfaces which come from some subgroup, discrete subgroup of SL2R, defined by a Quaternion algebra. So there exists such co-compact subgroups of SL2R. And if we have such an arithmetic surface, then Lindenstrauss proved that quantum unique ergodicity holds. And the key feature in this proof is that such arithmetic surfaces possess extra symmetries leading to so-called Hecke operators. So there is one Hecke operator for each prime number. And the starting point of the proof is that, okay, we consider some weak limit of these measures, or actually the lifts of these measures to the unit tangent bundle of the surface, the so-called micro-local lifts. Any weak limit of these measures is known to be invariant under the geodesic flow. And that's a basic fact coming from Egorov's theorem. But this doesn't give much information because the geodesic flow is highly chaotic, so there are many invariant measures. We do not have measured rigidity at all. So invariance under geodesic flow alone doesn't give much information. But the point in the proof of Lindenstrauss is to combine this with dynamics coming from the Hecke operators. And then in the end, it kind of proves the measured rigidity result for this situation. Okay, next. Okay, so that is one really central application of homogenous dynamics, a really important application. Some other applications have also been given for the question about eigenvalues, or eigen-energies of the quantum mechanical system. So recall Berry-Tabour's conjecture, saying that if we have a classical mechanic system that is completely integrable, then the corresponding quantum mechanical system should, at least generically, the eigenvalues should behave like points of a Poisson process on the real line. Let's consider this for just the archetypical, I'm using that word too much, but just a really simple integrable system, the linear flow on the torus. Then we are looking for the eigenfunctions on eigenfunctions on the torus. So torus is given by the Euclidean space Rd, modular lattice. And these eigenfunctions and eigenvalues can be given completely explicitly. The eigenvalues turn out to be just a numerator, we can enumerate them by looking at the so-called dual lattice. So the eigenvalues are now given by this sequence of length, or length squared, of all the elements in a certain lattice. So what Berry-Tabour conjecture says that if this lattice is generic, then this sequence should behave like the points of the Poisson process. But note that we have to, before comparing with the Poisson process, one has to rescale the sequence. If the dimension D is three or larger, then this sequence of points become more and more dent, I don't know what is the word, the mean spacing gets smaller and smaller as the eigenvalue grows. So in order to obtain a sequence which has mean spacing one, we have some kind of unfold or renormalize. So we consider the eigenvalues raised to D over two. Then we get the sequence which has mean spacing one. And the question is, does this sequence behave like the points of the Poisson process? So in particular, the gap distribution should be exponential. We don't know how to prove this in a single case. I guess I switched to next already. There are some other statistics, local statistics for sequences. You can look at the so-called pair correlation for the sequence. And that means you do not only look at consecutive pairs, but you look at all pairs of numbers within some given range such that the difference fall inside a given interval. And for this pair correlation, there has been some really nice progress again using homogenous dynamics. So there are results by Eskin, Margulis and Moses, actually the one which Jens mentioned because the quantitative Oppenheim for a quadratic form with signature 2,2 gives exactly this result on this counting function. So they can tell that for explicit lattices or explicit tori, they give explicit diaphragm time conditions on the tori, on the torus. And then the eigenvalues will behave like the Poisson sequences with respect to this pair correlation measure. There has also been a result by Jens for a slightly different problem. I guess I have to rush a bit. Sorry. Okay, so you've seen two beautiful, actually three applications of measure rigidity in this area of quantum chaos. Now let me just quickly go through another interesting set of problems which we can discuss in the tutorials a little more. As I mentioned, those who want to do some numerics and that's about randomness and sequence a modular one. So what you do is you have to try any of the range. These are these. What is the distribution? So what is the frequency of finding a certain gap size in the sequence? And then you ask, does this have a limit distribution and tends to invade? And in that way you measure how random the sequence is because if it has an exponential limit distribution you would say, oh, it's very random, right? If it has a very rigid distribution then you'd say, oh, it's not so random. That's the idea. And now the fascinating thing in this field is that we can't answer this question even for very, very simple number theoretic sequences. So for example, to be the fractional parts of J squared alpha. So this is fractional parts and then you look at these numbers from one to n and alpha is not rational. Say for example, alpha could be soft chalk, square root 2. And you do need these numerics. You see that they look like an exponential distribution and no one is able to prove this. Not even the simpler pair correlation statistics that I talked about earlier. And some very, very good people have worked on this so including Rudnik, Sarnak, Zaresco, Heath Brown. So really some of the top-notch analytic number theorists. And Andreas and I have a dream and you know what that dream is, right? To use measure rigidity to prove it but we can't at the moment at least. And it looks very, very difficult. Now, it's even more interesting that even if you don't look at something like this there are parts of small powers. So if you would take J to the one-third, for instance. Why don't you look at this sequence? That's the first histogram there. That's the gap distribution for this sequence. So in each bin you count the number of gaps that are falling at a certain interval, appropriately rescale. And it looks like an exponential distribution. Only if the power here is equal to one-half the distribution looks different. So that's a nice thing to program. It's a one-line or two-line program in Mathematica or Maple and so anybody interested should just quickly do these things. So when the power is one-half the distribution is not exponential. And in fact, Elkis McMullen, again two very, very accomplished mathematicians have showed that indeed this limit exists and is given by some new non-universal distribution which we'll call the Elkis McMullen distribution. And guess what technique they used to prove this? Ratness measure classification theorem. So again measure rigidity. Very beautiful. So that's something we can talk about a little bit further. A real fun paper, Andreas and Iro, which is great for undergraduate projects, if instead you look at the gaps in the sequence log j mon one so you take the fractional parts of log j and you look at the gap distribution. You see these pictures here. And the left one is for the natural base and the one on the right here is when you take base e to the one-fifth and you see it looks almost exponential and we can prove that. And it's a very nice, simple proof. It doesn't use measure rigidity. It uses vile-zacquard distribution theorem, a higher dimensional version of what we talked about before. Okay, so it's your turn again, Andreas. I don't get a nice piece of the cake. Okay, so just very briefly, I wanted to give an overview of the remaining lectures. So as we said, these will be on the blackboard with the chalk, all of the remaining lectures, not just breezing through as quickly as we did here. So we will start with several lectures going really down to basic things, just looking at hyperbolic surfaces and the group of isometry, SL2R, and then looking at geodesic flow, horocycle flow, and ergodicity and mixing for these, and so forth. Then in lecture four, we will start talking about the space of lattices that I said, the SLDZ slash SLDR. And look at the geometry of this space at infinity. It has a somewhat complicated geometry at infinity and flows on these spaces. And then we will have some applications to distribution module one, which Jens introduced a bit here. And then we will have some, I guess we can look at the next slide. In the remaining lectures, we will look at some applications. So for instance, an application to the Lorentz gas. This is a point particle moving in an array of spherical scatterers. As in this picture, one can study the statistics of how long this particle travels before it hits the first scatterer and other problems. And then we will discuss similar questions for quasi crystals. Yeah. And in the last lecture, again, other applications to additive problems, such as the problems about the Frobenius numbers and problems about certain random graphs, certain very special random graphs called circulant graphs. Yeah. Okay, just to conclude by saying that the next lecture in about 10, 15 minutes has been swapped. So I will continue with our first blackboard lecture starting with some basics about hyperbolic geometry and SL2R. And then we'll continue like this. And as I said, as Andrea said, it's going to be blackboard lectures which are hopefully easy to follow and I hope you'll look forward to those. Thank you.