 Okay, welcome to the final lecture of this series. And this lecture will not be on the blackboard there. I want it to be a little more light since I think everybody is exhausted and looking forward to the dinner, those of you who are coming. And so what I want to talk to you this afternoon about is another application of the techniques that we've developed in this course to a problem in discrete mathematics and in particular in combinatorics or graph theory. So as you all know, we now live in a world where connections, social networks, the internet, meta, and these are modeled by graphs. Networks are graphs and here is a picture from 2003 of the internet. Okay, so the thing you realize here is of course that these are huge graphs and things you would like to understand is how well connected are they. You want to find measures that in simple terms characterize if a network is highly connected or not. And so you're looking for simple quantities, a number that describes the connectivity. So here's the mathematical structure of a graph. It consists of vertices, these are the dots, and edges between them. And you can also put a metric on the edges by assigning a length or a way to it. And one of the numbers that describes the connectivity of a network is its diameter. And the diameter is the maximal distance between any two vertices that you can find. And distance defined with respect to whatever metric on the edges you choose. And these days you can just go to your favorite program here, Mathematica, and all these things are built in. So it's really easy to play around with it. So I've just generated a random graph here, the so-called, from the so-called what-stroguards model. And you see the red line is a curve that connects these two vertices. It is the shortest such curve. And it realizes the diameter of this graph. And it's well known and has been observed and known for a long time that the networks you see in the real world satisfy an interesting scaling law, namely that the number, that the diameter grows very slowly in the number of vertices. So typically like the logarithm of the number of vertices. And so the diameter is obviously only one of the kind of quantifiers that you can choose for connectivity, but it's the one that we want to study in this talk. And you see here just a numerical realization of the what-stroguards model that I had in Mathematica. So for each n, I'm generating a random, exactly one random graph. And you see that's the log n function and it looks pretty good. Of course, there are fluctuations around it. There's no question. Now the what-stroguards model is a very, very simple model that doesn't capture all the effects, but it's become extremely popular. So these kind of papers have, I don't know, tens of thousands of citations. And the model is very simple. So you just start with this circular graph. You have n vertices and you connect here the nearest neighbor and the next to nearest neighbor. And then you create a so-called small world model. So small world means that you have extremely high connectivity. So maybe you've heard of this law that everybody on the planet is connected with anybody else by at most six neighbors or something like six other people. So this would mean that the world has a diameter of six, right? Something along those lines. I don't know if it's true or not, but that's sort of the catchphrase as you see. And so what-stroguards model was one of the first to try to model this behavior, this logarithmic growth of the diameter. And so what you do is with probability p, you remove a connection and you reassign it randomly with equal probability to one of the other nodes in your network. And so as the probability goes from zero to one, you see these networks, these whole family of networks and the numerical observation of what-sand-stroguards was that already for quite small values of p, of the removal probability, you see a logarithmic growth in the diameter. Now there are rigorous results, beautiful papers. In fact, this one here, is very closely related to the what-sand-stroguards model. It's also, you start with an n cycle and then you add random matching. So it's almost the same thing. But of course it was written by some pure mathematicians and so probably it only has a hundred citations or something like that instead of tens of thousands. And they actually proved something. So they proved that the diameter is bounded about below by log n up to a small log log n correction factor. There's also a nice paper by Bollerbosch and Riordan. So you see, there's a big industry here of these papers. You can't really see it, maybe from the back. There are many other papers here which I put in gray because I only wanted to mention these two. So there is a whole industry of people working on this and it's a beautiful problem in combinatorial probability. Now what we want to look at is not none of these sort of random small world network models but a different family of random graphs that are very, very regular. And we want to understand how big the diameter is and for those guys. And these are the so-called circular graphs. Circular graphs appear in particular in computer science where people want to construct highly symmetric networks that for instance are used in parallel computing and that have a certain redundancy. So you want still good connections but you also want to have sort of a very symmetric structure. Okay, and so this is how they're defined. So you want to construct a graph with n vertices and so you pick numbers a1 up to ak that tell you how to connect the vertices and let me just go through one example to make it very clear what's going on here. So let me construct a circular graph with n equal to eight vertices. So let me draw the vertices here and now I'm going to connect them. So let me choose two connectors, a1 will be two and a2 will be three. Some nice colors. So these I'm gonna do in orange and this one in blue. So okay, and let me actually label my guys. So let's start here. One, two, three, four, five, six, seven, eight. Okay, and now you see the way to connect them, the rule to connect them is you connect two vertices if their difference is congruent to two modulo eight. We want to assume that these three numbers here in this case have GCD one because you can convince yourself otherwise you would get a disconnected graph and we don't want that, right? Because then we can always just throw one away. So it's a natural assumption that the GCD is one. Okay, so what we're gonna do, well one is connected to three connected to five, connected to seven connected to one and similarly, okay, two is connected to four, four connected to six, six connected to eight and eight connected to two. Right, so now we've done these guys. You see that gives us two graphs, disconnected ones but now we also have to connect with a2. So one is connected to four, four is connected to seven, seven is connected to, so seven plus three is 10, is two modulo eight, right? So we go to eight and then we go on. Five, eight, eight plus three is 11, is three modulo eight, six, yeah, thank you. And one, oh good, yeah, thank you very much. So this is our circular graph, yeah? Does everybody get the idea? Okay, so good. And now we want to know what is the diameter of such a graph and here's an example when n is large and we are interested in the case when n is large and when we throw these numbers a1, a2, et cetera, at random. Okay, and here's the result. Okay, so we'll now pick those numbers a1 for some given k and n to be random and how do we do this? Well, we have a choice here. We can take some set D, this could be a ball and then we enlarge it. So we've done this already very often, right? You say I can set D, you enlarge it and you take all the points in that set. And as I said, we want that the GCD of the coefficients of a and n, that that is one because otherwise we get disconnected graphs. And then we construct the circular graph which is Cnl of a. So I glanced over l, l are the length. So I can assign a length function here for all the blue edges. I want the length to be the same and also for the red edges I want the length to be the same. For now just take them to be one. The reason why I bother you with, you know, having lengths that are not necessarily one will become clearer later in the talk. Okay, very good. So this theorem now says that if you look at a random circular graph where the a's and the n is random and you look at its diameter, if you normalize it in exactly this way, then this random variable here has a limit distribution. And I'm gonna show you in a minute how this limit distribution looks like. So what you see already here is that the diameter is not scaled by log n, but by n to the one over k. So we don't see this effect of the small world network so when k is large, you know, it's growing slower and slower, the more connectors you put, but that's the scaling. And here is a picture of this limit distribution in the case when we have two of those connectors, k is equal to two and there is an explicit formula for this, which is again some formula that we've never seen before. Okay, and now what is the limit distribution? It is given by the covering radius of a random lattice in RK with respect to this polytope. So I have to tell you what is the covering radius. You already know what a random lattice is, but I'll remind you. So that's the answer. And there you see, of course, our random lattices come back and that's why I'm talking about this, right? Because what we've done so far all the time is we said we start with some problem directions and things like that and then the limiting distribution, remember this morning we talked about quasi crystals and yesterday evening. We looked at when you rotate a quasi crystal, you ask what's the probability of finding k points in a randomly rotated stretch set, that has a limiting distribution which is given by a random lattice. And so that appears here again. So what is a covering radius? So you take a lattice, so let me start with a lattice. Here's a two dimensional lattice and that's one thing that's given L and the other thing is k. So it's some set k. Let me think of the set k being something like this. The disc, let's say, right? And then what you do is around each lattice point you draw your set k. So this is a very bad picture because obviously this should be a periodic picture. And the covering radius is now, the factor are by which you need to expand or shrink your set k so that the expanded or shrunk copy of k translated by L covers all of our D. Now if you make R large enough then as of course will always happen. So the covering radius is the smallest R, the infimum over all those R so that you just cover all of our two in this case and generally our D. That's the covering radius. So it's a function of the set that you're using and the lattice. And what is a random lattice? Well, you're all experts in what a random lattice is, right? It's a random point in your space of lattices with respect to harm measure. And so the interesting observation is that in this limit theorem, let me go back to the limit theorem. This limit distribution here you see only depends on the choice that we have here, the polytope, which is a fixed thing. So in dimension two and k equals two, it would just be a square and then k equals three, it would be octahedron. And it doesn't depend at all on the choice of D. And that's something you remember that we have seen also before when we were averaging over our spheres and we were averaging over the sphere with respect to some measure lambda, the lambda didn't feature in the limits and this is exactly the same observation here. And of course, the way we're gonna prove this conjecture is exactly in the same way as we did before. Now, also the lengths, the limit distribution will be independent of the length. That's also a funny thing, right? So there's some universality here. Note that we normalize by the product of the length. So the only dependence is in this normalization. And this settles a conjecture of some graph theorist who wanted to understand whether indeed random circular graphs have a limit distribution. The diameters of random circular graphs have a limit distribution. Now, how are we going to deal with this problem? Exactly in the same way as we dealt with all our problems, we have to translate the question into a question of equidistribution on the space of lattices, right? And then pick the right test functions and integrate over the right thing. And this will be done in three steps. So first, we will show that the circular graphs can be identified with such lattice graphs. So think of this as the analog of the plane or RD that we always looked, right? And then we looked at a sub lattice and RD modulo that, not a sub lattice, a lattice in RD. And if we looked at RD modulo lattice, then we get a torus. And here we will do the same thing. We will take this lattice graph where you have Z2 and then you connect all the nearest neighbor vertices in this way by edges. And then you look at a subgroup of Z2 which will identify points and then you'll get a finite discrete torus in this way. Then we'll approximate the discrete tori with continuous tori. And then we show as before that the tori coming from circular graphs are actually uniformly distributed in the space of all tori, which is nothing but, as you've heard in Anton Zorich lectures, the same thing as the space of all lattices. Okay, so let's do it. So first of all, let's start with the lattice graph. That's just the vertices are at the integer lattice points and we are going to define a metric on there. And that's exactly done by simply, it's much simpler than it looks like in the formulas. So let me actually make the picture over there using already the right colors. So here's my lattice graph and I just define a metric by saying the vertical distance between two points is say L2 and the horizontal distance is L1. Yeah, and that defines a metric on this object. So that's the first line. And this lattice graph will then be denoted by Lk. Okay, and then we can sort of lift this lattice graph up to define some SLK plus one action. And then you can forget about all of that I've written there. I don't want to bother you with that. The main observation is this first lemma and that is that we can, if we take the lattice graph and we mod out by the correct discrete subgroup, not the discrete subgroup, I'm sorry, with a correct sub lattice of Zk. And this was what all this was about is just to give you a formula basically for which sub lattice in Zk to choose that gives you the right identification. You just trust me that this works and I'm gonna illustrate this particular example how to get from the circular graph to this finite torus. Is everybody with me? Right, you get a torus, a standard continuous torus by starting with a lattice and then identify opposite sides. And how do you do this? Well, the lattice points give you that identification. So now I wanna do exactly the same thing here. I want to find vectors that identify points on this discrete lattice graph and that will give me then a finite graph with finitely many vertices. And what I'm saying is that that graph will be isomorphic to the circular graph that we're interested in. And it's a little surprising that that wasn't used more in the circular graph literature to us. We found one paper where that surprise was expressed that paper was just a few years old. Okay, but I might be wrong. I mean, there's a huge literature there and I'm pretty sure that this correspondence has been used before. But for us, it's the key. Okay, so let me just illustrate this here how this works in this particular example. So I want to start here. Let's start with a vertex one here. So let's suppose this is vertex one. And then, oh, let's say this one here. Let's suppose this is vertex one. And now I'm just simply going, let's say this way. So I'm going to vertex three. So then this would be vertex three, right? This would be vertex five. And then where do I go next? Vertex five, no, have I done this correctly? Sorry, I'm doing the orange one. So I'm going from one to three, three to five, five to seven, and then I'm back to one. Okay, I need another line up here. And then I'm back at one. So what does it mean? It means that this point and that point have to be identified, right? So I'm going to just draw that line that identifies those two points. With a different color. Green. So this guy and this guy are identified. And then you see the periodicity coming up here. So this one should be vertex seven and so on, okay? So that's if you like one of the vectors that would give me the identification of the torus. Now, let's continue. So let's do the blue guy. So one is connected to, let's go the same way. One to four, seven, one goes to four, then I have seven. And what else have I got here? Three goes to, three goes to, I have to be careful I got the right way. Now let me do the whole thing here. Four, seven, two, okay. And then I should do that one as well here. So four goes, the orange guys, now four goes to six, right? Four goes to six along the orange line and then this will be eight, yes? What will this be? What? Two, very good. Why? Because yeah, we're taking things modulo eight. 10 modulo eight is two. What will we have here, two, right? Ah, so it works actually. Look, this identification also works here. This one is identified with this one, perfect. Okay, now what have we got here? We go along the blue line, three, six, and then, three, six, one, three, six, one, okay. Now we see one and one is identified. So again, I use a blue line, so these guys are identified. And, ah, damn it. Okay, so this should go, this should be the same vector as here, so there is an image of one here, right? So what you see here is that, what comes up here? What comes up here? Can that, someone tell me? Hmm? Four, exactly, yeah, four. Four, okay, and now what do you see? I have a fundamental domain here of my discrete torus, right? This is sort of the fundamental domain of my discrete torus. No numbers repeat, they're exactly eight numbers. So these eight vertices are the fundamental domain, and that fundamental domain tessellates everything, okay? So I've showed you here that this graph is isomorphic and isometric to this graph, because these lengths are exactly chosen, and they're all the same, the orange guys. The blue guys might be to have different lengths, but I've done it in a way that it does. And this works, this works, right? And full generality. And the proof is completely algebraic, that fact. Okay, so you believe lemma one. And so what we're gonna do then is remember, we are really interested here in the limit of large n. Large n. So when n is very, very large, what you can show here is that the diameter of such a finite discrete torus, now there are many, many fundamental domain, fundamental cell will now contain many. So here's a general fundamental cell of a discrete torus, so there are now many, many points in here. And we are interested in the diameter, right? So the diameter will be the distance, the largest distance that you can get between any two points. Let's say here, and the metric is sort of going along the edges. Now, let us rescale the lengths so that we scale everything to have length one. So we just have to multiply this picture by a diagonal matrix which has the L's in the diagonal. And that'll do it for us. And then we compare the distance that we get in the limit when n is very large with the continuous torus with R2. And you see when you just look at the diameter of a continuous torus, the diameter cannot be much different from the one on the discrete torus because you're just moving by something of the order one over n in each direction, right? And n is large, so you're making a small mistake on it. That's this lemma too, okay? And then the third observation is if you now look at the diameter of a continuous torus, that's the torus we've been seeing today in Anton Zoric's lecture. RK moduloel lattice. That you can show is nothing but the covering radius that appeared before with respect to this particular polytope, symmetric polytope. You just believe me, right? And you can prove it for yourself if you like. So actually the more natural geometric information is not this polytope but it's simply the diameter that appears here of a random lattice. So I could have formulated that in the theorem. The reason why I'm using this is because that's also a very natural fundamental object in the geometry of numbers. So both of them are equally, equally natural. And then, so that was step two. So we have translated the problem of the diameter on the circular graph into a problem of a diameter of a continuous torus. We said that's approximately the same thing. And now all we need to show is that the diameter of the continuous torus that corresponds to this kind of random lattice that we are picking. So the random lattices that come from circular graphs are not all random lattices. They are very, very special. In particular, the vectors that you have here, they're all rational. These are all rational lattices. And so the thing that you now need to show is that when you sum over all the rational lattices, so this is now the analog equidistribution theorem that we considered before where we were rotating our lattices and then stretching them and we were integrating over this rotation, right? You remember, we had an integral over V over the direction in which our lattice was going. We were integrating that and we would say that would become equidistributed. Now, we don't have an integral over rotations anymore, but we rather have a different averaging here over all those lattices L that come from those specific circular graphs. And what one can show is that this average over those lattices again becomes equidistributed in the space of lattices, okay? So now we are in the position of again, applying our trick as before of realizing choosing a particular function F here that will give the covering radius of the particular lattice that corresponds to this circular graph. We plug that in here. We again remember that we can choose a characteristic function that not just continues, bounded continuous functions by the argument involving the Siegel Beach formula or here in this case just the Siegel formula. Okay, so and then we do what we've done before and we get our limit theorem, okay? So that's the basically the main message. Get your equidistribution result, use it to prove the theorem on the randomness of the diameters that you wanted. Now this equidistribution result has been improved by Han Li in a very beautiful paper. And I should say the way I've proved this result is just using the equidistribution of unipotent translates that we had in the lecture. So there is a very nice approximation argument which is similar to this Margulis trick of thickening things that you can use. So I've simply thickened the neighborhoods of these little lattices by a tiny amount and then showed that you can then, once you've thickened it, you can relate it to a continuous integral and reduce it to the questions that we studied earlier in the course. And this theorem has been recently improved. So remember, note here that I'm summing over both A and N and there is a fantastic improvement by Einziedler, Moses, Shah, and Shapira who simply don't have to sum over N but keep N fixed. So in other words, what this means in this scenario here is that I had to also throw the number of vertices at random while they don't. And they use very, very heavy, egotic theoretic techniques that built on retinas theorem. Okay. And now let me come to the other problem in the talk and show you how this connects to the previous question. And I should say, as I've indicated, what was before that was all joint work with Andreas Strimbergsson. Strimbergsson. So now we are looking at also a beautiful problem in discrete mathematics that started over 100 years ago or even more than that on so-called Frobenius numbers. So you start with selecting a primitive lattice vector. You all know what this is. So a primitive lattice point. And then you ask, which linear combinations can I form here with coefficients that are non-negative integers? And what you can show is that from some number onwards for every fixed A, you can represent every integer in this way. And Frobenius asks, what is the largest integer that cannot be written in this way? And so this is just the formula for it. So what's the largest integer that doesn't have a representation of this type? This is called the Frobenius problem. It's also called the coin exchange problem or posted stem problem. Why is it called a coin exchange problem? Well, you think of the A's that you have here as the denomination of the coins that you have in one currency. So two cents, three cents, five cents. And then you have another currency and you want to represent, you want to ask, or let's put it this way, you go into a shop and you want to understand what things can I buy, what values can I buy with those coins, right? And you don't get change. That's because it's non-negative linear combination. So there's no change. It's a little bit like going into the cafeteria here. Okay. And this is a very old problem. Sylvester asked, what's the formula if you just have two coins, okay? And here's the formula, A1, A2 minus A1, A2. And there are no such formulas, believe it or not, if you have more. It's already extremely difficult for three and people to develop sort of continued fraction style approaches. And there's a beautiful paper by Brouwer and Shockley. So this all goes back to Frobenius. And then Frobenius had a student, Shure, who basically understood something and told Brouwer and Shockley about it and then told Brouwer about it. So this is, I think, Alfred Brouwer. There were two brothers, Richard Brouwer and Alfred Brouwer. I think Richard Brouwer was the more famous in representation theory, I think. And anyway, there's a beautiful paper here that goes back to Shure on this problem. And I'll show you in a second what this is about. And then there was lots of work. I got inspired to work on this by a paper by Arnold who called this Arithmetic Turbulent. So what Arnold's question was is, okay, we don't get a formula for F of A, right? But when you do numerical experiments and you put the A's in, you see that it fluctuates wildly. So it's really a random, it looks like a random function of A. And so he said, well, can we describe those fluctuations in some natural way? And Sina and Bourguin then worked on it and characterized those fluctuations. They were up and lower bound. So this is a very popular program in our whole community called Linear Integer Programming. Because certain optimization problems can be phrased in terms of integer-linear combinations that have to satisfy certain bounds and have to have certain solutions. Okay, so, what I proved here is that you can show that indeed the Frobenius numbers have a limiting distribution. Okay, so you take A random just as before. You take any nice domain D, you make it large and then you pick the A's in that big domain random with uniform probability and you get a limit distribution. And the limit distribution, as you will not be surprised, is given again by the probability that a covering radius of a random lattice, yeah, so that's the limiting distribution is given by the covering radius of a random lattice with respect to this thing. So it's no longer this regular polytope, but it's now this simplex here. And as I said, in dimension three, Burgan and Sinei, and also Schur, Sinei and Ostinov proved a version of this. And then everybody said, well, but we don't have high-dimensional continued fractions. And that's when you should say, ah, but we have the space of lattices and high dimensions, which is so much more easy to control, right? And that's how you can solve these problems. Anyway, so here are some images of the limiting density that Andreas Trümberkson produced from the limiting distribution. So these are, we don't have exact formulas here. And so the only thing we can do is we can take this random variable and devise and put it on the computer and make these drawings. Only in D equal to three, there is an explicit formula. In higher dimensions, Andreas Trümberkson proved tail asymptotics for these distributions. So that's very interesting because they don't decay exponentially, they have a heavy tail. Very good. And so now let me just say in one word, I don't wanna go through this whole slide. I just wanna say in one word, why do I talk about these two things in the same talk? Well, A, you've seen that the two limit theorems look sort of very similar, except that in both cases you get the limit distribution of the covering radius of a random lattice, just that the covering radius is taken with respect to two different sets. In one case here, and the Frobenius number, it's a simplex. In the other case, it's another regular polytope. Okay, and the reason is it can be explained in a nice way following this original idea of Brouwer and Shockley that as I said goes back to Schuyl. And so the idea is to first reduce everything modulo, one of the coefficients that you're looking at here. So remember, we're looking at this primitive lattice vector A that has coefficients A1, A2 up to AD. And so you reduce it mod one and you show that the Frobenius number can be written as the maximum over these FRA, which you can think of as the largest number that doesn't have a representation of this form, that doesn't have a representation with MA congruent to R modulo AD. And now what can you say about those numbers? Well, if you define the smallest positive integer that has a representation in R modulo AD, that's NRA, okay? So that is the smallest number positive, strictly positive, that has a representation as a non-negative linear combination of the A's equal to R modulo AD, then you can convince yourself, and this just takes some sitting down and thinking a little bit. I don't expect you to follow every little state. At least I couldn't follow this. Then you see that the largest guy that does not have a representation is exactly the smallest positive that has a representation minus AD. So you have this formula here. And then the key point is that actually you can compute NRA, and this is the formula that Brouwer and Shockley found. And so you've now lost somehow one dimension here, right, in this formula. So now NRA becomes the smallest of the non-negative linear combinations because you've taken that mod AD here that satisfies this equation. Okay, very good. And so now, summing up, FA is equal to that, and then you plug this in and you get this answer. Now have a close look at how this looks like. It's the maximum of some minimum. Okay, that looks a lot like the diameter of the shortest distance. Think about it a little bit. What is the difference here? The difference is that we're only allowing non-negative coefficients and only non-negative coefficients. In the diameter, we would also allow to go backwards and go this way. Here we're only allowed to go right and up, if you like, in the lattice picture. So that actually looks a lot like the diameter for a directed graph where you're only allowed to go in one direction, okay? And so the difference, oops sorry, the difference between the directed, the undirected graph and the directed graph is that simply now you assign a direction for each edge. So here we would simply say in our example, we would simply say I'm only allowed to go from one to three, from three to five. So I'm really saying A minus J has to be congruent to A1 mod, okay? And before I had absolute values. And so this means I also have to put now directions here on my lattice graph. So I will have directed lattice graph. And similarly for the blue, guys. So now I'm only allowed to go from one to four, et cetera. So I will have directions in this way. So now I'm only allowed to go right and up, okay? But I still can ask for the diameter of these things, okay? And then if you do this, there's a beautiful observation by Newnhoos, which is what I've just explained to you, that you can express the Frobenius number as the diameter of a directed, circulant graph minus N. The minus N is because AD now plays the role of N. Right? Yeah, where is my ND? So you see, this will now be the diameter, right? It's the minimum. That's like the distance, the minimum, the distance is a minimum, the shortest path. And I'll take the biggest of those, that's the diameter. And so I get this formula. Aha, but there is a length. So I have actually weights now. Why do I have weights? Well, because these now in this geometric interpretation, these guys here now play the role of a length, right? As I go through my path, I'm assigning a length here of these guys. And that length is exactly A2. And that length is exactly A1. So the lengths are not all one in this identification. Do you all see that? Yeah, right? Because you're forming these. If you would just look at the standard graph distance where each edge has weight one, then you just have ones here everywhere. So this is different weight. And that's why the weights appear here. This is the length. And this is the labeling of the edges. So they're both the same in this case. And so now there you see. The Frobenius number is exactly the same as the diameter of a circular graph with these particular choice of lengths. And so here you have the two limiting distributions. And now on the left, you see the same distribution you've seen for the Frobenius numbers. But now, just to make things a little more beautiful, here is I haven't chosen the length that come from the Frobenius number. I've chosen length one, which is the more natural thing when you look at your directed graph. I've just given every edge length one. And you see it's exactly the same distribution as for the Frobenius numbers. Because I told you earlier, the limit theorem that we had said that the limit distribution is independent of what length you choose on your graph. And that is a very non-trivial statement and it's reflected in this thing here. Right, so that was the last lecture. And let me just recap what we've done in the last week. We've shown you what measure rigidity and homogeneous dynamics is good for in terms of proving sensational problems like the Oppenheim conjecture, quantum unique, a Godicity, and some very nice observations on randomness and sequences like square root n mod one. And then we've told you some of the basics that we have here. Everything in two dimensions and the SL2R has a beautiful geometric interpretation in terms of hyperbolic geometry, SL2R acting via fractional linear transformations as isometries. The main game played by SL2Z as the stabilizer of the lattice the basic lattice. And so identifying the space of lettuces with SL2R modulo SL2Z. We've told you how to prove equidistribution results that then later can be used in the distribution of problems, in number theory, et cetera, that we were interested in. We didn't talk about how to prove the equidistribution of horocycles via Eisenstein series. Andreas talked about how to prove this doing representation theory and also how to derive it directly from the mixing property using Margules' proof. And we looked at the space of d-dimensional lettuces, how to understand when you go to infinity in a d-dimensional lattice using Mahler's criterion. And then you remember maybe we then use these things to re-prove some classical theorems in number theory on say the statistics of fairy fractions, which have a natural generalization to higher dimensions which people previously couldn't do without the space of lettuces because somehow even though it's natural, it's never been done in this way. People were stuck in the one-dimensional problems. The three-gap theorem for n-mode one interpreted in terms of space of lettuces. So there's a little handout on the web page which you can download. That was one of the tutorial problems. And so the solution is on the web of this thing. And then we discussed other applications, statistics of directions and distribution of free path lengths in the Lorentz gas. The key tool, the Siegel Beach formula. Yesterday we talked about quasi-crystals and this morning, and that was the last lecture. So if you're interested, here's the reading list again. It's I think also somewhere on the web. And I hope you've enjoyed it.