 I will begin by just finishing up what I didn't get the chance to finish last time. So let me just recall where we stopped. So we had one gamma, we call this X, and we had H mod over gamma intersect H. Y, Y lived inside X. So we had a picture that looked kind of like this. This is X. And so gamma inside G was a lattice. Maybe you're reducible. No, what this means, if not, it's not too important. And H inside G was called, well, it was a closed subgroup with a property. What was the property? The property was that G mod H was something that's called an affine symmetric variety. This was a setup and we proved the following type of equidistribution result. Yeah, so let me also remark that this was done by Eskin McMullen in dynamical terms, but it was also done in some more special, in perhaps less detail and more technical methods using spectral theory by Rudnik and Sarnak. This was, I was told to, it's good to mention this, so. At this point, so these methods and the techniques that I'll describe, so they can be made effective. So these guys were the first to get an error term, but the dynamical methods that these guys introduced somehow turn out to be, are more popular and have been extended more. Okay, so the equidistribution result was the following that for any beta, let's say, absolutely, well, continuous of compact support on X, the average over an orbit of Y, Y times G. Right, so Y lives inside X and we push it around using an element in G. So we integrate beta, then this approaches the average of beta over X. As G goes to infinity in this variety G mod H. This was the equidistribution result and how did we show it? Well, if, so what we did is we thickened Y a little bit. We made it a little bit bigger than, so, and instead of taking a manifold Y, we took an open set and we showed that the open set, we had this equidistribution result for the open set and we also showed that because of the assumptions, because of this assumption that G mod H is a fine symmetric variety, this open set really was nearby the variety itself, the orbit of the variety, so it didn't change too much. I should make one remark, which is that the paper, the original paper, really the argument they give applies only when Y is compact. If Y is not compact, then this is not, then you have to be more careful and this has been treated later by other people, but we're essentially working on their assumption that Y is compact. And now I'm going to explain the proof of counting, so, from equidistribution and the result is the following. So now we have a sequence BN of what are called well-rounded sets, oops. So I'll remind you what the definition of well-rounded is when we'll need it and these well-rounded sets are in G mod H and we had the orbit of gamma, so the statement was that the number of elements and this orbit was approximately the volume of BN. So we're assuming that the volume of BN goes to infinity, as N goes to infinity, and so this picture now is happening in G mod H. This is G mod H and in some variety and we said that typically this orbit, the orbit of a point is going to be dense under gamma but in some special cases, so if gamma intersects H in a lattice, then you get a discrete set and we were counting the discrete set as how it behaves itself as you take larger and larger, for example, open balls. So if you imagine that this is in three space, intersect this hyperboloid with a three-dimensional sphere and a three-dimensional ball and you will get a well-rounded set and you can get this counting. So let me say one thing, which is what we'll do next after I prove this counting and equidistribution. You'll notice that I write everywhere the volume of Y, the volume of X, so all of these volumes can be normalized to be one and so they can go away but in general they're interesting numbers, so they have interesting number theory, for example, or they're just mathematics is about numbers and these are numbers and they can be interesting numbers and after I prove this counting result, I'll come back to the question of what these volumes can be or what these numbers can come up to be if you don't normalize them to be one. If you use natural normalizations as opposed to just making everything a probability measure. Okay, so now the proof of counting is going to be in two steps. Step one is going to be an average version, so average version. We'll define the following function, fn of g to be the number of the elements in this orbit intersected with not bn but g times bn. Move this set, this ball, we can move it around a little bit and so this is well defined on g mod gamma so if you act on g on the left by an element in gamma, then you're not changing this count. So this is a well defined function and what we'll show is that for any beta, the integral over g mod gamma, so we'll show that as n goes to infinity, this function weekly, meaning that the integral of fn times beta approaches, well, let me divide them now by the volume of bn. It approaches the integral over g mod gamma of beta. So in other words, this function divided by the volume in the weak sense for any function beta, it approaches, yeah, so this function in the weak sense approaches one, meaning that for any beta this holds. And then step two will be that we'll use well roundedness, bn to conclude what we want, that fn of g, sorry, fn of the identity one over the volume of bn goes to one and this is what the count. So we're gonna prove first an average version and the step, so step one will be, is not going to use much about, it's not going to use much about the bn's, so it's not going to use this well roundedness property. So how are we going to prove this average version? So we have the following diagram. So we have our space g mod h on which our business is happening and we have g mod gamma, and we want to connect these spaces and the way to connect them is to take this intermediate space. So here we kind of unfold this g mod gamma, so we have this kind of unfolding and so what is the fiber here? So when you project this, so there's a natural map this way, so what is the fiber? The fiber is just going to be h divided by gamma intersect h. This is, the fibers here have finite volume over this thing. They don't have finite volume here but that's okay. So we have our sets bn here, so let's call the set, so then you can pull them back. So just take their pre-image and you have bn tilde and we'll set, let's say, oops, set alpha and tilde to be the indicator function of these sets bn tilde. So I should note the following, so note, volume, so I realize now that I think I'm off here by a constant and I think I should have here a volume of y. So the volume of this symmetric space should appear here, so I apologize in advance, I'm going to have to modify this a little bit but not too much. I just realized that I forgot this factor. So the volume of bn, well, the volume of bn is the volume of bn but the volume of bn tilde is the volume of bn times the volume of the fiber, which is gamma and this is just the same thing as the volume of y. Okay, so this is, so we have this function here, which I called alpha and tilde, oops, alpha and tilde and now we're going to project this function, so set alpha and to be this kind of push forward, so you just push this function down just like we did in the circle problem. So you sum over all the pre-images that you see here. So this function has finite, this alpha and tilde, it has finite integral, so when you push it forward, it's going to be almost everywhere well-defined and it's going to have a finite integral, it's going to be a decent function and what we're interested in, so we want, yeah, so what is it that we want? We want to investigate this function, so I claim that, so this alpha n is just fn of g. So alpha n of g is just fn of g because why so? Well, we're supposed to count how many, where is it? Where did I define fn? Yes, here it is. So we're supposed to define how many cosets do you see for a given g and so let's just check this at the identity, so this counts how many pre-images do you get here, how many points in this coset? So you have this projection, all these points will lift two copies of our group and when you project them, they will give you at the identity if you evaluate them, they'll give you exactly the number of points that you've seen in this orbit. Okay, is this identity clear or should I? Okay, nobody's objecting for good or for bad, so. Sorry? No, so now alpha n is the push forward, I've pushed forward at this, right? Yeah, this is without, oh, sorry, this. Yeah, so alpha n is just a push forward of this function. Okay, so this is what we have, so. Okay, so now I want to claim that this, yeah, so how can we think of this function, f and g or alpha n, alpha ng? You take this pre-image, so for every point in here, you have a copy of y which has been translated and then you push it down. So you have a, for every point in here, you have a copy of this h-model or gamma intersect h, so you have a copy of y and then you push it down in here and we know that each of these copies, each of these elements equidistributes, right? So let me write this. So we know the following. So equidistribution tells us, equidistribution tells us that one over the volume of y, integral of y and g over beta converges to the integral over x, one over the volume of x, integral over x of beta. And now by this, so if you disintegrate, or in other words, if you just do fubini, so we're interested in this integral over gamma mod g of fn of g beta, beta, so this is x. Okay, so this integral, if you disintegrate it, so I claim that, let me put this normalizing factor in front. So this integral I claim is that it's equal to one over this volume of bn and then now we're going to integrate over bn and so if we have an l and g in here, then we're going to integrate over y times g beta. So let me just say class of g because really, you see when you're in, so bn is contained in mod h, and yeah, so I claim that this integral, so if you just unfold whatever integration we've been doing here, so from the definition of fn, you will get an integral like this and so now we're basically done because, yeah, so the individual terms, so integral yg beta, so this thing converges, again, I have to do the right normalization. Because the volumes of bn grow to infinity, you get that, so in this picture, you get that as g goes off to infinity for this coset, you get this equidistribution, so you get that more and more of your set bn, it has to, since its volume goes to infinity, most of the set has to have these elements further and further away, so you get this convergence. So once you're averaging over a set for which you know this point-wise convergence, then you can get your convergence for the whole integral. Okay, you're averaging a function and you know that each individual term, as g goes to infinity, as bn gets larger, for most elements in bn, you'll have this, that this is approximately this. So yeah, this part you can't see? It's because of the light. So here it's the integral, so it's a double integral, it's an integral over bn, one over volume bn, so this coset g, you're integrating over this and then you translate y, the submanifold by this element g, and then you integrate beta over that translate. So if you go back to this picture that I had over here, for each element g, you push y around and then you integrate over y, over all g's that live in bn, and that coset, and you average it out and you know that as bn gets larger and larger, most of the elements have to have this equal distribution. Okay, so this was step one, and now for step two, we can do, we have to use the well-roundedness of this bn. So again, we're going to prove that for any epsilon, so for any epsilon, this function, yeah, so we want to show this for any epsilon, this function is one goes to one minus, roughly one plus or minus epsilon, and for this we're going to take this beta to be a small function in the neighborhood of the identity, so pick beta to be a bomb function in a small neighborhood of identity in this quotient, and so we get this convergence from step one, but what was the definition of well-roundedness, so remember well-roundedness meant the following, it said that for any epsilon, there exists a small neighborhood of the identity, which is what we would pick, which gives meaning to how small we have to pick this neighborhood identity, so let me call it u, such that we have the following estimates, we have one minus epsilon, the volume of, now here we take the union overall g in this neighborhood, gn, it's less than or equal to the volume of bn. So it's saying that if you move around bn, a little smear it out by this open neighborhood, you're really not increasing the volume of that set by too much, and similarly this is less than or equal to, less than or equal to one plus epsilon times the volume of, now you take the intersection of all of these things, so it's saying that if you move around these sets and you take the elements which are everywhere, then you haven't increased the volume by too much. So this implies that for small enough neighborhoods, fn of g, so like this one over the volume of bn times fn of g, and one over the volume of bn fn of the identity differ by less than epsilon. So it's saying that if you want to count in translates, so you have this set bn and you want to count how this translate intersects, so we had this set and this was our bn, and we wanted to count the number of points in here. So this thing, this aroundiness says that if you move this set around a little bit, you're not, and you're normalized by the volume, you're not changing the set itself so much. So the count for these things, they don't differ too much if you move the set a little bit. Right, because this fn of g counts the intersection, counts the number of points and g applied to bn, and if you move bn a little bit, it's the set, you'll get that the volume is roughly the same, and then you apply the counting to that. Okay, so you get this function near the identity, so a different way to say it is that this counting function near the identity is almost continuous, it's basically a continuous function, e fn of g is continuous near the identity, and now we have this average result, you see we have that for any beta, we have this convergence, so in particular, if beta is this bump function, this shows that fn of the identity divided by this volume goes to one, so I don't want to re-raise this, so from step one, which is the average version plus this continuity of identity, of these functions, so I mean continuity in this kind of generalised sense, I mean it's not that each function is continuous, but rather they have this limit has this property, you get that fn of the identity, so one over the volume of bn of the identity goes to one. Okay, so this finishes the proof of counting. Are there questions about this or? Okay, so in the remaining time, let me try to, as I said, my goal was to have different topics for each lecture, and I was hoping I could finish this last time, but it didn't happen, so in this lecture I want to talk about, so you can forget everything that I talked about, but I want to talk about these notions of volumes, so volumes of these manifolds of these symmetric spaces, so I want to tell you about something called Siegel's formula that says the following, so it says that the volume of, let me see if I want to caution on the left or on the right for this lecture, it's always confusing, yeah, on the right, so the volume of slnr mod slnz is equal to the following product, zeta of two, dot, dot, zeta of n, where zeta is the zeta function, even zeta function is the sum one over n to the s and from one to infinity, so it has this nice formula, and you see, so there's a separate theorem so the group slnr has what's called the higher measure, but it's only well-defined up to scaling, so of course you can always choose this measure so that this volume is one, but the claim is that there's a natural measure and there's an interesting meaning in which this space has finite volume and the volume is a number which is of some interest, I want to compare this with the, to show you what kind of integral this refers to, so if you look at sl2z, everybody knows this or hopefully you've seen this fundamental domain, they have this little tooth that is the fundamental domain for sl2z acting on this space and the integral, so let me call this domain d, so the volume of d is, if you do the integral with the metric that has curvature minus one, I guess you get pi over three and as you know, what's Zeta of two? You guys know this? What's the sum of the inverse of the squares? People are whispering, but can somebody say louder? Pi squared over six, very good, so it's off by a factor of pi and the rational number, so the rational number is because of normalizations, but the factor of pi comes from the fact that the upper half plane is sl2r mod so2r and this is the unit circle and the unit circle itself has a factor of two pi, so to compute the volume of sl2r, you would get here, for example, something like two pi squared over three, but one of the issues is this kind of normalization of volume, which I didn't talk about, so I'm not going to go too much into the normalization of volume and instead of, so I would say that rather than, so the reason why I chose to tell you about this formula is because it's a nice excuse to talk about a very useful tool which is useful for other things, things other than computing these kinds of volumes and this tool is what's called Ziegler transforms. Yeah, so the setup is this, you have a function from R, let's say from Rd to, Rn to the complex numbers, I don't like N to be the dimension, so let me from now on call D the dimension of the ambient space and so if delta, so recall that the space SL and R mod SLDR over SLDZ is the moduli space, the space of lattices, there's delta inside Rd. So an element here can be identified with a lattice, so a lattice is just a picture like this. So giving a lattice is the same as giving a basis for this lattice and if you, you can choose many different bases for this lattice, so you can choose for example these two vectors or maybe you can choose one of these or anything you like, any two vectors will give you a lattice as long as they're linearly independent and this corresponds to SLDR. The fact that you can change bases using a matrix of SLDZ implies that if you're quotient by this ambiguity then you just get the space of lattices. So any time you have a point in there you have a lattice. Okay, so the Ziegler transform is the following kind of thing. It's gonna take a function on Rd and produce a function on the space of lattices and so let me write it here so that they don't have to erase it. So the function F, the Ziegler transform of F at a lattice D is just defined to be the sum over V in delta of F of V. So if you have a function say in R2 in that picture you sum it over all the lattice points and this is clearly independent of a basis of a lattice so it's just a function. So this is a function. So I'm going to, okay so, and I assume that F is reasonably nice as a function. So one thing which is a little bit unfortunate if you look at various places that deal with Ziegler transforms is that there are 100 different versions of it. So I'm going to denote also what's called the reduced, this is not standard terminology. So people will call any of the things that I'll define as Ziegler transform but I want to distinguish them for this lecture. So let's call the reduced Ziegler transform the function where you sum over all vectors except the zero vector. So the zero vector is always in the lattice so you might want to throw that away. And there's another one which is the primitive Ziegler transform where you sum over again all vectors which are what's called primitive. What's a primitive vector? So V inside delta is primitive. Primitive one over L is not in the lattice for any L and Z or let's say for any natural number that's bigger than one. So if you cannot rescale your vector so what vectors are good and what vectors are bad? So these three vectors are primitive but this vector two two is not primitive because it's a multiple of one one. So one one is good, this is okay, this is bad. So one one is good, two two is bad and so if you write them with integer coefficients it just means that the entries have GCD equal to one. So if they have no common divisor then this is what means for a vector V primitive. So each of these transforms has nice, has good properties. So this one is the most symmetric and I'll explain why in what sense is the most symmetric the reduced one has very nice properties and this one is actually what you can compute with. Okay, so, sorry can you say a little bit? Unimog? Oh yeah, yeah, yes, yes, thank you so much. So lattice is such that are, let me say this. So all lattices are, so the volume of this is equal to one. So this is called a unimodular lattice and indeed everything I'm talking about is for unimodular, thank you, for unimodular lattices. So co volume one. This is equivalent, so this corresponds to the condition that the determinant of matrices is one. So this is the Z-Gal transforms. So what are, let me state their basic properties. So the properties are the, so the main, one very nice properties that this is SLDR equivalent. What does this mean? You see you have, so the Z-Gal transform, it goes from functions on RD to functions on SLDR. So I'm gonna start abbreviating as G mod gamma because SLDR and SLDR too much to write. Too much to write. So why is this, in what sense is it equivalent? It's equivalent in the sense that if you act by function on RD, so matrix acts on RD, so you can act on functions and if you do the Z-Gal transform, it's the same as just acting on the Z-Gal transform function on the other side. And you can see this geometrically because here you're summing over the lattice. So if you, what does it mean to act on the function by the, by a matrix? It means that you can change the function but this is the same as changing the lattice. So you can just work out this expression by moving the L and G from the lattice to the, from the function to the lattice. Okay, this is the main, this is the main property. So yes, so there's the following kind of very useful property and it's called Poisson's summation. So this is why this full Z-Gal transform is good. So remember Poisson's summation formula tells you the following that if you have a function, so the sum over f of v, where v is in some lattice delta is the sum, so the sum of the values of a function over a lattice, it's the sum over the values in the dual lattice where now you take the freer transform of f. This is how the function interacts with freer transform. So this is just a formula in free analysis. So if you take, so if D is Z to the D then D dual is just, is also Z to the D. But in general if D is some matrix G in SLDR, so if you represent it by matrix, you take the columns and you, this is a basis. How do you get the dual matrix? The dual, dual lattice is just inverse transpose. All right, so it has this duality. So in particular it tells you the following thing that if you, where is it? So if you take a function and you take a Z-Gal transform, you get some new function on the new space but if you take the Z-Gal transform of the freer, so if you take the Z-Gal transform of a function and you want to compare it to the Z-Gal transform of the freer transform, then this is kind of the dual where you have this dualization which exchanges. So you have this evolution on the space of lattices which takes a lattice to its dual. So freer transform kind of exchanges this property, this Z-Gal transform. So the remark I wanted to make was that this personal summation is only true for the full Z-Gal transform but the equivalence is true for any of these two, for any of these three because the notion of primitives doesn't have to do anything with a self-toward. So okay, so why are, so let me make the following boundedness, so remark on boundedness why some people prefer the reduced. So the reduced Z-Gal transform, it really goes from L1 of Rd to L1 of one gamma. So in other words, if you take, you can define it for Lebesgue integrable functions. So here I was assuming that F is maybe smooth and compactly supported and you get something nice but in fact the reduced, if you're not summing the value at zero, then on average this, so then this is a well-defined function which actually preserves essentially integrals. And the dual just means there's an evolution on the space of lattices and you just take the function you apply the, you evaluate the function at the dual point. So G dual at the lattice delta is G at the dual lattice. This, okay? Right, so let me kind of try to get to the punchline at least to end the lecture on something with content. So what's important is the following fact. So you see if you have an operation which allows you to produce, move from functions to on one space to functions on another then the dual, you know, there's always the dual thing which allows you to move measures, right? So if you have a measure on G mod gamma, then you can, if you have, you can transport using, so the, I'm gonna again, okay, so let me call it the transpose operator. So it goes from measures on G mod gamma to measures on, maybe probability measures, where just finite measures, measures on RD. Yeah, so this is what, so how does it do? So if you have mu as a measure on G mod gamma, then you define the transpose. So how do you define a measure on RD? Well, a measure is something, if you give it a function, it gives you a number. So if you want to apply it to a function on RD, you just apply mu, the measure on G mod gamma to the legal transform of the function, okay? And again, this is equivalent, so this function, so this, everything here is SLDR equivalent, meaning that if you have a measure here and you move it to RD, then you could also apply an element in SLDR to the left, move the measure, and then it's the same as moving the resulting measure on the other side. So now, let me keep the Z-Eagles formula here. So now the key point is that this is going to be a claim. So we have a favorite measure on G mod gamma, which is SLDR invariant. So which measure is SLDR invariant? So let's call mu R, this is R measure measure on G mod gamma, and so the question, well, so let me, instead of telling you the answer so that there's a reason for you to come back tomorrow, I won't tell you the answer, but we can ask, what is the Z-Eagle transform of this R measure on G mod gamma? So it is SLDR invariant, so the R invariant measure on RD, and SLDR invariant measure on RD is the delta mass at zero plus, so it's a linear combination of Lebesgue measure and delta mass at zero. So these numbers, A and B are the numbers we want to understand, so I will like to compute these numbers, so we have this R measure on G mod gamma, which is the volume, and we would like to understand how that moves, so I'll talk about that tomorrow. Thanks.