 Thank you very much for this kind invitation. It's really a fantastic seminar and it's always great to be the 100 speakers. It's my first ever time being the 100 speaker at any seminar so I'm very honored. Thank you. So I work on kind of the interface of economic theory and number theory and so recently there have been a number of results kind of to do with the values of homogeneous polynomials and quadratic forms and so my aim today is to give you some kind of exposition of these recent results. So I like this area a lot because it uses a lot of machinery from different parts of mathematics like erotic theory, the geometry of numbers, analytic methods and also that it seems essentially very widely open. So there are a lot of interesting problems and so I hope I can interest you in some sense. I'm just going to scroll because that seems to be the best way forward. Okay so we're going to begin with a very classical theorem which everyone, all of us have seen at some point. It's a theorem of mayor and it can be found for example in Sparrow's book on arithmetic and it said that if you take indefinite rational quadratic form in at least five variables then this has a non-trivial integer solution. Right so this is a very nice theorem and it's a consequence of the Hasselman-Karsten local movie present. All right so this is very nice. Now one can wonder what happens if you look at quadratic forms which are not rational. This was looked at by a British mathematician called Hoponite and he made the following conjecture. He said initially, so he said let's take a quadratic form which is not multiple of an integer form. So we call such a quadratic form irrational and let's assume that it's indefinite and non-degenerate and so he was trying to match it with the bias theorem and so he first conjectured this in five variables but later it turns out that three is enough. So under these hypotheses the conjecture was that the quadratic form takes a dense set of values at integer points. So what we have is that if you have a rational quadratic form that has an integer zero if you have an irrational quadratic form then you then approximate zero as well as you like using an integer. Okay so that's the that's the philosophy behind conjecture and here's an example of an irrational quadratic form. It's just any quadratic form which is not a multiple a real multiple of an integer quadratic form. So Hoponite conjectured this around the year 1929 and so this is what I mean precisely by rational quadratic form. So if you take a rational quadratic form and you have a multiple of rational quadratic form then it's very easy to see that the set of integer values it takes is a discrete subset of the integers. So it's some constant multiple of an integral form and the set of values it takes will be a subset of this multiple times the integers. So the conjecture at the time conjecture dichotomy was that rational quadratic forms correspond to the discrete set of values and irrational quadratic forms correspond to a dense set. So following Hoponite conjecture okay sorry I should first mention why the three variable condition in the conjecture is crucial. So it turns out that this is not true in two very good and it's quite easy to cook up counter examples and let's see one quickly. So what you do is you take a two variable quadratic form where you cook up this alpha to have the right diphenyl properties mainly you choose it to be a quadratic irrational and these numbers as we all know turn out to be badly approximable by rational numbers and so one can use the property that they're badly approximable by rational numbers to make sure that the quadratic form stays away from zero. Okay so not only is the conjecture forced in two variables in fact it's now known that if you look at the space of all quadratic forms there's a full house of dimension worth of counter examples to Hoponite conjecture in two variables. Okay there's an ergodic reason for this failure which I've come to in due course. All right so following Hoponite conjecture there was a lot of work in the number theory community using analytic methods. So the results of Chawla for diagonal forms came at least nine variables of Davenport and Heilbronn of Davenport and Redar and so on and so forth and so at some point I think the conjecture was known for all forms in at least 21 variables. So this the time that I'm now talking about is we moved from the early 1930s to the late 70s early 80s. This was a situation of that. So at this point I have to make a remark that in fact Hoponite conjecture is it suffices to prove it in three variables because you can restrict the quadratic form to a suitable rational substance and prove the conjecture. So the hardest case and only really the only case that is needed to be proved is the three-dimensional three-variable case. Okay all right so then what happens in the late 70s early 80s someone it's not I've tried to find out who this person was but it's not here was giving a talk on the operand conjecture at the Tata Institute where I work and Raghunathan was in the audience and when he saw this conjecture he realized that this can actually be turned into a statement about a group action on some probability space and so he made a conjecture to the effect which was equivalent to operand action. The story then goes that Margulis learned of Raghunathan's conjecture and how it relates to operand action conjecture and Margulis then proved operand conjecture by proving the ergodic statement. So now in the first part of my talk I want to explain how a statement about quadratic forms has what it has to do with ergodic theory and once I explain that I'll go on to the more recent so what is the probability space that we're working with so the space that we're working with essentially is just the space of co-volume one lattices in Zn okay so this is basically just in Rn so this is basically Zn and also Zn multiplied by matrices in Snr because we want them to preserve the volume of the fundamental domain okay we want them to be unimodular so Snr has a transitive action on these lattices and Snz is the stabilizer of the lattice Zn so one can identify the quotient Snr factor Snz with the space of unimodular lattices in Rn this is a finite volume quotient so the hard measure on Snr descends to a finite measure of this quotient and this is going to be the probability space that we're going to be interested all right so what is the dynamics so essentially the dynamics is just action by translation of subgroups of Snr on this quotient okay so it's easy to describe this algebraic but it's surprisingly rich and complicated all right so Rebonathan made the following point a very important point that Oppenheim's conjecture would follow from the following statement about the dynamics of a certain subgroup on this quotient namely one takes if you take a quadratic form in three variables and take its orthogonal group call it SOQ so Q is the quadratic form SOQ is the orthogonal group that SOQ is the subgroup of Snr and therefore it has an action on the quotient Snr factor Snr just by left multiplication that's it right and so what we have here is a nice finite volume non-compact space on which there is a group action and Oppenheim's conjecture would follow as well just by Rebonathan from the statement that any orbit of this group active in the space falls into one of two categories it's either close and carries a invariant probability measure or it's dense okay so if you're looking at I mean if you if you study a ergodic theory you're looking at actions which are chaotic such as these actions then this kind of behavior is very very rigid it's kind of surprising so it's saying this is a non-compact group acting on the space its action is ergodic and one expects a lot of chaotic behavior but instead one the conjecture at the time was that there are only two possibilities for this for organs so you can describe all the organs specifically okay so that's the remarkable thing so what does this have to do with ergodic front let's try to understand how this is connected all right so the point is in this particular case is that the orthogonal group is a conjugate of a group generated by a unipotent one parameter sign right so it's basically this group is essentially s2r they're generated by upper triangular and vertical matrices and so the conjecture that rather than made for the orthogonal group was quickly generalized to a very general rigidity statement for actions of groups generated by unipotent groups so there's a background to this which is in the 70s there were some beautiful results of Dani and Riech and Fusenberg about the horizontal slope on the modular surface and which provided evidence for this kind of rigidity so in the background of these results for the horizontal slope and these connections to quadratic forms we had a very general conjecture by Radunathan in the topological category and corresponding with a conjecture by Dani in the measure category so the conjecture the topological category said that the orbit closures of these group actions are very specific and have an algebraic origin and the measure conjecture by Dani was that the ergodic invariant measures are very specific and have an algebraic discrepancy okay so when Margulis saw this reformulation of Oppenheim he proved this particular instance of Radunathan's conjecture and in January of course as is well known the in full generality Radunathan's and Dani's conjectures were proved by Marina Ratner's phenomenal landmark book all right so this is some history now let's see how one can relate values of quadratic forms with their value right so consider the quadratic form q naught which is simply x squared plus y squared minus x squared and let q be as in Oppenheim's conjecture so q is some irrational indefinite three variable quadratic form okay so we're going to write q in terms of q naught so q can written as some lambda times q naught uh composed with G by G some matrix in SL 3 bar all right so the stabilizer of this quadratic form is just going to be the conjugate of SO 21 by this matrix G and what we're going to do is we're going to look at the orbit of SO 21 acting on the coset G gamma so this G the coset G gamma this G is the G which comes along with the quadratic form okay I have one fixed quadratic form I'm writing another quadratic form in terms of this quadratic form and this G depends on my quadratic form q so the dynamics is the SO 21 action on SL 3R 5 to SL 3Z and precisely I'm interested in what happens to the coset G gamma when I act by SO 2 all right so let's see what happens now according to Marcos's theorem there are only two or two cases it's either going to be dense or going to carry a finite invariant mesh so let's look at each of these cases in terms if the orbit is dense namely if HG gamma is dense in G then what happens is that when I look at the values of q at integer points this is just simply by definition the values taken by q naught at G gamma Z3 all right now since H stabilizes H is the stabilizer of the form q naught is the same as the values of the set HG gamma Z3 or q naught but this HG gamma Z3 is dense and therefore this is the same with the set of values taken by G Z3 which of course as we know is all of the real numbers okay so I'm missing a couple of closures here so there should be a couple of closures so it's these this equality should be interpreted little bit liberally it means equal to the closure of us okay I missed out the closure so that's what it is so if the H orbit of this coset is dense then as you can see it's quite easy to check just by continuity arguments that in fact the quadratic front it's a dense set of values the other side of it namely if the other alternative if the orbit is closed and carries an SOQ invariability measure is once so easy to understand so in this case what happens is that it's a fact that in this case a gamma intersected with G AG inverse conjugate of H by G is a lattice in G AG okay so it's a discrete stock group with finite covalent all right and there's a very nice term called the Borel density theorem which then tells us that the stabilizer of the autonomy form or the quadratic form is then contained in the Zersky closure of this lattice which means that the stabilizer of the quadratic form would have to be algebraic and algebraic group defined over the rational numbers which is only which is impossible if the quadratic form was a rational thought okay so this is how it works there are only two possibilities in the on the ergodic side of things either you have a dense orbit or a finite volume orbit a proper finite volume orbit and one of them corresponds to the rational form and the dense one corresponds to the irrational okay so this is the basically the end of the first part of my talk when I'm trying to give you some indication of how ergodic theory might enter into the study of quadratic forms so the message I want to kind of offer you to take home is essentially that one can study one can learn a little bit about quadratic forms by looking at how the stabilizer acts in the space of the axis and then study some dynamics of this action and try to translate this information back and forth on the very from the ergodic to the dynamic side and vice versa all right so that's the basic idea all right so now I come to the second part and the more recent part of my talk which is um so we are all everyone was very happy when margulis proved his result because of course it proved all the time to inject her but ergodic methods uh scientists you know suffer from a kind of historic deficiency so especially when compared to analytic methods ergodic methods are often not effective all right so this is currently I would say one of the bigger challenges in ergodic theory which is to prove effective versions of all these of ratness theorems and things like this all right so what what does one mean by an effective version of Oppenheim's conjecture and how would one go about approaching this so what I mean by effective in this talk is a very simple thing um margulis told us that given I can approximate zero by an integer vector to as great an accuracy as I like so now the question is where do I look for if I want to find this integer vector in other words if I want to solve the inequality q of x less than epsilon in integer vectors x can is it possible to get a bound on the size of x in terms of the quadratic form state right this is a very natural question and the approaches which I mean the circle method and analytic approaches always give bounds to problems such as this ergodic methods don't so far so this part of the talk is about some recent work where people have used a variety of techniques new techniques to try to get a handle on this question the first very important result in this was it is a 240 result of Linus Charles and margulis and they proved very beautiful the theorem that said that for an explicit set of irrational quadratic forms in three variables you can find integer solutions to the inequality q x less than epsilon and furthermore x can be found in a ball of radius roughly e to the one over x okay so this is a very important theorem because it's explicit so they give a diaphragm condition on the quadratic form and for that full measure for that set of quadratic forms they prove this explicit result and this was done basically by a very complicated somehow improvement of results of margulis and dhani margulis which they made effective and it's it's a very important very difficult paper so the question one might ask events when when you look at the result which is this is so what should one expect this is a very good result but is this bound really what one should expect and can one do better than this and oftentimes in arithmetic when one has a very difficult result with the bound what what you can try is to see if you can get a better bound by trying a random version of it right in other words if i settle for less than the discharge margulis in the i give up this expectation of having an explicit description of the quadratic form for which my bound goes but instead try to prove a result for say a full measure set of quadratic then can i do better okay so this is what was looked at that by myself in joint work with Alex Korodnik and I must go where we proved that for almost every three variable products you can solve this inequality with a much better bound so the bound now is x is bounded it can be found in the radius in a ball of radius one over epsilon so it's better by an exponential okay and in a certain heuristic sense this is the right bound to expect because if you look at you know lattice points in a ball of radius t they're in in three dimensions they're roughly t cubed of them and you plug them into a quadratic form which takes values in some interval of you know roughly size t squared so you'd expect the the least one to be roughly size one over t okay heuristically and so what we were able to prove is that in fact this is true for almost every product okay so it's a much better bound than it is not my goodness but it's an almost every bound there's this much stronger result because it's an explicit set of reaction. In fact in the work with Korodnik and Nemo we have many more results of this kind and so in fact we prove a result which is true for quadratic forms in all variables but the the quality of the result is not is somehow is a little bit it it deteriorates a little bit as the dimension goes higher. On the other hand it kind of provides effective statements for a lot of similar diaphanetic qualities on homogeneous varieties of semi-simple curves. All right so let me briefly explain how we went about this and so basically as we saw the Oppenheim problem could be translated into some problem about the stabilizer of the quadratic form acting on the space of lattices and this is the same idea over here but somehow here a different facet of the stabilizer is being used. So as we saw the Oppenheim problem can be translated into a problem about the gamma action on the quotient g mod h and similarly the effective Oppenheim problem translates to a quantitative distribution problem. One then does this quantitative distribution problem around by instead looking at so this is this is something called a duality principle which is an old and very very fundamental principle in dynamics so you turn the problem around from looking at the gamma action g mod h to the h action of g mod h and here somehow the fact that this stabilizer is a semi-simple group is what is the most important thing. So in Marulis's solution to Oppenheim's conjecture and in Ratner's theorems the fact that really is drawn out is that the stabilizer is generated by a unipotent one parameter circuit but here what one uses is somehow the spectra nature of the group action so one wants to look so what we do is we look at harmonic analysis of semi-simple groups and the reason for doing so is because one knows going back to the work of Neville and Neville and Stein that in fact the action of semi-simple groups on homogeneous spaces that admits ergodic theorems with rates okay so this is the beauty of semi-simple groups in Berkoff's ergodic theorem which is the ergodic theorem that everyone studies there's no rate of convergence but if one acts by a bigger group like a semi-simple one can in fact get very good rates of convergence and of course in the last decade this this kind of thing has been used to a very good effect by ergodic and Neville in counting lattice points and by myself ergodic and Neville in diaphragm time approximation problems so let me briefly explain what kind of statement one has so it turns out that this subgroup action on g1 comma leads to a unitary representation on l2 of g1 comma and then one can consider an averaging operator so you should think of this as a time average you know in the Berkoff ergodic theorem you have a time average along in orbit and the theorem says that almost surely this converges to the space average and so here this is similarly an average over a bigger and bigger part in the acting group and uh yes sorry please excuse me there is a few questions from Sergei Konyagin Sergei would you like please to unmute and ask away I have two questions the first question we say about a set of a complete measure of q is this set explicit or not and the second no so it's right it's not explicit it's not explicit right and the and the the theorem of ergodic at Neville says that for almost all q do you understand correct for sufficiently small epsilon Disney for two homes yes the theorem of my self-geronics in Neville says that if for a full measure set of quadratic forms yes you can find x and z in such that q x is less than epsilon and x can be found in a ball or radius one epsilon is efficient smaller yes is it okay to continue yes please thank you for the question all right so basically uh I just wanted to say that the main tool here is an ergodic theorem which comes out of this eight action on jima kama and one looks at these kind of averages over bigger and bigger boards in the group and then there is an ergodic theorem that says that these averages converge to the integral of the function over jima kama not only do they convert they converge at sudden explicit rate all right so this rate comes out of the harmonic analysis of age and the beauty of it is that it's it can be computed so sometimes it's not very easy to compute it but it can be computed and it matches up so that in this three variable quadratic form case that goronik Neville and I were studying the rate matches up so that it gives you exactly the answer one over epsilon up to a small perturbation all right so this is basically the idea of proving effective open hand conduct one way of proving effective open hand conductors for almost every quadratic form which is to use the fact that the acting group has frequently has some very quantitative ergodic theorems attached to it all right so at this point I make a small digression to introduce a work of Burgan so a reformulation of merguez's result that is the following so namely one can reform it to say that there are sequences n going to infinity and delta going to zero such that for all sufficiently large k this inequality code so you can approximate the real number c using q you know applied to integer vectors in the ball of radius k okay so this is just a reformulation of open hand conjecture and Burgan considered this reformulation asked well another way of making this effective would be to understand the quantitative relationship between the sequences n and delta okay so here is this theorem to be precise so he considered the diagonal quadratic forms namely forms of this kind and then he proved that assuming the linear hypothesis that this inequality forms for a fixed for a fixed beta he fixed one parameter and let the other parameter vary and almost all that for as long as this precise relationship so as long as nk over k to the eta delta k square goes to zero okay so this is a kind of a uniform version of effective open hand because in the the statement that I made about the theorem of myself Buranik and we were approximating a point c in in the real line but in in principle the full measure set of points could depend on this point the full measure set of quadratic forms could depend on this point here there's a uniformity in the in the full measure set so this is a beautiful result of Burgan and then uh kelmer and myself uh proved a result using ergodic theory which is the same as Burgan's result but holds for almost every three variable quadratic form not the diagonal form right so Burgan was averaging over a smaller family of quadratic forms we proved the same result for almost every quadratic form and recently Damaris Schindler has generalized Burgan's result to higher degree diagonal form so it's a very nice result of ours so this is a brief digression on Burgan's result all right so now I want to go back and so after uh so uh myself Buranik and we will use ergodic theory the athria and margulis came up with a beautiful idea to attack this kind of problem and this idea was uh to use uh Zekel's moment formula okay so I'll explain what it is a little bit later if time permits but essentially uh there's a famous formula of Rogers which kind of computes moments higher moments for uh in Zekel's mean value form for Zekel transforms on the space of lattices and this essentially converts this Oppenheim type conjecture into a lattice point problem and so athria and margulis used methods from the geometry of numbers but to prove effective versions of Oppenheim type theorems for generic forms so their methods kind of uh so as I said uh the work of uh Buranik Nemo and myself applies to a wide variety of settings this doesn't apply to as wide variety of settings however it gives you the sort of correct answer in all dimensions which we were not able to okay so that's a big advantage of this counting lattice point counting approach to generic versions of Oppenheim type theorems and this was refined and extended significantly by Kelmer and Hew in a beautiful paper in particular they also proved general versions of this result of Burgan that I just mentioned on the previous slide using these moment formulas for uh Zekel transforms all right so I'll try to explain what this is a little bit later but I want to kind of bring in some recent work and this recent work has to do with what are known as inhomogeneous quadratic forms so just a minute I have about 15 minutes is that correct Alina? Even more if you want all right so for uh sometime now I want to discuss some more recent work which has to do with the so-called inhomogeneous quadratic forms and what are these things so this is a quadratic form with a shift okay so I have a quadratic form and a fixed vector alpha and I define what is what I call it inhomogeneous quadratic form you know that q sub alpha as follows is just q of x plus f all right so these inhomogeneous quadratic forms have been around for a while and we study in a famous work of Sarnak and also a famous work of Jens Markloff who studied Bayer correlations and they have to do they have some connection with the Bayer-Taborg projector in physics so as far as we're concerned it's natural to ask whether one could prove open-handed theorems for inhomogeneous quadratic forms and effective versions of such results and this turns out to be a very interesting question so let's set the conjecture for us so what should be the appropriate conjecture it has to involve an irrational inhomogeneous form so what is this object so we are going to call it inhomogeneous quadratic form irrational if it satisfies one of two conditions the first is that it's either the quadratic form is irrational as we know it or if the shift is an irrational vector okay so uh and you know what is quadratic form can be irrational under two circumstances either the form is irrational or the shift is irrational and if either of these happens then we would expect that what has version of the open-handed conjecture in fact Markoulis and Muhammadi in the year 2011 proved a quantitative version of open-handed conjecture for inhomogeneous forms and so what does this mean so this means the following so set let's set up a counting function this counting function n which depends on the form an interval and the parameter d and it simply counts the number of integer points in a ball of radius t such that the quadratic form when you plug in these vectors the quadratic form the line is interval i okay and so a quantitative poppin iron problem is the problem of giving some kind of asymptotic formula for this counting all right so and indeed they proved a nice asymptotic result so they proved a lower bound in three or more variables and actually an exact asymptotic bomb in five or more variables I have to mention this point that this constitutes an inhomogeneous version of some very famous results of Donnie Margulis and Steve Margulis and Moses who treated who obtained asymptotic formulas such as these for homogeneous quadratic forms all right so this is what was known and of course this lower bound of this for this counting function already implies the open-handed conjecture for irrational inhomogeneous quadratic so now we come to some very basic questions so namely can anything be said about effective results for some explicit family of inhomogeneous forms like we were talking about originally and how does one define almost every inhomogeneous form because of course now there's much more choice there's the possibility of taking varying both the quadratic form and the shift or fixing the quadratic form and varying the shift or fixing the shift and varying the quadratic form and so can one get error terms in the quantitative terms of Margulis and Mamadi can one prove effective versions of open-handed conjecture as we've been discussing for this kind of quadratic form and this turns out to be a problem which is very interesting and by no means settled so I'd like to report on some progress which is joined with Duby Gelmer and Shuchen Yu and so the the answer to these questions it turns out to be quite complicated and so the first kind of regime one can look at is to allow one to vary both the quadratic form and the shift and this turns out to be actually kind of the easiest situation mainly one can follow the work of Atria Margulis and of Gelmer and Yu and get a nice bounce for the effective open-handed problem if you allow both the form and the shift to vary the situation changes dramatically if you fix one of these parameters and also interestingly the method of attack changes correspondingly when fixed is one of the parameters so here's a theorem it's a little bit complicated but let me just talk everyone through it this is a problem where we're fixing a rational shift and allowing the quadratic form to vary and here we are in a very good situation because we have a nice asymptotic count so this is a fixed rational shift allowing the quadratic form to vary you have a nice asymptotic count with a nice error term with a good error term and in particular this count with an error term implies an effective version of open-handed conjecture for these inhomogeneous forms namely it tells you that you can solve you can get a good bound this kappa is the same bound that was there in this effective open-handed for a fixed alpha and almost every two okay so uh this is the situation if you fix a rational shift and allow the quadratic form to vary you can get very good bounds in the counting of solutions to quadratic inequalities as well as effective versions of inhomogeneous quadratic inequalities um we also have a result for fixed uh quadratic fixed shifts which are not rational but this is not as nice as this result it kind of depends on doing some diaphragm then approximation and uh approximating this irrational shift by rational shift so I won't bring that up it's it's possible to prove some results but it's not as good um the second result is a result where you fix a rational quadratic form and allow the shift to vary and this uses a completely different method so this method here I'm going to come to this this method here is based on this philosophy of Atreana Margulis and Kelmer U and uses a kind of higher moment formula not on the space of unimodal lattices but on a common portion to lessen the time so I'll try to explain what that is briefly and this method here uses spectra technique which was used in the work of Gorandik Nehru and myself and here also we have this is a situation of a fixed rational form and varying alpha where one uses spectra techniques to prove an effective of an iron result all right so this is uh what the uh the results that we obtained jointly with the Duby Kerber and Xu Cheng Yu so in the rest of the talk I want to kind of briefly uh indicate how this approach using uh higher moments of secret transforms words by trying to briefly explain how one would prove this kind of bizarre and then I'll end with trying to highlight some more recent work of along these lines okay so basically the idea is as follows so this the philosophy is quite simple so let me just say so basically we have a fixed rational shift and what we do is kind of use this fixed rational shift to uh to get a hold of a congruent subgroup of SLHZ so one can do this in a precise manner but rather than kind of explaining everything as it is let me just say that essentially what happens is uh looking at uh counting the number of solutions to this quadratic uh in homogeneous quadratic inequality where the fixed where the shift is a fixed rational shift is essentially the same as looking at the counting problem on a different homogeneous space which is a homogeneous space with uh by a pommel institution okay and essentially the main tool here is the Ziegler transform so uh one you know many of us have seen the Ziegler transform on the space of lattices here it's written down in the space of affine lattices so essentially you take some uh compactly supported function on rn and a form of function on the space of lattices by summing over the non-zero lattice vectors all right so this is what it's called the Ziegler transform and uh Ziegler had a famous formula so let me just ignore some of this technicalities which are written down for the congruence quotient but Ziegler had a famous formula which said that the average of the Ziegler transform over the space of lattices is the integral of the function over rn okay so this formula here is a congruence quotient version of that formula and what we need in order to prove the result that we proved is to compute higher moments of the Ziegler transform because higher moments in Ziegler transform so the Ziegler transform the Ziegler's mean valentineal essentially is gives you the lattice point uh content the main term of the lattice point content problem so one side is the number of lattice points in the ball if you apply it to the indicator function of the ball and the other side is the volume right and so if you want finer information you need to obtain higher moments and actually you need to do a little bit more work because it's a kind of a complicated lattice point counting but maybe it's not appropriate to get into that but essentially you need some second moment formula of of this kind so the second moment formulas are actually very useful they were first as far as I know discovered by Rogers and then they were used by uh atria margulis kelmer and you and actually there's some very nice moment formulas for orthogonal groups in chucheng used thesis which are also have found very nice applications in the work of kelmer and you so essentially in this part what I want to advertise is that moments of these Ziegler transforms actually are very helpful in lattice point counting problems in a variety of settings okay so yeah essentially the main thing is to get a discrepancy estimate for a less point count using the second moment so maybe I'll kind of not go through this whole thing and I end by advertising some work of some early career mathematicians here the tata institute which is really nice so the first kind of series of works I want to talk about is a work of geong han she's a postdoc here so she's done some a very good work on s arithmetic versions of open hatching so one can consider isotropic p-adic quadratic forms or more generally s arithmetic quadratic forms and these were actually first studied by borrell and pastran who did versions of open hatching lecture and geong has done a lot of work on this and recently she has established s arithmetic versions of these higher moment formulae on the space of lattices that I was just discussing and also she has you know by establishing those she has kind of obtained p-adic versions of this generic effective open high and then using her approach she and I proved some results about moment formulas on congruence portion so some analog of the results of myself and kelmer and you for p-adic quadratic forms so she has a very nice work and finally I would like to advertise another young mathematician venaikomaras Swami who is also a postdoc here at the tata institute and so we enjoyed her work with him we looked at that kind of so I discussed this result of organ and mentioned the work of damara schindler who had looked at higher degree forms and so it's natural to ask whether one can get the inhomogeneous versions of this work and and so we were able to obtain some kind of results where essentially we fix a shift and allow the quadratic form to vary in some diagonal family like work at it and obtain some unconditional bounds and some bounds are conditional on little hypothesis so the method of attack here is basically to use is analytic in the style of so that's something I didn't discuss but it's a nice result so I thought all right thank you I'll stop here