 very much and congratulations on your birthday. Celebrating this was supposed to be a conference I guess maybe we'll do it at 61. That's what happened with me more. I had a secret 60th birthday in Israel but then another one in Princeton but for me the real reason is I didn't want to admit I'm 60. Apparently Zev's not shy so that's very good. Well I feel like I'm 16. Excellent. I hope that stays for many years. Okay so I'm going to spend half the lecture on maybe reviewing very in very brief terms only some of Zev's work and then move towards some topic that is related to what I'm talking so I'm only going to discuss the work of his that's relevant to what I want to turn to. Feel free to interrupt as Lior said I think it's very important otherwise I actually don't even know if I've been cut off and just talking to myself here in my bedroom. Okay so let me start. So I'm going to talk about I'll describe various summation formula and then how they might how they have been used and especially how Zev has used them and then I'll return to it in a discussion of a specific example. So perhaps the most important summation formula that every number theorist knows or anybody who's seen the zeta function knows or anything of modular forms is the Poisson summation formula. I have a cursor here which I'll use which is the only tool we'll have to communicate with you other than my voice. So you take a Schwarz function in Rn you define the Fourier transform with the right e to the 2 pi i this e is e to the 2 pi i that way the formula reads a little better and the sum over a lattice of the values of the function at the lattice is say the volume of the lattice is 1 as as co-volume as Zn is and Zn is self-dual is the sum of the Fourier transform at the integer points. This is a statement about the integers the discrete group Zn in the way that it sits in Rn and it's a fundamental fact and one puts in different test functions and learns a lot in various mainly analytic type of problems connected with the structure additive structure of the integers and another way of stating this Poisson summation formula is if we take the measure which puts delta masses at the point at the lattice points in Zn and that would be a tempered mass it's a point mass then it's Fourier transform equals itself and I'll return to this because I this talk is about summation formula something called crystalline measures which are of great interest in quasi crystals and things like that but certainly related to number theory. So this is the Poisson summation formula and it's my formula number one in the type of formula that I want to discuss. The second formula I'll just state the very simplest version of the Selberg trace formula as you all know the Selberg trace formula has proved to be one of the most important tools in the theory of automorphic forms in its use in certain ways but as I described Zeb's work the use of these formulae that he has made at least in print are much closer to the way Selberg envisioned you know used it and envisioned the way it would be used. So here's the Selberg trace formula we have a discrete subgroup of SL2R I'll assume it to be in a very simpler situation I'll even remove SL2Z out of the discussion so it's discrete torsion free and co-compact so that the any elements in the group which are not um the trivial element are what are called hyperbolic elements so they're diagonalizable over r and we form the quotient the upper of plane divided by gamma that's a hyperbolic surface it's also a complex curve of dimension one but it's also got a hyperbolic geometry and that's the part that Selberg simplest form of the Selberg trace formula involves we take the Laplacian for this metric on this compact quotient we write the eigenvalues as a quarter plus tj squared so the smallest eigenvalue zero corresponding to the constant function and the correct parameter for the for the summation formula are these numbers tj so lambda zero would mean that t zero is i over two and you must watch for that because we can evaluate the function in the complex domain unlike this Poisson summation where everything is real I'm following Selberg notation here so he we defined we make this definition of the Fourier transform here with a two pi and not with the e to the two pi i so let's just stick to that and then the Selberg trace formula in its simplest case reads that the sum of a test function at the numbers tj tj remember where those relatives of the eigenvalues is the area of this surface times a mass which is very important it's a plancherel measure t hyperbolic tangent pi t times h of t dt so this is a continuous part and then there's a discrete part which looks very much like the Poisson sum or what I will call crystalline measures later so the sum of point masses h at point masses is the sum of the Fourier transform at a discrete set of points and this time it's the lengths of the closed geodesics on the Riemann surface with a weight which is the primitive length of a closed geodesic divided by e to the l over two is a hyperbolic sign over here sorry and p goes over the so this is a discrete set and I'll be discussing how one oh it has ever uses these things the third is what I will call the Riemann very explicit formula so if you take the Riemann zeta function summation n to minus s you write the non-trivial zeros as a half plus i gamma j so the gamma j is supposed to be like the tj's and like in the Poisson summation formula so if the Riemann hypothesis is true then the gammas are all real and you sum this test function by the way in the Selberg formula g is compact support you can relax that condition but it's not symmetric it's absolutely not symmetric because h is evaluated at a complex number so g is compact support and the Fourier transforms entire similarly in this Guinean formula we're evaluating h so I'm using exactly the same notation as in the Selberg trace formula I'm evaluating h at the point i over two so g is compact support so the sum of this test function at these real numbers if the Riemann hypothesis is true we can get rid of this if we put a character so this is coming from the pole of the zeta function then there's the analog of the planche rail measure which is this gamma prime of the gamma h of r gamma prime of a gamma and minus not plus two times the sum of the prime powers which looks very similar to that very suggestive lambda n is a fun mangled function log b if n is the prime power and g evaluated log n again it's a sum of point masses discrete set but there's a fundamental difference here to the Poisson sum these two here and I want to explain that in a second okay these are the three formula then I will they have generalizations and variations set me the Selberg trace formula most notably but I will just highlight things with these three notice that if the Riemann hypothesis is true and I sum over these delta masses over the non-trivial zeros then this is a tempered distribution and if a distribution is tempered it's a sum of delta masses tempered distribution it's Fourier transform is also tempered but unlike in Poisson summation the absolute value here is never tempered because this h i over two minus this there's a big cancellation here which is making the right hand side tempered and if you put a character this wouldn't be a positive sum over here so it's tempered if the Riemann hypothesis is true but not absolutely tempered and this is an important point as we go along all right so for Riemann the explicit formula number c he never wrote it that way he wrote it in a somewhat different way it was his great paper where he gave you an exact formula counting the number of primes less than x in terms of the zeros and this is just an integrated smooth version of that for Gwinand I'll return to that a little bit so this is before Andreve for Gwinand this formula was about concordant sequence of what Mayor today calls quasi crystal crystalline measures and I'll return to that and for Andreve this was late in his life it was an attempt to try maybe get a handle on the Riemann hypothesis and he wanted to interpret the explicit formula that I've just written down in the way that it's interpreted that he interpreted or Schmidt before him many years before for curves of the finite fields in terms of the eigenvalues of Frobenius so as to suggest how one might attack the Riemann hypothesis now I know they have quite well but in his work he's led the way in applications as I said not in the way that Arthur and Langdon's use it in terms of comparison of trace formulas but rather in terms of using the formula one at a time the way Selberg actually himself used it for various purposes he and I and I'll get to that in a second wrote a paper about the explicit formula I don't know the extent to which he here he here means zev has thought about the use of c in the connection with the Riemann hypothesis if he's thought about it he's kept his thoughts to himself I don't know if he's had those thoughts but that maybe he can address after my lecture here but certainly others have looked at this explicit formula and I could give maybe two or three hour lecture on that but that I won't do over here I want to describe actual theorems surveys got his name onto this in trying to as I say definitely interpret the zeros in some other way all right so at this point I'm turning now to a few of the Zerb's works and I'm not going to discuss by any means all of his works I think we should leave that for his 61st birthday which hopefully will take place in a physical location somewhere so he can follow in the tradition of 60 and 61 birthdays and at that point somebody will be assigned to actually explain his works in in some serious way this is just as a make-up call for his birthday two weeks ago so his thesis zev's thesis which is not so well known but which I took quick notice of written under pieteski shepiro concerns the selberg formula that I wrote down there and something called the Peterson formula he did this for sl2z and the Peterson formula is a summation formula it's not a trace formula but it's very much related to what today is called the relative trace formula this is the Peterson kuznetzo formula and what zev's thesis is about is relating these two and deriving one from the other and making some other applications it's an important philosophical point to understand when you use the Peterson formula or the kuznetzo formula to your advantage versus the selberg trace formula and sometimes you introduce the one from the other as henry kivanich who I saw a little earlier come on uh it brilliantly has exploited anyway um his thesis actually is being referred to more recently so it's I don't think I don't know if it was published but people have referred to it more recently especially in connection with beyond endoscopy and you can see a letter I wrote to langans about beyond endoscopy where the this issue that zev was the first to look at is is central okay the second application that zev gave and this uh actually happens to be joined with me is the computation of the end-level correlations of the zeros of the Riemann zeta function uh a project that we started with some other completely different set of ideas so let's suppose that the zeros are half plus gamma j assume Riemann just for the interpretation here and uh in order so these are real numbers they the ordinates of the zeros of the Riemann zeta function and you want to make the mean spacing one so you inflate them locally a little bit like I have there because you know how many zeros there are up to high t so you make these numbers gamma j hat so that they normalize to have mean spacing one and then you look at the spacing's local spacing statistics between them and in a great and very influential as it turns out uh paper many many years ago montgomery computed the pair correlation in restricted ranges for the zeta function and what zev and I wanted to do was we were sure at least I think we were that uh the zeros of an automorphic L function would absolutely not follow GU and all this was just a red herring and a nonsense and the reason we felt that was that the distribution of the coefficients and so when you write down the explicit formula more generally like here with more general automorphic form these coefficients here replaced by the local coefficients of the L function of the automorphic form on gln say and then those don't obey those have very different distributions and they can be very complicated generalizations of the cytotate distribution so we were expecting that the zeros would also then have very different distributions but we quickly saw that the pair correlation would not change and so we felt we had to compute all correlations and we did that for all L functions and to our amazement there was only one answer GUE it was absolutely always universal as predicted by Dyson so this was a nice and we actually formulated the we actually I think the first to formulate properly what these things mean in the mathematical world and we computed these in restricted ranges I want to point out that there was something that entered there which held us up for quite a while and there was a combinatorial identity in order to show the answer was GUE that we struggled with we had to prove a combinatorial identity which we managed to do and then in the generalizations this then became quite a big industry in these generalizations these combinatorial identities were not so easy to prove but in a beautiful paper Alex Anton Roditi Gershon and ZEV in 2013 they at that point ZEV is heavily working in the function field which I'll return to in a moment made the same computations that you do in the number field and then made the great insight that in the function field there are other ways of proving the statistics local statistics that you want on average anyway and as a consequence the identity that you need to prove at the end is coming for free in other words you've translated and proved in the number field an identity by proving it in the function field only because it's an identity so it's not like you transferring the proof of the Riemann hypothesis from a function field to a number field that we don't know how to do but if there's an identity that you're trying to transfer that may be doable and that's very much how the fundamental lemma was solved because Hale's first to observe this and Vulture and Nagao of course proved the fundamental in the function field but it's an identity just like here and works in the number field so it's very much like that and gave a complete understanding of how you understand the identity which connects everything up at the end so that was a for me and for ZEV I think a an application of the explicit formula not by interpreting anything but by actually using it because it's got two sides and by doing analysis in the way that one does with such an identity all right following that paper of from I think 1994 the ZEV has made many many contributions to the study of spatial statistics he got very interested in the local distribution of zeros and eigenvalues in many different kinds of mechanical systems that you quantize so with on the zeros he wrote papers with Keating and Hughes but for eigenvalues he really got interested in the Taurus and I've always teased him why are you only working on the Taurus but the Taurus is a beautiful place to test all sorts of theories and it's highly non-trivial and his works with Bergen become obviously at some point become tricky harmonic analysis and they have made very delicate and deep results and the works on not just the spectrum but the nodal domains with Wigman and Kohlberg are also very deep results on the case of the Taurus but the Taurus is special and I think ZEV has completely led this study also for hyperbolic leplacians on hyperbolic surfaces ZEV has made many many contributions and he's made I don't I'm just not going to spend time going through explaining I'm just going to explain one example here and it's a typical rhythmic theorem so let's take the Taurus where he's done a lot of very interesting works and it's perturbed so we take the Taurus and the leplacian on the Taurus you can write the eigenfunctions down explicitly and the eigenvalues are quadratic form with maybe irrational coefficients so you know what the eigenvalues are that doesn't mean that you could answer these local statistic questions by the way and that's exactly where ZEV gets interested you can write them down but then it becomes a tricky number theory problem to answer these basic questions in any event suppose we take a rank one perturbation so we these are called CBER billions or in ZEV's paper with uber Richard they are called point scatterers you add a rank one perturbation you have to make precise what you mean by the self a joint extension with his delta mass uh so it shouldn't change your eigenvalues too much they actually have to interlace but the spacings between them are completely unclear so the theorem they prove by writing down an explicit version of this trace formula and make appropriate analysis uh and estimates is that if you want to look at the consecutive spacing by the way this is dimension two otherwise this analysis is not not true uh in dimension two the eigenvalues the energy levels themselves uh come down roughly uh you don't have to rescale them if you look at the eigenvalues so I'll just normalize them so that the jth eigenvalues as excuse me asymptotic to j and I look at the consecutive spacing for the Laplacian which I'll say a word about in a second and I look at the consecutive spacing for this perturbed this rank one perturbation they prove those two distributions are the same they don't know it exists even for a flat torus say with alpha equal to root two that's one of his favorite problems but if the spacing distribution for a flat torus with alpha equal root two is Poissonian as he and I expect then so would the perturbed guy be Poissonian and this is to me very beautiful and the typical clean uh Zerberlach's writing a clean decisive and short papers with a keen idea and a very beautiful argument that's one example uh I think it was in the 90s and then certainly in the 2000s that Zerb turned to the function field in a big way the function field's another place where you can test all these ideas but there's arithmetic and the algebra number theory that's separately interesting so the replacement here is you replace the rational numbers by fqt the rational function field fq is a finite field the integers are replaced by the polynomials in t of course if you're a serious guy here you don't just look at this function field you look at a finite extension because the zeta function of this particular guy is but no zeros it's it's one it's completely boring but if you look at a finite extension that's the zeta functions of those are exactly the curves of a finite field uh however when you look you can still look at fqt and much of what he does I'm sure would uh what I like very much about Zerb's work is he goes to the real action and doesn't worry about generalizing too much he'll take a curve or a field arbitrary curve the beef is here because I like to say and he's going to look at these questions of zeros associated with certain problems of L functions and their local spacing statistics and then also once you're looking at that explicit formula there are two sides to the formula the one side is you want to understand these zeros are the eigenvalues in this case they are actually eigenvalues and on the other side you want to understand the arithmetic of primes progressions and problems like that so uh in all in both of these he's made uh and others had well as well but uh I think in many works with Rudnick uh with Rudnick with Keating excuse me uh they have uh made decisive results at least when q goes to infinity that's the one thing they have to do because in the end they are equidistribution theorems for families and exponential sums that come from higher dimensional use of the lean's work that are often the decisive or the thing that doesn't have a lifting so for example they study the distribution of primes in short intervals in this function field and in short progressions and obtain sharp results that at least in the q going to infinity or certainly not known over the number I want to point out that uh Sarah gave a lecture here maybe 10 years ago and he looked over at Henry who and said uh the problems of a certain type like the charlotte conjecture the hoddy little wood k double conjecture are unmotivated and I thought wow that's kind of uh that's not a I still cannot recover from his comment this was about my work with Duke yeah I know I know he said it was unmotivated but what he really he had it was a deep joke it took me a while to understand what he meant is it's not a natural question coming from a motivic setup of the use of l functions so the motivic setting of using l functions means shabbat tariff the only thing you want to use is shabbat tariff and variations on shabbat tariff when you start talking about twin primes it's not motivic but he made the joke of unmotivated uh and so Zerv has looked at these unmotivated problems with great success and in particular I like very much his this would be a typical example his work with dam Cameron uh about the charlotte conjecture in the function field so he's looking at these polynomials monic polynomials of degree n uh in the polynomial ring f q x and you look at the correlation so you take alpha one to alpha r mu is the mobius function that's defined in the same way in the function field and uh alpha one to alpha r are fixed so you this is r correlations and you sum the mobius shifted by alpha one up to alpha r over these monic polynomials of degree n in the number field you have to sum up to some point because there aren't many uh numbers of given size but in the function field there are these big shells so you sum over these shells so you sum over m n one over the number and that should go to zero if the channel of conjecture is true uh in the function field and you shouldn't have to make this q go to infinity but that is the part that these works where one has a very complete results Zerv kind of leading that or involved in many of that uh this is a typical example this q goes to infinity you prove the correlations that alpha is a distinct of course uh you prove the correlations uh go to zero and you prove the analog of the charlotte conjecture I should say that the starting point is a great exploitation of the fact that in the function field in the setting there's a very simple formula for the mobius function in terms of a character and the discriminant and the degree of f there's nothing like this in the number field so the number field we don't expect any structure of this type unless there's a zeal zero lambda zeal zero then in fact there is a character mobius there is a character if you have a sequence of very bad zeros then you would have the corresponding character mimicking mobius and in fact you can then use it in this kind of way as he's branded many years ago to prove the twin prime conjecture if there's a sequence of zeal zero but of course there isn't such a thing while in the function field is it's quite different and it's the starting point that they exploit and that many people have exploited after them and before them I should mention conrad conrad and gross also use this kind of thing to end this little review of the function field there are some quite dramatic theorems in the last few years where q is fixed and you're still able to prove that and of course that's really what uh that's very impressive and these kind of things start with the work of ellenberg van katesh and west ellen who proved the coven lens for a heuristic essentially for a fixed large q and the big difference here is when you let q go to infinity you have a main term which is like q to the n and then the next term would be q to the n minus one and if you let q go to infinity then you've got your main term but if you fix q then the q to the n minus one would look very bad so you need to know that in your exponential sums or whatever you're getting your hands through in this kind of exploitation and there are exploitations of this uh in this work of saun and schusterman variants of that that they highlight and use uh one needs uh to have a vanishing range of comology and this was a big insight of this paper here and a similar thing is used in part here and we heard a lecture by saun a beautiful lecture just a few weeks ago of a proof of charler not for q a prime but for q a prime power as long as a is a little bit big they prove this conjecture over here but not forcing q to go to infinity but just letting n go to infinity which is quite beautiful so i said i would spend half the lecture on reviewing a very small set of zed's works but they are all using the connection between uh of course there's a tremendous thing other a lot of things that go into it but the explicit formula is used heavily in many of this is a starting point in many of his of his uh works i want to talk turn now for the next half of my lecture to an example which is very much along the line of what zev does and its joint work with pavel kurosov and it will return to the explicit formula which is why i started there and this is an example um which in many ways uh is much richer than i imagined before i started working with him and it is one where we can say something quite interesting and even solve some interesting problems related to explicit formula so the setting is as follows we uh went to start off with a poisson summation formula now the poisson summation formula corresponds to a torus but let's start in dimension one dimension one there's only one compact remanian manifold and that is a circle and it's parameterized by its length and if i write down the celberg trace formula for that setting the eigenvalues are m squared and if i take the square root of the eigenvalues and write down the trace formula that is the poisson summation formula it's a one case where the celberg formula and the poisson are identical and the poisson summation formula is just that the Fourier transform of a point masses at on an arithmetic progression is the sum of point masses on an arithmetic progression so i want to look at one-dimensional manifolds you see the reman zeta function has t log t zeros it's like a one it's a singular one-dimensional manifold so let's actually have something real one-dimensional manifold which uh we can analyze but we don't have to make it singular so we make it take a graph so here's a graph take any finite graph and we put lengths on the edges so it's got let's say it's got capital n edges and capital m vertices and let's put lengths l one up to l n on the edges and i now think of the lengths as a vibrating rods or springs and i join them the singularity of this one-dimensional manifold is where i add the vertices which uh if there's any degree two vertices then it's not really singular so we will assume they know degree two vertices in the graph and what is the eigenvalue problem we want to study we look at the Laplacian on the edge on this graph so firstly in on the interior of the edge it's just d two body x squared it's a vibrating string and on the boundary conditions and this is important we have to decide how we're going to resolve these singularities to make the operator self a joint and so this is what people some people call these quantum graphs i'll just call them a metric graph it's just a singular one-dimensional manifold and the boundary condition i'm going to choose here you can choose a little different boundary conditions but uh in terms of the number theory that i'll be using later not not the most general boundary condition but let's start with the simplest it's Neumann or Kirchhoff boundary conditions so the function at the vertex must be continuous and i want the directional derivatives of the function the sum of all incoming edges into a vertex to sum to zero for each vertex v you can check that this gives a self a joint operator and it has a spectrum so this Laplacian on this graph is on this metric graph this is one-dimensional the self a joint has a discrete spectrum k and we write i i'm one of the few people who still does this this is where you can tell that i'm 67 well so there was only 60 i write the eigenvalue is Laplacian phi plus lambda phi equals zero because i want lambda to be non-negative i don't put minus Laplacian and this Selberg did and i i followed his tradition so the eigenvalues are non-negative and we write them as k squared which is very natural for more points of view all right so i want to look at the spectrum k of this metric graph so it is convenient and this is extremely important for this to not define the spectrum to just be plus minus k so i'll make everything even but to adjust it at the point zero so the multiplicity of the eigenvalue zero the graphs seem to be connected is always going to be one but i'm going to actually change that definition at the origin because of a formula i want to hold later on so the multiplicity of the eigenvalue zero will be two plus n minus m n is the number edges and m is the number of vertices so that's basically one plus the Euler characteristic and that is the multiplicity i will insist at the origin and it will make certain identities true so for example if the graph has loops there's a figure eight graph with lengths l one and l two as shown there in the spectrum can be computed you'll see in a minute in a second this is the only guy which is different to everything else it'll be a union of three arithmetic progressions two pi k one of l one two pi k two of l two and two pi k three of l one plus l two and if i didn't introduce this multiplicity at zero to be three which i've done here so the spectrum is going to be the union of these progressions where if a number's hit more than once by definition it's counted the number of times it's hit so this is the spectrum if they if it's in and that is i haven't shown you how to compute this you'll see in a second but that's why we make this definition of the origin all right so there's a one-dimensional manifold the length is a positive integer or can not a real number positive real number thank you and i will be choosing them carefully later that's important these parameters l one to l n it's a they've got lengths and the lengths if they all equal then it's very easy to compute everything they're going to be chosen to be linearly independent over the rationals in a minute okay so but given any metric graph like this we have a vial law so zev has worked a lot on vial laws for you know hyperbolic surfaces and things like that but this is one-dimensional it should be easy and it is the number of points in the spectrum in an interval minus tt is like one-dimensional so it will be a constant times t and that constant is the volume of this with the length in this case which is the sum of the lengths times two of the pi just like you would see in any vial law and it's very easy to see that the remainder here is very small it's at most one so it looks smells and in in that in above case is an arithmetic progression so it looks like we are looking at something which is just a generalization of arithmetic progression at any final crude scale it is an arithmetic progression but that's the question we want to study here what is the nature of the spectrum of a metric graph and if it were this figure eight it would be very simple it's just the union of three arithmetic progressions all right so how do you compute the spectrum and this leads us to some very interesting diafantine geometry so on the edge is the eigenfunctions must be if the eigenvalues k it must be a sum of exponential because that's the only solution that it could be in the in the variable xj and then you go to the vertices and see how we've made the self-adjoint problem and we find a secular determinant this was first maybe done by Kotos and Smilanski and it's given as follows so I want to explain the computation of the spectrum by some several variable polynomials and some algebraic geometry so we make a n is the number of edges so you take two n by two n matrices indexed by the oriented edges e1 is an edge and e1 bars going the opposite direction and you make a matrix which is a diagonal matrix u as a function of n complex variables will be ufg so f and g are directed edges will be zf times delta fg so if f equals g it'll be zf and otherwise it's zero so this is a diagonal matrix with entry z1 up to zn so it's got n complex variables and s this what is called the scattering matrix and it's a good name for it uh it's got entry s little sfg with little sfg if the edge if f is remember f is directed edge if uh g follows f that's this minus delta fg hat then it's minus one plus one over two divided by the degree uh the degree will never be two because then there's no singularity then you just remove it so this is some number here if f equals g hat then you put a minus one there and otherwise you just two over the degree and zero otherwise if this is if g follows f and zero otherwise and you can check s is a unitary matrix this is extremely important and the circular polynomial or the spectral polynomial uh is the following polynomial in n complex variables pg of z1 to zn is the determinant of i minus this diagonal u times this unitary matrix s so let me say a few words about the properties of this polynomial it's a degree two n total and it's a degree two in each variable zj and if you invert the variables so if you take p iota of z1 to zn which is p of one of z1 up to one of zn so you invert each separately at the same time this involution then the polynomial of n variables this lauran polynomial it's best thought of lauran polynomial because uh we not interested in any of the z's are zero so but the way I find it it is a polynomial the polynomial p of g in the poly the iota polynomial are both what are called stable they do not vanish if any of the z's uh are inside the unit in strictly inside the unit disk and this follows easily from the unit unitarity so uh this is the notion of a stable polynomial and I mentioned that because that's one of the key ingredients that will be used in something that's coming all right now why is this polynomial interesting in terms of this metric graph is if I want to compute the spectrum of x and this was written down very clearly by by barra and gaspard who actually did a problem that they answered the question of what's the consecutive spacing distribution of the eigenvalues of such a metric graph and they found that you can write it down it's not universal and you can write it down in terms of some return map on on uh relative to this uh torus action this line in a torus I won't go into that here but they write down this very cleanly that the spectrum and this is why we defined the spectrum at the origin to be that multiplicity so the spectrum is the zeros of with multiplicity counter with multiplicity of the map the entire function k k is a complex variable goes to pg evaluated at e to the ikl one e to the ikl two up to e to the ikl n so the l's are fixed those are the positive lengths and you look at this function of k which is uh an entire function in the variable k it only has real zeros and if you look at the zeros and multiplicities this computes a spectrum of x and if you take the figure eight and compute this you will see you get three uh the zero set is going to be three hyper planes three lines and you get those three arithmetic progressions so from this point of view the algebraic variety the zero set of the spectral polynomial in several variables in the torus c star to the n is clearly decisive for uh understanding the spectrum and that's our starting point and now I'd like to explain the theorems that we can prove this is kurasov and myself and this uses heavy diapantine geometry which I'll end by just saying what the heavy lifting is so these are the theorems uh we have a whole study of the spectrum of metric graphs the additive or arithmetic structure of the spectrum something that I've never seen able one able to do in any other situation other than trivial ones and that is so the first thing before we apply the diapantine geometries we needed to understand the zero set so I'll assume here that g has no loops that's the simplest case and the case of most interest anyway so the graph g g's the graph and the metric graph I'll call gamma that x was the metric graph it depends on the lengths the g is just the combinatorial graph so assume the graph has got no loops then the zero set is absolutely irreducible over the complex numbers and most importantly the zero set contains no translates of any n minus one dimensional sub torus this is critical so the first part here a was conjectured by Colin de Verdi and we have a proof of that and that's important in how we proceed further and then the main theorem is a complete additive uh not complete but uh I would say quite complete understanding of the additive structure of the spectrum so if g contains no loops again and if the lengths are linearly independent over the rationals sorry the quantifiers always follows g's given there's a constant cg which we can compute effectively I'll explain what's not effective as we go along but constant we can compute effectively it's very large and way bigger than it should be such that if you take any linearly independent lengths and any arithmetic progression then the spectrum of x intersect the progression is got at most c of g points so there are no arithmetic progressions of length c of g inside the spectrum it's very far from Poisson sum nothing like in the dimension of the spectrum over the rationals is infinite as a vector space so these are that's the main theorem both of which show the additive structure of the spectrum of a metric graph is very different to the additive structure of in Poisson sum which would be should just be one loop now just to tell you let me make an outrageous statement we don't know anything like this for the zeros of zeta or the hyperbolic manifolds except for some trivial lifts I don't want to go into technical things so for all we know all the all the ordinates of all the zeros of all l functions in the world are all rational numbers at least for all I know let me say that I don't know how to disprove that statement let me make a wild statement to you every ordinate of every zero of every l function in the world is rational the only zeros we know are at the center so the ordinate would be zero corresponding to bsd type conjectures so those are rational but other than that I would expect them never to be rational but I'm saying the exact opposite you just produce for me one guy which is irrational so you can see that in the symmetric graph so the the fact that you might have a summation formula and you might try use that formula to prove irrationality is not in the cards at least in anything I understand so this is the strength here is obviously we're not using the summation formula in fact we're going the other way to say something about the summation formula in fact if we take this and write down the summation formula for the metric graph it's just a baby selberg trace formula and should read that way and it reads exactly that way so if you have a metric graph the summation formula takes an exact form this was not proved by the Roth from the Tuer-Ziegler Roth whose theorem we are using to prove the other theorem it's some mathematical physicist Roth and by Kotos and Smilansky and certainly Kurasov formulated in exactly the form that we need here and the summation formula reads exactly like the selberg trace formula is the sum over the spectrum of delta masses at the spectrum in it for any metric graph it's Fourier transform which say if this were Poisson sum this would be then or anyway it's Fourier transform like in the selberg case is a bunch of delta masses firstly there's a delta mass at the origin with multiplicity with weight with coefficient the volume over pi just like in the bio law plus the sum of all periodic orbits just like in the selberg trace formula the lengths of the periodic orbits l of p the lengths of the primitive guy with each p and then the coefficients and i want to explain those coefficients are very very important the length of p is by definition when you go over periodic orbit up to cyclic equivalence of course if you go over periodic orbit you just add the length so they positive numbers so the l p's are positive numbers everything is even here the way we've chosen it so there's l p and minus l p and when you go around the periodic orbit you multiply every time you go through a vertex you multiply by the scattering matrix entry at that vertex so the coefficients here are not positive they are on this side this is but not on the side the coefficients there's a lot of cancellation in the sum there's an exponential number of periodic orbits yet when you sum this something will be absolutely tempered i think that's not obvious and it's very important and it comes from s being unitary so the Fourier transform of delta masses at these points mu hat x is supported on the following set it's supported on the set of positive multiples the l's are positive numbers they fixed and we're taking positive integer combination so this is a discrete set so we have an amazing fact here that the Fourier transform of delta masses at points just like in Poisson is a sum of delta masses with coefficients at points and these are discrete and that's the exact definition of concordance or of crystalline measure that are of great interest in the theory of quasic crystals so we have as a consequence of this metric graph every metric graph the selberg formula for it the trace formula gives you the following object it gives you a crystalline measure so let me make an exact definition of a crystalline measure this exact definition is due to Eve's mayor it's a concept that's around in harmonic analysis i think for some time certainly goes back certainly to Guinand who was interested in these concordance questions so a measure mu is crystalline if it's a measure on the real line which is a sum of delta masses with coefficients and the delta masses the set where they supported is discrete inside the real line and that's not difficult to arrange but the Fourier transform should be of the same form so Poisson sum gives you that if the the lambda would be arithmetic progression and the Fourier transform is a sum of arithmetic regression if you take finite linear combinations of that they call this direct comb so the examples of crystalline measures are direct comb so they are like generalizations of the Poisson sum if I take summation f a summation a c f hat f of c it will be summation the same thing with a Fourier transform with f hat and you might try classify such things in any event these crystalline measures of great interest in quasic crystals and i want to emphasize again that the absolute value of mu x set here for the ones coming from the graphs are actually tempered absolutely tempered so it removes the Riemann zeta function out of this consideration and they're often called quasic crystals so we have these examples now that come from metric graphs and because of the main theorem which gave the additive structure we know that none of them are finite combinations of direct combs in fact they can't even contain arithmetic progressions of a get bigger than a fixed number and much stronger statements so they are exotic crystalline measures and that in fact leads to the solution of a number of long-standing problems due to mayor legarious and levolevsky perhaps the most basic unsolved problem that they raised and was raised for some years was are they positive crystalline measures so the measure on the left hand side is positive as else is because it's a delta masses at the spectrum of the metric graph are they positive crystalline measures whose Fourier trans sorry are they positive crystalline measures which are not arithmetic progressions and the answers they are all of these are every one of them is such a thing and these are very rich and exotic since our construction Eve's mayor has explained the construction in terms of cut and project non-linear cut and project operation which is connected very closely to those stable polynomials that I was telling you about and I should say that mayor and kordova and legarious and levolevsky have given theorems which under some extra conditions will ensure that a crystalline measure must be poisson some so clearly ours don't satisfy those conditions but for example if the both sides if the the both sets are discrete but if on both sides the minimum spacing is bounded by a fixed constant from below then levin or levsky show that you have to be a direct comb so one tries to classify the summation formulae as a harmonic analysis question and this shows that it's extremely difficult and they are these positive ones and the early such classifications I have to say this because it goes back to my advisor the early classifications use the cohen item potent theorem in the measure algebra that's how mayor and legarious prove these kind of things those are classifications of croissant some type summation formulae where you put extra conditions and you have to be an arithmetic progression these examples are very far from that I would mention a beautiful lecture that you should all go read by it's the einstein lecture given by freeman dyson about 10 years ago to the ams it's called birds and frogs with such a title i'm sure you'll go read it bird he describes mathematicians as either birds or frogs the birds are the ones like home and vile who fly around and understand everything and put everything together and the frogs are people like he describes himself as working in the ground and discovering serious things by hand and then of course the question is what are you while you read that you'll try answer that question but one of the things he speculates since he's at that point maybe 80 or so quite that the way to solve the reman hypothesis is to classify crystalline measures by which he means one of these he's called a quasi crystal and he speculates that that would be very difficult and of course this shows that it will be very difficult this way because they're just too many crystalline measures and that's not the way that I think anybody should be doing if they had the reman hypothesis in mind in any event what I think is clear there is that this is very rich and let me just end with the big heavy hitter dive and team input because this is one of my favorite theorems these days and it's been for many years it's the laying GM in its ultimate form this is what's used to prove the theorem of Kerasov and myself and let me just state it because it's such a beautiful theorem and it's uses the Schmidt subspace theorem so to a Ziegler Rothschmidt high dimensional dive and team approximation some of the deepest maybe the deepest methods that are known in dive and team geometry and that's a following theorem and this is a very uniform version of it due to everster schlicker by and Schmidt and that's a following suppose I give you the torus and I give you algebraic sub variety and then field of definitions irrelevant here this is also important in the application and I give you a finitely generated subgroup of rank r in the torus and I take its division group all points in the torus for which a positive non-trivial power lies in the group then there's a constant that depends only on the variety and this is used in the proof such that the all the points that are in this division group that lie inside the variety V are already points which lie in so there are finitely many translates of sub tori they needn't be by torsion points so finitely many sub translates of lower than proper sub tori t1 to t new they can't be found effectively in general but their number can be bounded that was what I was looting to in in the theorem there are finitely many sub tori such that the points in the division group which lie in the variety these tori are contained inside V are exactly the division points intersect the union of the sub tori so what is a highly non-linear problem of gamma by intersect V is actually gamma by intersect the sub tori and it's that that controls the spectra of the metric graph in this particular application and this theorem uses the absolute subspace theorem of schlicker by it has to bound small points large points it's a very powerful uniform statement about the intersection of division groups with some varieties of tori and that's exactly where the dive and team input is I think it's exactly noon so happy birthday zeb and welcome to the senior class you are the most junior member day of