 Okay, let me begin by, I think, the organizers to invite me to participate down. It's an honor and a pleasure. I was a bit hesitant because the last few years, my interests have decided to become more physical far from the austere world of modular forms. But there is a beautiful connection that we discovered in the course of our collaboration with Don and I thought it would be a good idea to present. I realized also that it's a very problem of yours, including some six-year-old and some other... So I will try to present a kind of overall perspective about the field and what are the interesting open problems and some of the things that are unique to everything again. But also another temptation was that a felicitation also offers an opportunity for a roast. So that was also a very tempting thing. So I will begin... So I will begin with that lesson time first, for the first few minutes. So I first met Don. So my first encounter with Don was through his classic book on the Coby forms and actually Boris Puglian and I, we had organized a workshop in Paris in 2007. Boris is also here with the audience. And we had invited a number of mathematicians just on the hunch because somehow the interaction was beginning to get important and actually Gritsenko, who is also in the audience, find Gold and Maxim Konsevich, we invited about Don and in fact, Maxim reported his work on wall-crossing Konsevich Salomon paper at that conference. So it turned out to be actually a fruitful workshop because also our collaboration there got started during this workshop and the main reason is that the counting function of black holes is often meromorphic to Coby form. Is there a USB stick in that thing? I mean, there you should put it in the monitor at the top. Is there a switch on the right side? Is there a switch on the left side? Okay, nice working. So the black hole counting function, which I will explain what it is a little bit later is often a meromorphic to Coby form f tau of z with poles in z in the elliptic variable z. And then dinner conversation about the poles fits our interests. I mean, so these poles were kind of bothering me. I was thinking about them. And Don explained that the meromorphic implied a loss of modularity of the theta coefficients in this expansion of this function and apparently this relates to some century old problem going back to Ramanujan and his former thesis student had understood how to restore the modularity at the expense of holomorphic and the connection with Ramanujan's mock-model from some century ago. So of course for Don, the modular is kind of a given thing. I mean, if you give him a, people say that physicists think when they think about chicken and they say let's assume that the chicken is spherical and then they add the head and the tail and so on. I think in case of Don, if you give him a chicken, he would say that the chicken is modular. But for physicists, the modularity in this case actually has a very deep physical reason that has to do with holography and it implied that modular symmetry was part of global difumorphism. There is actually geometric torus in this problem, which is not evident and it's the multi-class group of this geometric torus which is the SN2Z which is what is complicated in this modular symmetry and therefore in a geometric theory it simply cannot be violated. So this really bothered me and this was what puzzled me. This loss of modularity was very puzzling for me and so then Samir Murti, my colleague who was a post-doc at the time in Paris we thought that it would be good idea to collaborate with Don and maybe in a few months we will figure this out. And also I knew that Don had quite famous collaborators so I was of course familiar with Eichler and Zagheer. There was Gross and Zagheer. Apparently he had also collaborated in this. So I thought collaboration would be a good idea. And it was also fascinating that the black hole physics had connections with this mysterious mathematics of Ramanujan I had heard about in that school, there was this beautiful lecture of Dyson I had heard and I was therefore an inspirational figure for everybody from India so I thought it would be a good idea and with his help you know after all it's just a pull we'll hack our way through and figure it out in a few weeks. But Samir and I had no idea what we were getting into because this, anybody who has collaborated with Don knows that no paper gets done in a few weeks. In fact those of you who are familiar with socialist economies So you have a five-year plan. The budget is presented for the five-year plan. So this was our first five-year plan. 2007, 2012. It was actually very worked out very well that this long five years and many late night discussions over good wine in traditional restaurants led to very nice paper, 150-page paper to quantum black holes and multiple forms and then we have the second five-year plan which is about to end and the plan is to have an expanded version of the book in Cambridge Monograph and mathematics physics and don't we have to do something about this? I wrote a letter to Samir only yesterday only a few months ago to fulfill our promises. So the thing is that in the course of this work I was kind of reminded of this. Daisen has classified mathematics into two classes birds and frogs. The birds are the ones, so frogs is not meant as a pejorative he himself puts himself in the category of frogs. So birds are the ones who really like the reversals and they don't understand like some big picture and they are not interested in the details. The famous example would be like Rothenly who thought 57 was prime number because they have a story about him which is kind of a Rothenly prime. So they are not interested in the details of some particular prime and then the frogs are somebody like Nahanudin for him. Some unique place of a number like 17, 29 is really important. And I think and there are some of course there are both categories. There is no sharp distinction but I think Don I would say probably in the frog category and I realize that he is no ordinary frog. He is like a frog extraordinaire. And it really suited because this is also a bit like frogs in that they really wanted concrete answers and it was really fun therefore to collaborate with them. So in the process you know even for a Philistine physicist like me I really got an appreciation of the beauty of mathematics. For example something that I learned from him a sort of glimpse in my mind such as Ramanujan. I think that Don also works in a similar way. One thing that I learned from him is that if you know the first few Fourier coefficients of a modular form you basically know it because it's finitely generated. And if you secretly know some modularity of the objects involved you can prove all kinds of mysterious and magical identities. And I could see you know of course Ramanujan did all these expansions on a slate. Nowadays you can do it with Pari. You can work for hundreds of digits. But it sort of gave me a glimpse of how things work. Then you know in the old times when all sailors used to be men it was said that a sailor has a woman in every boat. Likewise Don also has an apartment in every country. So of course in Bonn but I had been to his apartment in Utrecht in Trieste-Hellicentry but in the apartment. Then I had been to his charming apartment just behind the knockdown in Paris and our collaboration would start like this. I would go there in the morning on some Sundays and then it would start with an accomplished piano concert. Then we would work for several hours and then we would go for a lunch somewhere in a nice restaurant together with this wonderful wife Zilke. So I have enjoyed many friendship over the years and many hospitality and many wonderful moments together. So I would like to thank him for that. And as I got to know him better I realized that beneath this exterior of an intense and brilliant mathematician lies an intense and precocious MIT undergrad. Because my collaborations really reminded me of my undergraduate years. One particular one I remember is in fact when I was in Bonn we started to work before dinner. Then we would work here in the institute and we went over to dinner, then we would tell the dinner, go online. Then Don said, okay, I will walk you over to your hotel. So then we walked to the hotel, then there was a bar so we started to work in the hotel. So it was like 10 o'clock, 11 o'clock, 12 o'clock, 1 o'clock. We were still working. Actually the correct thing to say is that Don was still working. One o'clock my concentration was being, it was 2 o'clock and then after that Don was satisfied with whatever it is that he wanted to prove. I was also satisfied that I could finally go to sleep. But then just as he was leaving I started to tell him about how I was in Korea and how the Korean script is similar to the Devanagese script. I also know a few Indian languages, a few European languages so I shared that interest with him. So I started to show him how the Devanagese script, the Hindi script is written and then he got very interested. So it went on to 213. And then after six months he showed me, he wrote me a message in Devanagese. So since he knows Devanagese, I will wish him a happy birthday. Those of you who know Hindi will find that Don is particularly respectful and particularly funny. So happy birthday John. I don't know why they are celebrating the 65th birthday because after all 65 is just a number. It's not even a prime. Maybe it's the 17th Fourier coefficient of some particularly interesting model. John might know. And as they say, 65 is the new 40. So I wish him many years of good health and productive mathematics so that he will continue with the same zest of the Manthi Anubhav for the years to come. Okay, so Passo will show Sirius as the same French. Let me come to Sirius' part of the talk. So let me start with a quote about the book of Manin and since Manin is associated with it for many years, it's kind of summarized that I want to stay today. So this is actually a quote from a preface by Dyson to the English edition of the book by Manin called Mathematics as a Metaphor. So incidentally Dyson puts Manin in the category of birds. So Manin sees the future of mathematics as an exploration of metaphors that are already visible but not yet understood. The deepest such metaphor is the similarity in structure between number theory and physics. In both fields he sees tantalizing glimpses of parallel concepts, symmetries linking the continuous with the discrete, and he looks forward to unification which he calls the Quantization of Mathematics. And to complement this, I would say that I will explain to you that the black hole horizon actually somehow knows about the number theory and mathematicians have become used to this kind of interaction with physics. For example, quantum field theory has proved to be very useful. For example, Charles Hamilton's theory is very useful in understanding modern variance and mirror symmetries. There have been several places where the quantum field theory has become important. But this is actually qualitatively different and I think it's quite a deep connection which has to do with holography and not related to any of the other intersections between mathematics and physics. And I think it's quite interesting and quite new and it suggests some kind of quantization of space time geometry. So this is sort of the kind of if you would like to say the big picture but one doesn't know how far we are from that. So let me now, so this is what I will talk about and in particular the symmetry in this case is just the modular symmetry and which relates for example, the Fourier coefficients of modular forms to some analytic rudder marker type of expansion. So this is one connection where this happens and in the case of the black hole, the black hole horizon is a very complex, it's a very geometric concept having to do with the differential geometry and the path integral on the black hole horizon is a complex analytic object. So it's actually very surprising that it knows about an integer which is a very numbered theoretical object after many steps of, involved steps of very detailed calculations that you see that somehow the complex analytic black hole horizon path integral really knows about a numbered theoretical discrete integer. And that's the thing that I would like to convey in this talk and how the black hole horizon on the black hole horizon. So this is the black hole, black hole horizon everywhere and you will really see that indeed they are there on the black hole horizon and not exactly on the black holes that have been discovered in black at the moment but these are some special supersymmetric black holes in string theory. So this is what I'm going to do, a string theory and a string theory which is what I will now turn to and to do that I will have to since this audience is very untrained in string theory I will start with a very basic introduction and I will present to you two views of black hole, the way it is to see it. So one is particle states in M-theory for example space time, like we did today, some six dimensional manifold times this one. So this is some eleven dimensional manifold by M-theory you don't need to know anything about M-theory what you need to know is M-theory like any quantum theory. So basically when physicists say a theory what they have in mind is a quantum Hilbert space and some Hamiltonian and related to it is some action principle. For example in the Maxwell theory the action is the Hilbert-Leyman's action or in the Einstein theory the action could be the curvature which is catered and so on. The Hilbert space is just a quantum Hilbert space so I will come to this action and I will talk in the second part so let me first focus on the Hilbert space and the Hilbert space has some self-adjoint operators like the Hamiltonian and that self-adjoint operator will depend by theory you need to specify some particular moduli of this manifold or six dimensional manifold. So K mu is typically a Kalabia tree form in our case we will take it to be K3 so I will specialize K3 cross T2 with some moduli, the complex structure moduli in particular and the torus for example has some flower modulars. And then given such a manifold I can consider a cycle in the H3 of TG and corresponding to it is an operator that you can associate with the Hilbert space which depends on the moduli. So basically we have operators H mu and gamma mu. So given these two self-adjoint operators in this Hilbert space of health theory you can find out their eigenvalues you depend on mu and this state this gamma also depends on each eigenvalue depends on mu and gamma it's a bad notation let me call this gamma. M is the energy eigenvalue of the Hamiltonian which in this case if the particle is sitting at rest if you think of a, so this is R3,1 and the state that I am considering is think of it as particle sitting at rest in a plane and therefore with some mass M so if I have the Hamiltonian acting on it since its energy is M the eigenvalue of the Hamiltonian is M but in addition it has some charges gamma which depend and at each point in this space it's k cross S1 and the radius of S1 is R there are two limits which we will consider one is the limit when R is much bigger than 1 and the other is the limit when R is much less than 1 and the descriptions actually simplify considerably in these two limits in this limit the same particle sort of gets basically Newton's constant becomes large Newton's gravitational constant is related to this R so basically there are two pictures of in the two different limits in one limit you can basically treat it as a point particle really like a point particle and gamma is some complicated cycle as you can see since gamma belongs to H3KZ it could be some complicated cycle and the physicists call it the brain wrapping some cycles so for example around the torus it could wrap N times around the A cycle and N times around the B cycle and similarly inside the K3 it can wrap so many times around some particular cycle of K3 it's a very complicated cycle in this but that cycle alone is not enough you have all kinds of interactions happening at the intersection points which determines the dynamics of this object the simplest example of such a cycle for example is related to for example if you take so mathematicians say physicists call this a three brain wrapping this particular three cycle simple example taking mind that if you take K3 and if you take of what physicists would call a four brain wrapping the K3 and N zero brains bound to it in the mathematical language what you should think about is you have this K3 and there are N points which correspond to the zero brain so it's one particular four cycle which wrap the entire K3 and then there are so many zero cycles however there is a modular space because these points can be anywhere on the K3 so the modular space of this is clearly a symmetric product to the power N and the quantity of interest for us is that in this Hilbert space in this Hilbert space H defined by M and gamma if I look at all the number of states in the Hilbert subspace specified by this Eigen values M and gamma I can calculate the dimension of this Hilbert space I can ask what is the dimension of this Hilbert space and this I will denote by an integer this is an integer it turns out that it's a finite dimension of the Hilbert space it's an integer that you can calculate and that counting problem is related to roughly speaking all the character of this see it's not the all the character of some cycle or something like that it's the all the character of the manifold of these interactive brains in this case this is a well known problem this will be related to the Thomasian invariance and modular forms actually appear in a very natural way in these kinds of counting problems for example in this case the the genetic function if you take the all the character of N plus 1 then it's going to be something like one of them 24 is related to the fact that one copy of K3 has all the character of 24 and to calculate it the all the character of N plus 1 symmetric product of K3 basically it's a kind of simple counting problem which is given by Q to Q Q to Q Q to Q ok so you can see that a very famous modular form appears in this particularly simple counting problem what is M and gamma in the last slide M and gamma in this case gamma is actually the number of so gamma is 1 4 cycle and 0 sorry gamma gamma is 1 4 cycle and 0 N 0 cycle that was the cycle that we chose it turns out that in this case gamma is independent of the modularity actually just the number N 2N so N sorry so in this case gamma is the number N and M also turns out to be the reference number N actually in this particular problem both N and gamma just become simple and related to this number but more generally in the problem that we are interested in this is generalization of this which is called the Donaldson's drama theory or Panderi Pandey's drama theory and these are similar similar invariance of this kind which you can calculate ok so I have explained one part of the problem to you we are interested in the Hilbert subspaces of M theory particular value of the mass particular value of the charges and that leads to a counting problem which is related to the modular spaces of some various cycles some curves and so on some kind of curve counting problem and if you can calculate the other characters of more generally electric generator of this modular species what you get is the number that you are interested in Fourier coefficients of these objects give you the degeneracy and the degenerate is simply because it is a degenerate Hilbert states where in fixed mass there are many many states corresponding to that particular state so it is just a degeneracy in the conventional sense the eigenvalues are degenerate so quantum degeneracy is of a particular state with mass m and charge gamma is given by these numbers what is this? now if you now rank up your r then you have to use a different description of M theory namely I will show you the action and as I told you the action is actually it is an 11 dimensional action so it is a 11 dimensional boundary form and the action is very similar to the Einstein river action which is standard and then there is an additional pre-formed field so the action looks something like this it is not a very complicated object and therefore one can ask what are the solutions of this which are obtained by a variational principle this action principle as a physicist call it and that leads to some equations of motion like the Einstein equation like R mu mu minus R mu mu R should be equal to 0 so this is some non-linear project differential equation known as the Einstein equation which simply says that some combination of the Ricci tensor should vanish or should be proportional to some values of this and so on and these solutions can be found solutions of this some of the solutions the particularly interesting solution of this is the black hole solution and not only basically in the first approximation ok so there are two the simplest black hole solution looks like this instead of now having R3 comma 1 you have some complicated 4 manifold M1 comma 3 and there is a region which I want to denote as black hole this is our best impression of black hole and the important thing is that every point now it's like a fiber thing it's no longer a direct product it's a key mu as a function of R plus S1 as a function of R with the radius as a function of R ok so this is the 11 dimensional manifold now looks like this M1 comma 3 it's just called the black hole manifold it's sort of fibered over now about continuity arguments from super symmetry which I will express but basically since I can vary R from very small to very large continuously you can see that the D the number D is calculated what happens to D we had some number on the left hand side and we continued it we were R much bigger than 1 so this point like state became like a black hole state and then the question is what happened to this number how does that number show up in this picture can you continue from M and G so when you take the D you need to continue and I can vary this and you can write it to this on the screen but it's the level T it's dimension so you need to continue one of the modeling you need to continue R keeping everything else fixed and the answer to this question is actually very deep and very beautiful and very interesting and the answer to this question has to do with black hole and which is why I sort of used this quote of in the beginning about the number theory and in physics the black hole entropy is really the window that gives us this kind of a window into the quantum structure of space time of this connection between number theory and mathematics so the entropy is a notion that goes back to Boltzmann from the 19th century and one of his really key insights of Boltzmann which we normally call as S this S is equal to log of D so I can of course calculate log of D since I have calculated D the question is how do I calculate S how do I calculate the left hand side of it and it's very similar so this is just to give you an analogy for example in the 19th century I think 1960 something I think it is something when you could not see atoms what you could do is entropy is related the change in entropy is related to the change in the heat that you have divided by temperature so for example in this room if you put a heater on electricity is consumed and how much heat goes at room temperature and Boltzmann thinks that should calculate change in this entropy and by doing this you can essentially calculate the entropy so this is one way to relate the entropy to something really macroscopic the macroscopic about what is happening in this room I put a heater on and the room gets heated up so this is a very deep connection between macro physics on one side and micro physics on another side and in our context this is sort of a connection between geometry and number theory you could also put maybe general relativity which has to do with space time geometry and quantum mechanics which has to do with counting things in the Hilbert space so this is why the blackboard it's a fundamental object that gives a connection between general relativity and quantum mechanics which is the big problem in physics is to identify the finding quantum theory of gravity it therefore gives a connection between geometry and number theory and it's a kind of a microscopic window into the micro world and it may be the way Manitz idea number theory is kind of played out in this particular context and I will probably mention to you one more complication and I will state that go back to our results so how do you calculate this s s is actually turns out to be the area of the blackboard divided by 4 and what is meant by the area of the blackboard this region that I showed you what I call a black hole it's called a black hole because it's actually this is a surface sitting inside 1,3 as I told you but it's a very special surface it's an s2 moving at the speed of light it's a bit hard to imagine that each point in the sphere is moving at the speed of light so what happens is that once you insert the sphere since you cannot go faster than the speed of light you can never escape and that's why it's called a blackboard once you're inside the sphere that's why it's called an event horizon there is a very beautiful equation by Hawking and Beckenstein that this entropy is equal to the area of this s2 which you can calculate purely from geometry divided that by 4 and that should equal this log of d for large and there is a okay so now the problem that is clear should be clear that you have to know if it's going to be from one side sorry I forgot to mention that that in the more general context when you take function just like you've got one upon delta becomes a meromorphic dipole form and these are the beyond the c's therefore are given by the coefficients of this meromorphic dipole form now one of the puzzles that immediately arises is that not only do you have these solutions but in fact we have other solutions which are called multi-centered blackboards so with the same asymptotic data, namely the same delirious asymptotic infinity what we'll find when we follow these equations is that we find not only these solutions but also these solutions and then the question becomes should we identify dm gamma with the entropy of a single black hole or with the entropy of the multiple blackboards and this looks like a pesky problem but it turns out to be actually very important and that's where actually the meromorphic function will integrate so how that happens is very interesting the question is that can we separate given this meromorphic dipole form can we separate the contribution coming from the single centered black holes and the multi centered black holes in some very nice way and that answer was given by basically that's one of the main motivations for our paper was from the physics point of view was to give an answer to that question and the answer is actually it turns out to be particularly nice and simple and I will mention how much time do I have, 10 minutes 8, 10 minutes so I told you that gamma belongs to h3 of k2 k3 cross t2 now for t2 you can choose an a-cycle and a b-cycle and therefore this is isomorphic to two copies of the gamma can be split into what is called a p and a q a magnetic charge and an electric charge this is called a magnetic charge and this is called a magnetic charge but mathematically speaking if you just think of it as one particular way to label the a-cycle then this h2 of k3 is known to be the self-divorant even if you know Lorentz in lattice so it has a Lorentz metric on it and so it belongs to R 19 comma 2 3 comma 2 so you can define p squared is equal to 2m q squared is equal to 2m and t dot 2 is equal to n and it turns out in this counter problem the degeneracy basically depends only on n, n, n and the model so instead of depending on so many different integers it depends only on three integers and this is related to a particularly simple nice Jacobi form psi m so n and l are going to appear as the Fourier coefficients of this that's a very precise connection between the degeneracies and Meromoff and Jacobi forms now the question over arises that the Meromoff and Jacobi form can even write very explicitly is the elliptic genus again n plus 1 copies of k3 divided by this Ramanian function delta times eta to the 6 this is the directly theta function Jacobi theta function it's a very explicit formula which you can put on a computer and I only would think except for a double pole and that was the pole that I was referring to in the beginning of my talk you could put it without this factor you could just put it on a computer and calculate the Fourier coefficients but with this pole you have to choose a contour but it turns out that the choice of the contour is very nicely correlated to the choice of the degeneracies sorry the dependence of the degeneracies on the choice of the modeling but that does not answer our question how do we split this into signal centered plus multi centered counting functions and that answer is given by our decomposition theorem says that this maromastic object form can be split as a mocked object of the form as an upper left thumb and we define this new object for the mocked object of the form basically building on the results of Vegas but it is putting it this way is illuminating also for physical reasons so it's illuminating many ways and in particular the physical interpretation becomes very transparent in this case so the decomposition theorem says that given the maromastic object of the form it admits a maromastic unique decomposition in terms of psi and finite which has no poles and an upper left thumb which has the same poles as this function in the complex z plane and it is given by this upper left thumb each of them admits a model of completion I am running out of time but maybe therefore I will now switch to the results and I will quickly run through the results in 10 minutes so the maromastic object of the form that I was talking about is in fact related to the EUSA it's actually the P expansion so one of the EUSA components appears it has this very explicit form it has double poles and the Fourier coefficients depend on their contour and choosing that contour and making sense of the Fourier coefficients is what leads to the maromastic decomposition theorem says that it can be always decomposed and admits a canonical decomposition this manner the maromastic object of the form has no poles the upper left thumb looks like this it's basically constructed out of the zeros and with some kind of an elliptic average and the contour depends on the modular in a very precise way the modular space splits into chambers in a way that is consistent with duality and so it gives a very beautiful physical interpretation in terms of black holes that the these Fourier coefficients which are pure numbers coming from number theory somehow are related to the entropy of the geometric area of these black holes and one can go further than that as I mentioned in the moment so this is the picture in the modular space depending on where you are in the modular some places in those regions of the modular you only have a single centered black holes in some other regions of the modular you have both these solutions are present the mob jokobi forms counts the single centered black holes which exist everywhere the apple lag sum counts the multi centered black holes when you cross this wall in the modular space it corresponds to changing your contour passing through a pole and the residue of the pole corresponds to the jump in the degeneracy which is the degeneracy of these objects which disappear going from one side to the other it's a very beautiful and nice picture and a very simple physical interpretation of a rather complicated mathematical story and the modular part of this is of course the fact that the mob modular forms admit completion so you can given the mob modular form you can add to it something non-horomorphic to make it into fully modular but that modular object there is not homomorphic and the homomorphic anomaly is given in terms of this very simple expression in terms of theta functions and the eta functions and basically in this way we could get all the known mob modular forms and in particular the one that Don likes very much has to do with the who lives running the class also appears for m primes the mob modular form that occurs naturally is related to this mob modular form that Don discovered many years before this so this is my last slide on black holes and number theory that the decomposition theorem is a natural physical interpretation part of the story which I have not discussed which is actually also very interesting is that the Rademacher expansion of the Fourier coefficient of the Jacoby forms has a beautiful correspondence with the path integral which is a complex analytic object on the black hole horizon and this was the origin of my title that somehow the path integral evaluated on the black hole horizon reproduces the full Hardy-Ramanian Rademacher expansion which is a kind of a relation between number theory and analysis analytic number theory but here it relates to path integral so it's a kind of by that indirectly it relates to physics and for example the Plusterman sum which appeared in the Rademacher expansion arise from some churnsamus not in variance in the geometry of the near horizon geometry of the black hole so this is a very interesting story there is another story that I have not had time I was planning to do it but unfortunately I am running out of time okay this is actually more general comment that the Mach modularity somehow related to non-compactness so the Jacoby forms arise in physics most naturally as elliptic genera of certain modular spaces and so for example here I give an example the symmetric product of KG that was the modular space of 4 frames 0 million pound states or basically the modular space I won't give the romance in Thomas's story now those modular spaces are compact because K3 is compact but what if instead of K3 typically such modular spaces are non-compact and one expects therefore so therefore I am saying that this mathematics of Mach Jacoby forms is likely to have much broader applicability and not just in this lack of context because generically elliptic genera in physics language that means a conformal free theory, super conformal free theory are non-compact and so whenever instead of once you relax the condition of non-compactness the elliptic genera remain Jacoby forms but they are no longer polymorphic they have some pores, they can have some pores so the meromorphic Jacoby forms sorry, wrong the elliptic genera therefore have to be Mach Jacoby forms they do not have pores but they are Mach so getting a variation so I sort of derive the Mach Jacoby forms indirectly starting with the meromorphic Jacoby form so I think there are a number of the ESOC problems which I have been thinking about often now without making much progress but is there a physical path integral derivation of the Mach Jacoby form and its shallow is there a physical derivation of the apple lurks and its shallow the second problem I think is probably easier to solve and is there a direct connection between the attractor geometry and the Mach one in the form this is something that I was planning to draw in view of the time I will skip it so I think I will stop here wishing me down once again have a good day here