 Спасибо большое. Я рада, что Антон Зорич за эту инвитацию. Во-первых, мы говорили с Антоном, если это возможно, что я буду объяснить что-то о том, о том, кто с номерами КП «Харархии». И это было предназначено быть какой-то привидной сессии, но потом это resulted in such representative conferences or workshops, so I'm very surprised. However, if you don't mind, I will try to feel myself at home and I will allow to myself some experiments. This is for the first time that I will try to present the talk using Mathematica program. It's not really a presentation, but rather some experiments, computational experiments with Mathematica, with some comments, and also I will supply some extended comments. So according to Arnold, Mathematics is an experimental science and the difference with physics is that the experiments here are much cheaper. So I will show you how it works. So I will speak on Hurwitz numbers and all related things and I will try in my presentation I will try to avoid consideration of geometry of configuration objects. So the reman surfaces, the modulus spaces and so on. Instead, I will concentrate on the corresponding numbers on the generative functions, on manipulations with functions like topological recursion, Kp hierarchy and so on. And I will show what's the geometry behind this. So I will look at the story from this point of view. Of course, a lot of things will repeat. I will not be able to avoid some repetition, comparing with Demos talk. And I apologize for that, but probably I will try to give some alternative definitions and you would not sometimes recognize it. The object that I am considering is the same. And especially this will concern with topological recursion. I will speak on topological recursion on two other talks, but today I will speak on another subject on Kp hierarchy. It happened that the theory of Kp hierarchy and the theory of topological recursion have some very striking similarity and I will show you how it works. But today I will concentrate on the Kp hierarchy. So I start with the Hurwitz numbers. Dima already explained the definition, the geometry of these Hurwitz numbers, but I will look at them from the computational point of view. So I am trying to do computer experiments. And the simplest way to compute Hurwitz numbers is to use so-called cut and join equation. It is a kind of inductive procedure. So I recall you that we have some ramification at infinity or complicated permutation and we try to represent it as a product of simple m, simple transpositions. And the cut and join equation says what happens if we increase the number of possible transpositions just by one. So just we should analyze what happens with transposition with permutation if we multiply it by transposition. Simple geometric consideration shows that if the transposed elements belong to one cycle, then this cycle splits into two smaller cycles. The transposition connects to points from different cycles that the product will, in the product, a great cycle consisting of elements of these two smaller cycles. So either two cycles join together or one cycle splits into smaller ones. Максим, I think it should maybe start from there because there are much more people than the morning. Okay, okay, okay. So this is some combinatorics or geometry behind. But as I said, I do not want to look into the details of these combinatorics. But just the result of this procedure is some collection of combinatorial data. And these combinatorial data are packed into a certain generating series and this is called the Hurwitz function. So the Hurwitz function is some generating series whose Taylor coefficients have some special meaning. So I have an infinite number of variables, p1, p2, and so on, which are responsible for the order of ramification at infinity and one extra variable beta, which corresponds to the number of transpositions. And I can constitute the corresponding power series. So the way I packed the numbers is slightly different from the way Dima did. So in the language of previous talk, this is the series we sum up these Hurwitz numbers with usual coefficients. Dima fixed the number n of primitives at infinity and used n independent variables, x1, xn. I use a slightly different agreement. I attach one variable pi to each index of lengths i and use right in something like this. So sum up over all possible gn. And depending on the agreement, probably I should put that so. This is either I sum up over all unordered collections, then I sum divided by n factorial. Or I use just partitions, then I divide by the automorphism of this partition. Is this the same? If we take equal to 0, p1, that's all. Just unique. I don't want to repeat the definition of Hurwitz numbers because it is not important for me. Once again, let me stress that there are certain numbers which have some combinatorial or geometric meaning, but for the moment it's not important for me. So by definition I define not the Hurwitz numbers themselves, but the generative function for them. And by definition, so define g by definition. The function in this collection of variables defined by this equation, so the expansion of this function with respect to the parameter beta is governed by this formula. The derivative with respect to beta is certain differential operator, which I denote w0, applied to this function g, sorry, for exponent. So this is the definition. And plus the initial condition. It's a formal power series. Both p and beta variables. So this is the definition. And after the previous talk or from other sources you may guess that the coefficients of this function has some special meaning. But for a moment I will not discuss this meaning. So what I'm going to concentrate not on the geometry behind the numbers, but in the geometry of the space of power series. So what kind of events happens in this world of power series. So this is just one of the possible functions that I will treat in my talk and this is one of the particular examples, but there will be some others. So you may treat this as a definition. So for each power b will be only finitely many monomers, yeah? For each power... If you fix this number m, which you didn't explain to an initiated people, yeah? Yeah. No, it really should maybe write formal for m. No, no, no. If you will increase g, even for a fixed power beta it will be an infinite homogeneous. If you fix beta and m it will be homogeneous. When you can increase the genus g. So this is a definition. I don't see how you will get anything. Because you see from the formula of double u0 when you apply it to q1 you only get the first one vanishes. So if you... And the second one vanishes also. Sorry, sorry, exponent of g. This is a question. Exponent of g is equal to e to the power of q1. It has terms of any degree. So this is a definition. The operator, the so-called cut and drain operator, has a property that it preserves the quasi homogeneous grading. If you have monomial of degree of the fixed degree n, capital of n, the result will be of the same degree. So this differential equation, it's a linear differential equation with coefficients. And it splits into finite dimensional linear equations. So in each subspace of polynomials of fixed degree np we have this finite dimensional linear equation. So it's the first, the second here. Students know how to solve this equation. The result is just the exponent. The exponent of this function computed as exponent of this operator w0 applied to the initial condition. Again, do not be afraid to take this exponentiation because if you fix the degree of polynomials, the quasi homogeneous degree np, it will be a finite dimensional space. It just takes the exponent of a matrix. So we have a linear operator acting in finite dimensional space and you take its exponent. So how to compute exponent? It's very simple. If you have a matrix, how to compute the exponent for the matrix, you should take the simplest way to choose a basis, the eigen basis, so that it becomes diagonal. It happens that this operator w0 is semi-simple. It has a nice basis, eigen basis and nice eigen vectors. So eigen basis, these are sure functions. These are the same sure functions that appear in the series of symmetric functions and in representation of symmetric group. And these are symmetric functions corresponding to a use if you identify this ring of polynomials in p with the ring of symmetric functions, p are power sums, Newton polynomials. Then these are sure functions. So the sure functions are certain collection of polynomials, certain polynomials parameterized by partitions and the amount of these polynomials is the same amount of monomials. So actually the sure functions form an additive basis in the space of polynomials in p and this is somehow an alternative basis with respect to the basis of monomials. Sometimes it's convenient to expand polynomials in the basis of monomials. Sometimes it's more convenient to expand the basis of sure functions. So for this particular case the eigen values of this operator w0 are sure functions and the eigen values are also known. This m is the same as w. Okay. So this is the way how do I compute this function in my computer experiments. So sure functions are also implemented here for a single partition consisting of single parts these are so called complete symmetric functions these are related to the PI variables through this identity and for multi-partitions these are so called how it's called well, there is a determinant presentation through the partitions. So this is exactly I put the definition and this is what do I get oops, sorry where is here so these are sure functions for small number of partitions for small number of parts so for instance there are 2-dimensional space of polynomials of degree 2 and there is basis consisting of monomials p1 squared p2 but we can use this to polynomials just in other basis. Powers of independent variables Newton polynomials, yeah. Okay, just so for instance this coefficient if I take so I use here the coefficient which is obtained from the sure function if we substitute p1 equal to 1 and other to 0 this is related to the dimension of the corresponding reducible representation and this is just the coefficient of the expansion of this function e to the power p1 in sure basis e to the power p1 is just a function as a function e to the power p1 it depends on p1 but we consider this as a function in the whole collection of variables and we expand it again in the sure basis the linear combination of sure functions they and these are the coefficients of this expansion so this is how the expression looks like here I did not expand the power series in beta the coefficients are here analytical expression in beta but you should consider this analytic function in beta also as a formal power something the individual who with numbers what we are very interesting in are the coefficients this small o is a kind of dummy variable this is just my own way to keep the degree yes so I computed it homogeneous degree I computed it up to degree here 8 probably 8 and here are some just initial number of terms some several terms initial and here are the this is the cut and join equation so in the first line I wrote the original cut and join equation just I checked that my guess about the eigenvalues and eigenfunctions of this operator are correct so the first line is the cut and join equation for this exponent of the Hurwitz function in the second line I replaced the Hurwitz function by the Hurwitz function itself this G is the logarithm of this and the equation after this substitution is modified a little bit in the unique term so instead of second partial derivative the nonlinear term appears here which is the product of first order partial derivatives so this is the 2 alternative ways to represent the cut and join equation now the main subject of my talk the KP hierarchy the main statement is that the function that I wrote here and also participated in the Demos talk is the solution of KP hierarchy and I will speak about this KP hierarchy this is a particular infinite system of partial differential equations in unknown function the bottom indices are partial derivatives it has constant coefficients you see it's nonlinear there are terms of different terms which makes it nonlinear and there is huge infinite system of such equations and there is some sophisticated way how to produce these equations and even the way to produce these equations is not so simple so it looks unbelievable that it is possible to produce at least one solution however I claim that the Hurwitz function does satisfy this equation so let me substitute and you will see how it works so it works and it looks like a miracle so that the main goal of my talks today is to explain that this result is somehow obvious these are the equations these are the equations where will be yeah so for each parameter beta one parameter family beta does not involve into equations so one parameter family of solutions so I will explain you in few minutes the theory of KP hierarchy it's a very nice theory and there is a special community of integrable system people who work on this they when they speak to, when they communicate to one another, it's usually they pretty well understand each other but it's almost importable to enter this domain from outside it's like a kind of mafia how it's called sector yeah so what I'm going to tell you I'm going to open some secret which is guarded very carefully by this community so it's a I will work like a spy so I will explain what is hidden behind these equations and this really the story is very simple whenever I talk to these people and if I explain my point of view they said of course this is the way we are thinking about this but they never manifested this in their papers so strangely what I will be talking is well known by specialist but it's guarded very carefully so that the people outside will have some special respect to these people so this is a situation approximately like this imagine some textbook on algebraic geometry and some chapter devoted to grassmanians and this chapter starts with the lines definition the grassmanian is a projective variety given by the folic equations and then you have the list of and then using this definition they derive all properties like dimension, intersection theory shuber, partition and so on so this is the way how this system how it works here in the community of integrable system so consider the space of power series in p variables and this power series contains one nice variety infindimensional variety which is so important and so crucial that it has no special name it is something like sacral object that never mentioned so this is a space of solutions of kp equations actually we speak let me consider let me name it tau functions so the tau function is an exponent of a solution tau function is an exponent of solution so this object this variety is very nice homogeneous space there is a huge symmetry group it has nice geometric properties and it's very easy to identify points of this space and I will show how it works and of course like any other algebraic varieties it could be doomed by its ideal, by its equation but this is one of the way to define the equations is just the corresponding equations but this is probably one of the worst way to describe this variety so what I claim that the original thing is not the space of not the equations, but the space of solutions this nice variety there is many many different explicit and nice and convenient ways to identify points of this the the equations themselves are not so important so for instance when you work with grass manians you know how to treat them, how to take planes, how to move them how they behave geometrically and you know that you remember that they are satisfied by certain equations there are certain plural equations but I'm not sure that there are many who will present explicit form of the plural equations but however you know how to deal with grass manians even without knowing the plural equations so the equations themselves is something secondary so what the original thing is the space of solutions and the similarity between between grass manians is not by chance actually this is the grass manians so called SATA grass manians and this is the ambient space of the pluker embedding of this infinite dimensional grass manian and the fact that it is infinite dimensional doesn't make a lot of difficulties essentially this is just pluker embedding of the grass manian so let me explain how identify solutions the points of the infinite dimensional grass manian the best way to understand this correspondence is through the so called boson fermion correspondence boson fermion correspondence is a very nice physical name which is nothing to do with bosons there will be no physics at all so don't be afraid so I have a space this infinite dimensional vector space and I have chosen already the coordinates there is a basis consisting of sure functions so I treat this as an infinite dimensional vector space with a fixed basis labeled by partitions and there is another basis another also vector space labeled whose basic vectors are labeled by partitions which provides a just another incarnation another version of the same space and this coordinate correspondence is just the boson fermion correspondence boson fermion correspondence is just coordinate isomorphism of two vector spaces whose basic vectors are labeled by partitions so let me explain what is this it's an infinite dimensional veg space so I start I start with some supplementary space of Laurent series in one variable z I will consider this just as an infinite dimensional vector space labeled by integers whose vector by sec vectors are labeled by integers so z this n could be interpreted just as super index top index it's not the exponent of z because I will usually not use the multiplication of these elements so just consider this as an infinite dimensional space whose basic vectors are labeled by integers and then from this vector space I will produce such infinite elements of this vector space I will produce such an infinite veg products so each term in this vector product is a monomial on z or linear combination of monomials in z and I will so this is an example this is the product which corresponds to I strong z to the power minus I and this is what is called the vacuum vector it corresponds to the empty partition then any other basic vector is obtained from this one by changing a little bit the exponents they variable in infinite number of places so for instance if I increase here the exponent by 1 let me put 0 this corresponds to this partition or this so you see if you change just this exponent it corresponds to one part partition instead I will increase something here so it's partition 11 but the order here is not correct so I use skew symmetry to organize this exponent in an increasing order and so all is the same I can use linear combinations and I use polylinearity and anti symmetry to expand any product of this kind the linear combination of basic vectors plus z say with coefficient a s2 plus b so using polylinearity and skew symmetry we expand this and for instance the sign could appear and so on and now the definition so we take any such an infinite veg product and by definition so we have this identification so here we have this space of infinite veg product space and the boson thermal correspondence is just this coordinate isomorphism and by definition we say that the power series in p is a tau function if it corresponds to a decomposable veg product to a unique veg product of some elements the definition is very simple so here we have something complicated but if you look at the fermionic side of the story tau function is just an infinite veg product so for instance here I expanded this is a unique veg product I expanded this and through the boson thermal correspondence I just replace any V by the corresponding sure function and obtain I will obtain and then take path to the logarithms this is exponent and this pass to the logarithms and if you do everything correctly this is an example very simple example of solution of kp-harrarchy so let me check so I substitute this function to the kp-harrarchy and this is the result of course the computation is not so simple because when you differentiate logarithms you obtain a rational function when you differentiate many times it becomes more and more complicated so the result is not very hard nice and it's not so easy to do this but however if you put over common denominator we get zero so the result is very simple so this is the definition is really very simple so the tau functions is just the object that correspond to decomposible veg products from the harmonic side so if you have such decomposible veg product then the linear span of this vi form a subspace of half dimension which is somehow complementary to the subspace of regular functions so we have space of Laurent series and there is subspace regular series and this the one corresponding to this is somehow complementary and all such subspaces form the grass manian and actually we deal with the open cell in this grass manian corresponding to those for which the coefficient of the vacuum vector is nonzero because we would like to take a logarithm to produce a power series so this is an open cell on this grass manian and it's very easy to treat the points of this grass manian on the fermionic side but when you pass to the bosonic side to the space of power series it becomes something very non-trivial and very close but the grass manian itself is very nice object and of course this this veg product this does not change if you pass to linear combination of fias so up to the scalar we obtain the same tau function so it really depends on the linear span of this fias on the point of the grass manian so let me make one some conclusions the first conclusion is that the grass manian is a homogeneous space any plane could be transformed to any other plane by linear transformation so it produces a huge group of symmetries on the space of solutions so having one solution one could apply linear transformation and get another solution but when you have some transformation you know how it acts on the fermionic side how it acts on infinite veg product it's very easy to follow but when you translate this to bosonic side you get something non-trivial but anyway you have a huge group so gl infinity and its algebra gl infinity small which is a group corresponding to the algebra of symmetries of the kp hierarchy so let me give you some examples so I will speak about elements of the algebra it means that if you apply algebra element to the veg product you apply the limits rule you have many many summons you apply it to the first element of the veg product to the second and so on and let's take this element z to the power m it acts as a shift but such a simple operator compute its action on the fermionic side on the bosonic side on the other hand we have something unexpected it corresponds to multiplication by a variable if m is positive and the derivative if m is negative what's peculiar here that this operator of course always compute the shifts always compute but here we have Heisenberg commutation relations they almost compute they compute up to a scalar this is because of the infinite dimensional space if you produce these correspondence between action on this space and action on the infinite space on the corresponding for space in the ambient space of this isomorphism there will be some scalar correction will appear so actually we have the action of not gil infinity but of its one dimensional extension which usually doesn't play a lot but you should remember it somehow even this correspondence gives a nice corollaries for instance it allows one to give a more invariant definition of this boson fermion correspondence so for instance let me explain how using this you can produce in an invariant way the mapping of these correspondence to the wedge product space so given a polynomial of p variables how to know what corresponds to this polynomial on the other side it's very easy we can interpret this polynomial as a result of application of this operator to the function one so on the right hand side we take a vacuum vector which corresponds to the function one and then reply repeatedly to this vacuum vector the operators corresponding to these operators so f of a1 a2 and so on so we plug to this polynomial these operators and we repeat them reply this repeatedly obtain certain combination of basic fermions this is the result how it works on the other hand there is a way to go in inverse direction given an element of the wedge product how to identify the corresponding polynomial here one of the ways of causes is to apply this coordinate correspondence but there is a way to do this in an invariant way for that we will use the Taylor formula given a function f how to compute this function using the Taylor expansion formula we know that every power series is identified by its partial derivatives at the origin using the Taylor formula what we have to do we take exponent of sum p i d q i applied to f of q this is the Taylor formula every function is a linear combination of mixed partial derivatives at the origin coefficients and the variables this is the the Taylor formula so now let me interpret this formula in terms of this on the right hand side how it works so we given some vector on the fermionic side what we have to do we apply this family of operators e to the power sum of p i and partial derivative correspond to a sub minus i divided over i so this is the when we apply this we consider p as a formal parameter and then we should substitute here q equal to zero what doesn't mean this substitution it means that we just take the free term of the resulting power series taking the free term means that we should take the coefficient of the vacuum vector so called vacuum expectation so this is the procedure given a fermionic element v we apply this family of operators and take the coefficient of the vacuum vector this will be result will be a function in this parameter sp and this is exactly the what we are looking for this is the way this formula is usually this is the formula that you will probably find in the textbook when you look for Bazon-Fremont correspondence so just looking at this formula it's not so easy to recognize that this formula hides just a very simple coordinate isomorphism between these two vector spaces this is this is just sophisticated way to represent the coordinate correspondence ok so this is one example of operators and by the way what does it mean the action of this operator the first line the invariance of the plukery equations of the kp equations with respect to this operator is just the fact that in the kp equations all partial derivatives are of order at least 2 it means that if you change the linear terms of the function f this doesn't change the equation so you can freely change the linear terms the linear change of linear terms does not affect the solutions so invariance of these equations with respect to adding a linear term is just a response to this am symmetries on the other hand also these equations also have constant coefficients so the shift of variables doesn't affect them as well so the shift of variables corresponds to the partial derivatives so the symmetry these two symmetries means that you can change linear terms of the function and also apply arbitrary shift of variables the equations are these transformations send solutions to solutions then the next the next operator corresponds to this operator in Z so you can take any differential operator in Z variable and you can treat it as a global linear transformations of the space of Laurent series in Z so for instance if you take the first order operator in Z you obtain this operator which I call lambda here and this is how it looks on the bosonic side so this is either it contains terms which have first order partial derivatives with coefficients depending on the variables then second order derivatives sometimes just products of two variables question what's the computation relation for the lambda n and the double n? it's a Verasoro so on the right hand side they compute but due to this extension what is the simple charge c equals to one word simple charge of the Verasoro computation standard one or twelve or one c equals to one case so this double dot means normal ordering so if you take combination of multiplications by variables and partial derivatives in such a way that all variables are put to the left and partial derivatives to the right this effect only the case when m equal to zero and also there is a similar formula for the operators of order two so order two partial differential order two partial differential equation on the thermonic side correspond to this complicated operator bosonic which is some combination of partial derivatives and multiplication by variables and in particular the cut and joint operators by chance is the one corresponding to m equal to zero so cut and joint operator is one of the examples of this family so it doesn't matter what is the origin of this operator so let me just show you here I just check by applying I applied these operators to short functions on bosonic and thermonic side to compare this is my best my simplest way to guess about these coefficients so if you of course you can derive these coefficients from combinatorics of power series but it's more safe if you just have experiments so now now let me give let me give the proof that the Hurwitz function does satisfy the Kp equation actually there is nothing to prove by definition the Hurwitz function given here is just composition of two symmetries of two symmetries of Kp equations we start with a function one which is a trivial tau function corresponding to the vacuum vector then apply the exponent of a1 a1 is infinitesimal symmetry its exponent is the honest symmetry of the Kp equation it transforms solutions to solutions and this is another transformation which translates solutions to solutions so any exponent of infinitesimal transformation is a symmetry of the whole system of Kp equations so the very form implies that it is a solution supposed to be W W W M I changed the time to a point I changed the notation M will be preserved for I don't understand, Maxim you say when you write the first line you assume its solution so you're saying it's a solution because it's a solution function one is solution but when you say that exponential G or J is exponential B WG the first line then you assume that it's a solution so what's the statement then you assume it's a solution then it's a solution you say it's a solution of the first evolution PDE no, no, no, the function one just the function yeah, for identically one, it's a solution no, but the statement here is that the solution of the first PDE of the whole system, why well, you define J solution of this evolution equation here then it's solution of Kp yeah, yeah so this function has particular meanings the coefficients of this function has particular meaning for the combinatorics of Hurwitz numbers but the very fact that its solution of Kp is much more simple it's not something special about Hurwitz numbers it's very special very general property of Kp equations so you can play here some other operators to produce a lot of other solutions some of them have physical meaning some of them have combinatorial meaning but just apply any symmetry apply any symmetry of Kp equations right now do you believe that it's really obvious not surprising so when discussing Kp hierarchy and tau functions I cannot resist to to discuss one more function which is called written conservative potential so if you put in alphabetic order you put conservative written but in Russian spelling written stands before conservative it's also a partition known as a partition function of two-dimensional gravity and again I will not discuss what the physical background what's the meaning of all these physical models let me just define the function itself and again I define it on phermonic side I just define it through the phermonic presentation as an infinite wage product the infinite wage product phi 1, phi 2, phi 3 and so on defined in the following way so there is certain differential operator which is called A the first order differential operator which acts on functions in Z and the first equation is the solution of this equation Up to a certain simple change is the same as the area equation the coefficients is quite easy to compute and these are explicit formulas for the coefficients there is an induction formula for the coefficients the second function is just we apply A to this phi 1 the next function number 3 is we apply A twice but when we apply A twice it's the same as we multiply by Z minus 2 and so on we can represent this phermonic vector in two different ways so the corresponding two presentations are the same up to linear change in the basis so the linear span of these functions are the same all different powers of the operator A and the operator A in each step it decreases the degree of the leading term by 1 or we just take phi 1, phi 2 then we apply Z minus 2 to the phi 1 we apply Z minus 2 to the phi 2 and so on so let me show you how it looks so I just checked that this function is the solution of the area equation and here is the corresponding matrix presentation of the corresponding point on the Grassmannian the rows are the coordinates of the vectors for which we take the wage product so the first row correspond to the phi 1 the next phi 2 and every second row is obtained from the previous one by shift by 2 to the left so we have such a diagonal matrix and then we take the wage product and this is just by the use, let me use it the definition so this is what we do result so let me list so this is just definition again my definition is just the way I compute it I do not discuss for a moment the physical background but just an explicit formula first of all by construction this is a solution to Kp equations we defined it besides it's a the corresponding play is invariant with multiplication by z-2 or z-k for integer k which corresponds to the power to derivative even variables so the result is a function which does not depend on even variables so the independence on even variables is equivalent to the invariance of the corresponding subspace with respect to multiplication by z-2 so the corresponding system of equations is called the Kdv hierarchy Kdv hierarchy is obtained from Kp hierarchy by an additional requirement that all partial derivatives with respect to even variables is equal to zero if function will be function in odd variables on the other hand this subspace is invariant with respect to operator a and also we can consider increase a to some power or multiply by z to negative even power so this produces also some some relation on the fermionic side on the bosonic side so this is what corresponds to this operator on bosonic side and these are so called virusoro constraints so this is partial derivative with respect to variable number 2m plus 3 and this is a linear combination of partial derivatives with linearly depending coefficients and also this is most second order partial differential equations and there are certain correction terms constant correction terms which are due to the central extension which should be taken into account so this just I checked that it does satisfy and the famous Wittens conjecture says that this function has an interpretation in terms of intersection numbers on modular spaces and the Taylor coefficients of this function are intersection indices of the modular space of curves and this this conjecture was proved by Maxim Konsevich in a very sophisticated way the proof is very nice but not so simple so after that a number of different independent proofs were given and now I will present a proof which makes this statement almost obvious so the same obvious in the sense that if you do understand properly the theory of Kp hierarchy then it really becomes kind of obvious so anyway if you wish to prove this statement you should know some independent way how to compute the corresponding intersection numbers so you should produce some explicit form how this could be computed and then to show that the way you compute fits into which satisfies the corresponding integrable properties Maxim Konsevich used some nice polyhedral model for the model of space which was reduced therefore computation to complicated quite complicated but elementary combinatorics of polyhedral I will use another approach through the ELSV formula this has an advantage that it is absolutely algebraic there is no it's completely done in terms of algebraic geometry actually to my understanding the ELSV formula this is probably the most direct way to relate modular spaces to something which could be computed more explicitly so at the time when Maxim Konsevich proved this conjecture the ELSV formula was not known it appeared a little bit later so now using the ELSV formula we can do this so here I check the conjecture some implementation of intersection numbers so this denotes the generating series which uses the honest intersection numbers and on the right hand side I took the partition function of responding to the fermionic presentation and let me compare result is different now ok let me check once again actually I am cheating here I am cheating a little bit because here when I implemented implemented intersection numbers of course I used the statement of the written conjecture which is mostly to demonstrate by the way here is how the KDV hierarchy looks like in T variables it does not generate the whole system of the equations it's called by Hamiltonian form of KDV hierarchy and it produces only to to obtain equations of the hierarchy we should integrate this with respect to T0 many many times then equations become more complicated but here it's very compact form now let me pass to the LSV formula the LSV formula it's a formula which relates whose numbers which are coefficients of the function that I considered in the first part of my talk to another objects which are so called hot integrals hot integrals are certain intersection numbers from the modular spaces of curves again I insist that I am not going I will not consider I will not consider the geometry the configuration space so in particular I will not define in details I will not repeat the definitions of lambda and psi classes from Dima's talk so this is in any way this is a certain collection of combinatorial numbers indexed by partitions so lambda and psi are certain classes in homology of spaces in homology of modular spaces we produce monomials take the intersection numbers and these numbers participate in this geratic series and again comparing with Dima's talk I packed these numbers in a different way again I use convention when I multiply the function corresponding to this partition to the variables labeled by the corresponding index not the power of the variable anyway we obtain on one hand some huge collection some huge amount of data corresponding to intersection numbers on modular spaces and on the other hand huge collection of numbers corresponding to the combinatorics of the symmetry group which are labeled Hurwitz function and the ELSV formula says that there is a way to translate one collection of data to another there is a way to translate intersection numbers Hurwitz numbers intersection numbers and actually this could be converted Dima presented the original form of the ELSV formula and let me just say a reformulation of this in the language of geratic series so on the left hand side we have a geratic series wait a minute let me do it for my experiments so here are some the corresponding functions so these are terms of homogeneous degree n and corresponding to the modular space of curves of genus G so in the notation of Dima this is homogeneous polynomial in n independent variables but I use the same collection of variables and instead of exponents of these x variables I use the index of the t variables so it's just another way to pick the same functions and when I use this agreement I can put them together to form a unique power series responsible for all integrals so on the left hand side we have Hurwitz numbers the ELSV formula doesn't work for the case of genus 0 and 1 or 2 ramification points at infinity because the corresponding modular space does not exist so I just subtract them and the remaining terms corresponding to Hurwitz numbers in this stable case are related to Hodge integrals and the result of these correspondence is just a substitution of variables so the Hodge integrals depend on t variables these depend on p variables we just make this substitution and also here the parameter is beta epsilon and here I am using the third parameter u which is just the same all these parameters are related by certain exponents so I just put this formula and in my computer experiment I just literally put this substitution this is the substitution of t variables this is substitution of epsilon and this is these are explicit formulas for the contributions of genus 0 and 1 and 2 ramification points so now how can we prove the written conjecture using the ELSV let me look once again to this relation on the left-hand side so if you ignore this small modification the left-hand side is a solution of Kp we should know we want to prove the right-hand side also a solution of Kp at least for certain parameter values so it would be nice if this change of variables would be a symmetry of Kp then it translates the solutions to solutions then we have the Horsch integrals genetic function for Horsch integrals does satisfy Kp hierarchy and in particular those corresponding to epsilon equal to 0 which corresponds to just Horsch integrals which do not involve lambda classes which those which correspond to the written conjecture unfortunately this linear change of variables is not symmetry of Kp hierarchy moreover this change of variables is not invertible because for each k we have this change is not triangular for each k we have all power series combination of pi with all coefficients nonzero so it's not so direct but however it's somehow simple so the idea is to use some intermediate function some intermediate change of variables I would like to use some intermediate variables q and to make some change of variables p goes to q which is replaced by q such that which is however symmetry there are certain linear changes which are which really symmetries and we choose this transformation in such a way that the resulting function is turns into conceived written potential for the zero parameter value so it's a slight modification it will be a slight modification of the generating function of hodge integral it will be agree with the hodge function for the original parameter value but then when the parameter changes it will be slightly different so in order to apply this I need to know which linear transformations do produce symmetries of kp hierarchy to answer this question so let me formulate once again this question so which changes of p variables produce symmetries so to do this let me look once again to this table to this correspondence oops here you see this lambda operator acting on bosonic space which is essentially linear vector field linear first order equation this part is the infinitesimal linear transformation of p variables linear linear differential operator is an infinitesimal linear transformation of the ambient space of course it has some extra terms which actually modifies the quadratic part of the function so we should check independently quadratic terms of the function but the main contribution is produced by this linear change of variables it corresponds to lambda m and lambda m is the first order differential operator on this in the variable so let me formulate the conclusion so I consider the linear span p variables just the linear span of p variables and let me identify this linear span of p variables with the polynomials in x variables so we x to the power k corresponds to the pk variable so any linear transformation of this space of polynomials produces a linear transformation of p variables and let me consider linear transformation of the space of polynomials given by just a substitution a change of x variable produces a linear transformation in the space of polynomials more precisely in the space of power series and through this coordinate correspondence this could be interpreted as a linear transformation of p variables so and the boson fermion boson fermion correspondence says that this kind of transformations of this kind are always symmetries of kp hierarchy and up to some linear terms quadratic terms ok so let me look once again to this substitution and I choose q variables in such a way that for k equal to 0 I should get q1 q1 is equal to t sub 0 to the power i over i factorial 1 times p i so if I use I have chosen already what should be the q one variable well these forces that my change of coordinates should be of this form so this is a change of coordinate of course you would up to if you ignore the dependence on the parameter u you would recognize the change of coordinates which have been used in the talk of dima I even use the same notations for the coordinates x and z so this is a change of coordinates that such substitution produces an automorphism on the space of power series and this produces a linear transformation in the space of p variables so let me let me show how this works first of all so this is our change well it depends on parameter u and it's not regular at u equal to 0 however it is certain infinite invertible how to put u in the denominator and this is the inverse series which also have been appeared in the talk of dima so if you substitute here instead of z this complicated series you obtain the x variable itself and dima explains how to produce how to prove this identity using Lagrange inversion or using counting trees but the result is is is this so now we have to consider series corresponding to the t variables which are given here and you see what happens how to pass from this series for a given k to the k which is greater by 1 we see that the Taylor coefficient should be multiplied by i times u squared i times u squared this is a response to the action of this operator so this series is obtained from the similar series for corresponding to i equal to 0 by repeated application of this operator this is some spring field vector field d is d o d x are you reading the d o d z here no no no just half a meter above the field d capital d here d x i am able this line corresponds to the comments and what's nice about this substitution that this vector field has very nice expression in terms of z variable simple computations show that it becomes the coefficient in z polynomial in z variable so here is a check this this operator becomes this in z variable so if you pass to the z coordinate you start with this original series which is just z by definition and you apply this operator which is polynomial and you get a nice conclusion that this series corresponding to the tk variable is polynomial in z so I just nest the command nest means the application repeatedly one is the same function so I obtain these polynomials so for instance if you apply twice you obtain such a polynomial which corresponds to this linear change of coordinates so if you what's interesting that the change of coordinates does not extend u equal to 0 but that t functions do extend they are polynomial in u so the linear term is z times u to even power and the free term corresponding to u equal to 0 is just z to the word power with a double factorial coefficient so this is the intermediate change of coordinates and for u equal to 0 it corresponds to just to just written concage function so this is the end of the proof so once again we use this intermediate change of coordinates this change of coordinates is on automorphism of the hierarchy so it also satisfies kp equations and if you the change involves negative powers of u so the change does not extend to u equal to 0 but the resulting function does extend its regularity u equal to 0 and for u equal to 0 it is for u equal to 0 it is a written concage function up to this rescaling of independent variables and that's all Maxim, but you use here the cut and join operator so it's not purely algebra geometric proof so it uses some combinatorics we cannot prove that this is a tau function without k so actually there is an interpretation of the cut and join equations through algebraic geometry it is done as follows let's consider instead of Hurwitz numbers and let me mark one of the one of the critical points and let us check what happens when this critical points tends to infinity this procedure could be described combinatorially but also it could be described algebraically as a kind of divisor as a divisor in the Hurwitz space there are some components corresponding to these summons in the cut and join equations and we should analyze the multiplicity of its reducible components this is done in the series of Sergei Shadrin so this cut and join equation has an algebraic interpretation and it can be proven it is done yes it has proved algebraically and it is an algebraic geometry so it is just a count of multiplicities of certain divisor under some degeneration and by the way these changes of variables correspond to certain transformation on the fermionic side corresponding transformation on z variables this allows one to compute what happens with the cut and join equations so the cut and join equations is also subject to change if you apply this change of p variables it is not so obvious to compute the result but the computation simplifies a lot if you make the computation not on the bazonic but on the fermionic side you need to deal with differential operators in just z-line so there is no infinite sums and so on so the computation is quite elementary so I do not present here the cut and join equation for the function g could be rewritten in terms of the function g tilde and then we could put u equal to zero in this equation to specialize to u equal to zero and this will recover the vera-soroconstraint of the Witton-Konsevich potential so this method allows one not only to to prove that Witton-Konsevich is a tau function but also to identify it explicitly okay so this is okay right so this is the end of the story about kp-hararchy so let me repeat once again the ideological part of the story is that there is a nice space of solutions there is a nice space of solutions which is really very nice there is a huge amount of independent ways to produce points of this space to find relationships between points of this space and so on and the equations themselves do not play essential role in this picture and moreover there is a huge symmetry g-infinity acting of the space of solutions so for instance starting from one solution applying transformation may produce many many others so tomorrow I will discuss the geometry of topological recursion in a similar way and to my understanding the situation looks very similar it's not seen very well in the formulation of Dima but however the picture is as follows again we have some huge space of functions which are so here solutions parameterize points of the grass mania and in the theory of topological recursion also we have solutions and here we have potentials of which are obtained by different procedures of topological recursion varying the initial data of the recursion but this solution also have if you just look the whole totality of all possible potentials it's a very nice object and it has also a lot of symmetries and it is parameterized by Lagrangian grass manian in particular again there is a huge symmetry group acting of this space and this space in this case is a symplectic group and this symplectic group is acting on this Lagrangian grass manian through so called quantization of quadratic Hamiltonian so this is more or less the same that was discovered or described by given tau but I will treat it in a more regular way so again by topological recursion explicitly for a given initial data on the other hand the same function could be obtained in a different way you compute just one particular potential corresponding to that you know in advance and then apply a suitable transformation one Lagrangian plane to another and this result will be the same so the picture looks similar but actually when you look at the details the details are quite different for instance there is certain integrable system so called BKP or CKP hierarchy which also parameterized by the corresponding Lagrangian or isotropic grass manian in the infinite dimension space and this is not the same strangely this is not the same so I will the details will be tomorrow but I will not come back to the KP hierarchy the only thing that stress that when I will be talking about topological recursion just have in mind this correspondence well I think that it's better not to start topological recursion right now I have fortunately I have two more talks so probably I will postpone it for tomorrow let me stop here there is a question so probably you will talk tomorrow about that is there similar correspondence in the bottom of that I don't know it does exist for the BKP or CKP but not in this case so here the correspondence will be somewhat different I am not aware of this