2/3 Mathematical Physics of Hurwitz numbers

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2/3 Mathematical Physics of Hurwitz numbers
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Hurwitz numbers enumerate ramified coverings of a sphere. Equivalently, they can be expressed in terms of combinatorics of the symmetric group; they enumerate factorizations of permutations as products of transpositions. It turns out that these numbers obey a huge number of relations represented in the form of partial differential equations for their generating function. This includes equations of the KP hierarchy, Virasoro-type constraints, Chekhov-Eynard-Orantin-type recursion and others. Only a few of these relations can be derived from elementary combinatorics of permutations. All other relations follow from a deep relationship of Hurwitz numbers with moduli spaces of curves, Gromov-Witten invariants, matrix models, integrable systems and other domains of mathematics which are often referred to as `mathematical physics'. When discussing Hurwitz numbers in the talks, we consider them, thereby, as a sufficiently elementary but highly nontrivial model of all mentioned theories where all computations can be fulfilled completely, and all formulated relations can be checked explicitly in computer experiments.
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but I missed a drop of this is a belt of the alleged infiltration can build of 2 some modification of the 0 . what do you Air bridges will last long the Victorian with the power of the Times India the wonderful or it was the duty of you if you your CSI stop for a while is a global cell phone and the 0 2 feature so if you have a therefore it will be by definition this is 6 I just come so of the come then you can rent the England this 1 . 4 4 this kind of thing because it's should been like this you works for the potential what I do you want to call it the serious side to keep the 4 20 I of Europe this so now we played a little bit these examples I think that your idea the the initial so now I you down the condition of the political precautions foreign start to 1st of all started some it is a fine dimensional community will also review and also acute business product which is rare for a move multiplication which is due by the functions so is 1 of the very little is known "quotation mark at 4 but I thought at the time for dimension of this spring In all my previous examples age was just the one-dimensional and the loyal considerable difficulties the typical example support signed the premier cells of which has been submitted for offered them so when of which I it wants to use them some multiplication like the fight square musical for the this is for the rest automatically but it's not the 1st of all the use of your place you want to the new awareness is proportional to the I don't remember well 1 the I don't remember the conventions prefer some may think variables which purport stop like you and also and also that it reduce the arches of regions this article built up please it is the 3rd part of the impetus for the advice of whatever this from expanding the base and that officials of the French all of this this is 1 of the typical examples in the case which responds to the precaution of friends and around them how it appears to practical is it most papers response to that at least get some or all of the Church of the 4 them all right but that happens in the petitions users who have so far as to the source Of the unity of the physical .period some Lodi so detention of capital of the so it's the convention than all the built-up Isaac equal to 1 is not the physically meaningful a typical example is called knowledge and call you go moldering cost some congressmen defaulted this is the way forward appears in Quantum ,comma 4 minutes ills the fix for once and for all considers that the formation of the initial date of political caution and therefore Minnesota but also known the deformed fetus at once and for all the next step is to consider the algebra the place used my examples age this is this spice of bouncy as you no 1 variable discussions I I hope for the sprays of water the serious work in and you can multiply multiple multiply using multiplication as of this spring the becomes it began the report also directed for this spread due by the following the formal so if you dig 8 x to the borrower can do what the physical demands 1 quarter of the year this case is a the planning of a and B at times the coefficient some extra coefficients regions president and the shriveled his father taught to plus 1 some because of this property it's skews future players can sign the negative side but there a more inward and white you have to learn to work with them the following is a due thank you you get a possible the boss was labeled as eurozone Booker plus 1 plus 1 different buyers I made his substitution my substitutions that affect people for the same and that it is primitive of these the primitive difference defined up to recall some of the constant doubled contributed residue the power to define the the basis of the schismatic working them the next fall and the spring the thereby away by the various this if you this was a vector for without Runkle produced many commendable resistors basis 1 of the possible of awareness of the following so 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subscribe you good so for these coordinates of where members but by the reference is just 1 of the most examples but of course there are much Romania more different .period mountains of spaces the a booming precise phony the basis of race CSI all the bases you know you and also some of the start of you ways the fall of the subjective many of these soft basis during the 1 another for the summer sympathetic some where the heroes of a deal with security the together former double always and then build space age the question the explicit form of this business is not so important but for the mother of computations for so to make some the competition and also the future of the bases no where it change 1 of the bases of the producers the changes of the other with for instance like users according this is the end the actual could do about it except for for all this there you the betterment total I means that of space protracted by quadratic forms solid from this and the following week folk space complimented players and all sorts of use the coordinated protests of 0 Algeria's according to spend it was you before for which could consist of money but not all of it so here we use which have chosen the way this CSI all 4 she overheard you always have which makes also there is a basis here which is also due police monitors but the bread and butter furious battles heroes of do not at this point the decision the difference between these 2 measures mutual what additional article last struck vertical so far although the sponsor the belongs to the so this is president of the would you be so you here None of analysts city the subways so claimed here on clay this opera 0 is assuming a single and all 4 rounds for all the barriers which he said the 1 thing that you know you can use different measures but not for a moment the business here and the was due to the spot but respect the differences of race that actually In just the before you T some of also some Utah some practical somebody told him .period of friend the system itself this is some specific cases of these 1 and it's 1 of the following 4 so the tube From time to time 3 the European Union on possible traces but this year the from you don't this have the year for infants for the cost potential was different for me then and 4 for maintenance for veil but the salute was like but further the harsh occasion cases was different the kasha now I From later the main percussion after 2 years this is a mail commissioner so what here while the potential of told lies with the addition of the bank's structures don't you also multiplication .period different the coefficients this specter parameters of the multiple Cup multiplied by this CSI of using multiplication image the question could In composite form like like must I don't think that I promised the Delhi was initially make of this all of this the caution and the initial date of cautioned acquire the port here are some special joy of this year you but the thing that I remember was leaving it to fix you on what you do for the child Church of it 1 support you ate he is a continuous Gold thank you we have to this year 1 this is because of the falling for some the people it's all so Excel for excited the boss functions therefore of multiply you have to to be still more often don't function of fuel increases that a assault would be above the law Of the functions but is getting greater and greater Fourier carefully defined 2 founded and later joined this contribute the Commission due to the human I will speak more about that for the of the initial data but now from this year so long it was just applied directly to the because all of these Church of also contains the definition of you 1 1 and you 0 the venue but applies this discussion you find your June for all Jews there for this you Aegean it is universe the 2nd statement is the falling so the the quality of this year's Eugene In uses itself this is the claim that the display on knew that there are only 5 men job here on final this summer's final action on the fundamental nature John and each individual coefficients Of this expansion is putting on to on for each gene of a heavily funded in real terms so there you for the for the conditional the point of his narratives it could it is a difference these properties depend heavily on defense some space is Lagrangian immoral this condition is the initial conditions also used where much of the proof of this condition of this property if you just but slightly initial convention conditions that this property for for a stranger to the British stopped all books were sure but the to the profit from difficult but this is so far as to use caution could be applied for any for initial conditions because Yanukovich saw it is the 1st of the 2nd landowners doesn't use it hundreds of property of but the Serbs properties in the property this is crucial since all this implies that you're going to recover have gym so as to truly huge uses differential for instance you can use them better form I what is important is also believed that 42 because differential function that can occur the function of of optical some of the examples of a convention before 4 in the form of accomplishment and potential this constant force holders 0 the purpose that we cannot assume that the soul of therefore session this summer break to recover the constant jumping on the cold here capital that this potential as many as it was proper to someone obviously look a the way so-called by so-called dual equation which said if you will defenestrated it all literature observers from Geneva to condition imposed problem because we're deficient ones so here are this is Jr -minus close by G so this call we already defined formed and little to do this year so it's all hold for the villages of well is defined this is going to hold together humanistic wanted the door jeez you in the same way so defiance In defiance of the company might be here to John mind soldiers who assumes this month in this isn't but the initial here reported here for would you mind to box and positive 0 0 0 2 0 0 1 2 0 0 2 and also want you're all of this year the no properties for the potential the Royal discussed :colon on the variation of the Lagrangian plans so long as there is a quantification sympathetic consideration of whether to on you because you tomorrow I'll will discuss his dependence on the choice of what to buy the knowledge in mentioned several on the property that's why we fall on the way but to what extent the Church of the day is important for the 1st probative of the different Utah leaves per shift all durables social aggression that all of the combination you might and to minus the offer was made so a different churches of EPA useful just shift of the timber will absorb the fact you the the death of the weighting functions the shift of the terrible says the Community tool there is a currency store the differences of the 2 at the center of the property this this is what happens if the troops a different ways drew different tiers of on the basis yeah war tho case findings if you look at his formula no it was meaning that the actual children of the business is not involved so this falls so so this constituted is all the there is a formal functions well defined and the the Bernardino of the terms of the baby the what you're the feeling in the event but developed a local officials to new diffuser coordinate so there's that particles just as the presence of the political coordinate system on fears of new defense bases but if you just consider the a formal functions so well defined on the and even is not too that's not the depend on the on the nature of anyways but more more all takes expansion of this function of origin expert so .period what problem I some of the functions that was a defined but In the what this is a space and we don't know how to expand this functions origin but not politics but this is the point of view when he had different shades of Utah but shift of foreign so this condition conditions on the community that if you change the terms their good responding footage of the functions of all defined the shift of the government and the direction of these notes is well-defined operations the function I think it's putting on his direction some of the children who knew that this potential has the burden meaning the it's essentially as determined by a Lagrangian point of essential part of the definition of the potential plan and all although think it's that just the coefficient which are used in them for the political interests of communities at the end of this the meaning of this actually is unique determined by then there the property of its creation the specter and also in the view of this I feel wall 2 hours thinking much b question no I think but to postpone the creation of the next book is the next stop will be mostly devoted to Huish huge amount of new different applications of foreign firms give and very few places not I think people want to believe that political leaders written so you're speaking on the eve of the Office of talking to kids Griffin's cautioned that these are the ones you want for varies from here's here because of what used to be that of the you won't it its close all the political clout of the you think that people be 1 or to times in 2 the fact that the rest of people over years and also professionals from different religions or what he said he had have the effect of general book few


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