3/3 Mathematical Physics of Hurwitz numbers


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3/3 Mathematical Physics of Hurwitz numbers
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Kazarian, Maxim
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Institut des Hautes Études Scientifiques (IHÉS)
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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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the combination of this regression of these forms of music the type progression said involves do you which solid of order 1 told some posted linear coefficients and sometimes service costs of gold futures and so on so far in the sense that this is another couldn't wait to represents topological recorder from some you may think that the politically cautious just and efficient way tool right down the assorted In many cases so this is the occurrence because it a form of the political caution what so thank you so much wastage of the now
this was just said the remarks concerning the caution and knowledge past was the main subject of 2 they talk that study the dependence of this potential to politically caution on the Lagrangian grace sorry there the space potentials all the political which is just the same as Lagrangian this menu excuse me for years as a 6 for this process homogeneous there is accused group Sumatra's saddled with precedents simply to yeah cellular the incidence of such a condition of locally when you make the salt also so you make mimic if still transformations and sometimes integrates and so on this simplistic the group also political formations that have reported victims also mentions the elder role for him in the infamous done that fuels and lyrical assignment on different fuel school begun by quadratic medallion function and so this not a result is the same as the algebra the algebra Of course redundancy friends of the as usual berated the lyrical lilt of operation so this yields are excellent USM expresses so the music and film potentials the question how to request that this action the answer is especially simple have especially Nice the is that this curable the concert because what's called .period a conservation of collaborating with onions many the falling that's going to commit on there this year span of the following Maloney also have died as a be-all you also Cucuta Delphic you theater there also be beaten so what the response to this little analysts so Monday applies a slight shift toward such space supporters following the fall of the the Abdulhamid on and that the field physicals for newcomers going on there but the social changes so it'll be the challenge good Britain as a differential appear at a hearing this case this is just multiplication all 4 of the virus the the same quadratic when I mills and lay idle youths here a few coordinates which are shifted coordinates the respected to as do dawdled shifts respect the interior most as origin to the points table created tasteful to eat up more beautiful to here source the 1st of the differential operator you Peter J. or and human health 2nd of different the tossed DQ Alpha distributor so this director a Paredes efforts have told conscious of fuel and the function of your visit just there the editors exponent of along the page fearful you and curious shifted so if you this correspondence of jobs solves function effort effort for themselves this be the described as follows so let this crap told slightest 1 by 1 so the Trust Law 1 thank you all for to Is this a parent or is that just modifies therefore 0 White good tactics jostle dysfunction and I said that I don't want to book the beard uh my sending this is all 0 so I don't want to consider the fills the 1st line of the correspondence all this on the other hand for newcomers dynamic changes most their way there is there as the articles of this substrates this golf view the devil please so it's probably real called it's countered the consideration situation that been all salt dome the Villagomez freezing and this is the old affect so-called collaboration which makes the difference between the 7th and the ones consists potential so just modifies the this was the toast of division of DVDs 0 but for a moment I don't use for the next line well so this is the alpha Cucuta this was to and these preserves advances of space so as does the change of the medicine Wheel already discussed there is the potential is there it doesn't change when you it is assumed by a relative advise the defined and the only way but we don't what happens if you to make a change of basis is just as produce this is also the just linear chain show all of this coordinate system for this is just the change in the romance also the problem is all the potential environment the survey was the last 1 which is more probably more interesting it all this is the potential buyers it is those infinitesimal changes so the potential which has the following 4 and it is the 1st time 1 this age all QB blasts during the cure-all for you don't to so here we have such a strange the summation of the potential omission all Sekimoto derivatives knowledge in explaining what does it mean it's true so let me from lists local artists so let me take a certain equal to 0 sorry that make just Wilkinson general 0 part of their potential John visit here says that we have just this summer if you know all you hear from you risks Estonia no 1 was sure this news that that was a Lagrangian there 1 graph all just 0 bottles of understanding of the potential is just turns falling victim Young is just move follows the for the moment you so it takes the differential offer of 0 is a solid Lagrangian will solve our activities not the uranium and then only applies sympathetic those formations electrons endurance but it remains Lagrangian saw Is it regained quoted presented as differential of certain functions of accuse President the way Howard Texans 0 auditoriums for call intervention use extension of this formula for the use of this summer the next coral is as follows that name chose the specific brazenly CSI so as defined up to treasure of the change of coordinated so that for example the particular common new case is the called the please stop divided by the thousands the to the floor of the faithful flocked to double total like went over to the standard Maine and also respectively CSI the new kid it wishful the Times cumin 1 kid was 1 of the poorest of the poor people saw the in this is now Our this the In court sorry memory the basic rentals might doubling the this is and now all 1st live reconsider the case when we have just the coordinates of space in 3 of the space 4 I understand Nevada this case is that and so-called to potential all of the political field here associated age the news of falling political skills you is a coalition of homomorphism Coleman fumes but in general some obesity the lawsuit June of semantics homomorphism I think used see Shearer stands for 0 dimensional homology College of about so that the political field here uh Excel still to go ,comma logical feels he refuses Kolko ,comma Lord of the model spaces but in this case for the book therefore home From results which is given by explicit formula jail 12 81 they end physical toll the U.S. scholar product of anyone of their products times this settlement occasions the continue the this is just such a number now we produced the potential who stole coefficients are these good fences at the intersection of our usual intercession number others with a modest pace the curently you currently have Duke conduct that falls with that is main consideration because of this topology glory of topological field it said that there is little consolation potential so that it doesn't give you an exact into provisional his formal the police blocked for all was there to charitable the it's incantation potential replaced by the following strange combination here are new year the James In new so now this element is element was New Age some replied black you to this potential the games of then with the element of then after southern this killing and paintings this revealed this gives you a number this is anyway that this is a potential for more logical of topological fuel serious so combination the area around the southern information off the same data from the social injustice of cyclists which participated the middle conservation potential Ceausescus expressed in terms of the written consent potential this state known is just another reformulation of the same Oracle's France it's just worries at the Czech slot much harder than the original it's the it's just occurred original statements of that in the case of chemical belongs the rethink onstage and so this is what happens 1 the holders Europe let me change should be so in general case the John Jessica really define his abuse of change of the vehicle did as falsely keep their people and the American just shift just days that there was there is no there there was contemplated in this direction so it could be that doesn't ejected from from the original but not until this 1 could be realized as a given volume flow give the full formal that's the theory that the ninth-seeded my fear was the story the potential although political occassional good buried them explicitly the following it's "quotation mark blamed frozen original potential of the political serious where is the the all the flawless of quantized gone as tho for dialog the Gladys medallions Beijing buyer samples all the details the old computer the altered the DEA 1 the key unity so so that it applies exponent of this settlement 2nd order differential and this prejudice there is a potential itself so what we are 1 the Orange part and I don't remember from tree with 1 structural problems with the management of the instrument so we started is 1 potential for each Reno's answer then the people I the flow of the Hambletonian and the obtains a final answer and that's fine is that loss able to implement this a showing the competitions at sites it very efficient formal for the for the potential so in general have gives the following situations that they have to alternative way to compute this potential Isaac applied this explicitly this formula or just applies step-by-step notables included cautioned that too race of competition of doing the actually Of of the parents of the 2 1st the Davis a definition that from this definition you can during the master questions Jordan aggression and so on only parties this is the result and I'll admit so this is makes some similarity is that what happened soon indeed the series of food fuel could the whole variety of solutions that it's folks realize there's huge an isolated group of doing all the group is where is so data and this is very convenient you can imagine what kind of the potential to "quotation mark Hicks obtained In this situation on the other hand there is nothing that isn't there is no local structure like the Bulls from co-sponsors but anyway it's action doesn't so just makes evidence the goddess of this fact so unfortunately them intermediates as a variant of the book the statement that for anthropological the precautions for any potential tuples according caution with coefficients are determined by the officials be also viewed the by the section oversold cyclists it's go observes that there is actually is a stranded Is it is a then shall there was evidence shows that potential the potential for it Bush generalized the cordial told logical views to field the the way ,comma logical feels news of falling further just any connections region section of the validation group In court is aged for all those know sumo funeral so that they want us to go all the way the field theory requires some additional for the competent so few of I don't want to pursue there that these bodies will properties no called this situation In good now idea neither decision that there was no no protection from from the universal care nothing but anyway if you just look at the formal there condensation of the plutonium than that observed that for instance if you take it norm "quotation mark D squared off all that it agreed to be done it could respond the the movement does the modification of these fields here by the following think you put should other the contribution of the devices which response to the across double .period which hit songs 4 something like that for a new ones the 1 new tool kit or so 40 1 the kid on the new 1 1 you sorry currently the Paul Casey ones the current fall to 204 this spring devices it's the model this assertion of silicon have devised consist of similar costs which goes points and cyclist frustrated because the land bundles of food In the same way so foreign also security you want to 1 do you or do you kit or just that please this 1 these principles projections from the universal cure insertion of couple yeah sorry I'm lost 1 the priority of services for workers .period and so on so we area over-represented participates it that the definition of this section of simplex a group called the modeled as a settle modification of this had the political field if you can call the company explicitness of this we looked at this for this for those who knows the result actually belongs to them the house call to the village according the this financial forward gunmen job presented computer explicitly in terms of summation autographs some the actions of the six-point could be their last summation um along graphs and the summation will and this formal test interpretation on 1 hand this summation autographs described this crisis is visited what kind of also on what kind of the I want to fight it was spilled Kurras's medical .period complicated mentioned cooperative and there and what about the the sport contribute so we have completed the configuration which cemented Boisgrimot appalled some of the Alpha Butte officials some combination of cyclists and of the of the group .period and integrate all together and this produces explicitly way how this exponential selects and the space of Cornwall illegal fuels you on the other hand the same formula could be applied at the level of formal with operation despite our serious the physicists and all the direction action of the 2nd or the financial operator of the schools and confusion squandered computed .period the summation or thereafter there is something convolution or graphs formal so this is formula is a moralists committed did good but didn't explicitly so reserves future the original which immediately called formula but not we have this potentials and some of the potential fall into this room most of this event Respondent journalists ,comma local filthy but there is some fire which corresponds to the economists mortgageable will fuel theory which allowed which satisfy all this excellent of gold will fill the area it is funny ,comma donors are so-called long-hidden hidden potentials and those all hidden potentials it's also a very muted tho described in this language of of this correspondence for the granting of potential so formal the potentials are described as in the unit Dulcia it said I still believe Lagrangian plans which satisfies certain additional signatures recalled the given substrates sluggish so by definition the good news is that preserved the title way Xenia yours this was a strange definition but that was the hole hit the Puritans but it was also given Gold understands this crisp understatement it has for this and this is incidence of what was performed so my case I used Lawrence functions take it medicine each Gonzalo all plus formed from makes this the same but this and that respondents as usual few have conceived to the club did use a society forever the extra support came to me and gave me some confusion this so this is just has a more fuel-efficient than spaces this complicated apparently becomes just the shift of the multiplication by that Arkansas so if you want to speak about the ball articles from the multiplicative structure and from there this presentation is also the convenient this case follows of course is not multiplication so for instance this interpretation although space is used to the proposal didn't volunteer preserved the notation of the variables disease stands for is the same but the paper solve goes sympathetic form has a different form Shearer saw in the jails of the function the simplistic form also something coexistence the integral should take careful neither the GOP and then take that there is institute consideration due to the so that that's the different form lot but anyway books like this no In response then the anywhere else in respect of these apparatus described to reasons so misconstrued or all Space View multiplied by the interviewers sorry the business of specimens from is that since got I mention such spaces Susan Circle of I'll probably the reason but the reason is that the not all yeah yeah yeah yeah have records so that it's more even tho I take Els and multiply by using the which contains this 1 and saw that Mitchell somewhere the new this is amazing so it's determined by serious Xinyi union serious video dealing in the 11 In the Matter system in the Baltimore fuels age the 2 of them are human well and also more tomorrow was that the fact that allies like France and with the the identity of mind is the latest car overseas dueled respect accessible to 1 this is near dull property and all this former group 1 these a announced and the whole of space because such brands and the boys you know that you know where the formed this so if it takes the this lakeside Newell Europe these this is the very exciting can you clearly just as you have problem once the CSI 0 up to so fucked the signs 4 this 1 so the call what grounds of substances determined by the this is a main and all defiant formal review potentials Our for associated his cold on the event in the history of Celtel consideration of these uh organizers substrates satisfying these additional property of them there the whole group of girls commercials reduces tho these small 1 for this as substrates as this a near arts for a subgroup the space of a group of simplistic personal exonerations Ridge is good that is as characterized by the conditions as they should commute with universal they preserved the substrates softened during the whole such but this is only a small part of the hall displays of Lagrangian and honest but a small part of those potentials that could be obtained by a the Baldrige politically cautions now all potentials they saw in general if you study general potentials of the political order cautioned the letter available Apostolos foreign doesn't is not very useful but it especially when you're dealing with a group of futile potentials that threatens inside could be your so this is smaller notice thinks so that new Member of the time is over remember if we want to see 1 of them other qualities of this although the 1 of the coordinators of the USA existence of Of the Velasco wants cultural series so-called property that Evgeny is a pretty long known that Tunisia is Morris explicit bonding for the video of this prolonged because Columbia it's quite homogeneous vehicle not be greater than the dimension of the model of space and Jim and some other properties feature moralists the which simplifies the fuel just look at on this side of the story we're live everything century-much so holds that you have the precipitated the compared to a different on different points of view that the politically cautioned that after the summit international get some patients have thinking thank you and that is it it was like all this is the move there is most bottled water to this small USA for now the or this in the virus or the search for special why you so there are of those that's right ideas as rises of good ideas and that the 2 units virus aggressions of the system Eliza although with concessions that the the politically cautions that stability is around local vet all potential to politically cautioned this is a very nice mark the leaders of the conditions of our true but the program was aware that 1 of the benefits of it the flight but that is a broader but there's still only broader acted within their rules then when I saw that make will comment on the offer still owns a special corrosive and you take a spectral can and she exists W 0 1 W 0 told they're not so important what's important all part of these and the accounting change freely is even part and those this gives you a huge flexibility miniatures of the spectral Crawford and it's not clear what for jurors to can in the global 1 so I can feel we have some freedom the justice of the spectral traffic and for some choices the converted to the global on sometimes not since the were on the injuries and the uniform he was of this that was just 1 fossil of the new US-Russia arrested among more than a single that don't here is not my invention but my interpretation what their autumn different products they the only inventions of the language the results themselves are not new I I think you can get loans through it the way you when you look at yourself over to the villages well maybe only of villagers here and there is the jury of basis sigh of relief and then there is this the thinking is that the news that extra leaves that is more than a year get them was granted on there is also a few comparison from of fraud and this is accepted the ladies In this is inhibitors and work with the lowest form of equipment for last for years it seems to me all these variations are monitoring the area no not at all so 1st of all my statement that this summation of what grass doesn't it I have nothing to go on this this is remove hidden potentials as could be done for it the political potential but if you win a specified for the particle 1 should you should take of this political basis and to describe basis you used "quotation mark of possible for him to In the event of former U.S. even take busses we want to do something and the of these reportedly replaces you by the powerful like losses and then the this pictorial double for the local officials of the Laplace transform the USS fallen given us form at the end of the students are probably a furor right then there you so in Europe so there is a lot lost souls form so for instance if you Texas you stop CSI the new this regional body CSI new clearly start annotation firms Anderson said this case .period sorry we just so you're probably right for this we will continue to use as little in the way of the political affiliation of the area because when I speak exist side by developed I tried to avoid to consideration of of particle based but if show diligence this practical 1 problem with this interpretation is the single-story belongs proprietor fury it was also the year people's or forceful some of us into this is not to be delivered by party at that it does exist as a collection of commentaries come to my computer programs I promised to on Poland title make a PDF file from this computer programs to help available so few and


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