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1/3 Mathematical Physics of Hurwitz numbers

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Title 1/3 Mathematical Physics of Hurwitz numbers
Title of Series Physique mathematique des nombres Hurwitz pour debutants
Number of Parts 6
Author Kazarian, Maxim
License CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
DOI 10.5446/16423
Publisher Institut des Hautes Études Scientifiques (IHÉS)
Release Date 2014
Language English

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Subject Area Mathematics
Abstract 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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information question here is that history of Ghana so what this function has particular meaning spectacle around the confusions of this function has practical meaning for the candidate of the numbers but the very fact that its solution of could B is much more simple and it's not the it's not something special about chorus numbers were especially general property of of keeping the creation so you can play sheer will to some other operators and to produce a lot of Father solutions some of them has physical meaning some of them have combinatorial meaning but just apply Tennessee symmetry it plainly see Materials Tapia questions right now I but I believe that this surprising so disgusting did you hierarchy Intel functions they cannot used to the discussed 1 function which is called recent consideration on potential if you put into alphabetical order you put conciliation meeting but the Russian spelling vehicle stands before concierge is also partitions longer the partition function after the greater the limits overrated and again I will not discuss what the physical underground was the meaning colorful this physical models that just defines the function itself begin I defiantly its on Fairmont excites adjusts defined throws the film money presentation as an infinite fish products is meaning this product few Phuong 4 1 2 feisty and so on the fund the falling inquiry so there is certain differential operator which is called a that just ordered differential operator Rich said home functions and the fewest question is that uh is unusual this equation after a certain simple changes it is the same as the area gracious the coefficient is quite easy easier to computer these explicit formulas for the coefficients there is a reduction formula for the coefficients the 2nd function uses just reply to these for while this is the next function number the use of a reply a twice but when they applied was the the same with the Baltic languages -minus minus-2 and so on who weekend the president this can only Finland refering include different ways so the case pointing to presentations the same after changes the basis for the sake and the extent of this financial services I as it takes all different power so the operator of and the period the aid in step decrees that has entered into on out the degree of the leading joined by 1 all or just a 5 1 for you to them the blazing minor square to the fruits of for wanted played in the minors to lose was a fighter to and so on so let me show you will cloaks so just checked the this function the solution of the reintegration I'm curious the corresponding Maddox around the room presentation of the case went point on the grass minds there also the coordinates of the 1 of the reactors for Richard Davidovich product so the show's stroll respond to the 5 won the next fight and every 2nd all users by obtained from the previous 1 by issues vital to the left of him Fletcher medics and then it takes a major products this is just by the use of music as the definition what
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jurors on this look this conjecture was frostbite by madam consideration we have very sophisticated way the boy but not so after that the number of different independent process for skiers and Nouriel present proof which merits the statement also of for the same in the sense that what this year 50 hierarchies and becomes so the if you wish to prolong the statement you should know all the best way to compute the corresponding section so you should produce somehow some explicit form called called the computed and then still shoulder the way you compute fused into the rate exteriors which justifies a .period integral federal property so much she would consider issues fell Nice are we Britain's role model for the model space which is which was used in the use of force In computational completed its quite complicated but his mentor combinatorics hold 40 she draws hope will use another approach throws the USC formal this has an advantage that it was absolutely failed to break the law the 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special summit produced no 1 else intersection on worsened this number participated in this jurisdiction and the game comparing views the dimmest book I had spread this number so different ways again I used British multiplies at their functions the point to spot
variables labeled by user crisp on an index will not among the power variable anyway looked at on 1 cents some future
collections of huge amount of data good pointing at all intersections number some models presence alone doesn't count Huge collective collection of numbers spawning tho simmered combatants of the Citigroup which enabled label "quotation mark and the U.S. OF formal says that there is a way does collection of data to another but there is a way to close late in the session numbers "quotation mark Islamists through intersection Boston's actually this could become worst good country in the office sorry zuma presented the original form of the USA formal only near just saying that if a moderation of the on the language of the ridicule I saw on the left-hand side we have had generate accused where TO I would it from my experiments for here also the disappointing John would do .period 1 of the girls Georgia is not much pollution independent variables but I use the same collection of wearable soon the committee instead of exponents of this expletive also use the index of the variables so just another way to protect Texas and functions and the value of the agreement I can pull them together goal for the unique power series for responsible for all intervals so all the lessons that we have chorus numbers the EOC doesn't call doesn't work for the gears of Jews I want to begin point of that because the pointing subject to them and the remaining dance gracefully number the stable cases elated Storm coach intervals the all this respondents is just the substitution of portables so the quotient of the parental until wearables visit the bumblebee wearables just made this substitution but also cheers apartment that his beeper hears parameters Epsilon and here are you soon so peremptory you which is just the same all over the world this parenthesis related by certain exponents so I just thought this formula funded by computer experiment they just literally this substitution this is a situation of curable this is substitution of Epsilon and this is that the the explicit formula for the contributions of the general 0 and 1 no
notification .period the forks
so all broth is a real danger chair using the Ulysses get near that must look at once again to this election on the left-hand side for or if fewer all these this small the modification left-hand side solution of gay people we should know we should we want abroad the authors said also at is solution of kph it is for certain parameter will it would be nice if this change of portables will be distributed all 4 of keeping them if it if the system it is evident that states there are solutions the solution then we have the horse intervals Rudiger function for Fortune the rules that such EP hierarchy and particle goalscorer spawning absolutely Coulter's Eurovision response to adjust the cautioned the Dalton both the colossus vision those which could respond to the conjecture unfortunately this year change of wearables this is not the it is not the initiative could be become Nora this change of animals is not notable because area for each carried for each it change is not turned for each character there there's a Hall of power serious combination of spiders this will all coefficients from 0 so it's not so there lots however it's somehow was simple so the idea is to to use some intermediate functions and some intermediate change of wearables I would like to use some of the region's political security and to make some changes of wearable spewed caustic fuel Mike besides that the however symmetry there are also so openly changes which are Richard Syracuse I'm real jaws this goes on much in such a way there there the resulting function is Johnson the conciliatory the potential for her 0 parameter review focused slight modifications pulled this slight modification of the generating function of course in the role it will be at the visitor who visit cost function forces or regional power and the read about them both the parliament of changes its attitude that's filled with slightly different sir In order to replace the sign of a new kind of rich the report also mentions produce those images of KB heresy ball falls this Christians promoting wanting indiscretions changes Of the wearables produced symmetry so at the movies let look at once
again to the stable the disk here year this land the director hidden home was honored spreads which is essentially linear reactor fuel of the the question this is this path infinitesimal your wishes to be .period In Europe the lid of differential Allbritton was isn't infinitesimally articles on which of the Indian Springs because has some extra terms majors circulated actually modifies the quadratic part the functions of a visual check independently of Howard some the quadratic Johnson the function but that is the main contribution is produced by 1 of its major change of wearables if Greece was the element on the end user it just under officials during the on this wearable so let me for what is their
conclusions so what I
consider the lean years founder off the variables does the leader spent a few wearable the pier but analysts in exportable foreign it would be a exclude Boracay kickers falls to the PKK wearable so a near the optimal solution of this of on also produces the best wishes of the wearables Honda L let me consider but he of of the space of criminals given by just the substitution the change most of the work of X variable produces only approximation this but also more precise and specify serious and through these coordinate correspondence this squandered job at time of the but in Europe but also of the wearables the book on you global after months of crisp said that this kind of creative Dawson wishes of these grants are all this see images of KB fairer him up to summon the courage of our collective terms "quotation mark did so the new look once again to the substitution and then the shortest curable since June variables in such a way that for do "quotation mark 0 I should get Q 1 you want is a global Q sub
0 or white pictorial 1 so if use have chosen already what should be the cure 1 variable well but these forces that might change of coordinators from this so this is the a change of coordinates well of course you would TOP stories should not the dependence of the peremptory you evil it organized the change of course demands which have been used to the talk of the Duma you and I will use the same notations for the coordinates x and z so this is a change of events that such substitution producers Baltimore fuel the space of Boris serious these producer selling a combination of also mentioned this space of the wearables show 1st of all for this is that Our change well the Bensalem Pa you and it's not getting but you equal to 0 however and it is certain In there serious if you allowed to put you in the around there your serious which also have built beautiful the Book of Duma that so few substitute here is that this complicated series of Europeans exportable itself when the Duma explains how to produce held across the identity using bloodhounds and rational you're using so not all we have those tool consider Sears crisp morning tho the key variables Richard arguing and you see what happens How to pass from these this year's fraudulent Kittles a case which has by 1 year that those coefficients from new multiplied by I you square at 5 times securities is the sponsor of the action of this sort .period so they this serious is obtained from the similar serious pH responding tho are able to 0 by the bridge of repeated implication of this separator is to he's curious to know during European Union Q and a lot of this the government all the he said these are here there your fortune today new edible I call it was nice ball this substitution that this are at Yalta peres modernize expression in terms of the wearable simple it is simple co-production show that costs of the coefficient of is the company belonging in the wearables full of heroes the this this varietal becomes the 7th in the wearable saw a few Bostel the as a coordinator you started this original cereals which has just seen by definition and you apply this apparent that just but in and there the 2 most conclusion the obvious there serious good pointed to the wearable brilliant on islands you so avoid just nests the ,comma best means the application 1 of the same function outside a player up there of the spot for instance if you play twice a day fetch a pretty this could it's possible this year changes coordinates so if you will what's the what's interesting is that there the a change of Gordon is doesn't extend to you will it will to 0 but the functions do extend to their their bottom line :colon so there it was in is there it is you times you want to you and power as a free John corresponding pointing to your local 2 0 is just His yakuza horsepower user double-fault total coefficient so this is an interim interchange of coordinates and therefore you recalled as Europe stalled just the total adjusted result contains functions but this is the end of the proof for once again the use of this
change of coordinates but this change of coordinates is long entering our ultimate fusion on the clarity of effects also satisfies KP questions around the issue the change involves involves negative forests fuel so is the change does not extend to you it will be hero but there is nothing function doesn't stand it's said at to Europe and for you year it is for your equal to 0 you deserve Middleton contains function to the Saskatchewan dependent variables that's all madam you hear this kind of thing you are so it's not viewed as where'd you solitude is there isn't an interpretation of the cut and run the Commission's strong-willed regimented it is done as follows and let's consider it a chorus numbers there is considerable respect and so read to is in that Newmark while also the critical and it does check what happens when this critical points Thursday this procedure could be described ,comma that already but also it could be different disgruntled to break his summer as a kind of divisive if there is a divisive in the Horace space it has certain components corresponding to to this summer months in the coverage on the questions that we should analyze the multiplicity of it's in the individual components is done the status of nuisance edition for this government question has a little break interpretations dedicated it's it's does not yes this is the incident has grossed about American government don't celebrate German right so it's good just called off multiple usages of sort of the visor under some the duration and by the way this change of wearables crisp want certainly does emotion was a excites the crisp winning the celestial 0 rebels the computer happens captain John equations the company and the question shot up change before you apply this change of the wearables not formula to compute the result but there is a computations the computer on the Mazzone but on the periodic sites From the excited need to deal the differential you just as you like so there is infinite sounds Walton source or the competitions .period so the dollar presented here at the cut joined the Commission for the function GE could be a good beer the 32 and jobs of the function Jedi older and then they could told you equal to 0 and the situation tho specialized to you will 0 this field the car 0 U.S. of the considers potential saw the man was not Dole brawl that isn't consideration is a tall function early fall down from his solution but also to identify it explicitly would take so the abuses OK all right so this is the end of the story about today to be horror here so let me go and do what it did once again the I don't logical part of the story there there is an space of solutions there is
no fissile solutions just Nunavut unless there is a Kruger the amount of the independent way store tho produce points of this pedestal for the tool find relationships between points of this display of this presence along and their aggressions themselves do not play a central role in the future and the moral or there is a huge synergies during period you have to impose a special solutions for Forrestal starting from 1 solution appointed the celestial we produced many many so the moral ideal dispersed there the geometry of topological incursion in a similar way through my understanding the situation as where is similar it's not the new Miss Universe well-informed patients of zuma Hi Huge disposal their functions the told me here solutions parameters for of the rest Monday and In this year of the political version holds or have solutions and here potentials all of to which our opting for by different procedures of the politically cautioned weddings the initial date of their percussion Michael Adams you also have walk of the the trial itself all possible potential so it's a very nice objects and that he has also made a lot of symmetries ended parameters by what around this minor the better aboard again there is a huge robot single displays and this person said that his days at this but Rob Morrow however this investor group existing holders this club has this monument through so-called quantification of quadratic amid going on of this is the moral is same that was discovered described by you but died or deleted at mall that always saw gain on one's hands bunker and blight the topological recursion explicitly for a dual initial data on the walls of the fans the same function could be obtained in a different way you start you go up you come to just 1 particular the potential disappointing tho that you pursuant to both goes forward missionaries the so far there the a similar but actually is a venue halted the details the details are quite different for instance there is sort of indictable system also called Bukit C however she which also parameters by Kazakh responding to a Lagrangian more as a drop address monument the fundamentals of space and this is not the same Center this is not the same so the details will be In tomorrow but come company on I would like to stresses that when they be talking about the topological recursion just current mines is Chris formed well I think so it's a better look at the start of the
political recursion right now here which fortunately have to talks probably will postpone for tomorrow the system here few the question the the company was it's interviewed does exist for the DTP Aussie GP but not in this case because here a case those who be somewhat different not of of the and few me
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