Hurwitz numbers, the ELSV formula, and the topological recursion


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Hurwitz numbers, the ELSV formula, and the topological recursion
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Zvonkine, Dimitri
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Institut des Hautes Études Scientifiques (IHÉS)
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We will use the example of Hurwitz numbers to make an introduction into the intersection theory of moduli spaces of curves and into the subject of topological recursion.
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at the end of the you can so I don't know how to call my job I decided to call its introduction to everything I have 2 hours and I will talk about I will tell you more as everything that I know about the 2 political repression and then much the mother of 6 hours to continue and I will also tell me about the Hermits numbers and they use the formula so the blend is topological recursion parents numbers and the TOC formula and I will try to stay as elementary as possible so that you will be able to follow them and the rest of the talks more easily so how does that political regulation work as the 2 political recursion is a method that applies to many problems where you have generating functions and complex 1 Lennox for any G & and surgery is usually the genus and then some number of Mark sort of something I will give you 3 examples of problems like that there are more but that the you 3 I again the 1st problem is unheard its numbers
Page G K 1 K
yes there are 2 definitions 1 combinatorial definition in another more geometric so the geometric definition is that it is the number from fight coverings that mineral adhere here is a that's ACP 1 here is infinity here you have fixed branch buoyancy and he wants to find 2 them to enumerate branch covering of this year with ramification profile K-1 K-2 Gary and over infinity so Infiniti has and bring images and there are branching and this is equal to a K 1 kayak and then simple branch fallen so all other points and there will be able to draw a properly something like this and I want the
surface to be connected of gene due so this is the number of brunch Savarese and didn't commit oral definition it is with permutations and let me notes that some of the key eyes by capital case and then it will be equal to 1 over key pictorial the number of lists of banned trans position I forgot to say that I am here should be equal to 2 G minus 2 plus some of the K eyes blasts and you can computers and if you want the surface of the equal of G 2 jet had to be of genus G again computers solid characteristic and you find that the number of simple branch voice should be equal to this this number so this is the number of lists of them transposition the sales lists tell 1 column Sigma such that 2 allies entrance positions a Sigma hairs cycles of lengths 12 lengths K 1 K laying K 1 and the group generated by them all is transitive yeah and evidence that the product I think that one-time style am Tennessee equal to the identity of permutations and identity permutation Solis is escaped In escape this this definition is equivalent to that geometric 1 because if you look at the modernism discovering you find and stress positions and 1 more complicated permutations for the monogamy here I here so this is 1 of its number and then Is it possible to get the job blackboards but pushing some but no matter how there is a stick to hunting there's a yeah I just that I was thinking of the 2 women since everything is filmed on camera should behave properly again and so now you can introduce the generating function yes I will hesitated to variables actually 1 set of variable is the most natural for Kent Alterman editorial work and that other works fine with you 1 supplies there 2 political retention procedures so I'll give you both this so-so some over and went over and pictorial some over K 1 K N HE K 1 K M here and over and pictorial Somerville writers plus plus some of the key allies pictorial at and here you again writes the guillotine right into the power some of the K-III 4 this product of capital excited the priority I know so capital lacks heat the our small acts the this is the way that will make a political recursion work properly and this is the way to end the usual later generating series it's a gay problem number 2 and that's the end of problem number 1 we arrived at a set of natural functions that for every gene ever and ever function of often variables once we get this week and try to apply to political recursion so the 2nd problem is the problem of wild Bettison volumes Williams also modulates spaces so if you fix so you think G and 26 effects and lengths L 1 and then you can look at that space of completes metrics of curvature minus 1
on surfaces of June as Jeter and bounds actually yes ma'am but didn't like this then it will be correct not so with hand-held Jenas Jeanne holes 1 to end and the troll has this geodesic here and I want them to be overplaying so 1 L 2 L and the geodesic bounds every you must have a prescribed length right so you get a space of all possible hyperbolic hyperbolic metrics overall a complete metrics on on surfaces like that again on the space there is 2 forms Gold wild form synthetic forms let me call this mg L 1 L and then again compute the volume of the space so the integral mg although 1 will all there exponential-of the well Peterson form and this is called the volume Jeanne of fell 1 so once again we have for every gene as union for every and would have a function of and once again if you once that topological recursion apply have to performs some strange changes variables so I actually had to ride to the hats Jeanne 12 X 1 X and they will be there a lot less transformer Fijian so that integral Over you to the power minus 0 acts the Aegean L 1 element this will be a 1 1 some and the topological recursion actually works when he applied to the to these functions I can finally the last example at Golden Crest stated triangulation here is a picture of an increase stated strangulation the averaging the surface mostly triangulate but there and in stations that are some other polygons of any with any number of sides here 1 garnered to run a triangle so 2 years later strangulation as GM K 1 K N is the number of triangulation health Vagenas G surface with CapitaLand triangles and then more polygons and more polygons With do 1 of 2 gay and sides and here you also can collect these numbers into generating serious adhere you elect to have 1 extra variables because 1 extra variable because of this capital and so I 1 had and you who will be equal to the sum over and CapitaLand 1 over and pictorial stumble over K 1 K N them Tax went to the K 1 the extent of the K you to the end I think it's over and that Chile will almost sure to GM K 1 but so once again for average originals and at every number and have a functioning and variables
you will bid on a variable that will not be involved in the 2 political recursion In I want to do more of this no because them it said the about the exported last fall you know that the dimension and that you know that this is not the same pictorial and here the thing is that you have the sum of all the key eyes and here here is just that control yeah I think yeah I will know and I'll have to check it and I think it's like this and In the end I such related to the to the to the exponential throughout locators and mature but you wish I think I think this is the correct formula would elegiac a case and no what do you do where does the topological recursion do with all these functions so I have to warn you that the political recursion is a subject and progress so it is now pretty well understood how how it works but not too wide to work or in what situations it works so there are lots of examples sometimes it is proved that it works in a more-or-less standard way and it appeared from some set of examples where it works in a standard way there are some examples where it is proved that seemed much more complicated with and there are examples were still a conjecture so I'm going to describe the procedure and that for every example you will have to work hard sissy if it works or not and why the procedure is as follows you take the function of 0 1 effects for the simplest 1 this function will give you something that is called a spectral current scene and then using these both things you will get although others so you will actually get and forms W G and on ceded power and so you take this curvy take its and spyware and have and poor and 4 and forms of symmetric and forms symmetric informs that when it sends power and these will be equal to do 1 d thank goodness radio expected 1 12 the and affects 1 it's girl who sees at the end of this season actually in the plane in the city in C 2 10 this actually 1 of the cordons of the plane so if you look at the examples and that a gay sometimes have 0 1 is really the easiest generating function to computers for the Hermits numbers we're going to compute of 0 1 right now directly and it will give us the spectral curve on the other hand in other examples at 0 1 may not be defined at all for the wild Peterson volumes if you take this year with 1 hole there no no hyperbolic structures at all so there is no modulates they said nothing to integrate and the theory works the works very well uh this function should be defined as so well in the ad hoc way of history so sometimes is user-defined sometimes you may have to be amended to guess at my father mother methods that is alleged to affirm its numbers but it for the various numbers would have to compute page 0 K which is the number so 1 overcame a vectorial times the number of list tell 1 talc a minus 1 these are all in escaped ultras positions L such that style 110 so OK minus 1 is a case cycle all right so you have to compute the number of wasted a composer case cycle In take a minus-1 trans positions it's the smallest possible number of trans positions and to get a case cycle and that it is automatically transit we also see its geometrical associate few if you draw a fired Roy again my my picture with the fight covering so here have just 1 Greenidge and assurance that maybe so if you if you draw if you draw a star like that on this plane that that fluent in yellow and red if you draw a star like that on this on city 1 here the 3 image of this stock will be lots of stars there will be all good together at the perimeters of these points of the perimeters are here you will get a picture like
that with lots of stars all that together and no to only to wait and then again erased the unnecessary jails handy gadgetry so this treaty will have gained gave artisans and gay minus-1 ages so this is equal to the number of trees with kit numbered vertices divided by the pictorial 4 the number of trees no just just skin number you wore a number of trees With minus 1 numbered edges he won this 1 numbered edges and divided by Cayman minus 1 Victoria I so let me let me tell precise sincere asking about the 120 take a tree with numbered edges it has a plan a structure automatic because around every verdicts you can order the edges and the in the increasing water right so when you take the Prix image of this you can number that the points here 1 2 3 you can take the prima Jarvis of the started to get a tree with numbered edges form Bush it's so of if making use of the money I exactly what's going wrong you have to be a long way to go so want no no no this is this is this is correct and just that in you visit and his courageous and in the generating serious so actually what I what I wrote here is age there OK divided by minutes 1 Victoria and this is equal to em 2 G minus 2 plus and plus some of the key allies In this case the sequel to minus 1 it was in the end it was in the generating serious so I number a number these market that I have a tree with numbered edges and then I divide by K minus 1 pictorials are in some way in some sense they forget the numbers of the edges and if I want to 2 good to see like that it's to it as permutations I also have to number the perimeters of this .period there'll be Cape free images numbered from 1 to care and then get trees with numbered edges numbered Virtus is and I will have to divide by gay Victoria to forget the numbers of the on the vertices I think it will run away allegation out does anyone know how many trees with a cane numbered is the key gain numbered vertices there are those that gave formula it says that the number of trees with K numbers numbered divergences because prosecutor the park a minus 2 it's actually not so not a trivial formula the answer looks so simple that you would think there is a and it's an almost obvious statement there are lots of proven is that 1 of the most famous Canada formula so there lots of proves there is a major active proved it but none of them is particularly simple and 1 of them so 1 of them but merely a miscarriage 1 of the proves because only the parts of it's in the end later safe I take rooted said trees the same thing rooted trees with Gay number advertisement the 1 2 3 4 and 1 of the registers as a route and I call while Faxon generating series then control of the more complicated 5 6 7 so that I can erase the roots and obtain several trees all of them with you and all that will be rooted the routes will be the purchases were those whose ages were attached and from there so well against the quality that this series is equal to an accents to the power wife while so acts stands for this route that diary moons and you to the power wife so one-plus y plus players squared over 2 and so on every time said so and this war 1 stands for the simplest rooted tree with no other nothing attached why is 1 the route of of degree 1 and there is 1 tree y squared over to his when the route is of degree to win the right to trees and so on see get an equation like that and then there is the Logrono inversion theorem inversion formula that allows you to find the coefficients of a series that satisfies an equation of this kind and you find that the coefficients of why are we came to the park a minus-1 over gay pictorial facts of the case so this will be an important power series 1st but it's
anyway for I have 0 1 Gullfaks I found that it is equal to the sum of 2 the power came minus 2 over Cape at Oriole you do that by working small acts From this I find W 0 1 effects as I said it is equal to D of 0 1 effects so remember I told you that's what you actually find with the political recursions on forms rather than generating series to any of your froze generating serious to take a differential and you find some of 2 the Boracay minus 1 gave Victorio into the power acts DX so here you can recognize the power series and this gives you the equation of the spectral curve so the equation of the spectral why equal some over K to the and to the K minus 1 Ricky pictorial to the Boracay and if you look at this equations you can also write it as acts equals y miner's wife something like that this Fraser a fire rights why equal facts you to the power y then capital axes equal to why to the power minus wife so this is due to the power small Axe equals wife times to the minus why and then I take the and gets back logarithm why minus wife which gives me the equation of the spectral curve and I had the 1st differential form that is now just wide DX right if you look at this those just wide the X so this will be the starting point for the topological recursion and there topological recursion procedural give me all that differential forms From dollar from this information but so now here's the recursive procedure supposedly have seen yeah curves in the tomb do you have only 0 1 equal to IDX and then so for the recursive procedure you also need to know W 0 2 so w 0 2 in many cases it is uniquely determined by this data so suppose that suppose sees a rational :colon is a rational curves with global coordinated disease global coordinates the quoted then I can write the falling to form C squared poncey squared I have DC 1 DEC divided by Zee one-liners into squared and they claim that this is a to form the date that is unique in its kind it does not depend on the on the global coordinators 3 invariant under if I change the variables of replaced the but a Glasby oversees the blows you can and make the computation you'll find exactly the same thing so it's a four-month C-2 to rates so all seem to have accorded 1 and accordance into this is the last thing you want to yes yes yes but right so actually no examples that I personally know their crops he happens to be rational and this is that the 2 I know that there are some advance people who get examples were the Caesar fired units and then there is some choice in this case you can also require you can also ask for it to form that has the double bowling this on the diagonal but that it is not unique and I think it is not very clear how to make the choice so there is that as in the original paper by a and although they make some they specify some choice of W 0 2 but it doesn't look particularly natural they have to choose a basis of of differentials on the curve seas of the trees depends on the basis and it's not very clear how the choose it so I think it does that I think it's still not that elucidate his question if if the girl is not rational I don't know what to choose exactly and then there is the Huge frightening formula that you will see I'm sure In might seem stocks or maybe he will hide it in 30 so this is my 1st season but not big at critical points so I don't have a critical point they have local and
pollution seeing my hails the directs yes this is next year and so I have a local inversions in oversee that exchange the 2 sheets it's not defined and that on the whole ,comma but indeed it is defined in the neighborhood of this of this and critical points so from this they define a colonel colonel Gary holds the insists the prime OK so it's the integral from signals the disease of W 0 2 of the injured Brian divided by 0 W 0 1 of so the type of this thing is something times easy prime divided by something times easier W 0 2 has a DEC and it is the prime their taken integral overseen so dizzy goes away rights because agreement DEC goes away said David easy prime and this is a 1 form with the with the disease the game now wants to have this kernel I'll take a fresh board hurricane some over critical points residue Zedillo's this kernel is easy prime but it's easier my plus 1 actually at the hearing however some overall G 1 G 2 because gene and I union disjoint Union Jr peoples 1 2 and and so you splits the genus and there set of Mark points into 2 parts then here you're writes W G 1 still I plus 1 z eyes W G 2 J plus 1 Jr position blasts some over In here there is no sense of the WG minus 1 and plus 2 z 1 0 is easy and actually in some depending on the papers sometimes against instead of against the single disease in some of these places it does not change anything everything is a city and put 0 single of the year you have to change assigned before our kids you know I know so the Latin said decided not to put it and saying no I think it's more beautiful level of since I have had this know so yeah so this is equal to 0 and this is actually a triangular system that determines all the WGA hands a recursively because if you look at the big so there is nothing there is new there is 1 main terms here if you take the 1st time in the 1st semi candidates G 1 equals g and I equals 1 so that here you will be left with W 0 1 2 W 0 1 8 2 0 it seems so here you will have narrators that merit somewhere they claim that this this choice will lead you to just WGN plus 1 of 0 1 museum plus 1 and you can convince yourself that this is true rather easily so if you take this choice you will have WGN plus 1 because the 1 pizzeria and disease then he here you will have W 0 1 fuzzy and then you have to multiply by the kernel and the colonel will In the colonel plates and 0 2 is easy and plus 1 divided by 2 0 year end due to a W 0 1 of zinc so that other 0 1 of the simplifies right and you just saying knew I just left this thing and with some residue to compute for this W 0 2 as you look at the W 0 2 on the diagonal it as the end of the residue from there is a need and you just get 1 yes it was no no I can't explain where they yet yes so the formula the formula comes from a region from the Matrix integral Batavia and and again to talk about this now so this is the maintain main German this formula some people put it on the left-hand side and then they have to that to do something to kill sermon the summation so in this case they usually by convention that puts W 0 wanted equal to 0 instead of my choice which is the most natural trace there and they say that this is just some special form that is used in the colonel and W 0 1 is equal to 0 for them in the rigors of recursive formula and I think it is the unnatural choice for her numbers for example again
computer the function of 0 1 and it gives it a W 0 1 to be equal to the to this so you don't want to change it by force and that iterated that more beautiful tried it in this way a game have this this term that is there but say the main term of the formula and all the others editors other Germans have smaller uh too numerous to and smaller characteristics so once you know W 0 1 and every 0 2 we can compute all of them 1 by 1 using this formula the government has also yes WGN has fallen diagonal at all of the day as you see as against Nelson knows WGN also have opposes at the branch .period section With this yeah the case am I so I know that section of the only 1 that has a poll on the day it's already here OK hopefuls city and they should be simple so now again now there isn't there are developments and now it is possible to generalize this procedure if you have multiple critical point at the formula is different that in the end the answers to the question this formula only works for simple critical points and W GM's WGA are there more fake metal more where polls only at critical points only at critical points the truth on the side of the political waters which portable so was so what was should the region is this form of all who is business also called America's sense and this is the whole point of the amendment but they did not address that that they will have all the critical points ensure you it you you will see an example for 4 numbers a conceivable so only at critical points and W 0 2 is an exception W 0 2 has a poll has a poll on the on the diagonal L C squared and that's the only 1 10 minutes later I didn't know about it I this I there I was surprised me OK that's fine this the OK so now that is the Alessi formula 32 a local I can't talking about something else but actually soon after that I will come back to that of political recursion and you will see some their relationship with their with the university formula so here is the formula it's a formula Herbert's numbers H U K 1 equals and pictorial products today I to the priority I overcame advertorial integral over engine block 1 flounder one-plus landed 2 since along plus or minus but this 1 to the Jedi plumbed G 2 1 minus K-1 PCI 1 1 minor skin side and the U.S. newsstands work taken Lendl Bashir here Wednesday staying but as I have already explained what the nervous number is now there are lots of funny letters here that they have to explain to and bombastic to include an introduction to modulate spaces so I will spend some time and explaining all the letters here this em again is to you minus 2 plus K-III plus and that's that the number of simple branch Florence so this is and block is so it's the modulates space modulates face of stable Jean GE
curves put then more points In Royal but that the bigger so that means still every point here is associated the curve of juniors Jr With marked points was 1 3 so I heritage Units 2 and 3 mark points this so this is a smooth curve but again also have an curves with simple self-interest actions stable curves are allowed to have singularities but only 1 type of singularity name a simple of intersections so here you have some devices with normal crossings and if you a point and there's the wiser you will get take care with a simple self-interest action but 1 2 3 see so In this case I decided to end take a with non separating nodes if you normalize the Kermit this nodes the curve stays connected and then this is a self intersection of these devices when a curve has 2 separating 2 and then there's another here you can have a separating node that so the other actually also devices began because against that the government in different ways you can't have 2 components of Rivera's distribution of the genus and Mark points and so here the intersection of the 2 we will have once operating node and 1 None separating them thank you but is so over every point of the modulates space have a stable curve and all these together form what is called the universal curve so MGM and what is what is called an order falls you can think of it as a manifold that usually doesn't dozens manifests itself in any in anyway To that properly it's in order fold so far ,comma budgets complex 12 dimensions so complex they mention the three-year minus 3 per cent so this 1 will be of dimension 3 G minus 2 plus 7 1 dimension more and then you have the projection whose sheets was so his fibers are stable curves there's 1 tiny piece of yellow tulips probably and it has tried to use this 1 but over MGN I will define and line bungalows and 1 vector bundle let's take the coach engine blind to the 1st stood to the stable curve at the 1st mark .period site I can do that in every fiber In every fiber at the 1st mark .period Abaco the tangent line Analyses over each point of energy and water I have a complex lined the coach engine plant at the 1st mark .period as to the curb at the 1st mark .period it's all taken together they form a line bundle that is called L 1 so I have line bundles a wonderland if you prefer formulas ideas the ball back under the eyes section of the universal curve all the relative go tangent line bundle To the universal curve again then there is 1 vector bundle Gold the Hodge bundle since the rights of Frankie and its fiber it's fiber or were poisoned is the space far homomorphic 1 on this curve abandoned differentials so you acts it is the space of a billion differentials differentials for more frequent forms 1 c x right to take a point at sea have a siac's you take the space of 4 billion differentials it's vector spaces of right of and dimension G so over every point in the space of dimension and a former rector runs this is the 1st all the mean about that a stable currency they want to know how to extend the Hodge bundle the to state had the same workers again so so the rule is when they have a stable career it has simple South intersections the rule is that set this point you allowed simple bowls that most simple polls with
opposite residues when there is a reason of course for this rule but let me just give you an example if you take an elliptic curves you know that there is exactly 1 a and differential up to multiplicative constant if you take a lattice of parallelograms again just take the differential Desi and descends on them atletico and then he again you can take a stable career like that right that just fear was to identify points that season and infinity C P 1 0 identified with infinity so you would like To have exactly 1 a billion differential here Donna there are no other than differential if could take a bed there were differential 1 form 161 A polls but if you take this form oversee it says a simple bollards our the simple Pollitt infinity with opposite residues so with this new rule you still have 1 dimensional space of an of abusing differentials and it actually works in the same way for energy is indigent degeneration you always get the space of a space of dimension and you can actually see that if you take a whole Moffat 1 form and you take a family of elliptic curves tends to the stable curve you can see how the agreement differentials degenerate into something like this again so now that we have to these line models and the vector bundle I can take their characteristic losses so I can define sigh I used to be the 1st turn glass of so that's a first-round loss so it lies in age to G and water With rational conclusions and so this you here is actually the only place where you the fact that it's a normal folded and that makes appearance usually the 1st shot glass has integer coefficients on a manifold on or before they can actually have rational confessions and then London I live the luggage will be the Sharon glass of the Hodge bundles so that's an age to years both MGM were you and here I goes from 1 to and because there unmarked points and Geragos from 1 2 2 G because it's arranged the Victor rumble and finally there there is 1 less thing I should tell for you to understand this formula it's that when I write 1 over 1 minus the I'd say I would actually mean is 1 plus the I'd say I glass OK I squared I squared plus and so on you and your job yes so so here we have a product of and plus 1 factors the factory in the numerator and the factors here that I expanded in this way they look like infinite series all of these but actually the dimension of this so as I said the dimension of this is equal to 3 G minus 3 Boston so when I perform the integral I will only extracts the terms of the degree and expression all this thing is actually a polynomial falling on health in Game 1 again and it is symmetric with what dubbed degree equals the dimension of the modulates space 3 you plus and lowers degree Lois degree equal to 2 three-plus so I get top degree if I take 1 year and maximal greatest possible power eyes and why I was grief Friday plundered so the lowest possible powers of size and of the eyes but then if you want to have been financial coefficients of this pollen on all this is much harder because then you have to actually start intersection theory on the modulates faces and computes the wall compute the integral of all these characters classes it's 1 thing to define them in and much harder to actually computer anything so now I'll give you an example a good example plastic genus 1 and 1 market buoyant so I rewrites the university formula it says military-age 1 is equal to OK minus 1 gate to the power over gave reptiles as they gaze at gate plus 1 and it was 1 factor 2 Jiminez ticketless 1 4 2 times integral over and 1 1 born 1 minor slander ones 1 minus case I 1 so I do notice gate I write gains that of K 1 since there is only 1 marked .period I don't want to write the index every time day so this
thing here and 1 1 bar is equal to political participation of the this modular figure could you take this thing I need on the left-hand side to the right hand side and this art to this stock and then you add 1 more point at infinity to compact to we'll get an 1 1 mark safe you imagine what what happens you will see it said something like this so it's homomorphic tears here but let me rewrite this integral the integral and 1 1 block 1 minus slammed 1 times one-plus capes I 1 and I don't have to continue because it's a series of companies that mention 1 so all other classes will be equal to 0 the next powers will be equal to 0 so that's equal to the integral over and 1 1 bar Paul Gates I won -minus number 1 I drop all the terms that are not of the correct degree so that's scary integral over and 1 1 1 "quotation mark I won -minus integral program 1 1 block of Columbia 1 let me do notes this acts and this by wife it is possible to conclude this integral but I don't want to do it now I want to use the UST formula that will compute them for me With that with no effort I now made but is it's better to use the LSE formal and I have just 2 numbers to determine acts and wife I know these X and Y I will be able to plug them in here and get there the final answer so to get X and Y I just have to computer to have its numbers so let me computes H 1 1 and H 1 2 a each 1 1 is is the number of coverings of this year by a Taurus of degree 1 right here my sphere simply 1 here have a Taurus yeah infinity and Fiji must have a pre-merger of multiplicity 1 primitive multiplicity came right this is the definition of the it's number now I decide to take April 2 1 so how many covering so many covering their I was fear about Torres said of degree 1 and say 0 but it is not possible to if you have a map of degree 1 if this 1 has to be is here to attend the BIA Torres if you want to use the combinatorial definition you will have to count lists of uh let due to transpose transportation To address positions In 1 whose product is equal to the identities of permutation while it is 1 that I know just positions so you can then and there no list like that again at another stage degree too that's already more interesting now you have the agreed to and then have 3 more marked points that are simple branch points right in this case In this case you have if you fix the branch points have exactly 1 double cover of this year With rifle with branch points he writes exactly the picture that the dead wrong moreover there is the hyper-elliptic and pollution so it is actually countered with 1 over at one-half I think I didn't tell you that they should have and when you count the fight coverings of recovering should be countered with efficient 1 over the number of automorphisms In most cases this number is equal to 1 but in this case there is the to afford amorphous and say have to divide by 2 and if you use the combinatorial definition it is that it gives you won't have a demand that automatic it's 1 over to vectorial times the number of lists lists of 3 address positions 1st positions being assumed such that their products is also transposition right Ernesto only 1 transposition so there is no choice there is just enough 1 2 1 2 1 2 3 times the same France position the product of this mystery cross positions is also equal to trans position so that's exactly the monotonous of all that of discovering is just the permutations of the 2 sheets and then you divide by 2 photo I also get one-half again is that is that fine rights and I have a system of 2 linear equations equals 1 gets 0 equal to To pictorial 1 to power 1 over 1 pictorial times fax minus wine the "quotation mark Suai gets 1 have equal to 3 factorial 2 to the power to over to Victoria 2 X minus 1 so the 1st equation tells me that is equal to y and the 2nd equation tells me that taxes equal to wise equal to 124 you can check 6 sent students to its fine writes that
so the USC formula tells me that this integral is equal to 1 hour 24 and the seat with integral is also equal to 1 over 24 and I got this may just counting permutations in the in the group's S 1 and S 2 and now I can write the final results while simplified the plus-one Victoria and gave a our liking I'll give K plus 1 times gate the Cape k over 24 -minus 124 or they squared minus 1 claims cater the gay forward 24 a case of and this is the full power of the fearless formulates its computers to insert roles that if you don't know how to computer pretty hard to computer and it also sold for me non-trivial combinatorial problems so this is actually a number of ways to factorize case cycle into a product of plus wondrous positions if you try to solve this problem by hand it is extremely hard once again it looks like dancers and rather simple if you try to do it by hand I think it's a possible but here's the answer and you can see this 24 in the denominator that is particularly surprising OK so before but it's now I'll I'll tell a couple of words about how the OSI formerly proved and not going to give any actual proof but just an idea so the idea of the proof is to look at the space of stable maps From Jean as surface goes to see the ones with and bowls of degrees K 1 again so there is that this face of stable maps mg K 1 K see the 1 and then to come up like that you can assign the center of its so you can we've never met like that you can assign doodads set off branch points so that will be something like C to the power am here but to draw a picture this is 1 here have the stable manner for the polls so when have a stable market can actually have some contracted components and then here you have the brunch wines and there is a fear and that tells you how to that's this so the contracted components they also count as branch multiple brunch for once there is a fear that tells you how to extend this maps of how to and how to sign a multiplicity of every connected to every contracted component so that this this map is well-defined only smooth maps but only on also unstable maps so as soon as I have a contractor's components it counts as a it's the image count as a branch .period with a multiplicity and now you can you can try to complete the degree of this matter so this is called the branching morphism and you can ask What is the degree of the branching morphism and the degree can be computed in 2 ways the 1st waves today but a generic point in the image so you take simple French wines and then there do you will be just the number of smooth ratified coverings with simple branching over these points this is exactly the Harris number but the coldest and the on the branching morphism so branching tree image of a generic .period he is the head of its number if you take simple simple branch points can the number for images is exactly that by definition is the Herbert's number but then he can also take a free image of 0 0 0 0 0 so I decided to put all the branch points at 1 place and in 1 place at 0 and then there's branching morphism has an infinite fiber In this case the Prix image so the maps that sample their branch .period 0 look like that here is the 2 the Boracay 1 and so long you to the Boracay so you have a bunch of you have a bunch of on which the map looks legs into the power something so that there are no branch points here at all and then have a curve of gene is due with and points that is entirely contracted to 0 so that all the brunch points accumulated accumulate a reserve and this is an infinite fiber because this curve of gene is due within marked points can have any complex structure and then
algebraic geometry have methods to compute the degree looking at an infinite fiber it will be some integral of characteristic glosses over this engine block when you carry out this computation you get that the right-hand side of the others you formula and so it's a rather common situation when he can compute the degree of a map over generic point at a finite number of parameters and then over some special point to have an infinite fiber Sept introduce a characteristic class integrate over the special fiber most of the people who did you want him what the generalization the political 1 you want you see is that yeah it's what is called the relative relative stable maps reveal infinitives the device areas there I think but the reason I have 2 small discussion about this fall in a mail it in so in my opinion and the fact that this thing is a polynomial is the most mysterious property of Purvis numbers this evening numbers can be defined commitment orally by enumerating transposition and southern complicated definition you would expect that most combinatorial properties can be easily improved from this definition just by studying how trust positions can give you a permutation with given life of cycles so the property here is that when you take this number and divide by some simple combinatorial efficient you get a polynomial Inc 1 K and for a very long time the only proof of this property was Viadeo's so you had to introduce all spaces of relative stable maps and modulates faces in this line Monzon characteristic classes there was no other way to prove this fall in a mail it and the spontaneity is also contained in the topological regression procedure and so much she wrote a note several years ago already quite a long time ago the showed that could actually approved both the LSE formula and the and that a political recursion From yes from combinatorics but assuming this fall in mailing property and this plan and had property no 1 could prove independently from the OSI formula so finally last several months ago there is a joint paper by 5 Waters a thing try to dilute the Kowalski look speeds by senior guard Alan Belzer gay shattered rates no and they finally so they finally found a different proof of the pudding that is pretty pretty complicated but purely combinatorial and Angela myself so if you all of head and then from the from the pollen analogy can today where they were able to use the the topological recursion procedure and the others he formula simultaneously so now how does this ample in immunology appear in them in the 2 political recursion so let me please let rights once again age median 1280 X 1 X and its sequel to some more than 1 over and for the total sum over the 1 clear and the agency K 1 K N program Victoria X 1 to the power K 1 Banks and to the Boracay and so this was my definition of the of the air power series of the of the generating functions age G that enumerated the carrots numbers so now let's use the LSE formula and blog adherents see what happens so I claimed the following some older and 1 over and pictorial some over 1 Indiana thinking the integral over and Julian Bond say 1 to the D 1 say and to the DTN lounger 3 G minus 3 plus and minus some of the DI is it's the efficient and in some cases products from 1 to some to the Boracay I close the Over factorial next to the Boracay it said trio statement I just blogs the expression for the Hurwitz numbers and the from the UST formula to this to this summer but I also did something else so I actually expanded the denominators this saved when I when I write 1 minor I'd say I was 1 plus the I'd say I lessons and so on a the guys squared I squared I can and try to seperate all the terms so let's look at the term where sigh eyes that say eyes raced to the degree DI and his race the DVD and right I fix this fix the powers of the Natick some overall all possible powers then I have to leave and helps to the power I will also
have a K to the power DI coming together with its so here is seeking to the power DI and then there is the height of the gay Iowa gave up again pictorial that comes from the From the USC formula and am vectorial heroes in the fight to them pictorial here and the thing I just it's that do you read it is necessary for this sum to be equal to somebody of the who really should be 3 G minus 3 plus which is the dimension of engine parts have to put this and if I want another integral thank 1 -minus there were shot out the others carries this now a minus 1 to the head to this power is thinking but and now I once so now the question is what is the relation with us all this power series right this power series to the Boracay placental some of overcame gate to the Boracay Paluzzi over cave Abdullah lacks the power came and the question is what is the relation was with my power series while facts which was some of Gator the Boracay minus 1 over gave Victoria elects to the king you can see that the series look similar but it's not so clear what is the exact relationship this was the power series that I used to define the spectral curved and now I said that all the answers should be uh neuromorphic forms on the spectral curved sides on so I should be able to express the series In terms of these axes in terms of his wife but is so here's what you have to do 1st take in other policy areas the effects it will be more practical some of the anarchy to the power Katie gave a total access the came and I claim that this is equal to y over 1 miner's wife who is is also the reason functional excellence and some all in 1 here is a list of some of the highest again over the years you know you just yet but I think you and the US so in this case so in this case there is probably not even there is probably not even there yeah see you were asking about them Victoria I think the thing in this case there is none Victoria yeah I think we should erase all of us yeah thinker yeah that what I think a digital that every time I I wrote this power series yeah I think I didn't offer enough you have to do that all and every time you where you were asking about them pictorial at the beginning you were you were perfectly writes that there is no seminar and Victoria so here is my spectral :colon why is a global Cordiant on the spectral curve but is more it's more practical to have a global coordinates z defined as while over 1 miner's wife right this is a rational curve so this is also a global accord the rational curve its advantages that it has a poll here so this point responses equals infinity as you see here is white walls 1 so the pose here and that's the global coordinate that has a ball at the branch .period and as I said to them there Mayor amorphous forms that we get have polls at the branch for but it's secure I already had the 1st claim How do I prove that this so this power series and this power series right this is called the this is called Why and I'm saying that because he is equal to divided by 1 minus Y this actually so if you write it out if you write out the power series you will get the falling identity some for the last few equals and so and vectorial people Q vectorial the 2 the power P due to the Q minus 1 is equal to around minus 1 and 2 then minus 1 it is called identity and again it looks like a trivial combinatorial thing actually it's nonsense nontrivial at all you can do it only if you know the can formula for the trees I tried I gave this as an exercise to international so your allotment inversion any proof that works for the care for their came formal work here but it's it's not and I gave it an exercise international math Olympiad winner and he wasn't able to to prove as they that's not a simple matter simple problem you can't if you know the Kaylee formally can actually uh approved by cutting out the energy of a tree from a tree you see here you have the number of rooted trees so here the here's a roots and then have a tree with unmarked and markets voters and numbered Virtus is then you began and minus-1 choices and when you remove the said he needed to dream 1 of them with too much clients so Peter the party and the other 1 is just 1 1 route so it will be due to the parking minus 1 if you know the game formerly it's easy to prove and then by proceeding in the same similar way you can prove that all these power series so some to the power cable as the overcame Victoria lacks the power came he is not degree polynomial but degree polynomials all in all in z so when you raise heat to this series you 2 different powers and and by taking polynomials of again construct see all this power series that you get here and this is what the
topological recursion does so it starts with the power series like that's the W 0 1 and then my some and some combination of fresh is it gives you a sequence of polynomials In In several variables so WGN if you look at them and is in the local court and then in the in accordance the these are and forms under on this underscored years :colon and forms on the and power of the scurvy can write them in accordance you liked so this is the Here is the most practical coordinates and then these things are falling on sales right so now we have to pull a manager properties that actually equivalent to each other but another nontrivial weight 2 equivalent Paulina manager properties so the first one that is obvious when you look at the university formula it tells you that a that permits number divided by some for multiplied by some trivial combinatorial coefficient K 1 pictorial is a polynomial Pincay eyes In the 2nd properties that the generating serious AGN 12 6 1 and it is a polynomial he is the 1 the end where is the eye is this power series so some of the gate to the work a pictorial excitedly the case so that they could lines between the 2 is actually not a trivial thing it uses the Katie formula and Zobel laid into his but nonetheless it is true that the that both falling Ameritech properties I equivalent to each other and you have to know well you have to mean that you have to know the 2nd property if you want to prove that a political so this was this is their crucial the crucial fact about the about this power series that's allows you to pull that a political and that the 2 political regression formula works its action it actually gives you a very strong very strong constraints on the on the power series if you want them to be if you want them to be global polynomials on this curve yeah thermal only moral standards in 15 minutes earlier the schedule but my knowledge of the 2 political recursion is limited so say told everything you know if you don't know if you you have any questions you can ask them not to all of these things will be on the rise so for so for the uh about Peterson volumes it's actually quite this statement is very simple divided Peterson volumes themselves so this is a polynomial polynomial Finnell 111 and then you spoiled spoiled this property by taking the there are a lot less transform so in the end what you get is notable in on little around honor with the personal properties of hidden In this at all yeah that's a good question this no no no no of for this 1 for this 1 is something like X equals to sign why they think yeah 1 why people sign of square root of acts OK if so if you know the answer by cards that give area In another year another power the probabilities for them .period function for energy and water so it's a little bit series this 1 also foreign would involved the glass couple 1 when you can the center which remove the couple 1 you will just get the problem In this case and for the triangulation saving didn't know the spectral curve yeah for the triangulation so there is the matrix integral to enumerate the generations and then you have to compute the spectacle of the of of this matrix model but didn't know by heart with the result is the the creation of future home I think this is a very on you ministers you know what you think of this the country so there is a part of yeah so there is a formula that is called wireless unit is still a conjecture that was it's my conjecture from maybe 10 years ago and is still not rooms at that involve some different service numbers ends pulled permits numbers and here you have an integral over the space of forest and space of Western curves this is the last 1 and there but about both operative long service that it would be a topic for different and there was article the notes of the and the case of the so articles 1 you get back the year-earlier loss who form I IPOs 1 the was to which I equals 1 Stewart and I don't know now says that well actually so that she wouldn't let me think about I think for articles to I can prove this for actually have a method that have a method to prove this formula for any for any given hour by found among finite amount of computations so for IPOs to again proved as formal and for any any oddity give me again computes welcome viewed many examples and show that history but the proof is in the end but the it's anyway for and for these numbers there's also there is also a topological recursion With the spectral curve equal to the text the to be Long y -minus sway to the power are here you have so for the ordinary herorats numbers why minus Y here is slow why miners way the power
tenser with sedition and and Luke speeds finally proved that the 2 conjectures equivalent without proving neither of them so that's the state of arts In the end there is there is the pollen Amalita property that in this case nobody knows how to prove what please all of the questions that you can compute these concerns along with computers he's yeah yeah everything everything can be computed there is no problem in the future want actually computed major problem but theoretically all of this can be computers the Hermits numbers a defined discriminatory so they can be computed the sensor will you know everything and everything is computable that this year or in the case of a solution to the nuclear industry will so and that of logical regression works for works for any number of any number of simple critical points if you look at the formula there was a summer critical points already erases only simple yet but that this could discover of a simple critical points there is no problem with it says it has ah assemble critical points yes soldiers have refused to go along with the 100 please read in construction the Commission the construction this book she grew up yes sir years of course you years get so construction is quite simple you go you see 1st computer age 0 1 of facts then you take W 0 1 Halifax equal to deed page 0 1 facts and then you decide that this will be equal to wide EX they just take quite equal to d page 0 1 effects over DX and this gives you and this gives you a spectrum of this is the question of the spectral curve so it looks simple but you should be warned that you have to choose the right accorded acts it is not at all clear a priority why should then take this to be equal to so his age 0 gave some of Asia OK over rights while 1 over and pictorial and then there is this needs to the power kayaks so it's not at all clear why should take heed to the power kayaks rather than thanks to the power K for example and for this I have no answer if you know if you know the rights the right to researchers the variable and it's just a simple computation and if you know that if another 1st the 1st power series the implication of and another warning if you want to do is to apply the same method for the webpages and volumes you will fail because there is no the 0 1 for the malpractice some volumes can only define it in the stable range so you don't know what to do yeah 1 more political correspondent for the original was partially funded full policies in this language of violence in movies during the collision exodus the by working here if you wrote the wishes of the people who have had a lot still this is real change of the new considering what the global finance coordinator the French forward this is far from the origins of the responses of ordinary life this small coordinator of the state yes the work market but only as used you start with a power series some of it to the Boracay minus 1 over the pictorial explication right this is a power series at the origin if he once to that so this series has a finer traders of convergence equal to 1 over and if your rights as a few draw the the graphic again something like that this is 1 over here and after that it doesn't converge and then again extended stood so in this case accused him and then again extended to a global curve with the global twice this is what you were saying rates is surrounded the use of particles so there were no longer there is good news also inspections .period I want was there is a function of the that's all there is the always WGA ends are calling on those in the variables elite that has a poll here and the 0 here as nothing and it doesn't see it as any particular value at all but if you want to find a hermit's numbers have to take the expansion at this point you can thank you conservation where do you have to take filibusters forms invested in this example let me see if so I have some sort of an answer I don't know if you might again to speak about EUR syntactic formalism years here yet so we assume months my seem has a formalism for this topological enter political regression that looks similar to given Intel's formalism for uh uh form of wooden potential didn't know if you've heard about them but also their similar but they acts into to go from 1 to the other half to perform a lot less transform 1st so and so and if you if you want to look at the 2 procedures there and too formal too big formal machines to compute something 1 works well in 1 set of cordons and other works well and the Laplace transform of these coordinates as so this thing here is well fit for a given transformers and if you want to do the 2 political recursion said to have to do that the Laplace transform I don't know if you're convinced by the sense I think Phuong


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