1/3 Mathematical Physics of Hurwitz numbers
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1/3 Mathematical Physics of Hurwitz numbers

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6

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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. 
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2014

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English

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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, Virasorotype constraints, ChekhovEynardOrantintype 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, GromovWitten 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.

00:00
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07:09
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11:09
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13:07
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14:58
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18:58
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21:47
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26:07
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31:41
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33:17
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36:34
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38:40
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42:15
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43:50
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45:22
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49:34
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53:59
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56:49
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1:01:48
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1:12:00
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1:15:50
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1:19:47
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1:21:12
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1:29:49
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1:39:50
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00:03
the top more than the figure much in the ritual to enclose rich was this In addition actually use originally and we discussed it isn't thorny for disposable that will explain something about the chorus numbers Skippy Hershey and it was I assumed to be a son kind of private sessions with them for this resulted in such a representative conference of all foreign troops and that surprised however if you don't mind turned to me flow to myself some experiments and this is for the 1st time that the get try told President talked with my pickup some experiments competition experiments because mathematical some comments and also Oriole supply to some extent of Clemens so according all our ultimate Semitic mathematics is the experimental science and the only difference is physics at the expiry mills here are much cheaper will show you current forks song I will speak on Horace number 7 to hold related things and we'll try In my presentation I will try to avoid no consideration of we have John material also configuration of the surface of the water spray along this comes trip on could spawn a a number songs underrated functions manipulations his functions like the border collie cautioned her Arkansas on and I show what's security generated behind the soul of real look at that the story from this point of view what will be the real available to avoid some repetition competing stock I apologize for that but probably I will try to told due some alterative the Phoenicians and you would not sometimes referred to as a result of the timeconsuming covers the same and especially this still concerned is the politically cautions :colon I will speak on topological caution him on on the the talks but today we will speak on another subject keep the hierarchy it said happened that have to be that of the Board of course where is striking similarity and will show you how it works but the concentrate on keeping so I started chorus numbers the Duma already is explained there definitions adjournment of this quarter's numbers but I'd look at them from the computational point of view referring to I'm trying to build computer experiments the simplest case the simplest way to compute chorus numbers is the use socalled gotten joined the Commission this kind of and Isaf procedure so far I recall you that we have behalf there are some of invocation at the Cheney to or completed the retention rate and we tried to award presented as a product of the simple now Cymbalta spies editions around the gotten joins the cautions that injuring aggression serious what happens if you increase the number of possible to decisions just by 1 the source the justices should analyze what happens this position with multiplied by his firm attention if you multiplied by transportation a simple genetic consideration show that you've their toes post it men's belong to 1 cycle cycle than this cycle splits into 2 smaller cycles and the disposition connects 2 points from different cycles the product yield in the property built a great cycle consistent golf elements of the 2 small cycle so as it goes cycles drawing together or wants cycle splits into small ones "quotation mark Jake educated at son I thought this is there some combinatorics so trumpeted the current but as I said that I don't want to Gold all right into the details of this combinatorics but to all of this but I couldn't and this could be held parked included itself and generating serious and this is called the "quotation mark function for the full chorus function the sum generated serious Rich "quotation mark "quotation mark fusions have some special meaning so I have an infinite number of variables B1 B2 and so on risk ritual of indicated that a major responsible for his order of the implication of infinity and 1 extra where it will be established response to the number of 4 of those positions In a kid constituent because spawning policies so as the main hypoxia number this slightly from from from the language "quotation mark the rest this this shares there is some other this was numbers
07:14
this "quotation mark officials and that the Duma fixed the Duma they must all he fewest fixed as there is a number of that offspring registered at infinity and the gun used an independent variables it's 1 itself I use a lot of different agreement Dutch 1 variable to each index will flanked by and he was right end frantically disease some of the were all forcible G tell them that they called the agreement told should foot that's all this is Missouri sum up for all a lot of collections and some divided by and pictorial or use just the petitions and divided by like the ultimate fueled this petition is necessary if you take all if you want that's just unique Bloomberg is in to with on the what I yes yes yes yes I don't want during the the definition for its numbers because it is not
09:19
important for me yeah "quotation mark duties once again let me stress that um very certain a certain numbers which have some Jamaat as some some combinatorial dramatic meanings but for for the moment it's not important for me sold by the had defined not their poorest themselves but they're generated function for them and when the nation's define gene by definition is a function in this connection for also defined by these equations was the 6 is there expansion of this function is that of the parameter beta is governed by these formula their duty for suspected Utah is certain differential operator which I note W 0 replied to this function G 34 it's funny so this is a definition of environmental initial conditions was initial conditions as a final power series both pm beta variables so this is a definition and after that the U.S. or from other sources you may my guess is that the coefficients of this function has some special some special meaning but for a moment I I will not discuss this meaning so what they I not the Germans behind
11:10
the numbers but in the German off of the specify our serious so what kind of happens this new Walter flower series but this is just vinyl was a possible my as 1 of the examples of what to do he faces up to the sea well on all those who will show increased energy use and the Sicilian for a fixed by our that he'll be an infinite homogeneous it until it can it's beautiful and midfielder homogeneous where can increase the geology so this is for this is the definition I don't the thing there's a lot on this following you see on the export of forests unexplainable exponent of
12:54
duty there the question responsive due the global please leave to the party won his his don't suffer for this
13:08
definition arms the operator the call socalled cover covered in Britain has a property that it deserves the poison would you bridging here when the the figures and it although the result of the centerpiece this differential equations that
13:31
selenium differential this course of coefficients as free simple refunded finer dimensional the question in each substrate itself but in the annals of his degree in these we have this final dimensional clear Corp so the 1st secondyear students smallholder folders aggression the result is just the exponent the exponent of this function is computed physics .period of this separated don't use your declined Tuesday to Jose initial conditions a you don't the afraid to take this exclusion because he the men is the 5th gives a degree of criminals of it is that it requires a homeward induced these until their final dimensional space just exist .period from automatics so heavily in Europe operated acting find dimensional space and you take it 6 1 saw halt all computers .period of very simple and few paramedics counterintuitive them to the point where the medics you should take the simplest way to 2 do Joseph basis hygiene basis so that it becomes diagonal but
15:05
some has a nice where is this item basis Simitis I so I did menaces is that these sure functions that the use of essential functions of that period the city of Semitic functions and that in the presentation of cement to group and these are so far could respondent doors use if you identify this think of political Millson is that Semitic functions be our power sums Newfoundland then these are yeah sure functions so the sure functions certain collection of Blumenthal's no these of the kind of person analysts franchise by petitions on the amount of the Berlin also the same as amount of money wheels so actually is a sure functional forms energy base in the space of Bill Nelson the emphasis is on hold and noted that will almost Aldermaston basis respectively on the basis of some sometimes convenient to expand but animals the basis of my model sometimes it's small but we need to the is official functions so for this political games they ideas eigenvalues of this event double a double euros partial functions of their solve are also known to this and is the same as W by so this is the way how good computer's function like computer experiments so transfer
17:07
functions also implemented here for a single petition consisted single parts of the use of socalled companies some of the functions of the is related to the PRI wearables this this identity and for multiple repetitions is called homeschooled "quotation mark beneath Georgia well there isn't determinant of presentations from the partitions so this is exactly pulled the definitions this is what where is she this sure functions for a small number of petitions from small number of flats sold for instance there the twodimensional space of political subdivision talked and there is there is a basis consistent from almost was clocked at but because use these the professionals Justin was amazing he had all of our some powerful from the ability Newfoundland "quotation mark but he's just of sufferers of these coefficients if you take sales he shares the coefficient which obtained from function the substitute while equaled 1 and other through 0 this is related to the dimension of the corresponding the irreducible representation under this as does the coefficient of them expansion of this function it is a popular 1 in
19:00
interim basis so it does a party 1 is adjusted consider just the function as a function need to the power 1 independent you want but considers this as a function of the whole collection of variables and expanded again in this short basis the linear combination of 4 functions there "quotation mark holiday expression looks like this year ideas that expenses it is that is a follower serious appliances the coefficients of Europe political expression beautiful but you should TUC's considerably the music functions and Peters also a formal power some of the individual "quotation mark Islamists what we we sinking of the coefficients this the whole that this is just my own way to keep to cause code that goes with the flow by computer who commented that the computer doctor degree here they probably it and here are some number of junction Thompson and here 1
20:34
of the couple agreed to the original copy of the Koran just check that's mine it's about the Reagan Edison High and functions of the Supreme Court going for the trust land is that got injured creation for this exponent of the "quotation mark function the 2nd line and I replaced the exponents of Zamora's function by the quarter's function itself Visagie is a lot going for it and then question after this substitution with want modified it albeit in the union jobs or instead of seconds partial derivative of their knowledge Europe John appears here as a region which is the product of host of a partial decorative To alterative praise president now the main subject of my give the hierarchy in the main statement is that the function that the stock
21:54
and then I will speak well the Skippy Hayashi this is introduced particle Hinton assistant partial differential equations among function what partial derivatives trades at his constant coefficients this year it's snowing Arizona at different jobs which makes it the moment there is a huge infinity such liquidations of such equations and there is some sophisticated way how to produce the equations the even the way to produce this aggression is not so simple so it looks unbelievable that this possible to produce at least 1 solution however they claimed that their quarters function does the Commission so the net new substitute and you see how it works it works like this but the that was the main goal of my talks today is Dole explained that this result is phone call from and the latter questions will this is the year of the footage for to people from 1 part of the favorite the development involvement in creation 1 1 Premiership and solutions right In explained you few minutes the center of KP Paris you it's a very nice here and there is a special community often terrible system people more consistent and now there when they speak stories they committed to 1 another's usually is a pretty well understand each other but it's almost a portable is from outside it's like the kind of things we must hero set the song and what I'm going to tell you something going to open some Sigit which is guarded very carefully by this community centers and will work with conspiring so I will explain what is hidden because these equations this is really the story is were simple wrote but never did the talk to these people and the fully explain my point of view of this said of course we know this is the way they're thinking about this but they never manifested reasons and their papers so sensitive what I don't think but what is going it was the father people also said to have some special assistant people so this is the 1st detention of approximately the legs is amended uh some on hold to break German and some chapters and this chapter starts his alliance definitions address mining a projected for writing yuan by the folic equations and then you have the list of broker creations eventually the definition derive all properties like the mentioned procession fearful of prostitution on so this is the way how this system how it works here in community of all this is the sort of consider the space 12 more serious
26:10
incurable on this for serious contends 1 Nice were infant dimensional variety which is so important and so crucial that it has no special name it is something like Central over the memo mentioned so this is a special solutions did the Commission's actually read the speech and that they consider the total told of new Member power functions so the dollar function 1 as an exponent of solutions so that this object of this Borat is modernize homogeneous Perez a huge signature role with the prisoners dramatic properties and its really Egypt or had points of these verses of this space and Mariel stroll hard work the and of course like any other not a break urges school but it could be by its ideal basic question but this is 1 as a way to win the question is just because funny equations but this is probably 1 of the lost weight due to describe his variety go the orginal thinkers not the spice of another questions about the place of solutions that last variety and there is a Romanian many different text explicit nice and convenient ways go the the
28:19
Commission's assemble themselves unlawful important for instance but you work through this money as you know how to all 3 of them to take plans call to more than call the have dramatically ahead all that you remember that there is there a satisfied suddenly questions the certain political questions even most mathematicians over over all the existence of expressions but I'm not sure that there are many people will present explicit form the polluter creations but however you know how to deal with Grossman and his even outlawing the political questions so the question themselves is something secondary so what's the original thinkers the space of solutions and the similarity of between but the real gross margin is not by chance actually for this he is aggressive took all Sotiris monument and this is the end of explosives curriculum Beijing concept of this infinite dimensional many of the fact that it is infinite dimensional doesn't make a lot of difficulties essentially this is just returning where embedding calls against menu so let me explains how identify solutions as a point of some of the fundamental Grossman the best way to understand these correspondents is through the socalled Muslims was entourage good respondents the buzz on Schumacher's bonuses urbanized this was not the therefore sprays this infinite dimensional vector spaces and I have chosen Florida the coordinates there is a basis consisting Co for functions so I needed to hear as there is an infinite dimensional that expresses embraces of its brands labeled by partitions and there is another basis another also riposte brands labeled was basic vectors are labeled by partitions rich broad suggestions another incarnation allows version of the same space and this coordinates good respondents is just there was on the road respondents of the both automatic at bonuses just coordinator I more fuel tall story of special schools Mexican leftists enabled by partitions so let me explain what these it an infinite dimensional this space so I
31:43
started stuff summer and that's supply mentally who
31:51
expresses very soft Paul Laurence series in the 1 of where Rosie the considering these just as an infinite dimensional labeled by individual schools victimize a grip of soluble by introduced for the good of our associate this and could be interpreted just of this sobering because it's um index it's not export the rosy because usually not use the multiplication of these sentiments chose just because you know this an infinite dimensional space whose basic reference labeled by into 1st and then from there little especially produces such incidents from elements of this vector I'll produce such the infinite this the products so the strongly in this script the product is In 1 of Milan's Armenia combination of animal models and the and I the Associated this is an example
33:17
this is the problem astronomers the Zito's of our mind's eye this is what is called the marketing director of the city's most of them to partition then In either by the basic character is obtained prop from this 1 by changing a little bit the exponents of this year wearable and invited a number of places so for instance the increased Shearer's exponent by 1 of people to use your this could dispose of this partition were there associates said they feared change jobs this exponent if you want to at them except without the ship for this petition 1 of me the 1 1 but on here is not couldn't so users Kusuma Medicare bill organizers explaining the new provision inside all as it is the same can use I could use some new combinations and the use Bloomberg coupappointed only sky onto symmetrical expend any any product of this kind of the basics in the earlier combination of basic directives blouses saying this coefficient minus b we 14 so you think about it .period inherited and Canada's consumer to be expanded for is the science good appearance alone we know that this information so that they can you session to its products and by definition yeah something have descended and edification securing
35:49
his face of infinite fish from explains at the bottom for muggers 1 this "quotation mark and by definition it said that the power series of peace in abuse of power if it's crisp on uh become possible this productivity tool a unique niche products for some elements the definition is a very simple saw Shearer rehab something complicated but if you look at the film at of the story so for
36:39
instance here I expect that this is a unique fish product the use and throw the Muslimcontrolled
36:50
respondents cited just 2 places by the corresponding sure function In the middle of day and nite the path to the export of the body parts thing example really simple solution could be so let me check 1st the substitute dysfunction to keep horror Of course and this is not so simple because the new definition glory from Europe than the rational function renewed finishes wintertime's becomes more and more complicated so the result is that we got a lesson it's not easy movies by hand but however if you put or common to the further the result of a very simple tho this is the definition is a very very very simple so the the the could respond become possible which products will the exciting so if you have such become possible this products than the leanest spent this very form a this Ospreys cover dimension which is somehow complemented the lips of space also at functions special warrants
38:41
serious and there is such status of their more serious abuse the point of view is somehow complementary and also says such spaces form because my aunt and actually is the whole point so as reverse monarchists warning tho both for reach the coefficient of the marktomarket upturn is there is a known here because it would like to take a lot reform and produce policies so this is a local cell holders Greece 3 of the point of the money but you but I'm excited to the space of power serious has become something real None jails and miracles where there is money in itself was nice objects that year and of course there a point of the this money in this there there this this this problem this is that the changes view possible the linear combination all of the the ice stopped on the scale of the characters in the film Dole functions so it's really depends on the on the the this shies from the point of interest money so let me make 1 some conclusions the frosting over on his that 0 the rest Munyaneza homogeneous space and plan called the Bulls form to other player whaling articles formation said produce is still a huge group of signatures on the space of solutions so carrying 1 solution what 1 called for the people imply a miracles formation and you get another solution but when it vessels of Munich some hinted at such products it's very easy to fall what venue falsely this good to was unexcited get something nontrivial but anyway you cut Kluge Group sorry June the and is it's also were GM in future I was reaches a group of friends that this would not proximity of the date keep horror so let me give you some examples so I will speak about it limits of the algebra I knew that if Europe lies in the elders really meant this product he replies in there its role you you have many many someone's you apply to the 1st element of the rich problems of the 2nd the this state Of these elements it is used the borrower it's the
42:06
but such a simple operators reusable computers its section of the Tuen Mun inside on the was an
42:16
example of other countries have something unexpected it's because possible multiplication by the horrible if it was for the difference is due to if it was negative now what's the bureau here is that this operator of course always computer commute the shift to shift oral those computer but hearing there hasn't been a mutation relations of the almost complete their commute up scale this is because of the infinite dimensional space that could produce this correspondence between action only home will displace space action home wasn't on the inferior to the space of whom the disappointing Fox stressed in space of
43:17
this that is a more there will be some scholars correction do appear so actually recover the action will not June futures but also the whole its onedimensional extension which usually doesn't play a lot about you should remember some In this case follows the coastal used a nice coral orders for instance of
43:52
the walls 1 you forever the more inward and definition of
43:57
this but on the American respondents so for instance that may explain how given this you can produce very their medical histories .period and those of its products space so you have put on all the wearables held all what this belongs to With this pretty on on the other side if we could put on it addition of these but they're identical the function 1 song that consent to take a walk on which corresponds to the function once and then the boring they didn't been to these but after the .period Oscars .period this operators so careful they want me to water and so on so bloc to this but in the mail blazes as could attributed and hope
45:26
certain combinations of how it works on the other hand the
45:37
there is a way endures direction and the dimensions of its products call to identifies a corresponding but in the mill here 1 of the ways of all this is to expand its and social awareness movement apply this great group good good at this point but there is a way to look this important ways for that too little USA the former New them this function expansion formula no power serious is identified by is near partial derivatives his origin using 0 Diller formal wants to but to have to deal with the exponent of some work if you flights to and from Cuba this is a tale of formal right the functions is the linear combination of Olympics next partial data notices origin this coefficients motherland's of variables this is a as there was developed for more so now he has never let me interrupt this formal and Johnson this it was right photo so if you want 1 of the few many sites what you have to know we these family of Fabregas you to the borrowed some of the PRI but could spawn summer claims by the way it worked fine so this is it when apply these because Europe d is a formal parameters and then characters the substitutions that me that is just a the free Jaume of this of dissolving power series that is taking the free during meals that the should direct their coefficients Of the markedly after socalled Larkham expectation but this is the procedure if it on its head and interviews we apply this family of liberated us and take the confusion of the victim this will be the result will be a function this Parliament of speed and this is exempt but silicon for this is the way this formal Europe this is a formal literal probably find the decibel the Bills search for wasn't among correspondents so just looking for this formal is not so easy the governors this formal highs just a very simple Gordon has a more fuel which insisted that the space the recorded correspondence case so this is 1 example
49:36
of operators and by the way what does it mean the action of the superior now this shows lands and bonanza Of the printer questions of the the KPA questions respectively separated is just that the Fed that In the did aggressions all partial do you would've thought off at school that means that if a chance at the evidence of the function of this doesn't change their creation so you can freely chosen your chances of in a change of scenery Johnson does not affect the solutions so in of this the Commission's that fighting continued Germans just responsibilities they have signatures 1 of the founders a equation also of regions are also have constant coefficients so the shift of wearables doesn't affect them as well so the shift of variables cosponsored to the partial for the synergy this to assume it is me was that we can change the terms was a function and also apply Herbert a shift of wearables the Commission this transformation found solutions to solutions when the the separated this for this the alternative you so you can take many years differential operator considerable whom contributed as the global the linear transformations of the space of foreign citizens and so for instance if you take the 1st order the presidency obtained supplement director image as I call and the hero this is how it works were there 1 was unexpected so this is I convinced Johnson which just on partial that it would just be a school officials the pinnacles of wearables them 2nd or that they were just so and sometimes just products sold to it's a sort of Soros although that I conveyed computer but you to this and there is the extension of the suspension along from routes the 1st 2 years of stunned of 1 or 2 12 1 you want to know so doubled up news of normal ordering so if you take combinations of multiplications by variables and the partial derivatives in such a way that all variables are put to the left and partial derivatives to the right this affair will lose case and we hope to 0 it also there is a similar formula for the apparatus of fourdoor and so on the bill partial differential what about abortion differential equations on the Monday said respond to this complicated but director was 1 of the richest some combination of partial video decimal take petition by rebels and in particular to the Tufton joint appearance that operated by chance if they want is for them to at was 1 of the so it doesn't matter what is very origin of these parental so that need just short you will hear it
53:53
just check White Plains
54:02
write to me I'm glad this
54:14
apparatus tool for
54:16
functions on was only confirmed were excited to computer This isn't my best my simplest way told about this coefficients of gossip during this confuses from combinatorics will follow series about what's safe if you just think experiments 40 so now for instance knowledge New year give me let me give old was approved the borders of chorus function dissatisfied the creation the question is 1 that is nothing to prove by definition course function With during here just compositional storms signatures Of those images of KPA creations stock is a function of 1 of which is that you Dole Fonseca's pointing to the remarkably then applied the export of a 1 1 the symmetries exporters is policy the signature of the KPA aggression against those forms for the usual solutions and this is another which this solutions to solutions certainly exponent of infinitesimal transformations suing of the whole just progression so the very formed implies the hazard solution because there is no W yes yes L change that Samper .period chins and patient and will be preserved for his music that has a history of the status of the religious wireless solutions function a solution on the status of national jeans and his men the Museum in the 1st year was among the issues with solution to what's usually a solution that is as far as its history of the universally known and not the functional Mom does the function the fund industry 1 is a solution
56:50
of the state here is that this is the 1st in the Will the whole system well you to this
56:58
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 minus2 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
1:01:52
to do things that result so list for this job initially again my definitions that does was aware computer tidal to discuss for a moment the physical the ground but just an explicit former just default by construction this is it the solution tho give me the questions right the defined for the continuation this products it's in the case .period playing isn't Brantley multiplication by without using minor stores reminds for diversity this response to the power of that he would have been even variable souls of result the function which does not depend on even variables so the independence and even variables is a prudent all the variants of the case .period substrates by respect multiplication by the minus 2 there corresponding system of equations is called the KGB hierarchy did obtained from from duty hierarchy bye 1 is all part of the respected even variables is able to this year so the functional building function in what variables the homes of hand is there it was a substrate Susan Graham is a specter of director agent also loses we can consider you you can use it to some our multiplied by zeal to negative you power this producers also some relational the film site all the buzz on excited so this is what the response to this separator 1 was I'm excited under the socalled streams for this partial derivatives respective wearable number the delay and glossies this is the linear combination of partial derivatives is linearly dependent coefficients and also that this is malls 2nd order partial differential equations then that there are 7 correction terms the constant correction terms speech which are due to ruler the central extension which should be taken into account for this just check that this is successful the famous those considered conjecture says that of these functions present interpretation in terms of intersection number some models president that belong coefficients of these functions are in the Social index indices of psychosis on the models please 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 it's it's completed also hold break German that shouldn't be too white the Senate is the U.S. you formalizes this is probably is the most direct way late interception serial model espresso still something which could become pure more explicitly so the time but some consideration brought this conjecture the UC formerly was not known it appears to be a little bit later so now begin on the security checks there there is a conjecture I have some implementation of intersection number for this do loads the divergence shares which is their partition function of the of the responding to all the Vermont presentation let me compare on result is different the 1 OK to quantity what works actually I'm cheating she arrived and cheating little good because Shearer when computer without implemented implemented intersection on loss of course I used to think that the state delivered to conjecture had sought constraints that is mostly to demonstrate by the way this year the call KGB however books like you the wearable it's not the whole now these aggressions socalled by the middle informal of Kittery hierarchy and it produces only total stopped and the creation of the hierarchy which to integrate these respect is due when you went times the create a questions become more complicated but here it's very common for now let me pass for U.S. a the U.S. former is a formal image relates chorus numbers of issues which are coefficients of the function that they consider the 1st part of my talk for another order which socalled fortunes roles for such opened the session on some of the models face of cures the game I insist that I am not going to ideal not consider overlooked considering the German other considerations space on particular defined in detail the awarded by the repeated the definitions of flounder from the so it this is in any way this is startling collection of all I collateral numbers image by partitions London a certain class volcanologist based on all the order for the 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
1:11:47
variables labeled by user crisp on an index will not among the power variable anyway looked at on 1 cents some future
1:12:02
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 lefthand 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
1:15:42
notification .period the forks
1:15:56
so all broth is a real danger chair using the Ulysses get near that must look at once again to this election on the lefthand side for or if fewer all these this small the modification lefthand 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
1:19:48
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
1:21:08
conclusions so what I
1:21:19
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
1:23:23
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 coproduction 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 doublefault 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
1:29:52
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 strongwilled 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
1:34:50
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 wellinformed 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 socalled 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
1:39:51
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