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Introduction to wall-crossing, in physics and differential equations

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I'm trying to think and got so it so it's not great pleasure to have the opportunity to be here to talk about the subject think tank thing for much the organizers but OK so so my coalition said reductions so Wall crossing in arms in physics and some application to 2 differential equation but so so in the last and last year's winner last pioneers so I this bidding resurgence of interest in this problem of war crossed but how you would use in just a moment but did but the thing is when you 1st hear about it on his wall crossing selling something that you would really rather not think about it's kind of it's kind of annoying details the messing up whatever you were trying to do you knew you had some quantity which you were saying were trying to count something you were trying to count volume minimizing cycles in Colombia 3 folds in physics were trying to count some supersymmetric particles and you thought I was going to be invariant you thought it was an independent parameters at all and then when you look closely when you try to get all the details right you discover that there's a little problem but the thing is not quite in variant on when you buried the parameters in your system under the thing is kind of piecewise constant but there's some critical moments where it might just so so rather than anything this is globally invariant you 1 up with the thing that has this kind of piecewise constant dependence on parameters I'm mentally sound like something would rather ignoring you hope that someone else will come up with a better injury the doesn't have this problem but but but it seems that that had of the wrong .period you so so sold this this phenomenon this phenomenon of the jumping of invariants on it we show that a lot of context it turned out to have a kind of rich a here structure to it but which turned out to be connected to a lot of subjects so it so it should have been quantum field theory but on the other hand it's it's somehow should have also in symmetry and hybrid killer geometry will systems cluster algebras on an important 1st today the theory of the theory of differential equations the it doesn't very obvious endowment is also this the general phenomenon of war frosting on the here and there about a particular kind of war crossing so here like about this particular bps crossing which is somehow not the same as the war crossing doubts in theory at least but I haven't heard any connection between and between the 2 so that's why I put that here and and so-so so my main goal today is to talk about something relatively simple examples of the war crossing phenomenon and the ones that are somehow connected to the agencies and I'm in to explain how the related in particular to and to differential equation and so this is part of unity said that I say anything original in this talk is part of a long-running joint work with Davydenko and and agree so that we ultimately a pure mathematics story but when you start by describing on what the problem of war crossing is actually from the point of this so far as I have a son physical system and something which if you look at it from a long way away what looked like just a particle so many it's like an atomic nucleus but when you look a long way away you don't see the structure you just see a particle and so what the basic questions you when asked about the thing is always it's stable or unstable I will live forever will eventually decay into some more basic message well I had the answer might depend on this the parameters of the system so that for example and you imagine taking this particle and putting in a box full of particles and then you would change the density and of other particles in the box and forge a more ambitious city you know if you if you're high-energy theories you might imagine on the European Union vary the kind of fundamental constants of the universe and change a court masses of the coupling of all the interactions and maybe the question of whether some particular atomic was a stable will depend on the answer to that question so here so you know so as to vary the parameters there might be some smokers in the parameter space where the particle changes changes from being Seybold unstable and the other way around and so those was signed you know there evidently there real coordination 1 in the whole premise business only called on the walls and so on 1 side of the wall things stable I among In 1 of the something's there there's some things that are really absolutely stable but inevitably get the latest the latest particle that's carrying even charge and that particle really cannot advocate even in incident and yeah for electric that is a good example so it really tries that the electron is is stable but if we would change the parameters so that some other particle carrying the at the scene charges the electron became wider than the electron then it would become possible freely from the decay into that other particles that going good example of this army it's an arm and will be studying summary special field theories for which there really is a notion absolutely stable particles I'm so a narrow we might be interested in the
history of this view and was his goal of that and with the costs and the yeah I was really proud of the way he did and I will go daily about you knew led by him in the gaps where he is not those of the 1 in the world this year :colon I'm going to lose it all In instead the people and the inclusion of this year's morning GE In the meetings this will cost he was the only 1 of the most of it the location of the elimination intermittently from the cold snap over the course of all of the as for the good the British interchanges on this is the reason why all remember where we were filed by the overseas but yeah maybe may be the sort of mathematical insufficient institution of the silicon but a but so so so narrow and we may have gone missing physical system we might have many different species of particle and would like to say all of them simultaneously as a correspondingly there might be many different walls In our parameter space across with something happens so here 1 particle decays another particle decays and so on and so do the kind of question you might understand is 1st of all we're in your premises are the walls and 2nd known and I look at the center of altered absolutely stable particles and UBS how said changes when you move from 1 side of the wall to the others the country we imagine you knew all the people particles on 1 side of a wall and finally turning wearing the all evil particles on the other side and but a question like that now probably too hard to solve exactly the use of this is what you call a man with a view of the law "quotation mark you hear people talking with original lush would love to see a long history many of the city's physical examination the losers on the day of the season by Pentagon number momentarily on a making number momentarily then some of the numbers satisfactory said so and so so the problem that as a formulated so far is actually is too hard to solve exactly for most of the physically realistic and on the other hand there is some kind of way examples where it is impossible to do to do much better in those systems where the underlying the major reason however especially that the so-called supersymmetry on and these these supersymmetric models out superficially forcing more complicated you take a look at kid the particles in a at roughly speaking when you particle for real particle the service superficially more complicated but actually the tragedy much much more nicely behave this sort of random and arbitrary system he should think of the differences somehow like the difference of the real analysis in complex analysis on complex analysis you throw in a new double everything you make all your numbers complex that actually makes us very much wasn't and in his had systems there's a there's a kind of privileged class absolutely stable particles so-called BPS particles and by restricting our attention to those with a problem that we have a chance of actually sold so so no more precisely known as well I have any quantum systems say a quantum field theory that describes the physics in our universe for some other fictional universe so then on this class system should have a Hilbert space in particular we could focus on the whole the space of just the sensors inside that Hilbert space consisting of just the 1 particle states I'm mentally drained by some lattice on the lattice of global charges in the it's even think of for example electromagnetism in this manner will be literally out of a rank to let us and just providing electric and magnetic charge meaning in part so we have a Hilbert space is divided up into the are into these spaces on and so for a 1st arraignment physical system in each of these things would be representation of the quandary algebra just be as cemeteries of space-time underneath were saturated the representation of something a little bigger will be called super pondering algebra I'm and then what 1 defines 1 defines integer number so for everyone in the covered spaces 1 defines the integer number which you would think of roughly as being just the I'm of version of the of the dimension and a more precisely that exactly the dimensions but rather you have this disease famine you decompose under the supervisory algebra and you keep track of the number of times that some special small representations roughly the dimensions that precisely which something fit I'm so that we can formulate questions so now the question becomes so all the particles which cities small representations they have in particular the property be absolutely stable I'm and so now you can ask I suppose now that I vary the parameters of mine supersymmetric system how does this collection integers and how this collection interviews change OK I so that's the kind of that's the kind of problem that are that arises in physics who like to be able to solve they said that the
loans allegedly came to the farm yeah I think maybe he won't be satisfied until I translate to the intellectually the actual mathematical example so so come on I uh so his problems the 1st cases where this problem was solved were II system isn't just 2 spacetime dimensions the same with the early 1990's by a Chekhovian but in that case the gives her the most perfect possible solution of the problem they had a formula which tells you if you know all the particles on 1 side and what will be the particles on the other side but I inform mentions but switches somehow the more physically interesting case but that the problem also use is much much harder but so was solved In a few isolated examples in particular in 1995 salary would solve this problem in the case of 1 very specific needs and the so-called purity was to most and what they found was in this series so this theory of a primary speech which is just a one-dimensional and 1 hand which dimensional space and there was 1 wall inside of the wall they were exactly to move this thing stable particles outside of the water was invented to invent family In and so every that no further progress was made in distance instance the examples are but to solve the problem incomplete generality but I've seen just stood at just hopelessly out reach yeah but did does so fortunately the the sort of inspiration from all the new progress in the subject is that mathematicians have came to the rescue so they formulated a theory of the theory of generalized oftentimes invariance they're finding them in a totally geometric way and in many particular physical systems and you can formulate value formerly the generally thousand Harmison area they they seem to be exactly the is only a game so exactly the 1951 which studying physics and on the same qualities were studied by the time positions particularly in the work of considers Simon enjoys song and they found it totally general solution to this problem of of war crimes but OK so now I so now I wanted and switch essentially from physics to mathematics so I'm I want show you a few actual concrete examples of this war crossing phenomenon and an end of the war pricing formula that consumers will so so and so let's fix some of the algebra of the type I'm mistake contact agreement service their armed with much more points and so for every such choice for every choice of the AT the algebra and her with more points and there is actually a corresponding four-dimensional quantum
field theory on how many in that particular client field during the questions which so far described to you in a kind of eerie physical way on the questions particularly questions about these Omega gamma become now completely concrete and geometric question so it's OK and so on will specialize a little further and when he specialized in a case where the wheels Aridjis it is a wonderful words and and so we're gonna do construction and the construction of I was going to be isn't involved the more quadratic differentials on the so was a murmur of aquatic French alleges that English so thing vitamin which locally in terms of some local Gaudenzi on curve is represented just as ever these eastwards a summer market function tenants so if I fixed the If I think the singularities so there there were more than Cingular is exactly the point z that I want on Minifie thinks that if I fix the problems the water of the singularity at each other's work .period Anderson might dimensional space of these neuromorphic quadratic differentials and that only my promises I'm so we do with the credit different well sold so doing these criteria differentials and I also a phase and there's an induced the affiliation of 2 ways and pollution so I think it to describe it is to describe what looks like a local court and so we have a thing which was Evans easy squared up locally at least away from the zeros of this quadratic different weights away from the zeros in the polls are magnified a local coordinate W where just absorbed a 70 in the definition of doubly so that he was like being in the role of the swearing armed so final coordinator which recorded differential just looks like the view squared up and that local coordinator of immediacy with despoliation as you guessed take the pollution abide by straight lines well with straight lines there's a choice of what angle this reminds that this this primary believes elitist rewinds and whose inclination is stated in the report Over the Hills legal this is the 4th time in the office supplies what really want relief aid from so so Talbot so it's Indian army defoliation and yeah I know I yes I'm I guess that the way it every here I we have to do every here is slightly ambiguous because I could always hit W W W 2 might still be without changing this so yeah we write a little more precisely to dealing with enemy duty but Alameda to army I'm so what is the lesion look like I'm but well is it away from the zeros in the polls just a regular affiliation and at the zeros in the polls some of the squad ready differential and we have the foolishness of singularities I don't describe example what happens at whenever the double pole which is somehow the most basic but added double pole that you just In work at the local leader of the trajectories around that pole on what you find is that the victory standard spiral and ending the exact way they spiraling depends on the year and the relationship between this program which is the residue on them and the year the premier fatal for the fine defoliation arm but anyway except as an exceptional case and the territories are always the spiraling into these people on and it for Testaverde a technical convenience from now on I was going to assume that I'm that the year that microarray differentials have at least 1 poll which is double or maybe fireworks on entire order it still attracts the trajectories just innocently differently so it would use all of that is that it can control the global behavior of this despoliation so you think the way define despoliation in general you might think that the trajectories even every crazy way that they are on the move like a got a clear round the surface and in the case where we have these polls that will happen on as long as premier is generic on what will happen is that every trajectories of art has always 1 in the polls was eventually attracted roughly when it comes close enough to where these polls will be attracted involved I'm into 4 that we use some for showing struggle studied in great detail the trajectories of differentials on from his results yet because the suspect but so here so that's what happens so that what happens the time at the polls on monitoring and you also have singularities at 0 than the quadratic differential and the simple zeros and but if you work at the local behavior around a single 0 you find a singularity with just a three-pronged singular so symbols era as always with this cross and women assume finale in fact actually from the start and when he assumed that this year are only simple 0 so for the generic differently only have simple 0 and so then we have reversible error we these 3 distinguished trajectories I'm going to go right into the 0 I'm and so on the focus in the construction of a tell you we actually essentially don't use the the generic trajectories at all we only use the directories that are coming out of the zeros on and for every 0 in these 3 things coming out with the union of all those directories arm all call that the the the spectral network In this case said it also has another name it's called the critical wrath of the of the the French arms already W of fighting I'm and so generic value of data on what happened so that we have the zeros but many zeroes emitting 3 trajectories on occasion trajectories travels around the services uncomplicated way eventually ends up on 1 of the polls and so as liturgy something like this on the surface has divided up into the into the cell so this this is what happens for a generic for generic values in the face I'm so now we will reward understand is how is this thing theory I'm as we bury the premier's is very acquiring fragile and very this face pressure for :colon I so well from the white conservatives that it's it's reasonably clear that for a small variation of vitamin this network is just by a by an isotope and but as Meghalaya variation on are we may reach some critical plays with apology of the network actually suddenly jumps With the picture
that so this is headed the simplest example of such a jump so here taking the current BCP 1 and I'm taking my climatic differential to hear the quarter differential which a singularity infinity but nowhere else I mean it has to zeros when it was when minus 1 and what you see the time we see that as area and as you very this parameter theater announced showing you holding fight to fix it is bearing the premier theater and as that parameter on as you
worry that parameter of the thing of this picture this critical reference spectrum our just inches varies by it'll be accepted it was a critical moment so here this 1 critical moment where right there that apology suddenly changes so we had to victories they were going out here they suddenly jumped and then we had to victories here it's an army so HUD and Wydad if you look at the moment where the jump happens on if you look at
the moment where the job happens I have and what you see is that the don't haven't exactly the moment when something on happens with limited entry which comes out of 1 0 and goes just straight into the other's 0 on so that it have special trajectory which occurs just at this critical value in the face so obvious aware the special
directories so I told you that I bridging value the phase of an average victory has no at least 1 and on a pole newer travels it might start to 0 but it travels around the service initially in the portable on but I but that only the Jack statements over some special value of Faiz Ahmad might happen that it never trajectory which has both ends on and some calling them a special directory and those things come in 2 flavors you have either we call a cell connection which is what we saw in the last picture is 1 which travels from 1 begins 1 0 in India's on another 1 I'm or even closed trajectory so that's a victory which the news it is 0 travel around comes back to the the same of because so so that the moment when this man is a critical Graf suddenly jumps ,comma exactly the moment when the special trajectories of appears OK so now we informally questions so so consider the following questions on suppose I now fix my quite differential and then I allow this these parameter to vary and then advanced well sort course question UPS here's how many special directories will we see and so very theater just from from you and from 0 . 4 0 pie and how many times was picture John Pemberton Valencia special trickery and OK well OK so what's startling about this question so 1st on the grid preliminary comment about it is that the special directories they can occur from too much they only occur at most accountable many phase phases will why is that I'm it's because as explained in a 2nd I'm the only have counseling any possible topologies arm and the end of the perimeter of the area where the thing occur actually determined by its apology I was in the following sets and others say with apologies but but but yeah roughly means exactly the position on the surface but but measure that a little more precisely so on and so on so we have this quite ready differential and didn't want any differential and particularly canonical double cover of the current attached mainly to take a 2 square roots of the quadratic differential so as you move around on on her fight you look at the square root of vitamin C and E because the thing was it back with a climatic differential on I it's where at and actually died in the content and bundled and so this spectral cover is on his living inside the quotation bundle of her I mean so this is the obvious to 1 reductions is free of the square root on an ad covering is a branch of the zeros of fighting those the places where the 2 square roots come together so case we have this rash covering well really over the complement of the of the polls and never mind the polls for a 2nd important thing is that this thing is Branch Branch of 0 I so now but let's define what is Indiana and to be the 1st homology of some of of the spectral answer before Indiana abstract physics part of the Tucker find some lettuce which is indeed a lattice of charges the thing which kept track of the charges all the particles in the theory in this particular theory that taking 2nd the charges adjust the personality of this spectrum and Anambra dealer each 1 of these special directories canonically can be lifted to a 1 cycle on the spectral cover so here's the picture presenter connections so if Ibisevic engineered something which exactly run between 2 0 0 I wanted to branch points so voters think the union of its 2 lifts I'm about to recover from that becomes actually a close on the cover and is a wave even determining the orientation of our so we get a corresponding homology class so that homology classes 1 this is measuring the topology of mine and my the special stricter OK so then now and so case at every 1 of these things and associated homology Class-A World Color Chart and now OK so I now I come back to this question of some and what will the thing up there and what once we have 1 of these and whatever quality class on the cover we can define the most obvious thing to consider recover which is sitting inside and the content and level the because bottle has this tautological illegal 1 form on anybody's Aquila that 1 form around rounded around around 1 cycle of well what is this 1 :colon can really would you evaluate on the spectacle that would form the really is the square root of the cleric differential so bad for 4 panels which lie on this sector cover you think of the Santa Rosa in you entering the square root of the quadratic differential on Brazilian way you're integrating living at the variation of the scoring what I called that W on and on particular if this game is not random hardly class but it comes as the Charter when the special directories Maryland that's targeted directory up and the central when that is guaranteed to be it's not random number rather is valued in its status as the army OK so so did the periods for special directory to occur in a particular phase Ahmad period as they have exactly that the announcing only counseling many homology classes in this equation can be somewhat for Karoly many of the and and also you know if you know what charges looking for and then the corresponding faces determined soul and windows and where to look for the 4 special victory with any particular departure OK so OK so so now I can formulate finally what exactly are the sort of dolls Harmison variants of the DPS counting and variance in this particular problem and mainly written the following things so we fix a particular quite differential 5 to on and we fix a homology class in the spectral cover calling him I and we asked what we reformer would would presumably had a funny combination so we think the number of cell connections which appear with that particular homology class and and we subtract 2 times the number of closed loops from which it had several close pairs here that's because an editor that this close-ups research the these close ones when they occur they generally don't occur in isolation surrounded by
of 1 close your foliations and that is where the neighboring means of of defoliation like but it turned out that there they also have been close loans and so on until you reach the 2nd of a 2nd break point the picture like this it was basic budget analysts say particularly at this to close with on the boundary analysts and calling that a close comparison this this whole structure when it occurs and will count that with display number will count that as minus 2 and so were attending the celebrations include 200 were counting them with this very particular
and in this very particular way many coefficient really crucial so for the whole wall crossing story that the 2 of them itself and that it essentially you have the things in exactly this way OK on the use of members of the number set up collection Sosa literally you know I'm at that right at the rate at this critical phase so we've got something that is damning here as it would take the liberty the argument of the gamma and you just look at the full initially exactly that moment by end of so yeah you would swear would usually expect is that you get the 0 or 1 at any rate stable at our age we looked at you always get here at 0 or 1 of the principal I the quadratic and differential people whether it's possible to have more than 1 year a complicated enough to a complicated of surface and then they told me anything examples and so were you have 2 things
but 2 things that occur simultaneously in virtually the same logic and so in that case you if they're really talk about that occur simultaneously that I would have too Aberdeen generally held in every cable show you the number all Adidas 0 or 1 seat you think 0 if there's not anyone if there is
OK other great so and so that's so now nearing variants but I
selected examples so that they can get a occurred to me to see the 1 in our study inquiry differential with a slightly worse singularity infinity before it eastward was 1 of the things each year when is the city's 3 heroes here OK here's 1 they won't sell connection but which connecting the leftmost too zeros and and then in a moment we're gonna see a 2nd set of mentioned here that is the 2nd
and so on the
I so in this case and that slavery was very data from Europe I on weekends just exactly to settle connections so they got a lift in this cycle gamble 1 and lifted his other cycle sitting the 2 loops on the on the spectral cover on variants are various only Gamal 1 equals 1 only a game 2 equals 1 invective Ivory data from Friday to apply the same picture again
repeated but then attributed the orientations carefully and get my scanner one-liners at so according to the definition is areas will only give customized ever 1 was 1 only a personalized came 2 equals 1 and that all the other insurance adjusted just program and that's simple enough all this now I now I've
tried this is a very similar picture where I've changed my parameters just a little bit but the beforehand the Q minus as the nowadays aquatic differential to be just Xingqiu you minus 1 and so we moved a little bit in the printer's space aquatic differentials and now we see but is that to the sea received 3 of these celebrations
and so and you know it by
the corresponding families federal currency of very adequate differential correspondingly we buried at the special curve moves the family arm and if you compare the analogies so now we have so this cycle here Is the cycle that I used to call him at 2 AM this 1 is that no 1 would deny her that was heard him 1 +plus came into is finally finally there is this 1 which has charged in 1 I'm glad to
refuse a little bit now we have 3 of the center connections that have to be on missile 1 of these
variances jumped on injury which has jumped is this 31 only Indiana 1 was which used to be 0 now it's 1 and so that's it that's OK me and so that's that's the picture so what we say so as we know
as uniform aquatic differential from this way this 1 and this is it said the invariant and change from 0 1 and so this is really the most amazing example of this army of this while crossing phenomenon parameter space was the the spaces specimen were acquitted differentials some fixing your behavior but on 1 side we have this summer collection on this and we don't have look at for the year slightly
fancier example so so here again I'm thinking the curb BCP 1 on but now where the study heart rate differentials which have singularities both 0 ended Infinity on a single area worth 3 in both places guides are represented as a cylinder so zeroed in the new player on they have symmetrical roles In so so here in this picture of
within the endure to celebrations as they did varies from 0 to and Fortuyn occur at the very beginning in Brandon Miss Indonesia so here the 2 branch points and here there's always the chance victories coming together that right there the 2nd will be the end To this represented way around the cylinder in both directions 1 it was around this way and 1 that was around the other on the correspondingly here
we have just 2 variants to Nigerian thinkable won all the rhetoric Eliza bananas was
a format 1 so now I'm I'm again on the same Grandma's visitors deformed this "quotation mark differential despite any a constant but I after making that information well sites of Soviet arms the picture has become much much more complicated so now so now either read the Phase I and II Hannah actually infinitely many saddle connections in filling a set of pages which are wanting more more times and around this summer and so soon again
on so so here is the first one the 2nd 1 the 3rd 1 and so on this day of a visit in tower of them winding more more times I'm in right at the Marine right at the very little of that picture is a critical moment where you actually have a close trajectories where is a church religious begins at at say the 0 went around the cylinder and come back to the same 0 and so down here I so so right at
that moment I rated that
moment when disclosure directories and a correspondingly here and the invariants have become similarly much much more complicated so here we have this in a collection of non-zero invariants to rainy no European this is equal to 1 refiner cell connections and then we have also this 1 critical moment where we had the close factories and so there's 1 Omega which actually people's minds 2 according to this definition there at that time any and all the others equally your arms so took OK so-so
and so here we adjust to
Mandarin invariants here we had
this in collection of 0 army so now the sort of amazing thing for us it was amusing ways we 1st learned about it is that these warm rising phenomena that that we just observed are completely determined by the single universal formula which was written down in the form that I'm going to describe it was written down by consumers oilman and my thinking analogous thing was written down possible address and so on and so the claim is that just knowing the spectrum of trickery on 1 side in other words just knowing about these 2 trajectories here using this well rising formula we can actually
determine what would be this complicated in that sector mature trees on the side so when I tell you have this formula works I'm 2 finally this formula union ingredient which at 1st looks strange social model technical grade differentials in our injuries something which at 1st seemed to have nothing to do with the quota for each on consider it I was considered as a was just the start of the end but already is the harm from now on the letters gamma in indices so this is it this is a Torres who's sort of the woods yet day I was as loud as it was the 1st apology this federal on so this is sort of way of thinking this is its representations of part of what can do with them as if there was the 1st margins I'm so this Torres it's the start of the end of the where the way we presented here it's sort of canonical cornered functions are labeled by traces of Damaso for every game albeit for a motor won the support seas are a function of the step by step by evaluation OK so it wasn't a thing about the stories and and now consumers and Solomon invite you to consider a certain automorphism this story and the unity morbid ideas that they tell you what happens to these weapons in accordance of tourists so Bregier mobile devices or more physical Acadiana arm and the winners on Mormonism acts ,comma takes accorded X will go into the enterprise inadequate axiom trying to get him a prime multiplied by some factor the signal Alabama is traffic he signed enters into the story so it would be better if we ignore it for now just imagine that this is the user fees are not always using federal letter this on with your work and so it is often with force is considered as a complex of you know this is is the year at the Baltimore Maryland United are more of a group of armed of irrational map of the tossed aside Justin Justin by this formula so here the Sigma Gamma SimCity signed on and this appearing here is just the intersection carrying on homology OK so so Oman say consider their amorphous took and now what we do OK so soon so now know where drug picture so each each curve on this picture corresponds to 1 of the saddle connections are close trajectories that did too and that I should you in the pictures before and so can't really so here is the axis were were bearing the quadrant differentials use beef the space of these Quebec differentials premises and things will fade as if everything that we had before and then you know for some quadratic differential with celebration that appears at some particular thing that would die out here so that's what this picture it's all fight and phases in which the the armed this adaptation disclosure trees occur so I just like the ones for which only is not equal to 0 so just the ones which you actually have 1 of these things but and now the fact that these lines across the branching like this reflects exactly this phenomenon of Wall crossing so on the moment when you the moment when you reach the wall I mean actually lot it's a consequence of this formula also you directly on the places we reach the walls are exactly the moment when the phases of 2 objects from the color line so here we have 2 objects cities to be the guys with charge Gamal 1 Indiana to and on the other side of the wall and the other 1 the to also a Newmont has been morning was charged again 1 of the dispatcher look something like this I'm OK so In the mission settlement hosted draw this picture another 1 crossing Miller says now to do the following things so you look at this look at this picture and considered as any close path on the picture Dominique intractable closed that I'm and then as you go around the and every time you cross 1 of these 1 of these walls there with every time you everything you need 1 of the South actions were closed factories but using a factor but many of these automorphism Acadiana and raised to the power only to him were exactly possibly again after going up a miserly images going down to resemble understaffed here I would have Acadian so can and I was the automorphism that we considered just a moment ago decided by rational transformation toward building up there or building up summary complicated looking more amorphous but some were complicated looking transformation of the Tories into itself these are not commuting and so we have to take part in a particular order in the mentally of order we take it as they were going around the peso's start somewhere around that have carried on more period transformation to understand the worries of world but that right well OK for me just mentioned that the Pentagon somewhere in the middle of well it'll turn out that doesn't depend on Greece will this is a of I think you always what to get months creation right so it's so oversold so familiar it will just suppose that begins somewhere in somewhere but end so we go around this past summer while the body of abuse these transformations of the Taurus and now with the war present formula says is that whatever you do this for any deductible have whatsoever when you come around we don't want that to be back to the site and that this is this is down there is no upstairs here so this is definitely of the standard like intractable doubt that this is not a pat on the curse is not the only current this is path in the parameter space of quadratic differentials and cross was her it's the point where I'd love to go right to this .period
well I mean In a sense the formula will say that you are allowed to do that favor a prayer I would not allow that on because exactly at that moment the Omega vehemently deny quite well before over let's avoid that that's in your situation and I will restrict through arms so that the war Miller is that whenever you do this the army multiplied together these all these translations of the Taurus and what you get when you come around a little bit is usually a betrayal of the history of the world these things will depend on the arm by years then there is the reason for the use of the yeah well formulated for now only only 4 people I'm sorry OK so long as they the local utility with ongoing review so I'm so OK so so here is at least a formula is not clear why this is a war crossing from 11 years 8 divisions but I think it is I told him this that's cares about here a former army which indeed follows from the previous divisions but hearing obvious way I so obvious Why is this is actually what crossing from arms so it won't receive because support that might have PDB actually had a rectangular pan like this term focusing on 1 particular of war crossing where some some another address on 1 side and some others during raids on the other side just wanna go around this path but I don't trust anything here here I cross some Honduran variants at quarter differential fight now I go around this and here I crossed but some lines responding to the None 0 invariance at quarter differential by 2 prime different from fighting and so that won't analysis in this case is the transformation that I need a composite transformation that I need by crossing these guys show me is equal to the composite transformation that by crossing these guys because a around this happened translation here times that affect the orientation of the Congress of the transformation that I got here so another word about a final product available define the products and just going from the 2 1 defeat at 2 take care of fate in their prime and I think the product of the transformation that here it should be equal to the product of a transformation that on the side of the wall rising from notices that these 2 illegal arms and now the year on the miracle about this is that this model of this condition is actually strong enough to determine all the only news on the side but in terms of in terms of the only as on on so so in that sense this thing is war crossing formula but you might ask why does it not simply determined to be equal to what is and what isn't this formula just say that the only the salary equal the signed on and the reason is exactly that I'm the product I think on this side is occurring at different water from a part of it was so here for example of this is never won this is Unit to hear everything yeah 1 before in its here I'm taking to 1st Indiana 1 and to compensate the fact that they don't have to assert some new things in the middle and to make those 2 products equals on some sometimes of the is that trade is a kind of 0 curvature condition for some for some connection that the miniature I'm OK so so so now they say so on these 2 1 crossing phenomena that we've that we already observed are really just the governor by instances of exactly this formula so so 1st of all I have to come to charges whose intersection product is just 1 that was what happened in the 1st example of the way to do this the question is what the conviction that used to be notice that this is not a connection all over the curtain exactly this is so this is a connection which looms over the parameter space of arms over the premise basic already differentials in on so now it's something a little bit to the literature so the following his connection is said to be the 1 who read the letter and the hallways that's right so there is such a connection the very but for years yeah I'm OK so so bent on means a much simpler so so so suppose for example that I have come just 2 charges who simplistic views on the intersection product is 1 and so then if I look at that these 2 or more and the composition just taking a 1 game 2 and so that's what I have I that will have on 1 side of a wall and then I look at a wall where the that this intifada Ziemann ones become the line was he given to on the other side of the wall on the piercing the opposite order only taking in the 2 1st pitiable 1 last time and then by Tamil Hindi don't but In words to the effect that documentation you just have to insert 1 more the transformation of Acadiana one-plus at I'm so this is identity once you know it you easily triggered by hand just with the force with the definitions of these transformation and so what it says is the on what it says is exactly but if I have to settle collection on 1 side of a wall who's enterprise is 1 of them and then on the other side of the wall at exactly 3 settle connections to resolve 1 and 1 more in at exactly what we saw in this example of the nations of the world on is whole because it is only bilinear maps so you know it is absurd and so you know with the watering is controlled by the phases of this as Zenia Mucha arms sale linear map the
from to see Parliament on this side the ordering that comes from the fact that it is won estate here and Indiana Tuesday here then because one-year 1 was never to and will be sitting and say here and so on the order is controlled by the face of the of the faces of the city faces come in this particular watered down 1 facility would comes between them 1 Indian so that's all that's that's all going on here and that would control the ordering of all these products on the freeze to the greatest who leads the league the world and always will be trying to hear what I wear this perhaps you that after break the symmetry of the house told me at a New
Year's looking a little arable we're taking this contractible pats on that little there's a definite ordering and ordering on that he said if you do want to have to find these lists this year you don't know what the the right and then use the right about that .period thing is that yes you can deduce yeah I'm so it is uniquely determined by the time that this form of any use more complicated example of the same thing I suppose I have again to invariance would report 1 so here in this case onto the power 1 caregiver to our 1 and only the changes that the army the intersection between the the 2 charges should be 2 instead of 1 arm and in that case you find the same thing you take this part of the Federated is a product of things In the opposite order you find again can be done but but the formality of the right is somewhat more complicated enough as infinitely many of these transformations in the Tower of on the by hand another in tower and multiplied in the opposite order and then in the middle of you have to put 1 so all these occurred on Dec . 1 in a million to put 1 special thing which occurs with the export minus 2 on so this is something this the formula is exactly governing what happened in the ah 2nd example where we had on 1 side just to settle celebrations on the side which in the tower at several connections on plus 1 the annual close trajectories which we recounted with the specter of minus to it's OK I'm and the claim in fact this is this is completely general doesn't happen in these 2 examples but it happens every example so everything was really formal policies can rest rather than in the Indonesian case some of the results that we can understand say that the rest of the nation is the the result of the condition of the economy of life outside I have no way of these programs and the formation of the unity of the people on union had we had we know to make this particular definition of the question before the court of the government will go a long way to go in for of writing letters with all of you already know that some of the people of world of what you did on the 2nd day of the trial of people who regularly hold on well they also have had is 1 no the arrived this formula I'm not just as a purely algebraic question have you drive this form of New year and the year fury I don't know that I have the bomb so these formulas and made it into a you can you can add on checklist formulas on a computer so 81 want turn this formula at any finite water just truly out a request to have finding on right side living on the west side of the standard of the affected people usually only the legislature the level of the I'm going to have to remind of of the I would like losers would know it's not the part it is not the positive and negative territory they're all positive they're all positive and that the it's more like the following this is the 1st of all you don't have not lived there all upper triangular that's exactly what arms and so on and that this sort of finite dimensional and a lot of this so this is taking place in debate over which is a group of somewhat
amorphous but you want the kind of client analog when it is fully in the development of new adding years here's the hears that sort of thing isn't what it used to that changes the global and there are more than that the Taurus but actually they they also preserve some syntactic structure which I haven't mentioned the yet but I'm going to so the ones who are going to hear the views of future for 2 seasons will also use of personal completion of in the school that are normal legislation in what they play the this policy is that and yet if you want the sort of finite dimensional analog of the army is really the following kind of thing so plan so suppose that I consider this product but we're here I took the lead in the 1 using before the 2 3 things in recommend in the opposite order and we'll have following many in the middle but but but but but but so so that these 2 things don't like you and I and the dates and is mind I had these 2 things that don't genuine to correct liver failure to commute you have to insert this extra that's given by you know some of the students on so that the army that have identified a lot of what we're doing it for the tools for you you get the sense that the writer said that he had to leave the release of the the last line until the will I'm always right questioning that should be the progeny from 0 to 100 enough from 1 of the things the this is the head of unit why caved in with this particular formula so that's OK so so that's what I'm going this in the 2nd part of the talks so so so far I've I've just told you this formula and I've told you that you know what looks like a miracle that this formula this totally out to regain I'm actually you quickly captures the war crossing behavior of the strict requirement differentials this solutions I let's say I'm not aware of a connection with the British so I don't think so and so so now you can ask you know why is it right why did this on why does this totally algebraic I thinking is moving translations of Torres Winery have anything to do with the special federal Quebec the French of itself OK I'm so there is by now a few different ways of understanding this but when you choose 1 particular waived its right in here and so will take what personally giving the torso arms reused aquatic differentials now in totally different ways and so were and think of them as having something to do with the asymptotic soap solutions of 1 parameter families in different and so will take is still there is a little while to get their sofa while you won't see these Kagan visited Britain something else with quadratic differential OK so so 1st of all actually were not getting quite just use the Quadra differentials were in around use money so that slightly cancer objects and so on so small I have again I have this currency and a fixed of some rank to homework the ecology arms and then as a let's consider our memory accession Indian workers in the tender with the canonical bundled because you will see men will take it to be Will they continue traces circulate work is that is a vector bundles of a terminal volume formed within to be valued Minnesota and so you know locally locally what is it can't really means but it just means that were making a matrix the major 2 1 4 and some major Samarra market functions on like this but that's what it looks like in every local patch trivialization and so that so appear like that because it's OK so that the pair of them and whole working vector bundle plus this while 1 for metamorphosis that Perry's colleagues now so given 1 of the things bundles and we can recover ,comma differential and we do it we just take the trace of the square of this of this this field to be called fire so I was a 1 former valued thing and so it's square will be quadratic differential would if the trace of that then that the globally defined thing that is independent of trivialization on August instead of quite differential will think about these things these things models are and spectral curve which we used before so before I was looking at the year on this double cover connecting the 2 square roots of the quadrant differential arms and out the way of another way of thinking about that is that those 2 shih-tzus to square root of the quadrant differential are just the 2 values and of my of my hands I took it to the point of the special are the agonizing OK I so OK so that you know so that England was a little bit more information the library differential and morally speaking with 100 differential about the agonizing now we're also introducing we're introducing this fine arms would not only unite denies that it is finding in space and then there's the space of his levels but is not just equal to the specified rate differential but rather the vibration over the space of quadratic differentials so bypassing from fighting but from 5 to its trace where I forget a little bit of information and 1 way of saying the information I forget his guidance bases the Yankees basis form exact a former line 1 over the spectral her and so will we get here is really a complex internal system roughly we have is a Jacoby ends of the circle groups which support I but today were basically only interested in this phenomenon that takes place just over the base just over the space of quality differentials Ahmadzai won't care much about exactly where I am on the on the Taurus fibrous sadistic any Higgs so any Higgs field that and Krystyna competitive book
on OK so it's so army I just remind you what instilled in me a religious not scale but also in flight connections over the curb on so and it's a necessary for this audience that remind you what it was it was a flight connections again locally and what sort of like a connection is something unless you to differentiate sections of to differentiate sections of your your bundle to locally and a 5 trivialize the omnivorous locally trivialize my bundled up and then I could write so that its section adjustable to wait to some Ooltewah 2 vectors of functions and to differentiate 1 of them according to the section well I think derivative its partial derivative neutral sense and I and Army some trees was matrix acting on my section and so locally connections is represented by the pair appraisals matrices easy and is and we say that the connection is flat exactly if there exists you know if you give locally I can find no basis for my bodily on which is annihilated by the connection arms so your connections words getting apply connection is equivalent to giving up some system of linear differential equations and would address these equations compatible system of linear differential equation look as you know
if I'm given army of flat sold to see connection but you know 1 of the things I can do is I can think of felon transport by which I mean you know from the start that Isabella transfer really say that modern farming operation so you know I started in some neighborhood and the local solutions of this equation Nablus equals 0 but then I II continue them and news equation now as it was 0 continued along some paper come back to the original .period and I'll come back to potentially some different from some different solutions but they they differ from the original solutions just by an element of so what we get is a is a representation of some of the fundamental problem occurred see and that to equivalents and In Hammersley you can recover on the connection from representation of technicians and workers that he got his wish up those amorphous army get up to equivalents of connections from so I'm going to have the map you know concretely really but if I'm given these matrices representing the connection known locally and even given major I want to construct the corresponding representation at the sentences are complicated met to go from the on I had to go from the explicit nature seems to be To the corresponding monitoring representation OK I'm OK so I was I was working on the 1 hand but his models on the other hand flight connections and now there's this miraculous smacked of which is I guess this is due to the kitchen thousand college and in anarchy in the cases in the cases the singularities it'll come to in a moment it's Simpson and the bulge and the remarkable fact is the following so divorced someone when you see 1 of the things but also without acquiring virtually shows anything but I suppose someone gives you 1 of those on the corresponding only canonical and 1 parameter family of flight connections and bundle and so different by about privatization and the Swiss this idiocy family farms and what that family looks like and take this it's of it's of the simple kind of a three-term former army so here we have the original the original Higgs field and here we have some connection which is actually you relative is an electric in the bundle on and here we have the on the edges of existing army so this is what this is all of this is not a simple thing because if you look at this issue of this family connections and to require that connection is flat and to require that it's flat for every value of privatization never acquires some equations you know roughly the 2 flats as the bracketed novel without was 0 the that imposes some equations on this firefighter and the quite understand complicated partial differential equations there the pitch the questions answered during the use of defamation this is kind of a hard day and is a hard to the banana was true so so yeah model there's this associated and 1 parameter family of flat OK and understand as what rising phenomenon would understand some property of this family of attention well they well actually but here were considering and that that we consider the music the whole market but rather Mayor Moffat because there the quandary differential had some singular isn't so correspondingly that they're making so that we consider will have some will have some Cingular's army and so corresponding to that you know we don't make connections but on the whole we tensions with Cingular's but at the same point so you know if there if the Higgs model that we consider these villages simple polls and then this family connections each member of the family will be a connection with regular singular and so in that case ,comma reducing our representation on the nite of Part 1 of the original curve or other part 1 of the curve with points to lead this monitoring around the singularity arm and if are middle-aged has higher order polls and then the connection has something more complicated singularities so-called irregular but in any case the the representation of Will you still have a moderate representation on the Somme under-representation of Montgomery presentation doesn't capture all the information about the connection and even on the agenda information so it's replaced by summer something a little more complicated the so-called Stokes the on I don't want to say I don't wanna try to say anything in detail about this spokesman we're going to talk a lot about different strokes phenomenon which occurs in the asymptotic of this family connection disease goes to 0 but in that event there's also this the other is the founder of the Christian army so this kind of this kind of example sounds harder but actually it turned out that the most Tractebel examples are part of this kind of example on because here you can get something unfamiliar you occurred BCP won by the moderate wasn't it well effective exceeding 1 even which is a single singularity Stoudamire representation would be absolutely trivial but nevertheless you can get something interesting and so it's OK so now we can formulated question so I so we have a Higgs model were given its 111 corresponding family of connections but according to this according to the it's not doing hard and for each 1 of them but we have a corresponding representation on some of the final approach to have a family of representations of the fundamental group on and it is very cold water with within resiliency son and then we can ask the question how does this family behave and as the privatization goes to 0 advantage is that everyone understands family completely and you know because it's because of presidency's our understanding completely without just understanding what happens as it used to 0 or is it goes infinity but there's symmetry exceeding 0 infected so the 0 is essentially enough understanding this is like walking Watterson army OK so what's the idea I welcome the idea is that this family connections comes the witness explicit form and ensuing particularly you know in the women as it goes to 0 but what would you think so you look at this connection and what it looks like it is dominated by the 1st term I so the vinyl you would think that there defined a solution might be enough just to deal with this just to deal with this 1st term and someone is thinking of what Our army so what it suggests is that you know in order to try to build a solution of this equation what should you do on the key thing would be to just Analyze this the 6 field and because if you want to solve the equation In that case is the races army so if you want all the equation on you know the drill press people's on some functions and that's that essentially easy you just at that time as is the Interreligious functions easy on eye so you know if we forget the difference between that Didier and the partial derivative operator for just the 2nd on and if we diagonal analyze the lead in terms of if I have over the years and correspondingly the interval before receiving here so it so so if we if we forgot that the not the partial derivative and if we do that underlies the leading terms then you might think Will you might look
at least for a solution for a for a basis of solutions of the sport so I just take the various eigenvalues and read each item that you come to this is what this is where you might try now welcome so so that's the 1st thing you
say is will it find a solution which will to lead acid tactically like that's what strategy that while the more
precisely you know uh you if you take a solution like that of someone who's leading terms what I wrote and put it back into the equation omnipresent Yukon disillusion what exactly be given by the would you want tries to make a series of a series in Zadar on who's leading turn is given by the thing that bothers me with arms sellers try says when he tried at so we're going to have a read on the basis of that candidate solutions on franchised by the item values of the of the Higgs field army solution alleges a series like this well OK so so so indeed you write this series of new substitute the series back into the equation of and you get a quite interested irritably determine all the higher terms so what so that's right so it so it looks like you manage to solve the equation and you at least locally raised that unless they still locally you get the basis of solutions and I have you completely solved itself in particular would would mean is that no 1 all the dependence on a basis point is only here only in this interval and so I would say that you know in order to all the equation in order to work out monitoring all the do is evaluates into roles of island in the roles of these forms of all right so that would be pretty good and so it was great the switches but but actually cannot be true why can't teach well the so-so so suppose that there existed actual solutions of this form so we have a locally on every domain where you can dial rising spilled so so then you could study the peril transferring leaders bypassing the solutions together and little overlaps here would be given by some in the rules In the rules of this kind of thing OK so that would be nice now the local
solutions as guest of the label but I do well but I values of the things field army and so and that we moved so we move around on the curve we move around 1 of the Braves points of this covering so the ringleaders of the zeros of the quadratic differential on there were around 1 of these points on the 2 I'm values what I'm calling here when the 1 limited to those joined ideas are being changed that serve the leading term of the series but also true for all the submitting terms are determined out directly by the leading to so if you had no actual solutions in this form the Pearl transport around West Point would change well OK so that says that the Liberal transport operator here on the exchanges this these 2 solutions that contradicts the fact that did this in this connection with what the different connection not only on the region were we that analyze the thing that if I collection on the whole are on the whole crop up to here were finding In this approximation which by the way this is usually called the new hit approximation of in this approximation under we find we find money from me but that moderately I cannot be there can't be there for the actual solutions so so what went wrong and we wrote this formal serious solution then we just seen that based upon the real they can actually exist they can exist because this instruction from monitoring so but there defined by a totally new procedure the only thing that can go wrong is that this series that we rode is really only a formal series so history has a radius of conversion and so formerly it's fine but actually is not find has this problem OK well so then you might and you might say OK was aware only a formal thing useless in the actual world on the understanding that the approximations is something that people use all the time and if it does seem to have some use and so so how do we make sense of it well we can try to make sense of this series not as an actual formula for actual solution only as an asymptotic expansion of a solution to this series we don't have converged delivered for anything not exhibited the asymptotic was an actual solutions well OK so so as it turns out you can you can make sense of it is that as necessary expansion but as usual with asymptotic expansions of this has that expansion is not going to be valid globally so we're not going to find solutions that have the hasn't addicts have uniformly as the day goes 0 in any direction and that's only ask for something a little bit less I suppose we has I suppose we asked for solutions which can be asymptotic I'm only in some sectors affected it seems that there been news that has been so and so what sticks so fixed some angle feeder the same faded appeared before us will see I'm fixin angle theta ,comma unless they ask for solutions which of these asymptotic Zaragoza 0 Johnston may have space page on and so then I and the following claims that this is this is on the part of the joint work with and get to work and the thing is that if I ever had a generic pointed out of this terror there are genuine actual solutions and so now announced it again depending on item value of these field depending on the state of their genuine actual solutions which do have this kind of asymptotic expansion as it was durable nearly half claimed only in some help OK I'm so explorer little that the consequences of this spring so you know the I said we have these local solutions but but the local solutions on John that she goes smoothly and if they did we would have the same moderate problem that we had before saying that getting have space that doesn't get rid of that problem the remedies local solutions on these local solutions are jump on many junk that something can coordination 1 will sign in and wishes that this so-called Stokes lines and so here sometimes for the Zaragoza 0 and sometimes arm and the sometimes for these these local solutions of the equation we've actually seen already in this talk and so if we go back again to look at this quadrant differential on the Sox lines on the slopes lines for these local solutions on exactly the lines of the record before the section to the kind is now they were studying the sense was in this family of differential equations we're looking for solutions with good hasn't Ataturk's many have space in the kind is that you can find such solutions and that you can find them only in the complement of Bill of the on of the walls of the spectrum network arms and then there's some jumping so there is be some matrix that tells you when you cross any 1 of these walls will be some metrics that tells you have a solution misdemeanor related solutions misdemeanors and those of the so-called Stokes vector OK so OK so now I don't even come back to the year to the original question and the question that we started with was the 98 the original question war crossing but the question of setting the asymptotic surveillance and monitoring accessible added any close link with and expose that we have any closing on a curve on and we look at the matter from I'm around that loop than ours they have family connections we study that might remain in some his family but
so we're getting some elements of 2 see for every the choice of primary data army and now you can't really try to that monitor made by the same strategy was doing before you just patched together you have these local solutions and pension other but now I have that you take account of the fact that the you have jobs on the walls of this spectrum on to you do that you find the following picture of you find that you find that the trace but the trace their interested in so he could hit by itself it doesn't answer somehow well you find the trees can be expanded in this form so you have a son of over pieces which are labeled by the homology classes by Monday classes of this new spectral cover so speech we have some some functions later standing here and they appear with some of coefficients I'm actually in the coefficients the picture it as it goes 0 out of the tent space on the essence of the asymptotic of these functions so these are the function which would get by served as naively doing the beauty the procedure that I told you at the beginning so and you just you just take this take the 1 on the phonological 1 former stockbroker industry new around some cycles on and we're going we're going here is that the actual traces given not just by a single thing like that matters by single and roll around some cycle and even the summer to do will you might think if you have 2 solutions responded to I'm going on but rather buy some solid pieces on the each of which has each of the pieces has the kind of naive asymptotic so you expect from the UK and they come to you in some they come to you in some complicated combination and then coefficients the coefficients that appear here are modest at odds determined despite taking a path and looking at how intersects with that With the special network so you start out trying to trying to study the asymptotic so its only function if I was a good way of studying asymptotic arm is indeed to do this the DVD procedure usually at intervals over the spectral curve but those minerals are a little bit more complicated than you might have thought because you have to keep track of how your path intersects with this and the spectrum of the critical analogy that this fire name can recover the Zaidi was 0 sometimes you're interested in and there is dominated by whichever term has the largest coefficient of this exponential I didn't know I was thinking about this and thinking that this construction so so for a moment and the lining another about evanescent Alexander's look at this formula as it stands I'm sure saying is that if I if you have a if you have an insult to connection over and over curve on medicines a useful way of organizing its organizing on its invariants organizing the the whole nominees I'm around some have nearly the whole underpin the trace of the holding around some panic and get expanded In terms of some similar objects of these things here these things here the whole armies of some flat Aflac connection which lives over the spectral cover that exactly what the WP naively said that you could do great it said that you could do you can relate the whole if you're the process for the connection To some into roles in the new cover on and here we have a hand and we have a shot over him that I'm the only track is that I given any path here you may have to expand in terms of a bunch of different hands over the over the spectral power and so this is kind of the slogan for this it's kind of unionization so on you're interested in setting itself to see connections over the curve and related to something similar you relate and the cease-fire connections over this over this cover and now worth it not literally like push forward is not like you have a diagonal collection you pushing forward that would give you something that all downstairs but the thing is that you take the pushed forward and then you do some cutting gluing along the along the walls his cutting gluing is exactly the nature of the spokesman and the fact that the local solutions could be extended the global arms so the starting line just can't really speaking what it involves just in some new putting matrices for the parallel transport every time across a wall if this Dickinson important matrix everything else is that so it's so OK so so in particular you know what this says is that if you have a vessel to see connection and this is this is us to be actually won 1 match so the data Monday sold to seek an action over the curve is just equivalent up to they section of 2 equivalence is equivalent to the date of the architecture to equivalence over over the cover and aware that says that this system on line from on 1 Wildlife life legislative connections just identified with that of the model line of C stock elections over this discovery and all I would say is that In this way we recovered some known interesting Coronet systems the so-called plaster cornices and characters just wrestle to see connections and so those are also called complex Evanshire Court but the higher rank which I'll hopefully get to at the end of higher-ranked you get in this way some really new exotic-looking Gordon systems some of which have been studied by by her arm OK so that is citizen intervention is thing which you can talk about without talking about the UK the has enticed but what I've said here that arises very naturally if you think about getting taxes are forced to think about this immunization the Ferguson
announced so so now we can come back to the question of the were thing that antics and what happens now if I try to vary the primitive so I move a little bit in the kitchen base right move my my the status of motivated I'm worse that he has enticed just as some have Space Invaders controlling which have space where are we looking at I'm still only remind you what happens so if we very different respond the adjustable bed and then the 2nd over Justine is by an isotope I'm on a variation like that on me don't
think anything about how you have I consider close same disclosed but that the government to have things but this value the bee analysis is all about how the pair intersects this specter network but it I just change the network by lawyers it'll be I won't change this list stability the euro and analysis particularly these coefficients and don't worry if I just during the inspection work a little bit or a defiant yes but on the other
hand over so as Ivory this this premise further on my crossing the critical phases were special victory appears so these were exactly the things that we want to cap on and that at that moment the UK picture of the analysis of his clothes paths changes or in other words the demonization I have had 2 different ways of ability of humanizing objections are 1 involving the spectrum work before this crossing and 1 involving a certain award after this crossing in those 2 structural the difference of correspondingly I had 2 different ways of expanding the horn only in terms of abelian hole the 2 different to different extensions so these coefficients Omega changing army but I have the original SL to see connections there level to seek connection we were studying doesn't jump on interest various you know the legislative agendas collection at all if I move imagination Movers connection by moving continuous work and so it these coefficients jumping I'm in order for the whole of 30 consistent these abelian hold means the whole numbers on the cover I have also jumped and we can calculate that jump and so the weather these only change it is determined by the topology of how the spectrum the workers the generates and even really calculate that jump you can calculate directly from the from the topology of the generation and now what you get
and so when you get well so 1st of all of the specter network the generates in a way that just includes 1 of these general connections on so the picture is on the the picture is that on we have something like this
and then after the degeneration of something like this so there's 2 different starting works with 2 different corresponding reasonably unionizing your connection on your ass out those 2 new innovations differ so here I'm getting some functions yearnings among genetics pioneer motor the hold Elizabeth civilian section of the army and the standards that those 2 words of the different but the different computable way and the way that they differ there is a big never exactly by this map which appeared in The Wall crossing from lots of this transformation so in this
case ex-prime people's backs him at times 1 one-liners it's never not mn not is the logic that's the thing that appeared in the middle and so did the job of the villainous connection is exactly given by this indicating get so here's the moment were on His more were were affiliated explaining are encountering the same formula because there are only so that the relatively easy habitation with a little bit harder is what happens if
you have this annual so close directories arm but there so this broadly we're working and you find that the jump there is exactly the Kadima now reads the power lines of so so in general I'm and we the 1 that will run this speculative forms are men the jump of these functions the which before they were formal coordinates on some Torres now they have a concrete meaning is the holding of abelian connections are made the statement that should be thought of as the Taurus civilian connections and the these transformations are did jumps that occur when you try to Apulia eyes single fixed as OTC connection using different circles on so also in
particular now if I come back to the soul you know in each domain of this picture now I have a particular spectral network of dies it'll be a particular wave of unionizing the connections on Our so in each 1 of these domains you know if I let my not really connection vary continuously mind this scammers and Providence civilian collection on half the job and jump by exactly the stadium on the other hand we go around a close look it would travel around a closed loop and come back to the original point on this game a defined by using develop procedures so we have to return to the original on the phone to the extent on another deadly the wall pricing formula which we want to explain and so the way we explain is defined some concrete objects which actually transformed by this funny-looking transformation a scam and then because those objects are you are you going to find priority but they come back to the original objects that but OK so on the whole point of view
in this village of 1 of the the more
fort on whether it could be that the where is the statement of the the the trade deficit if they intervene if you study at our as if you study a single not doing connection not believe insult to see connection with varies continuously on the procedure for relating at 20 billion connections and that procedure is have forced on you if you try to study the the asymptotic but in things get deleted this entire there is this procedure that procedure depends on a choice of specter network in and around in the parameter space on the course when spectrum jumps up and Johnson Circle work exactly the stadium is appeared in the Wall crossing for yeah that's statement it's OK
so when I said so and so we were considering some accounting problem that had to do with the the special directories acquirer differentials and what what what we found is that 1st of all obey this considers settlement this kind of universal war crossing Formula One and moreover that if you wanna explain the formula reviewers see a rise in some natural way 1 with that arises naturally is when use try to study the asymptotic of these families vessel to see connections that come from which inquiries army so that natural problem which is you know what I want uses Nestle to whatever the president SLK 1 or more generally in a community leader but to start with it's OK I forgave bigger than to army and so here following the exact same head of strategy and you need a new kind of accounting problems which so far hasn't been much exploits of these the to the special directories are kind of classically said think people take more seriously than what happened but I a rank analog which now that briefly tell you about the timing has been relatively little exports and interesting income on this OK so it's so it was just starting and so so now I understand that things and so on and now make it SLK valued suggested rank a matrix is an incentive right to on so now the finest that occurred to again just pressure the eigenvalues is just 419 values of this Of this so you know locally 1 way of thinking about it is that while the covering is the same as the recovery is the same as getting K just hold Warford 1 foursome were really behind still just at the agonizing Higgs spilled that is UK 1 forms which vary homomorphic with now globally that modern remains so it will be a key global things locally of K 1 on and now so so far so good it seems so in must work with that on more we've tried to generalize the the whole soaring that I just told you to this situation arm and it seems that it works on but the structure of the stocks lines so the structure of the the analog the spectrum network will more publicly than it was in the case cables I don't tell you when you know the definition the definition analogous to this definition of the trajectories of quadratic differentials in the higher higher-ranked burden trajectories I'm don't before it a phase and then I will define a network I so the network made up of soaks once it 1st tell you with the local structure of those social and so before they were that they were the views of despoliation but now this rather than differential equations and they're not anymore Leasor global foliage and now they're going to be carrying some label and labeled by 2 she'd signed Jason Edwards to widen values of this excuse tissue that into I denied it and then you can write this differential equation so it says that what is this so you take when I might inflame J but that gives you someone form it that 1 form evaluated on attention vector candlemakers tear the reliance on should be you should have faced data so that the equation that further the fines for you again locally locally affiliation using the leaves of this foliage but now globally arms this is the different kinds of wine connects but then so is now the local structure around the branch points isn't much as same as it was before the age range .period now brief winter will by 2 sheets of the coverage so I'm and again when assumed that in a situation of simple ramifications so we only have places with 2 sheets collide another Nova wage numbers that occur because because they're here in the day there no reason despoliation depend on a choice of true sheets that you like it could be on some price Metropolitan ever heard of anybody's directly on on this I'm so each each of the branch points is a bridge .period with army labeled by 2 sheets and defoliation defoliation with label IJ around this branch .period armies in much the same as it was before it is a from Cingular and in particular has 3 distinguished trajectories and we shouldn't be among our are so clients so there the stocks either and the solution of this equation where I and Jr the particular that met at this cover at this Red .period and now a course globally I would have a problem of the trivializing discovers already explicitly new during a special potential branch could then the Sox lines this was little that I J this 1 is JI this was I again so the alternative national across the country and so is that local picture each branch .period Unit 3 the Sox and then there
is 1 further thing this is the 1st thing which is a 1st surprising so now you got these lines and just like before you would travel around the curve do whatever they want you army but sometimes they're not anymore Leasor global full lesions that may have been that they would actually cross each other and if they cross each other if I find I J 1 and J K 1 the intersect and then there wouldn't be that they should give birth to a new oneself the average period J K and they cross each other in addition were generated new wine on the line of type but I came out and so this is in a sense this is heaven echoed something that occurred in the story of two-dimensional while crossing a long time ago it is essentially by that formula exactly so so yeah somehow in the interview division procedure is exactly this formula which which is forcing universities 1 arm but is actually recently we learned that this had actually been studied before so people interviewed said they had noticed this phenomenon ,comma OK so so it's
so so sexy so so so that the definition of the spectrum that workers are priorities you just should be streamlined and each grade point and then you let them do whatever they want you but when they cross so if you look at evolve according this differential equation when they cross if do Britain wants on and on so here is an example of
what this procedure might look like so here I took it on I case of just threefold cover so this is a Villanova ranked 3 models Ahmed at the curve to be accumulating 2nd thought really takes 1 so here it's start recorded also and we put all the singularities so I think that with the new areas of the securities those 3 bluegrass and shows so that the recovery has of 6 break points of the 6 with 6 exits the army and we
you see is that the thing at least in this particular example armor the
behavior is it's defined by a variety of totally crazy procedure but at least here the behavior is actually reasonably well under control so the trajectories travel around when they collide they can give birth to new trajectories on but eventually I are all the trajectories attracted down to these singular this year the more on yet the real as a the as it did write exactly the S L 3
sphere with Pramod points and with singularities of the most not generic kind of those reports of foresaw polls ,comma generic agonized 3 points and so on docket
so-so Army Army so this at least in this particular example the behavior is a lot like what we had for the trajectories of quadratic differential but now is an unfortunate fact that not every example is his arms so well-behaved but at least there were some in so in general it may be that this procedure defined you actually tents said of trajectories arms in that case we have to we have to live with that but
at least there are and there are many good examples in which is really like what we
had for the quadrant differences and so this is the object that we as the Everclear study this is the object that gives rise to the today to the Sox phenomenon and in this higher higher-ranking and higher-ranking I'm OK on
the the 1st sources who are I think it's I
believe is true so I think there really no theorems about this army experimentally I believe that's true that if you have this if you have the polls on that every today April 1 about poll the problem is that the collided many many times with other guys on the way they may give birth to a new things and you have sought infinite regress of these new things being created we 1st thought that this can happen but it now seems that it eventually on it does happen but still this opens at the priorities were doesn't and you may just consider those open sets as a way of getting
started on interesting examples
which it never happens there is some progress were doesn't happen anywhere on a lease on life and before just before coming solutions of the equations the corresponding families of flight connections and just before the asymptotic so I'm already controlled by a kind of unionization so you express the libraries of SLK connection downstairs and in terms of some similar quantities which are holding isn't in collection of upset over the it the that was it was this is the best prerace OK where it will be the key to the whole the world over last year while we are solving exactly the same that we derive from this are exact statements I needed the statement this formula of the expresses the monitor me in terms of the moderation dealing connection that exact formula One I mean it's it's you drive that familiar with lead you to consider that that particular construction motivation is the existence of the beauty asymptotic but but noted the point is that we don't really exist actual solutions that have a dozen topics so the formula is exactly that this relation between moderate downstairs in the library upstairs exact relations but so and then again we have the same kind of story that we had in the army In the right to cases of when this network so you have this network that gives you the this network of stocks once on controlling the jurors products arm and whether there were jobs so now again we can we can ask to slavery parameters however that would change army and the network topology may jump at that moment these X Games also jumped on me but on before the cool thing is you can study Avery generally what will be the form of that will be the form of a many fine just as in the ranks case and there always controlled by this abuse kind of transformation that appeared in the considers so Formula One it is the same it is the same characters and after all what this came as it was defined by consumers and some woman was on more visible the Taurus there is a united by rational transformation Taurus Avenue right now we have a case cover but again were just sitting seaside mansion there were that people cover so that still Torres Justin Torres the larger dimension and I know it is really exactly the same formal structure as there was in the case of rent took connections you know it was the 1st time every position is very rough procedure is viewed from history was as well as of now and yet so I'm sorry that includes a more about exactly how you know how you do this and mean that I have said slightly rough way on but it's actually precisely and its use of resistance the rest of the year was given suggested that we've got to go with you to the miners legislation and was released that can also probably OK so there's something there that something like that will happen if you consider it there .period bitterness which is very special select and ending you consider like a case of In the end went into that you can reduce the thing the jewel in water we have to do is considers summary special point of Hitchin days where the spectral cover has had in 1 sheets together words they were it's not things like and 1 copies so that there is a kind of stepwise version of the things that only so-so arm so now so that the full story is that it just exactly the same as it was before the army except so now that the jobs that you get before I can totally classify what happened the classification was you could have the saddle connection to disclose into nothing else on now in this hiring situation on the classification of possible past apologies somehow much much more complicated so when me show the 1st new thing so here
situation this is again in rank from Tables 3 and this thing this little cluster of 3 branch points and here we see the following thing happened so right now with apology jumps on and so the the the new object that appeared at that critical moment was an object that looks like it's something that these 3 which points
a each of West some is coming together so here is the pair to intimidate a 1 2 introductory and at the next 1 is pair 1 fair 2 3 in the last 1 of the 3 1 and and there's some critical moment the critical moment the moment when these 3 trajectories come together a common point in this kind of 3 string network and that string network is the object of have responsible for this jump in that apology so now work you know we're finding a new kind of invariants the variance the county's jumps and this is the kind of contributes
this is a new kind of albacore contributes to these invariants arms but not the only have that so once you once you allow that apology and you have to allow much more general well interested things he will you get on but it does include just met on it includes you know certainly could have even a more general and treating you have something like that you have something that 5 vertices you have something where you you some have something where you attach a loop to at this moment came from behind so you have something reattach to it like this and this and this too arbitrarily complicated structure in this problem but it seems to be just forced on you and if you want that version of this story that's what the story is coming into Indian particularly 1 particularly sort of crazy things but earlier so this is it said his addiction adjusted so here's the necessary before and after picture
of this inspection network under this 1 if you work it out but I like this 1 contributes just 1 injury only equals 1 so it's really just like this summer connection on but in doing so in general you get a sort of much much more crazy things here a sort of exotic thing which happened already for cables itself three-story this brought on the cylinder so we should identify the left and right sides and you have a combination of trajectories which which looks like this so this is the locus along which the directories ahead hitting each other head-to-head on in this case you go away you calculate the know procedure for calculating the job of the demonization that arm and you were you calculated in this case and what you find is that it is indeed given in this same formula you know Tania much the power only again but now it contributes not just to only gamma so here correspondence picture this summer logic says we should get eaten by lifting the trajectory suspect occur and where can hear the very it's not just for that holiday quest also for all the multiples of that logically which are exponentially growing so so the story in this higher-ranking case and seems to be seems to be some how much more interested in and much less explored nevertheless know the same sort of formal story that told you before approving the walrus and formula works equally well in this situation and so this thing is that how we extract these various army and that way the warm frosting behavior of these new numbers is still completely captured by this war crossing follows in particular this sort of complicated sequence of events that I that I just ago I also could have been derived from principle just from the from more pressing formal so here and it was crossing between 2 things that have the product 3 on a city of 2 things :colon them wonderment to you this product of this simple product of frustrations and I try to rewrite it is a product in the opposite order you find incredibly complicated thing particularly in the middle of that thing so there's a lot but adults here for things that I'm not writing in the middle of that thing and you find this product arm of transformations with exactly these coefficients and so there's something to this summer new accounting problem here on which is still part of this whole it doesn't is invariant structure and but which gives you which is able to give you users fairly complicated stuff I'm OK think it's a leaders summit so so I told you in the beginning with this war crossing phenomenon is sort of word came from physics on Monday March and and then I and II this handsome particular examples you know for a particular physical theory on the whole story becomes very geometric on and so on we 31 pressing for the special should quadratic differentials army and then more generally forever for the generations of these new object which recalling spectrum network army so they both come from the the comes from equally naturally from commentary on and going in each case you know 1 way to think about the objects which are trying to count is there something to do with the the some moderately for this system of linear differential equation with a small brain army and from that point view the war crossing familiar once you sort of absorb this this way of thinking about the story of the war informal overcome something which actually does automatic asserted consistency condition for the whole story work army so that so this has said that the special directories of primary differentials are kind of well-studied subjects although I think the thing that this war crossing phenomenon of this thing they will be the walrus formula hadn't been noticed before and so now that story is somehow connected to the hysteria thousand Thomas variants army and then there's this new theory for higher rank about which essentially almost nothing is known so far which I still hope to every household interesting things believe that it's thank you I don't have many comments suggest that he last the that Japanese that you really have to study this kind of thing the place is always of the great so so I know that there is a case of great so it's only right that was good for the this is the only 1 great Southern practice so so I so far don't know very much on it by connecting with these desperate yes thank you In the aftermath of the war In who was following is the question of whether the parliament so far I don't understand the relation between this story is going to year of your question is yep yep so I mean so yeah no doubt these things have some role to play in the story of the conversation on that so far I don't know what to do we have analysts noted that you really have a small brown rot that is not the the new house in the previous week usually the the pair of you held police generally welded at request so so I mean the thing that arose 1st in physics but can automatically came in this very particular form these very particular families that have the symmetry between 0 and the net worth the structure that we find look a lot like the structures that people in the UK the have studied before and there they city without that restrictions on so so I don't know whether here I was describing a pretty young thing about you know precise formulas for the asymptotic since the end of the monitoring and I don't know so the proof that did so we have an argument for why these local solutions exist on L In an argument we used affected the thing has this symmetry so it's a prayer possible that that the results like told you really only hold for the special class of things the confrontation systems are more it might be that it works for every family and I just don't know the good thing is that you know that this is also 1 of the exhilaration thing isn't totally general thing which you can do it as soon as I wanted federal network problem as soon as soon as someone gives you a way from the user when you used in the year all the time yeah I think I need you to formulate that formulate that whole procedure you could indeed formerly dental procedure without the structural Infinity from a book you would use the government said it would a using mafia on if you decide to move to London case you just journalist is not part this I know we work at that site so that I mean that final only has to do with the attributing interview could so I I I tell you that the when you have a flat tension you view it as a can over the spectral cover but actually I should have said so the the most canonical thing to get is not exactly a flat tension over the spectral ,comma where there is a flight connection which is minus-1 around the year of the pop the branch points so it and it's it said those things a tosser over the set of flight connections but it has this little minus 1 now In many examples are able to avoid that Europe would get rid of that sign on but the process of getting rid of signed introduces this Sigma Gamma somewhere else on securing work with these twisted the twisted the blood actions actions would have this minus-1 1 everywhere and everything is totally canonical yeah exactly so that you gave me this is known as a well-known thing bait occurs generally in the civilization process if you think about is that it could be that sign just Armour if you want get excited distilled water reflect actions even knew about the price you pay as will signal the end of the year when local men events have you started by saying that the schools would be forced to leave the the EU would you actually I read to review that essentially officially wishes he can be bypassed this more more efficient way of completing the loyalty of these releases will hold and that the use of mobile lack the ironic are interesting structures was of course no partly motivated by the interview would probably not on so a victory that this stuff is not necessary for the WPP that apparently had a bad if you do you have a history of the city was the scene of the blast a hole in the wall of of the relations between the 2 nations of the education the Florida Lottery for for the use of force yourself as you are armed with a hunting estates around we've done a little bit and we certainly did study that particular case of toss with 1 puncture on wrestled to result of the problem is that this is the sort of thing ,comma how well the simulation and in in the limit where you take the residues to be some specific residues for this for this year with 4 punches to just the losses on the aging eating regular 34 for both but then I think you have to some for the relation before they become the standard that and that what the matter which is that the model line of flight actions a vessel to see fluctuations on the 400 sphere the insecurity can really is like an avalanche of its surface on but it is more precisely if you fixed line moderately around 4 points you hear the views of the future of the right to call hula releases the foundation of the state of course yes I'm right if you put that if you put some restriction on the I'm so many units available for this for this Taurus with 100 also services special parameter so in that sense the moduli spaces the same I haven't thought about what legalization procedures also the same legacy probably and sometimes it's possible find he tries some on the 1st of the year and I think in a sense in the sense that this the Stokes lines that were drawing are exactly the kind of not perturbed corrections the you need now I don't have that I don't have any direct comment but but In the formulas that we write ultimately answer enough attributed formulas and for example so maybe 1 thing is that for example His example that had not perturbed of correction so summering here it will raise their formula for the for the trace of the around some now at this point this formula on this formula is actually an exact formula so it's an extension of the trees in terms of these simpler objects which on the whole a means of this civilian connections now if you like he might leading Internet service for a particular quality differential were pretty good .period originates mishandling term which is the 1 where the real part of this is the biggest arms and like you might say that the other terms the subletting terms which are coming from all the groups on the spectral curve are kind of non-petroleum corrections that 1 so maybe that's a moral the these users are now in this case and and while the series for this for the for the section the dysfunction DEC's name of and it goes found the game of you're standing around 0 they have none of it they're not given like evidences of the daily as invited nodded yes and he's just a little too little he he used that as police week the fastest in the last I'll focus I fully sympathize with that point of view of pound and euro deserves in these restrictions was the exact formula but but amended strategic stand around equals 0 interesting thing is you do find this divergent series the
Stabilitätstheorie <Logik>
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Singularität <Mathematik>
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Invariante
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Körper <Physik>
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Physikalische Theorie
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Singularität <Mathematik>
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Abstimmung <Frequenz>
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Mengentheoretische Topologie
Momentenproblem
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Wurzel <Mathematik>
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Parametersystem
Multifunktion
Homologie
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Klasse <Mathematik>
Physikalismus
Abgeschlossene Menge
Zahlenbereich
Ikosaeder
Bilinearform
Kombinatorische Gruppentheorie
Trajektorie <Mathematik>
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Physikalische Theorie
Überlagerung <Mathematik>
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Homologiegruppe
Loop
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Weg <Topologie>
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Flächentheorie
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Einfach zusammenhängender Raum
Kombinator
Ultraviolett-Photoelektronenspektroskopie
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Singularität <Mathematik>
Quadratzahl
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Loop
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Gleichgewichtspunkt <Spieltheorie>
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Stabilitätstheorie <Logik>
Punkt
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Einfach zusammenhängender Raum
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Singularität <Mathematik>
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Trajektorie <Mathematik>
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Polstelle
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Differential
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Momentenproblem
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Trajektorie <Mathematik>
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Trajektorie <Mathematik>
Turm <Mathematik>
Faktor <Algebra>
Polstelle
Phasenumwandlung
Randverteilung
Punkt
Momentenproblem
Gruppenkeim
Kartesische Koordinaten
Komplex <Algebra>
Raum-Zeit
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Eins
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Homologie
Frequenz
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Anpassung <Mathematik>
Gammafunktion
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Transformation <Mathematik>
Trajektorie <Mathematik>
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Ausdruck <Logik>
Differential
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Einfach zusammenhängender Raum
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Orientierung <Mathematik>
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Transformation <Mathematik>
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Metadaten

Formale Metadaten

Titel Introduction to wall-crossing, in physics and differential equations
Serientitel 3eme seminaire Itzykson Wall-crossing in Hitchin integrable systems
Anzahl der Teile 3
Autor Neitzke, Andrew
Lizenz CC-Namensnennung 3.0 Unported:
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DOI 10.5446/17056
Herausgeber Institut des Hautes Études Scientifiques (IHÉS)
Erscheinungsjahr 2013
Sprache Englisch

Inhaltliche Metadaten

Fachgebiet Mathematik
Abstract It has been known since the early 1990s that wall-crossing phenomena are crucial for understanding the spectrum of particles in supersymmetric quantum field theories. More recently, the rich algebraic and geometric structures underlying these phenomena are beginning to be understood. I will give an introduction to this story as it is known today, via examples, focusing mainly on cases related to integrable systems introduced by Hitchin. In this context, the story turns out to be related to the WKB analysis of families of differential equations.

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