Spectral networks and harmonic maps to buildings

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Title
Spectral networks and harmonic maps to buildings
Title of Series
Number of Parts
3
Author
Simpson, Carlos
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CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor.
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Institut des Hautes Études Scientifiques (IHÉS)
Release Date
2013
Language
English

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Abstract
This is joint work with L. Katzarkov, A. Noll, and P. Pandit in Vienna. A boundary point of the character variety gives rise to a spectral curve, and a harmonic map to a building. The differential of the harmonic map is the real part of the multivalued tuple of differentials defined over the spectral curve. Gaiotto-Moore-Neitzke have introduced the notion of "spectral network" associated with such a multivalued differential, determining the WKB approximation of the nonabelian Hodge or Riemann-Hilbert correspondences. We have tried to gain some insight into the relationship between the spectral network and the harmonic map to the building: basically, the spectral network lines are located where the curve intersects the codimension 1 faces of the building.
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as you don't want to In addition would like to thank my team for organizing this state which is very interesting so selected projects which were we've been working on for a little bit of time start computer thing here yes so so that we can work on trying to understand the diets of more nite sky considerations at least some of them with no account of Alexander Nolan born of money as well as other people in the Indiana that would be able to speak here about did you want to understand the special networks of die tomorrow nite from from the perspective of Euclidean buildings so this is what will be the same and that is that should generalize the trees which opened the SLT did so and likely for me and I hope I didn't actually include any discussion of the SLT the likely from you we've discussed the SLT case a considerably already this morning and was this was 99 emphasized that way you can see is that in the cell today's were talking about affiliation defined by the real part of the 1 form and we have affiliation that corresponds to a treaty which is the space that we use them for years so not that many the Morris somewhat more long-term goal but this is not really my zone of expertise is to understand the relationship as was being discussed between the the pictures by phone networks and the space majorities of stability condition In particularly in the case especially words the important role was played by these transformations where where there's some jump between those in the top quality of the spectral network and in the case of cell to a 1 way of thinking about is that the space of the treaty which is the space it leaves a void nation has undergoes a mutation when the when you cross over the years so what we like to thank my team and also that another student in the annals of United for important conversations about so what I I don't have any pictures in the 1st part of the file and the files some inventory of his 1st it is so close together on service acts vector bundle of Simmons has determined at 1 point in his field should apologize to my around here is what NBC known around the world and my mind than on round he will become the ground fuel become known the use of his office today and morphism value 1 form and let's think spectral and will be that's reduced because we haven't logical form and we just think of that as multilaterally differential form on sex so we can write locally is the 1 part of the ah uh conditions critical 0 the German 0 for the connection Fraser of the conditions as the son of identical 0 and the condition that rental car is reduced to the same as saying that the the I should be distinct 4 general points in it start about the size of this was admitted full-screen so it was just consider that the device which is the focus of which the special couriers branch and once was a story that account so you will find extra special purpose is a tall covering next door so now we like to think of 2 W kg problems associated with this data the 1st is the real complex wkt problem so just take it as a base connection Nabila 0 1 assumes that exists and look at it a connection at Woody which is now 0 plus 2 times the Higgs field and use of steel large parameter so too is 1 of rage market so we get
a moderately representation for the connection and it's also true fixed matches on the middle east on this set up were just fixing a single bundle so I mean I think Andy and 19 were actually discussing situation which is more or general in which the man who can also varies functionality and effective in the curtains on various options now we get a flat structure in which an unwanted given by the flight so the transport so if we didn't know got a single 1 which depend on we fixed single matches on this model but now we have a transfer function which transports back to the base .period depending of course on a path that's why Extel universal coverage so transport for the metric mile the flat transfer back to the base .period then we get a mention on the fixed base on the fixed the fiber of the bundle Over the base .period reflecting trivialization here I can't think of this as a map to the the cemented produces as much space as saucy model as you are the map from HD from still then to match its as Rudy so this is not necessarily harmonic map but is supposed to model supposed to make you think about like map basically as energy sources and so over the complex that we came probably don't understand there's another behavior of the of them representation roti and H or just the transformative itself as a function of the those find things this way solid-state 2 points in the years ago since we've taken to point the universal coverage is indeed path going from 1 to the other but fled to you of 2 the from the fiber you over the .period keyed the fiber overcame the transformative notes defined WPP expert to be the limbs soup of 1 over to the log of the normal transforming the normal being operated along with respect to the fixed matches this is supposed to be basically the highest ideal value of the transforming wishes of roughly this is roughly speaking the biggest term of respect and honor the Justice found 1 now that the problems so actually know this I think now and then and then I heard talking I realize that we have a slight misconception about what you were doing users who he had a whole Moffat family of I have connections depending on your parameter which according which all of which could be bad for the people 0 and the people infected which is actually exactly the same as those guys but we we can do whatever he said here too this is not actually exactly what you did when will you as more women were closer to the to the complex that would be because he was here will not depend more on parameters ways we can still do that anyway so I suppose we have stable and bundle then we pigeon equations so we have generation angles ventured on the bundle depending were worried multiplied by the number which is a large real number analysts said the associated flight connections and take the Associated representation and the
scarcity of a family of actually harmonic maps from universal coverage of the cemented space these are the amounts given by the solution of regional and so 1 can also ask us to understand the hasn't Onyx the solution of the generation in this sense we can define the transporter and and the explanations on your using magic which is which it also depends on T 2 major I don't actually have a theory for the moment of so many mourners to explained that the but the export the various In a way of being cured .period on tax arrives in a way which is dictated by the that so let me maybe a pause here to see you know what is basically saying so I'd like to just give the following example the so that more nights he said 1st of all so this is kind of a classical differed over time .period here for example to point here these services this is my picture of the special network so these guys are the major affiliations labeled something like 1 too many wayward but this is and all 3 cases so the labeling of 1 2 1 2 3 and we get some good lines if we go from the point Peter the point here than the prescription because we just as there as we crossed from here that is what want across the 2nd 1 of these a major resettlement work wine then we can no longer and calculate transport from here to here as just a simple in the role of 1 of these forms from PDQ John and the collision phenomena is the same thing the Clinton fine phenomenon says the same thing happens but it's all the more so here so if we try to break from the point P the . 2 we consider it what is the transform interest motorcycle and from the point gain .period keeps doing what happened here was that every draws sort of the drug use look like that here we sort of
went over a mountain pass and then back down again and when you go over something and back down again and then you have to count the back down again negative with so for example you have to integrate the 1 up to here the 2 years that you get the point pin .period Q then we haven't actually for example here we over this matter into the upper part the has that we can actually go out because you here and similarly and for this guy he's already inside the upper part of the mountain pass here and then it goes down .period however as it what going on in the state which is somewhat more the most unexpected industries is actually have to take into account is closed and we went over a collision line the back over the conjugate collusion but you cannot actually calculate the transport from PDQ as just a simple and role of 1 of the forms from people like where you would be able to from here to here or from here so that's just a brief roughly speaking the rougher explanation of
what I mean by dictator that were dictated now hear the remark so we will this was well maxim was discussing the comments that we made the case we can actually the transfer function in terms of research functions and you know it it was transformed was transformers and roll from there wouldn't be any of the function times the money also on the inside of my privatization is that this is not the end of the year is innovative the area north and function defined for for larger and mentality continuation with infinite politically find branch however that talk about things we can't willing to say we can describe the possible locations at the ranch .period and turned out as far as I can tell you this is going to discuss this description is roughly speaking compatible with what is going on more nite sky said the I think it's relatively precisely compatible in the sense that I think we can show that that we get exactly the illegals from points while repeated point on her work you on the spectral occurred good question is not the a way to actually go you get and so I have a way of saying that in general he is the key to 1 of the things that the slight difference by biopic makes too much difference so that I think we can deal with whatever team in the in the perturbed at essentially comes down to putting 1 over tea in the perturbed piece of it the connections they want about table at different so you can so the 1 question what this particular thing for example anywhere you want to hear here great something like the 1 of 2 years and the 3 needed to hear .period when you get this particular value their spines of some other and so we would like to look at them terms of harmonic maps to build and when he also give for 1st look as to why white buildings might be involved among other things that you could really see what was happening in terms of the monument to the tree in the case of solitude already suggested that buildings would be involved but we can draw a a picture of of the following features it was the 1st the 1st thing I said the Man going from P 1 2 1 by going over a mountain pass please showed strong just this the lower part of the function what's wrong like this years and possibly and we have .period P on the Net for actually the point you the other action the other differential that 2 3 differential is roughly speaking cost of doing the .period pinpoint if we don't vary that the history of financial too much Arco and principal shouldn't really vary too much from from a straight line drawing PDQ so In terms of the US and so this will be the function of 1 too the total duration function 1 2 longest pass is actually sort to big it's not what we get from the prediction however the prediction tells that what he's actually do is go up to this point here than and back to you but that's the expert from the point .period we can that's what you get by looking at the distance in the building to another 3 answer service to building which is just a three-legged tree thing Cross the distance from 1 . together in the building so the point of that whole part goes to the fraction and all partners to the back of the on the same shade of the trees to go from 1 would just have to go up to here in and back so this picture was sort of a basic picture which coincided with the the British and the India just in the simple case where we were not looking collisions anything and yet and so our goal was With the people Indiana was to try to understand what's going on With the conjecture that in fact ah ah delegate gets money is actually going to be the distance in the building between 2 points this can explain the sort of cancelation phenomenon which is sort of try to integrate the differential equation but little pieces from
PDQ there and get a much bigger answer them the correct answer and the cancelation of I'm sorry to say cancelation and says that all the pieces cancel out the real explanation is just entering up to here and that is the beauty of the the
world he said that he he thought that that's the subject of the talks so there is that so here's the basic idea of what is wrong with again a at least I don't know to make it correctly the idea but the basic idea so that if you want to look at the feel of functions of germs of functions on all terms of the functions are what sort of support that we can define evaluation given by the sort of exponential growth rate of the function then our family 1 of the representations constitutes a map from Taiwan said 0 into a store of the future so far no 1 actually its building and we could just try to acquire harmonic maps from the universal coverage brought its building following promotion unfortunately doesn't seem clear make this precisely because of the fact that we can't really defined field of functions because it wished to culture like scientists signed pacts that still stand exponential growth rate equal to the expert equal to 0 was to have a growth rate constants however contributed 1 overcoats and knowledgeable also the said that it has abandoned so far at least to me at least a dozen seems to make this precise that's kind of the idea it's not likely turned out there and AT & Power has actually developed exactly this type of theory based on work Kleiner so turn that we can justify workplace maybe some slight modifications are necessary of course so that is to look at a map of HT it's been absent semantic space but we scale the distance on the metric space by dividing by I I don't think the global women of the century spaces with very skilled distances that just means take metric space and sort of look at from really far away Of course it's sort of sense it's kind of infinitely big even if we get from far away it's still infinitely better and as you take this sort of aluminum that you exactly that some kind of building modeled on our blood it's building you get a building are building an arena were spread over some kind of going out and doing what but we are actually in the definition of will have a definition here but which will be country's so we need to worry about a thing but with the construction of the choice multiple drive you know inaudible Travolta is should should 1 day I hope no 1 is managers and it at roughly speaking means to sort compatible sequence of choices of subsequent whenever you need 1 and then you go and which is due come on major factors ,comma made of the brother of the group the effects of these sport and so
important paper she discusses this limiting building ,comma and the answer is that you have a lot of the action of the group 1 on the building and that it turns out that you can do equally well get limiting map each made from the universal coverage of the In and the importance for us this whole situation is that the distance between 2 points in the in the building the distance between the images of 2 . is exactly what we want which is to say take the distance and cemented space between the 2 .period we scale by whatever team and then take the way but that's an Travolta limits so that so many of the women some subsequent switches but that's the limit some
Cyprus subsequent which is sort of cleverly chosen you might now it is
that we can be more precise because on the building so let me just say this is the limiting building is a building model on the that find space for the cell to or for this brought its buildings which all ominous they will be actually yet
so going to get that will be the fact that the good .period but before but let me just say this 1st so in fact on the building of several different ways of measuring the distance of the perhaps standard ways to measure the Euclidean matches that was just about the Wikimedia distance between 2 points in building is that 1 of the actions of a building is any 2 points are contained in common apart apartment just an image of a map from here into the building so occurred
points and put them in a common apartment just measure the distance in the apart from the 2 points depending on what kind of distance you put on the apartment a distance function on the bill the standard was just the accordionist squared sum of squares of course that's not actually have a choice for what really what we're interested in there is a better choice which is the defense sawdust the it's a distance the log of the operators as are the circumstances just the maximum of the course variation of the course and more refined which all summer floods In the course of the last 4 years ago actually on altogether different Gordon's altogether and get what's called vector distance at which basically means that take up take all according to arrange them in decreasing order and these guys got caught all come from the limits of the corresponding differences on semantics space you could innocent comes from the usual distance citizens comes from the guy which is just a lot of the operating arm of the military and the vectors and comes from the full collection dilation exponents between the 2 matches on the center's but and if so what we're really interested in to get kind of full information but the detritus and so are space so that our faces point or 2 the or whose son is 0 . 2 0 minus-1 1 development adjustment measure backed by for court and a distance between the origin point is obtained despite reordering the points therein decreasing he said he
recognizes this he doesn't trial and voted for him if you refuse me after entering you had alter my now it has something better than this but argued points in a common environment then use the vector stands in that apartment now I suppose now wants the corresponding on matter just put the distance defined the distance the vector distance between 2 do you and want to indicate where basically 100 is equal to the and on the other hand on adapted comments from because of the relatively fast and soldiers opened this will be the 1 that export what is termed the transformative matrices the 1st acts on the 1st distances just lot of the norm of major the operator matters we can just use operating arms to get hold of all these guys because we can take the operator of the majors on the Keith exterior power now with the some of the 1st so this is just step that looking at factory guys just basically the same as looking at the at the operating room of the transport so we can just basically think of the land 1 distance that's the France interest but were only
interested in these distances on this match is facing a lot of the past and went have to be scaling and distance in the risk distances just for the concerned I just 1 of a lot of the operator so what's defined all Travolta expert to be the same thing as the except the whimsical later replaced by the Ultrafilters and that means it's some subsequent In regard the limit of sequences in the last 3 "quotation mark limbs soup but also called on the me chooses some cleverly chosen system and subsequent and and so that'll be less than whimsy and don't the question maximum asking is this really the same as well so that will be the checked the case in some cases so the 1st observation is that if we fix the choice of points P and Q then we can choose some Travolta such that the old Travolta women is equal to the swimsuit for that choice of P and Q choose subsequent because wasteland Superman as all Travolta to that subsequent at what time it is whether we can really do this for all pairs of points P and Q at the same time the example treat that actually will
follow posters after going but what I got was the 2nd observation is that some of the problems that was actually equal to women then of course is the same as the author of older women so this is the case for example use local wkt approximation and it would also find the complex wkt approximation for generic angles of the of over after we buy generic angle if we knew that the loss boss transformed into an essential security so that's but I'm not a I don't general non-approved approval boss transformed a don't have central securities as initial singularities that the likes of having several different security all glued together at the same point to any be sort of like having walls and then there might just get some kind of weird behavior if the lost-and-found from doesn't have essential singularity and if you twist things by generic angle so that there's only 1 security plus transformed which has biggest real part then everything is governed by the asymptotic expansion that real hard and as an extension of the leading term and so you can just see that the same soup is actually equal to 1 the virtuoso was so we pretty much expects that summer there will be able to prove that will be the case Our main remark that just the classical wkt approximations which says the as more than this but this service which is supposed to have a short pass it the XL the stock index still there but not going near the scene of a point and that the non-critical so critical that means that if we go back to the path the real part of the differential forms the all distinct all that means that we have and reordering of different forms and up to the reordering we can assume that the person is bigger than the 2nd 1 and In that case the jury that's just complex that we be serious is my understanding that did that just sorry that just classical wkt vary from like 19 20 years and in that case you can't really describe dilation explanations for the transform and that we exactly did approximation like that so the odds are here story CAN nations not quite the same as in the talks when I the what was called land died before you which is here the and here when eyes were on the off field so I didn't police yeah this is going to be really sort of a fairly good approximation which is to say he added that tax moments ruling and I am but not less and qualities that we have and limit we always have the the distance and the account this is equal to the family of the exponents of given by that's a statement for the complex that would include people think we conjecture that the same actually true for kitchen that would be wrong but the fact that this is a pretend this is a misconception about what you guys were doing we I thought we were actually claiming the statement had been evacuated actually more of a complex wkt problem even tho has won over to you terminals also it's still at home market so you are really but as far as I can In hidden casework were actually solving regional but but we think that that should also be true OK anyway growth in the complex case and also of course in the agencies that turned out to be true if we have any non-critical about even not even a long 1 not necessarily short then if we use that to the building or perhaps in a manner that everybody in the world in the and the vector distance the apartment is just given by the same intervals remember these land all eyes are either the
intervals of the real part of the forms so this is just from the classical wkt approximation we can get sort
control and the dancers from short pass France's is using the triangle inequality for the actors and the case of equality and triangle inequality I should also say that if we have 3 points such that you have some relation for actor distance then three-pointer are all 3 in same apartment usually building any 2 points are contained in the same apartment but 3 points are not necessarily that apartment if we have a rich like that and they are contained in factory office Chambers centers .period maybe the strong mentioned here a what is normal life we have 3 different
families of reflection I replied no 1 2 2 3 1 3 and the comic told for example that the same in the fields were distance from this point on this point for example something looks like so back to the action of the good segments to look at in the building at the actual Euclidean segment is not all that useful in this context that the real estate market at this point this point just the told these 2 points but in the end so distant that means we go like that means any profit looks like that sort of non-critical and calculate the correct answer stance through the so now of growth is the fact that this this and this actually implies that map from excellent economy is the harmonic
female the sense of promotion what that means that means that any point In the company there's a discrete singular points which could which of course singular point or have already they could also in principle contained some kind of other bad .period but In the compliment that that any point of the neighborhood which maps into a single apartment buildings and furthermore the map to the to the apartment is given by just integrating the 1 so I think this is this is the very similar to and he was saying that we have sort local systems a fundamental solutions on the on the sectors I don't think that we can go from the statement to what you're saying goodbye to and from what you were saying to this statement on but it's OK so this is what we can say about the general situation which is that we get this harmonic female agent book which depends on the choice Travolta the the using that may happen in the Saudis it is smaller than are recall to WK expert and for that we can choose old villages so that that holds the quality hold for at least 1 try came and also because it holds in the local case 1 P and Q close by now we actually started 1 should be be able to choose Omega which gives the building which works for all .period cure may be almost all P and Q for the sake of points set a bad points so that isn't which might have wrongly taken in years subsequent should have sort of small enough major at different points of view on the sector the don't cover and this is just the start of the tunnel and from there to the value of what so what does that mean that making here are for the case to the complex WB KEB would so you were taking a model and an additional connections that never 0 I will hear the case you can use the and so this has ended her mind accident and I'm not only on the realities little will also always pages you right now the only 5 and the U.S. has also know the governmentwide don't work for example might be that the has different behaviors that for us please special amount that will not
then we lost we would we would sort of imagine that we get a better
control next month to produce generic values that went not an at so so so I'm saying now is for this case because it this is and this is the case in which we know how the the local wkt approximation so what 1 could open the way for the actual equations but where we take we would to hold the 3 case it would take to we don't treat the story takes this can examine all solvents equations 40 times the and what about or you are already talking about more general things were lot the bundle the spaces everything a at what is it about the cost of the hope the abstract to be able to have a statement for that kind of situation also tossing
principles for the the complex that we would be what is the only to analyze the harmonic In and that's the main observations just know that if we take the reflection of applying the bill that means that the IRA planes in the buildings along which the different faces a joint those guys are just everybody that Carroll 1 of these 3 guys in there In the apartment in
this altogether now if we pull of the to to still but on we just get the affiliations which therefore includes Michael network because the function playing an apartment in our case In all our case just the
it agreed that such calls constant went to the different at Siemens extra the fullback to curb Nextel that would satisfy equation real part of the budget
equals 0 you're exactly the earth the the equations for effect on workers the 2nd curbs on the special ones that you get sort of starting out from similar points and doing collisions so so what we were trying to do is to understand geometrically why these inclusion guys OK so so that that sort of what we can I don't notice that will get the original work Nevins Roberts example of someone in the eyes of the he this was the original
example at the Japanese guys referred to somebody else also changed the margin but we check up not that many so users seems to be the 1st example reactions the spectrum this is the 1st original place so diluted that labor and the paper that I actually have to franchise family of equation so the hypothesis is that they're actually setting each Oracle to 1 which physicist who attended the hearing that Maxine also notice on your side so you can the paper but it's OK actually residing in sprinkled among which plates were back in person you want to put it in a way which is homogeneous you know going with that and then we just take it for us as a team has won praise for his 1 2 and again we don't care about our eyes on 1 someone can actually the thing that really got me about the vapor is the family different which looks like this then we use the command
matrix and so on so it's but a single instance of the equation for 1 functionary transform that into a matrix equation for 3 functions you see that you get the field and battle for her for the field was given by the thing that you get from the equation by replacing by the content of so why is the corn on the cob and among the options X on our remind and that may be a sign terrorism and got the start purpose given by the nationwide minus lots of sex equals 0 help from years we wanted to also have
the that it's just a really
trying to adjust the real point of view appears this is an act of war but I think that it's part of the unnatural comes in the property take some of the images inactivity red of plus or minus 2 if you take the parameters on In terms of what of these points in the polls minus 1 to the right .period and also minus to so if you want to do some calculations by pulling back to the spectacle itself and using that as an equation the formulas are not too terrible as at the probably are not all that many hospitals and see the description floor because of the way in the fall this point their involvement in the contributing the supplier is all that and say how they calculated that and how they calculated anything in the example but maybe that's the way it is but I don't think it matters too much for what we're doing but it's but most of all year they gave an explicit description of this collision from it's so that are different from wanted to convey all form while on the DAX 4 1 1 1 2 1 3 3 solutions of Austria branch .period Act which are at Tuen Mun Tsuen was often picture and the imaginary such elaborate is in the following picture the scheme so this is that this is actually much of the media rotational questions that have so here we have the 2 but you think these minus 2 and 2 spread 3 lines of liking and video but expects the reason but needs to find you go up here and they collided these collusion .period and create and send out a clear line here and some of the direction we get a clear mind going on here so to close so it
was just Apple's seeing pictures of the 1st thing is is that we have to collision .period of clients and vertical line now the next thing see in the tournament the 2nd workers divide the 2010 range 4 regions and should tropics on top of of the line I think that there's a sort of vertical social network line which actually continues on the other side is the same for the 2 curves and that's point of general phenomena and the spectral were divides everybody into into 10 regions regions on the outside of the 1 2 3 4 5 6 7 8 and then to interior region here's the colored yellow and pictures of the it's
time mail defense with some arguing with local that became approximation we can actually conclude that each region is snapped into a single file sector in a single apartment the building and the interior squares yellow dye maps into a single apartment with a full line along the cost the cost because this is a girl like that in the words of with differentials crossover and In doing this discussion we just use the actions of the building as they
were discussed but part of the people that's yellow region that's the image of the origin inside the but you can see for herself and the fault lies not it's not it's not this sky before 1 the cost extra these 2 .period
actually go to the same point but not just arguing by the general principles maps to buildings what we see start another 1 for the piece of data which is that it turns out this case that the images of of these 2 points are the same .period so the 2 go to the when the building and everybody goes to sectors which start from that point the different sectors and that those guys go to so we can make a graph of the sector so this is this is that this is the actual this spherical building which corresponds to the graphical sector in and if you if you have enough I'm going to have a bunch of sectors which all start from the same point then you can just represent the that hysterical itself 3 buildings which is just a graph the statement press correspondent sectors and let me just quickly finished so that the remaining part of the argument says the once again configuration like this to try to put inside a building the amounts to trying to put them that this graph into spherical building a spherical building something sort of covered by accident and no 2 points or to be distance more than 3 apart very tiny segment of length for should do what he should join together so for example we should add a segment length to joining this point to this point 2 to close up for into acting pretty much aid to Paris opposite .period should be joined by a segment of length we've already done this with the yellow what basically happened so if ever are assigned so that the collision phenomena in the BNR example comes from they .period piece here and there .period you here it was found that the couldn't farmers says if we tried to the transforming it from the point here to the point you here that we can get the export just by doing an interval of 1 of the forms along this path you have to integrate this you to integrate form like that if convicted pollution .period we don't .period it's going back like that and then like that to be the 1 and the 3 all over here if you want to feet 3 on the collision phenomena and what that translates into saying here is that these 2 is these 5 the adjoining has 1 2 3 4 5 cannot all be the same that's just
a path is not going to go into a a single apartment building and actually just from the actions of using additional the statement you get from the local WTA venture which is that as you cross over look point for example here when we go from here to here the image of these 2 sectors inside the building his house was twofold local but we think tells us that in a matter of time as we go into a single and the man and the environment is given by form that's was too 4 if you don't have anybody to fold on acting on behalf to start you
then you can put I think basically because what's the menu if you if you try but these 5 guys and they are more than that if we get this guy and this guy in the apartment then that means that there is another factor going directly from year to year but then we would have had 1 we have started cycling for not so as to the building is supposed to be negative weaker hotspots have only 4 sectors to join together the point that these 5 sectors included in the same apartment these 4 sectors now looks like 1 2 3 4 the sector in the world then that there has to be only a single remaining sector joining from here to here but then we would have 1 2 3 and the remaining 4 sectors so that can happen and that's basically phenomenon which forces these 2 guys not and stuff so only just finished with strong this is in the image of the sky was falling we have to be the government's drawing I I think going to joint use of paper together on something which is you 1 of 2 sectors the image that goes into this set pieces just
success sort of and Scott you get
universal buildings with a feet from excellent the building such is for any other the factors through universal an example from 1 of the property that the
factors which is not necessarily summitry on the on the Universal building itself it's a commentary on the part which contains the
point and expert so the
distances and CBI .period enacted the same as those in the universe Goldman that coupled with remember too that we could use this a mega depending on GameCube which gave us the right the next the defense is independent of the choice so now deleted the dilation is calculated as the distance in the building of a fire so that's what we're hoping to say is that generalizations in higher-ranked case of the fact that an insult to taste you can look at the tree which is the space it leaves of affiliation and In some cases under generously of others such as the special curving speeds the W action is equal to the distance transverse defoliation in the space of leaders of the voyage so the hope is that we have a hope that the hope is that this the we will conjecture that something like that will exist in general we're going concern that in this example of something like that would exist in general and that would be some generalization of the space leaves the voyage In our case were doing itself through the pieces of have dimension to service has dimension to so most of our sectors actually cover their corresponding apartment except of course that is the only 1 which only covered a small piece of the pie course hiring case the room service so
as mentioned to formatting and some higher dimension building artificial going so we don't have any idea right now what what kind geometric description you could give of the points of the building which are not In the image the that's the main question is to know if there is some intrinsic description of the points of the building this can be example used for transport to solve the question of the moment what about what they say in the paper that give exports meat that may with a description of pollution and you can see some of the systems phenomena
on reform the health of of the US so right so I would say the spokesman commenting on inclusion Jurassic come about because of the fact that interval the rest of the eastern Gulf called the beauty you have to to worry about the deletion of a so more complications have had no confidence you have to worry well for 1 thing we were just supplying this totally general theory for why haven't we take the women of some some semantic Space Mountain dismounted but we don't necessarily know anything about them and so for
that general theory you need to use the old but have that apply our case but in our case the the WP that the functions of something like you know maybe he they won the plus the EU aid to them if the real part of 1 and a real part of a tour of the site then this might look like something like you on I the it was not authority which is just "quotation mark years this function is equal to 0 for a sequence of point-to-point and so far alter filters had the wrong choice and we chose the points workers and the Eagles 0 then we get 0 rather than 1 that would be severely dented his hopes he can still use the land rights here he would say he student the sum .period switcher account generically chosen so that the proceeds something others approximately usually over half the problem is I and I don't know maybe maybe you not produced by not approvable boss transformed does have an essential security Jonathan securities kind like adding about 1 2 terms like this this and infinitesimally close to each other 2 and I think people have no control whatsoever on what the function of Woodside and 0 function so that you can hope to try to that the places where the function goes to 0 or 2 small and have small arms or major and then try to argue that lawyers and like them or try to prove that the supports transform doesn't have essential singularity the system needs to do so in the spirit of the season of the year .period presidential solution for the soul of because the actual proof that there is no but you know I this strategy it's going to take it on and tell us what happened it's something about this expo which is the size of the lamented it was a it's currently done here and in the course you'd hope that maybe some summer generalizations about that but really give you the exact solutions in some ways but I don't see how they you like Bureau were just talking about you know what's the biggest was the biggest export offenses Russian the worst thing you can write down the vector of dilation explanations saving for example about what would have properties that the bases the I increases
you may wish something which is also for the sake of the this environment no I mean there were claiming that this is true for the vector distances that just because you can think of dissent was just a bacteria powers of that actresses given by working the exterior powers of our regional system and so he and operating but I think it might think is explicitly stated Paris paper that the vector distance on matters base maps to the actors distance implementing building and
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