5/5 Contact homology and virtual fundamental cycles
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Title 
5/5 Contact homology and virtual fundamental cycles

Title of Series  
Part Number 
05

Number of Parts 
05

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License 
CC Attribution 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
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Release Date 
2015

Language 
English

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Subject Area  
Abstract 
The goal of this course is to give a construction of contact homology in the sense of EliashbergGiventalHofer. I will begin with an introduction to contact geometry, pseudoholomorphic curves as introduced by Gromov, and some target applications of contact homology. The main focus will then be on extracting enough enumerative information (i.e. "virtual fundamental cycles") out of the relevant moduli spaces of pseudoholomorphic curves. I will present a general framework for doing this, and then discuss the specific application to contact homology.

00:00
Point (geometry)
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Computer programming
Complex (psychology)
Group action
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Translation (relic)
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07:47
Point (geometry)
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Group action
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Link (knot theory)
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Maxima and minima
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Glattheit <Mathematik>
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Local ring
Spacetime
16:33
Axiom of choice
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31:44
Point (geometry)
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42:26
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Summierbarkeit
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Module (mathematics)
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Doubling the cube
Glattheit <Mathematik>
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Family
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1:00:27
Point (geometry)
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Potenz <Mathematik>
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1:07:15
Complex (psychology)
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Uniformer Raum
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Term (mathematics)
Operator (mathematics)
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Nichtlineares Gleichungssystem
Set theory
Weight
Projective plane
Physical law
Mathematical analysis
Algebraic structure
Variance
Equivalence relation
Vector field
Vertex (graph theory)
Glattheit <Mathematik>
Fiber bundle
Exponentialabbildung
Nearring
1:20:50
Logical constant
Equaliser (mathematics)
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Amenable group
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Potenz <Mathematik>
Gamma function
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Order (biology)
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Glattheit <Mathematik>
Local ring
00:02
now do you know the old and the members of the crew group members who were in the room her up a and we don't want them to do and the government in only said today I think on both the crumbling so that is found where 1 Chavez describes the and the local structure the modular space near arm curve suggests but Kearsarge provision broken at the airport finally the images recalling the seller so we have our target supply of Nevada contact form and you have our almost complex structure on Oct Crossland around this was the following the in property so it should be on R J vote yes it's includes the Butterfield and Jason's accede to itself and restricted Dixie is compatible with the latter on this land and we didn't by Peters set both closed orbits of and premature as and now I told the gluing is simply a statement about 1 fixed much less space and so I made sure that fix once and for all the hourplus film ideas and have walls of Kurds poured into the the whole of year bites so on In Italy we fix this data and then much less space and bar it just a set of curves Chris analysis the the program on my lips and it's not yet assessed 1 now and Britain's Got plus and these pheromones and that's closest each of these is up the whole Norfolk curved mapping to our cross and on it's consider up the translation each of these individual pieces is considered up to translations modification and on in hotspots that so From now on all just forget about job plus demo myosin beta a enough annotation to adjust interest this exports less space and no bite and paraded the locusts where there the nearest operators suggested so than those me the there said the 1st Anderson open and here any point it's found there exists a local polymorphisms the party had crossed over to the and case 2 the heartbreak sending 0 something 0 2 the debates .period and were on it's a virtual dimension and bar case in a number of nodes of this at this point the and the and and that's compatible With the stratification that is the lefthand side of stratified by which of these 1st take auditors 0 and about on the righthand side of stratified by which at which places did we resolve it's compatible with the stratification and that compatibility with stratification says all he comes serve to really from the construction of course it's it's crucial to include an actual statement because the only efficient type of this depends on whether K 0 possible stratification is important data keep track of their movies to know you little yet and like so have the 1st remark is that In action in the elected to to establish contact multi we need something slightly stronger this stronger than this what we need is actually need to establish the same statement for for the stick and much less racist remarks it
07:49
this the addition of these serve thickening terms in this much less space so it doesn't really change the agreement on and so forth its simplicity and explain the proof of this some Kansas moreover we will not only do we have to deal with the 2nd
08:13
guys we also have to prove on the submersion property ,comma we have to be suspicious of this embarked paraded inside emerge a rate is cut out transfer facilities like this we need this to be a need to be a topological submission 0 so all will let me not say more about this right now it's mean is exactly the submersion action emphasis was definition which I check it right but the point is just that of both of these are are really only slight extensions of the proof of that and
09:05
other markers that come we don't need to establish any sort of compatibility results on compatible the properties between these local :colon charts under just local honey morphism come probably if if 1 works hard enough on news including profiles Our 120 1 can show that we do we get a smooth structure on the space as a transition mastodons charter smooth but on but this is not needed for further that was 1 of the most people here in the United States we something about it it and that was for more people when we were in images of the of the situation and on you can think of it like that I own mean but I think in the end the same and you can you can write something is being hedging in homogeneous equation Oregon write something as the true whole Moffat curb equation on the bigger thing and I think the thing that when it comes down to the of an injection after after chakra the same no matter which prospective right so and that the closest to result in the literature there was apparently come as far as I know this is result clearing the of James Thompson leave in 2009 in the construction of embedded come technology so on what they do is they they provide link them for embedded curves In and that it isn't it is very important that they they blingfor as if physically on In Barcroft of 3 metaphor on the fact that there the result of somehow more much more involved in this statement because they have this this extra thing that they do that they stick on on press coverage of trivial cylinders in between the Kurds that the and they also have this obstruction model willing to happen so the thing somehow the techniques which go into this sector of basically the I come from the same body of work that this license will serve originates from the gage theory literature with top salads and income hammer out and they eventually will serve but applied the whole market curves and by Florence attitudinal 1st so I just want show how these techniques are applied to prove this theory in the case of contact mulch kills himself still conversely have to fix at this point which instead of calling American can collects civic service .period 6 0 In Oregon and that really is is map Israel from some curve C 0 2 x 0 so so this this is the domain perhaps this is this target which is the bunch of industry Union a bunch of copies of our Arkansas I'm just going call expect that on the new notes the index index of this modular space that is the the virtual dimensions 2 0 will note but the dual graph 1 of his 0 the real draw for example so you take this example so many users see 0 the next x 0 hat is the distant union of for copies of all across separate copy for each of the components of the curve on the map was considerably after translation is more than an empty 0 it is the dual grab this "quotation mark on the left the 0 this is just the sort of irreducible components of C 0 and he says the interior edges are really really here really mean interior edges so this is the intermediate work you don't have to go along with of ESL In the standardization are crosswise exact so there will be a good time but but 1 could want to also add a bubbles on in that case I'm not going to keep track of the bubbles in the inventory just because just because we don't need that stratification to and you can call it that yeah yeah yesterday and for the purposes of this argument really doesn't matter the genus but but yet Symington practice an hour goal is this to construct a
16:43
map or local army officer during a mail from this too here while respecting specifications and landing on the regular so this Tom right into the German homey morphism that means it covers an open neighborhood of x 0 and since lands and Bahrain that means and contains an open neighborhood of x 0 in Rhodesia and is open and to since assist you forever it's as tho we have to show it's so so 1st set clear ideas on domain stabilization so home we really want to that was the purpose of this step Over a neighborhood of this . 6 0 in the much less space we want to their fix canonical coordinates on domain the problem is before this and this these these 2 components but there they are by whole Moffat with cylinder in the plane respectively but the not canonically self and we have to get rid of this regret this extra freedom disengage fix In order to be able to produce local bowl on parameters for the much less space this cell will do is fixed .period skew I can see 0 and who are invariant devices DI inside our crosswise on such that you 0 is transfers to the acts you up I don't care whether it's transfers to DI anywhere else just at that to us such this holds and c 0 equipped with these points is stable it also has the support here's the really see zeros already equipped with on its marking by the positive and negative function then so yet so it has to stabilize the look something like this since the I and you are beaten so now since you zeros transfers to the eye to eye this property Is this preserved and the small deformations of the home of a curve that is over an open neighborhood of x 0 you corresponding points in July in in the domain of all those other home and therefore but you've done this gage fixing over underground and so 2nd introduced include coworkers on target this is something that will still go forward growing problem of all :colon element of this the state's including prominent on we define this include targets for the obtained from 6 0 by truncating containing the ends in the horror the and stands look like 0 inventory crosswise to trumpet them 2 0 it would come from swine the same thing for negative and then including fire this the 2nd so thanks but other described as to whether he was the the idea you ask for and so were latest credit it's not entirely accurate so I suppose for taking this curve so we have x 0 6 4 copies of our wide let me just draw and has distraught factor looks something like this in supposing glue all of the parameters using exactly this recipe I get the this 2nd extra hot X she had done the success I've had this 1st piece 2nd piece I good both of these here to get something sorted 9 houses like this Of the fallout of the Sept . 1 1 House of target lead the service of an canonical way of disclosing a backer 2 . so as X again the exchief of course again just that a big union copies of our cross but somehow this particular description is the 1 it was the review year yes
24:15
if if if if we had something more more general than we have to the then that there be some constraint on a space and promise pretty so not it's important to keep track of the 0 sections here so for the in mean that there is a at Verveer useful component of the original curve on 11 0 so the inside expires 0 at X had not on the notes for a corresponding 0 section hands and left the corresponding bite but the corresponding 0 section it's the center on sex preservation canonical yeah 0 0 Crosswhite here at some points for each component of the original Kirk all I know that finally this is going to cost when I was 6 sections Q The Prime of this universal curve over on but billing month once space of curves of June's 0 pound cooked points marked by the Input Output edges and these extra market points to us police our product comment earlier that by choosing these devices the we identified the domains of all the whole curves and all .period amounts of space neighborhood of x 0 with fibers of of this man that was the purpose of adding these devices to gage fixed the target but was the domain of the it's 6 of them well I do now get fixed a target and gage fixing the target uses the sections so sections QB prime such that when you were evaluate them act this at C 0 on the land each she lands in the corresponding irreducible proponents of Caesar so there is 1 point this year the prime In every irreducible component of of C 0 what the the "quotation mark it this follows you wherever you go the result I'm a little surprised it is on top I just pieces the she's the 1 yet yesterday half of that again it's 0 rail so that a needle and did the visitor sections defined in the neighborhood of teases me obviously that maybe this whole footbridge that section we interval locally yes and a little point of affixing these sections QB primers to gage fixed a target for going too now only look at maps which sent QB prime to the corresponding 0 section 0 Christopher good not be the age the case fix that the main fire on these which are essentially put the inverse image of the and the gage fixed targets by requiring the use of primer From what I have jurisdiction so really what that means is that at . close to X 0 I get a unique including prominent so few Mac Kostelic's 0 you're new domain is cannot be identified With on now with all the fibers of this universal view these points QI from infecting with the eye I gives you .period skew the crime by this arbitrary choice of sections and then this requirement that you the prime the maps to them 0 0 section I told you at the moon Prada has to be just this length here is exactly that OK so I think this is all gage fixing we need new systems good In you could say here have a look at just for all the decision how of to use the on great
31:56
so all come from right so were going to furthermore fix trivialization of and the universal curve over this because of my exactly universal curve here in Iowa it's such realization something slightly different so and I'm going to do not by and bar 0 chairmanship a compact find much less basic residues 0 with that was unmarked points but the 1st and Mark flights some are in addition currently equipped the marking of the tangent space by ice eh so this is a question forgetful map and this is a principal G 1 at the end is classified the victims classifies on curves see with on the 1 she was she hand signal From Sea to the attendance based upon us the years the we will decide it would be in the subs so come and lower parentheses too this was what I care about this In curve within with extra mocking him so when you have on agenda so if you have your mouth that's what care together the whole Warford maps they you from Standard have cylinder to our y as entirely who Ray Warburton on means means that you are as comity is equal to length of the rape or attends s plus some constant Ganic told us of TV of course 1 as as as close to "quotation mark however tell Prime is privatization of this the report its asymptotic but the point here now is that we but the that the cordon it's on our cross crosswise on give rise to marking of the attended space of the come by sea need namely by serve requiring that Gemma told Beazer not rotated all and B B 0 that is somehow this is the way I want think of on these curves I think it's C 0 as equal to the marking of the attended space at that found that each of them the punches and with respect to those Coronets I want to B 2 B 0 and genital to be not protected I think so you know now we have Blix cordon local trivialization of this small expects 6 local diffeomorphisms J and J B L denotes C 2 times "quotation mark just mention this much was based solely for each irreducible components reach and we fix a local diffeomorphisms office of this process based on where you have the points Q I and you have attended space market agents the punches still right now I also affixed to her Coronets connects 0 . 6 0 that is a fix apartments like this before Of these these are valid for press officially lodged on neural the punches and punches on such steps on with respect to the square we have you 0 Estes asymptotic to the length of the corresponding rate tends asset down the course show has asked its fairly
39:26
and 5 per cent I mean that rateable the point here is that the discordant B 0 and that this is not not listed the myself so finally think the last piece of 7 is we have to go there the curves with a domain so the ladies who incriminated gene g lying in far agrarian 0 the this should be thought of as the gloom premise for the target but the going parameters for the sauce bill would you put 6 G on the to basically we do exactly the same operation so we truncate the 0 problem ,comma from which ended 2 0 GTE conversion and we need a factor of the length of the report also it is this is 1 reason why right because of trivial soldier right has the specter of an hour In itself so the glue that the cobordism by truncating a translating by GE universe with better good by by g inverse times help I pretzels always was there and identified this before so picture looks like this we have the see c 0 this is a puncture never ended that's 0 on the trunk at the ends to cut off here the man who couldn't get out this since seating now I want to come
42:31
pregnant and the rotational corner is chosen on then fell by by you have the asymptotic matching of the other 2 ends the Belfast enticed the same the so early it's crucial introduces fixer .period To the prime In a center each glued edge entrances to the double crime in CG exists for every E which has a possible and this is this is needed for some gage fixing the problem somehow as you have threatened so what happened the money a woman who is the GC say it's the point in the middle of inexact middle of the the group the neck in this fix that aimed just that takes such a point on In the best 1 corner doesn't but just as long as it takes some the defining so what White why do we need this so but the point is that there is it's a france so I have these families of almost from the trivialization of modular space on the beach of each of these components but that somehow only tells us about curves on this stratum Wilson incursions on the other stuff too so so what happens the fixer G we give about come from this product of Jason to come to what well the product over all vertices cover of the of the glued treaty 0 so it can look at with eyes which lie hidden the vertices which which meant there Qantas to redouble crimes in the edges which collapsed plus some punches and they have they have double what what just the point attended is also so this is a local film office which and this this would not be true if we did not conclude that the date of Q double price when that the dimension wouldn't even be correct that's we if we just do this hearing if it we just do this link instruction look at all possible almost complex structures J. we 2 extra degrees of freedom because but basically because of we have now will note the marking of the tendency of the tender space peace but the sentence was refused it is this African some served keep .period on but a local differ markedly the idea of of the emphasis this only a homing office Donna it smooth that I think that only smooth method local homey workers and I think that's all actually all "quotation mark the solution you need to have to do you use girl what the report states that the cost of you and the shops this is the I only need local park near the particular fervor could you don't know who is the local yet I mean it's a glowing is akin to create local state so that there no more than a year yes for the construction the happenstance the yeah I don't need if it's not it's not really use either come yeah that the some right In certain setups you might you new might Serbian just in the question of the much like the look of this much less space I know you don't look at this much speculative just doesn't have a section this is basically saying all it is is there for global way of choosing asymptotic market this is I think this is not true believers in a section .period but guess for 1st this in the summit we don't we don't need that section but there I no I for work the case along with no defined the gage fixed everything in the near .period on we the find cobras prisms to good curves the the final thing is that we have to define the good laps so of course 1st reading that created perhaps Yuji itself it takes ,comma function 5 and officials of the 2 steps so 1st define how they all call flattened that you 0 Laboratory on from 6 0 at 10 0 so what is was the picture so you Matthew approaches a trivial cylinder the travails on the here that's what methods 0 this cylinder and real and that is going to cut you cut
51:28
off you 0 To materials on the solicitors all hand in symbols of this so we take you 0 4 Richards a capital ass it will be notes they don't Jeanne biased 1 side of the strength instead will do no back now I remember I said Michael S. on services for and here we interpolate but this taken an exponential trivialization of served diffeomorphisms yeah right take existential Coronets near this rebarbative and and just cut off this is for us 160 this is yes something like that yeah yet it does doesn't mean gathering things things will happen Over the NEC and this is the 1st thing that happens a lot of things happen the other things that happened or will happen serve only hammered away on the part of the 0 G which is travails on the it's going to get you he said the ideas alive ideas even begin Gamez from interest orbital never fix whenever something from tries the top precious on we get to this preloaded maps on the UGC from 17 exhead GE is just the center of his 0 g map agrees with descriptions to search so know that we have so is the skills maps for every I still think we map from forever going from where we get something this map sort out Malta right here it's Of course I can write embarks this is matters homework but will we wanna do somehow correct this map to be whole office so that lands in in embarked and we also I want to on somehow had is extra so members on the left and then we are today new minus the and Chris once the cradle of the linear separated at this point 2 1 2 hours I'm going to give him some actual analysis here so the ruling the 1st time I need to tell you waited so a Lawrence you so we need some waited Sobolev spaces of functions and maps sections over C 0 2 0 credibility as bases like W K P sections Of the pullback of attention bundle of xg happened CG so define where hoping head the dog learns on here in the end to the next but the following in his hand ,comma something like this so basically me and we just added a factor of the 2 the Delta to the function for most part of onto my factored into the darkness is the functions here at the Catholic heated up the on average undisturbed writing the contribution to the peace power of the norm over the sand and away from the ends it's just the usual teacher long and then In the next something like 0 2 capital last SES 1 of the cost something similar another weight is the hope times the minimum of tests and capitalist practices so it found that the way it looks like this he felt as and enact tho it looks like the EU to the delta and minimal as To this somehow that the norms which are relevant for this problem so you dealt this is small positive number and by small uneven you can say precisely what you need mean by small arms to specifically I felt there should be less than that small side in value of the asymptotic nearest operator some of the report In in In in each of the and Yemen this home analysis this is where we really need on the ants were where we see the need to have and speed on the job and rated as 1 of the places the other places in the Lopburi Aston at if you have a and the trivial cylinder over 90 generate but then if you take a C 0 another 8 Holmer Africa which is 0 close to trivial cylinder over a montage everywhere with the creek decays exponentially for that not that I'm travails on the exponentially on meeting at least as fast as eating the minus Delta work for any Delta less than the 1st
1:00:27
positive items this fundamental after estimates of but also that this is the fines for us always somewhat spaces in this Over over on each of the glued courtesy subject and why it would give it suffices to take people's 2 for this is that it's come this argument we don't need any harm anything else but of course we we could call me everything and anything this is true for any pianist range those on Wall both of you not for this is Chris Waddle has this blog post about this a little of the need for the "quotation mark whether at any point you need to use this the people the sofa from her for this it's it's really not not not needed on I think that was the point yeah we somehow given all the opera estimates which which you know which which sometimes require people with which up Bodmin maybe so maybe they'll never required equals 1 we somewhere directed me can you pickles 1 things necessary the solutions to some of the injuries were the main square requirements I would be here I think it really doesn't mean people that equals the really fine I played you already know you're only dealing with curves which you've already prove after estimates for and me we take a could can take a very large yeah I think the difference between peoples yeah the weight of the issue with those the you might take over the In the last few years I year it is a killed large right of isolated should summarize since we reviewed various his saying the wrong thing so it fervor for this for this schooling argument all the curves are we have were lurches doing this argument after we've proved all the upper estimates and we know our curves are very highly different Seibel just fix a large on Kansas case largely can take people's to there were 1 maybe we the people if you wish to hold 4 this was the year the the island regions and beginning I mean I don't know anything about but this the a little it's unusual to structure will use it to make calls you it would the world is used to reduce size of the well them you know the rules of origin the price also millionaire what we have readily 1 understands the modular space of whole market curves near precluded and so on what we do is of any kind such matters exponential of some exponential of some effective field and on the most complex structure on the domain although actually were Nike's all vector fields so we have to preserve this condition of intersecting D I X Q I still only conservative of space where sea of Q I'd estimate tens of space at the eye and we also may consider some space Delaware this was the same conditions at Shui double was so we have this point futile prime now and then In order to gage fixed the domain again we we have to take that the specified to redouble prime
1:07:16
on opery specify which you double prime on domains of all the Kurds in the modular space near near x 0 so won Wednesday's you double prime is that the inverse image 1st tried in the image of the all of this space under the exponential because regret that as a local devices the inverse image of that under the map will be treatable the story of a vector field under most complex structure and from vector field and almost comic structure we can ask followed bar and this section I J on the on the source policy were varying jailers and I think if he raised a banner day with the idea was this from set of almost complex structures on the on the soft yeah never lived almost comic strip targets always fixed so I want to be Rachel is this now as the so selected takes vector field and almost public structure I'll do know by J. supply on there so sold reading so X U G see that's the map on we take differential take the on the whole Moffat part suspected this J. while most complex structure and jail target this the section Of over CG not of this bundle but rather this it is in the the sections W. payments when Delta sections this part all of all maker 0 1 CG with the chair while most complex structure and with X U G C pullback of Extended Fund so to play here on Billy's parallel transport he promised part is Carol transport along part use some some fixed identification that this doesn't does really matters some complex anew no so apparel transfer for 4 8 j Woodmere connection so many of them there was a tense Republican sense of jailing your connection on no 1 except those are across states on variance to the new connections just for to be safe and I attend services just fixed identification that doesn't really matter what what we do here this summer any map which depend witches see linear and smooth and depends mostly on as the identity for clues alive on the undone with the definition because we also this other component with target so he added and so what is with this component telling us this tells us that vertical positions of all of the the images of all these Cuvee Proton's the Trenberth constraints we did this gage fixing words were now interested in in maps which map QB crimes to the corresponding zeros so we take the next day you'd see see holes in the prime of why so read it to the crime of why because at this point in In this season In C 0 mcg really depends on almost complex after 1 yeah I was going say Q the promise a section of the universal curve so they have this and then we come to this project are the settlement of our cross that dark and we we take we do this for all the ins and outs of all the QB crimes so now with this complicated definitions on for much less space yeah near x 0 the Naxera are much less space to the union over G end of and I'm writing like this but of course I really just mean small small Chrysler's due to the germ of identification on of and this local this gives a like a local snapped from 4 M bar too the space :colon premise but this is not all canonical mappings
1:14:02
depended on traces of of devices and sections QB press and so tight as far as I know all of this should should get works for ratios another comment about the case equals 1 versus larger case should apologize for a string up this discussion 1st time on yeah this this operator on what makes sense seems to make sense for any doubt any K 1 in any peace between 1 and and if you have case 1 and p 2 worlds smaller or if it to and he equal to what I guess we'll see about that and then sections here not continuous which is all but disconcerting on that yet race of course does not and of course yes the need you need this right because this we can do this expansion Texas was only makes sense for on indicate these 3 April 2 and the greater than 1 work April 1 and the greater than to press the differential does not make sense for any and but in takeaways from the and we action so for the glowing Arkansas doing right now I was just whole Norfolk with adjustable forfeiture curve equation but we could really do with any and any of these that we want to it's certain somehow the Fed home analysis is a bit easier With the people's too on and but now when we do this the same thing for the thick and much less spaces it's really important to be able to use large cash because there's an extra terms investors part of the equation the thickening terms and depends on the places where this exponentiation curve intersect certain devices In this mismatch I'm from the space succeed 2 the CG given by take the inverse image under this exponentiation map of affixed devices of this map is a regularity class something depending on cat something seek a minus 2 months of the 1 the United smooth map we can apply inverse function there and to understand that 0 said we can be large a lot of of work so you all you want you she did not the you have to go yes yes I mean if a thank you great so another few fundamental estimates which could spend up collected in the calculations for but not from Saudi just right on something which 1 can prove about this matter so 1st of all estimated the norm of 2 0 0 sold them off with a lot of 0 that is so this measures how far the printed matter is from being whole markets refusing using the law understood 0 as the justices hyperreal although the first one comment about the norms I mean you can define like norms with like actual explicit formulas including the trivialization of of these bundles and in the end but but all that really matters is the Comanche ability class the unified commands a growing class of these norms uniform in jail take big norms which is controlled the commenter Billy constant which doesn't depend on GEN the fine to the point is just to have estimates which are true for all the business and why the grantee fixed gene under this Miss North defined by this weight is controls the norm defined by not doing anyway but 1 that is used to goes to 0 per cent not uniformly commentary also the tune it's little 1 form part of the skies 2 parts the 1 former member of the 0 1 4 vertical Coronet part of someone from part of of active 0 is supported in this region he recalled a definition that was just saying Where Eugene not we're not J. Holt Norfolk on was a splicing together the whole Moffat diet trivial solution the only place that could it's might not be homework because in the screen part of the spice
1:20:52
on on an elevator now tower just use on the exponential convergence of unity to trivial cylinder and the fact that Delta is less than that I don't so or we have a waiting function which is divest that but this difference that we cut off following In the case by even faster than children's topless lost the party and then the 2nd part for this part is just the statement that the prime of CG 2 0 approaches the descent both are this DVD crime Caesar J 0 on just because as as chief of the 0 because the sanctions I got into this so there was some calculation to do but this is so the other thing we really have give a key part of understanding the 0 set of FG of 0 near sea ,comma Y equals 0 is the falling the contract estimates and what the should be really thought of as a uniform bound on this this modules of smoothness of attempting something on basically saying at this in like the efficicacy 3 map from prospects fastpaced and moreover were giving an effective bound on its C 3 Norm independent of G all I know we actually don't need to be C 3 we needed to be we need something considerably weaker it's it's and what we need is this the proposition for the House take derivatives the derivative and 0 is close to the Martin was a derivative of its 0 minus the derivative people and there astounded by constant said so some statement which is very similar to stay derivatives the uniformly for leased 4 C all is capable of at least 4 condition no 1 is really only needed when you have the thickening terms in a sexual devices for for for this guy's it's no should be true for any case also I won't professor good even so now we come to power which has and most of all of the content and services to construct a down the right inverse so so let on the GE the note this derivative of FG at 0 on the mission there in the direction of a 2nd argument the best was all it is sorry about this is terrible notation it's Zeta and overhears a and C are both elements of this hope the new world order the yes and miss this notation pyramid of measurements taken at different 0 is an element of we really 0 ,comma 0 KSE usually CSE ,comma y traders they become "quotation mark president of the state as soon as the driven that act 0 ,comma 0 this operation so it's a linear map from the source to the top of the market but the derivative of FGD at so stingy socalled organized by assumption in 0 is subjective on fully want 2 shows that DG is also rejected on although we that more clearly show that 1 although they write verse for D 0 constructor right the strategic and and it'll it'll have to keep properties 1 is that on 1 at the exact given the Chretien version for the 0 we will construction security here she this is the right verse GE such that huge and have uniformly bounded independent of GE and the image of G for this and "quotation mark varies continuously in safe precisely what for them then in in what sense varies continuously in gene that means it the spaces don't very nicely in terms of gene so it's not really is this knowledge well defined the obvious notion of various continuously but also need I need to see what I mean by that process later it said so on this construction is is really a very handson construction given given the writing verse the 0 we simply sort of down of the right inverse for the 1st 4 DG so that we have the key diagrams the following people look at this and say 0 see you I didn't in the kind of job so this gets parallel transported with W K 2 Celtic Sea 0 and flattened guy again with this map which would map which at its peak Texas traced to secrecy and finally phobia calibration that takes this and that satisfies this extra conditions at 2 the double Pratt supply RTS posts are as but I know we have all these In the nearest operators this beginning of transport there's a man in the other direction :colon break and this is the same as they walked right again I would be in the calibrate the racial calibration services that this is something extra which have had which has happened on 4 continents tumultuous which doesn't have a need to happen on when you gluing just nodes and Agrofert theory or even Hamilton the theory on yet the pointers you know we between 10 service this condition ornithologist sketched the definitions of these maps the final great so here have seen 0 and here we have seen the so did I. the gloom man so for glued so including map exposed to giveandtake section over C 0 and give a section of CG so it going to do credit off here cough and a cough here so these at for the sequences series of complicated operation so this is going to happen at something like or 1 3rd in the region where where we've already made attributes under and then there some some come with a school some of them on this map of income because another map break is as we pull back we cut off at the center I thought that was cancel the way we can cut off and for the glue that is not really well use whatever come from the way we cut off for the break that this is crucial peace come on so forceful has happened center and secondly meet these these color functions have to be symmetric that is when you if you break mn serve take the son of fibers back down should get the same thing is this cut functions 1 minus this kind of function all of us and finding finally finding calibration mapped so here's because how picture love scene genes amazes vector field which is just constant combat call off so we look at this ,comma function so this is kind off High Times the so here's you double prime and this is vector field so it can calibrate on simply adds cracked multiple see all of this so that this condition projection about director Phil becomes now 1 observer both of these maps all 3 of these maps of the accident all the maps and diagrams are we are uniformly bounded for for good off check for cover the calibration of 2 it's important here on the Delta is positive for this to be for this to be uniformly bounded In the portend a Q U double Prime Minister in the exact center of the Of the next phase of the plan with a maximum weight so this constant vector field here on which is In the colonel d up the the operator as costeffective field in here the usual Norris has constant or 1 but but some are not the right not to think of it and we should think of it in waited and In the wake nor does the norm Of this constant vector field looks something like this it reaches a maximum at versus a nonMuslim accident this 1 In in in the center both Of the sky too he double and then decays exponentially as we go 1st to set boundary of and this is somehow white wine is an OK thing to do ,comma cut off here because whatever CSE is at that you double prime it's very small it's like to the minus onehalf capital Delta times the norm of CSE so this thing that we added is actually very small and a and the what so he faces key diagram does not commuter but almost commutes and misses basically the property we need to use it to construct this down the dreaded workers so he digressed from he meets the high so statement is following the 1st repair will transport the statement very simple commentator the lapse approaches 0 in March so severely 0 B statements as just 0 a 2nd statement concerns the bottom this goes to 0 and finally the statement further for the middle squares will be complicated because and bragged on opposite directions on and that the statement that you can is both efficient and can actually be as the following this norm is little told 1 times more data only where break quite a agency quoted DE Reggie C ,comma than what a lot of fun treatment of doing the sort of the 2 an the 2 upper squares here sir combined into until 1 I'm history comments not considered this flattening I you 0 party on but I think it's a good serve intermediate Parkinson's divides 2 constituents he said switches were different it's also a very important that we coming off this happening in the in the region where you 0 jeez is a trivial cylinder which makes the estimate much the maker doesn't make all that much easier than this there is less to worry about we would like to hold to the book he was also the year for refer all yet so it's all the housework so so see also csce lies here and it was here and but not to the same thing here an understatement when you it's a them this way it certainly takes see this way all of this that not to the same thing here and statement that we'd take see around this way it's recalls service strange sounding statement by I'll know any of the formulation which is both correct and simpler the state me that you could ask that like I don't like going this way and going this way I really close for the century for each of you here it's not true or it's not enough to show what we once 1 of the 2 I also on behalf of all so mean many of the same words like this is true that mean when 1 can some explicit just explicit calculation on for this point is again just you you 0 did the decays exponentially on too trivial cylinder and I the only really differ over the part where you cut off you 0 2 travails on the on and over that part on now we have assessed that that use Iraq which real cylinder for at least the exponentially with any Delta strictly than Delta Gamma and are chosen Delta's less than gamma so so we have a system on the calibration won the point is just that this vector field the SCG is it is in the kernel of full nearest operators just corresponds to translate into real sold a rotating materials on the on and off they would cut off all but the point is that cut off happened really far away from Q G and so the nor that use product will expand this year the list of services for small again like the demand something comes Delta so it's crucial that Delta's positive from following well I finally this such this yes men may be made of sponsors such amenities in explicitly analyze what's going on .period is somehow that unfold the weight but the key point is somehow to compare weights on the 0 you have weights which look like this and CGU social like this and what he turned to Chapter of the error terminus difference turns out to be on something supported here this type of region but the normally care about is this norm a balance on its Norman in the snow and so that the fact that the ratio between these 2 norms responsible and 0 and justice the case analysis justifying use abounded Friday versus itself what I mean varying continuously and G this is server crucial for the event will not be continuous in this this image really has the very continuously Angie persists and and this is somehow a nontrivial state because the space glowing promises to look something like this and In continuity surfeit for this glowing primary and the school imprimatur complementary so Sony continues GE itself will just 6 some linear map from the source of the UK the Delta To some vector space given by like a Dennis on evaluating at some points sofas so for any any substrates on such that the L 0 restricted to a colonel prompted the 0 isn't my small the fix some points and look at this evaluation map Christopher C. pick some pointers evaluation that is is injected on the colonel curls fundamental so you can certainly do this exam .period on the domain C 0 substances Ganges fight dimensional can do this common and closing down by suitable subsidies the still it's actually on itself this of course gives us a corresponding math LG on on the domain of etched in operating began just no Our going to fix on right 6 Q 0 with which is the writing verse so the 0 0 steadily and the image of Q 0 is the colonel of El 0 Sincell 0 restricted to the kernel of the 0 1 is an isomorphism on the kernel all zeros a compliment for the of the 0 and given futuristic complement determined that the just probably term uniquely in just about every close complement is this continues map of the closed on racial this the slammer implies moment diagram implies all this guy GE which by definition is power creation composed of gluing composed power transfer composed of 2 0 "quotation mark transport and breaking it it is the approximate break inverse 4 you meaning that this difference DG in G Minor said there is less than 1 in fact the goes to 0 as he goes to 0 I thought I mean what happens if we fix our are right in verse In the year 2 0 and we define our approximate reimbursed by just going around and that's that's the thing Of course it is a diagram commuted on those on this would be a true writing versus a profit connectivity gives us that some approximate the inverse Pompano on Wikinews the standard geometric series 2 2 in the office just get true writing verse future with the same covered by construction the image of T G is contained in the colonel Kovalchuk From the space just falls by definition we so T followers boy passed it is some loss of maps are we apply to 0 and we went in the the cradle of L 0 every would've been definition of power transport calibration the somehow only happening on the on the next these evaluation points is for sufficiently small Jia not in the next themselves this not sort composition commutes with the Elf this is our conclusion right now an index calculation shows that this is a nice office so when contracted the index of DG and it's what will be the same as the 0 on so and so 1 concludes that this is an equality population I'm I guess really fervor combined with this races so we end of really we also get up the colonel precluding maps colonel :colon isomorphism from Colonel the 0 2 the :colon the picture of Stephen by simply applying these maps and taking the projection onto the colonel induced by this particular justified this is just projection off of chronology election finally we can the the cleanup which gives us so we want to do so so the founded right injury Terrence local diffeomorphisms between the Kremlin of the DG and the image of 0 so this is basically the implicit function there all had implicitly also of course using crucially in here goes the estimate of enormous Beth FG of of 0 come and the mountains of this writing verse and the quadratic estimates so the picture is like this colonel the chief on the image of queued uses that complemented hands there's some manifold here which is f g inverse of 0 on the map here is just projection it was in the kitchen so this is the man and this gives us a local diffeomorphisms rejection right so the moreover this is valid in a ball of some fixed size independent energy and that's the through the boundedness it is important so so combining these so for combining we get combining for 4 different values of GE forget this map from the past cruelty 0 2 and far X using a chosen ice more efficient use of of the crime up to now claims the is 1 continuous 2 objective meaning its restriction to some neighborhood of 0 is injected and surjective meaning justices rejected means that on the face of a germ means that on for every neighborhood of 0 In the source said there exists a neighborhood of x 0 on the target but which is covered by this map restricted the neighborhood at the source always the least through properties taken together applied at the local army morphism since that sources local plastic on so was the content here on surjective leaves the content here is using these he operated estimates of offerors Huskies under about next which are C 0 close to trivial so others injectivity on essentially follows from the definition of this stuff was talking about gage fixing at the beginning of the continuity of the contents Is that will somehow this in the image of kinky on the various continuously on and what I mean by that is just the image of is given by the the crime of this matter algae which hello how how redefined varies continuously ship which we should be satisfied by this evaluation appoints map on the content here basically is that we have as much from Colonel teaching to pray this is the right we have scrawled glowing from curled a 0 rolled each canvas 10 and ended the images the and image should be prescribed so this this is approved thanks for seems