Fredholm theory and DeligneMumford spaces for witch balls
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Title 
Fredholm theory and DeligneMumford spaces for witch balls

Title of Series  
Number of Parts 
36

Author 

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. 
DOI  
Publisher 
Institut des Hautes Études Scientifiques (IHÉS)

Release Date 
2015

Language 
English

Content Metadata
Subject Area  
Abstract 
In workinprogress with Katrin Wehrheim, we aim to bind together the Fukaya categories of many different symplectic manifolds into a single algebraic object. This object is the "symplectic Ainfinity2category", whose objects are symplectic manifolds, and where hom(M,N):=Fuk(MxN). At the core of our project are witch balls  certain pseudoholomorphic quilts with figure eight singularity. I will discuss recent progress: toward the construction of the moduli space of domains on one hand, and toward establishing the Fredholm property on the other.

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FredholmTheorie
Group action
Spacetime
Multiplication sign
Sheaf (mathematics)
Theory
Infinity
Quilt
Frequency
Category of being
Lecture/Conference
Ideal (ethics)
Resultant
01:03
Point (geometry)
Geometry
Standard error
Group action
Symplectic manifold
State of matter
Variety (linguistics)
Image resolution
Correspondence (mathematics)
Multiplication sign
1 (number)
Manifold
Sheaf (mathematics)
Student's ttest
Subset
Morphismus
Sign (mathematics)
Manysorted logic
Positional notation
Lecture/Conference
Operator (mathematics)
Energy level
Subtraction
Graph (mathematics)
Product (category theory)
Prisoner's dilemma
Polygon
Mathematical analysis
Infinity
Incidence algebra
Set (mathematics)
Sequence
Euclidean vector
Local Group
Flow separation
Category of being
Field extension
Funktor
Symmetry (physics)
Duality (mathematics)
Mathematical singularity
Dependent and independent variables
Right angle
Object (grammar)
Arithmetic progression
Resultant
Matching (graph theory)
12:28
Symplectic manifold
Diagonal
Correspondence (mathematics)
Multiplication sign
Connectivity (graph theory)
Sheaf (mathematics)
Manifold
Theory
Hypothesis
Pi
Lecture/Conference
Boundary value problem
Energy level
Circle
Condition number
Focus (optics)
Product (category theory)
Graph (mathematics)
Projective plane
Set (mathematics)
Functional (mathematics)
Bilinear form
Quilt
Category of being
Order (biology)
Dependent and independent variables
Normal (geometry)
Waveform
Untermannigfaltigkeit
17:36
Confidence interval
Multiplication sign
1 (number)
Function (mathematics)
Mereology
Expected value
Explosion
Manysorted logic
Velocity
Circle
Physical system
Process (computing)
Product (category theory)
Theory of relativity
3 (number)
Sequence
Category of being
Funktor
Chain
Mathematical singularity
Right angle
Heuristic
Figurate number
Pole (complex analysis)
Resultant
Point (geometry)
Computer programming
Observational study
Symplectic manifold
Event horizon
Hypothesis
Prime ideal
Morphismus
Frequency
Lecture/Conference
Boundary value problem
Energy level
Subtraction
Condition number
Addition
Neighbourhood (graph theory)
Element (mathematics)
Mathematical analysis
Algebraic structure
Counting
Mortality rate
Line (geometry)
Limit (category theory)
Sphere
Local Group
Quilt
Doubling the cube
Object (grammar)
Family
32:35
Point (geometry)
Collision
Correspondence (mathematics)
Multiplication sign
1 (number)
Infinity
Food energy
Frequency
Lecture/Conference
Term (mathematics)
Operator (mathematics)
Boundary value problem
Circle
Subtraction
Series (mathematics)
Spacetime
Theory of relativity
Product (category theory)
Concentric
Expression
Algebraic structure
Counting
Infinity
Sphere
Quilt
Category of being
Summation
Funktor
Order (biology)
Chain
Right angle
Figurate number
Resultant
41:54
Collision
Beta function
Multiplication sign
Embedding
Sheaf (mathematics)
Insertion loss
Function (mathematics)
Parameter (computer programming)
Weight
Expected value
Manysorted logic
Strategy game
Meeting/Interview
Length
Injektivität
Compact space
Spacetime
Theory of relativity
Coalition
Sigmaalgebra
Generating set of a group
Closed set
Moment (mathematics)
Curve
Infinity
Price index
Bilinear form
Functional (mathematics)
Sequence
Category of being
Proof theory
Cylinder (geometry)
Order (biology)
Right angle
Figurate number
Resultant
Directed graph
Point (geometry)
Sequel
Connectivity (graph theory)
Image resolution
Drop (liquid)
Inequality (mathematics)
Regular graph
Number
Frequency
Lecture/Conference
Operator (mathematics)
Boundary value problem
Associative property
Subtraction
Alpha (investment)
Condition number
Multiplication
Standard deviation
Military base
Neighbourhood (graph theory)
Algebraic structure
Sphere
Estimator
Formal power series
Equivalence relation
Vertex (graph theory)
Game theory
59:13
Logical constant
Differential (mechanical device)
Multiplication sign
Correspondence (mathematics)
1 (number)
Parameter (computer programming)
Mereology
Protein folding
Casting (performing arts)
Mathematics
Manysorted logic
Homotopie
Analogy
Square number
Length
Compact space
Process (computing)
Spacetime
Coordinate system
Staff (military)
Hecke operator
Functional (mathematics)
Sequence
Category of being
Funktor
Order (biology)
Mathematical singularity
Normal (geometry)
Right angle
Figurate number
Point (geometry)
Geometry
Image resolution
Manifold
Translation (relic)
Event horizon
Hypothesis
Gaussian elimination
Frequency
Lecture/Conference
Operator (mathematics)
Energy level
Theorem
Subtraction
Condition number
Addition
Diffeomorphism
Military base
Forcing (mathematics)
Expression
Mathematical analysis
Algebraic structure
Independence (probability theory)
Rectangle
Local Group
Estimator
Ellipse
Quilt
Field extension
Quotient
Vertex (graph theory)
Object (grammar)
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so as for the time and we will and the and the 1 thinks of the organizer for organizers writing meets pleasure to be here so I'm talking federal period and leading Mumford spaces for something you haven't heard of cold which falls and all of this is joint work with action there have had any can see what we've written so far in these 2 people and so on and so here's the plan so can idealism motivation and didn't tell you what it calls and correspondences all tell you the blueprint for the Algiers would come up with for this thing called since this to category like this and finally in the last section of the new result about Fred was for quilts with certain kinds of singular so for could be called the cheese sections "quotation mark so Sony
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called for you that and if we have a supplied the command of all then under favorable circumstances leading find the category associated it so this is an infinity category the objects are selected Manafort accident the objects are certain kinds of Lagrangian inside of them depending on your situation will be required to satisfy some hypotheses and if you take 2 more prisons to around you can see them as long as these intersect transversely morphisms are defined to be the free vectors which generated by the intersections in general is equal to the complex OK and the last thing I want to tell you about the is that their composition operations that need D inputs for D at least once which is something that infinite categories required to do and the Deri composition operation as defined by counting still Norfolk the plus 1 God so take the images of these pseudo Moffat polygons in an look something like this OK so this is sort of the Maine the thing we care about the toxin from 1 any questions about this all right rights so several years ago but the Hyman would redefine something that they called quilted and on the point of called 3rd is to build functoriality into the the agrees to relate the guy categories indifference about to grand Grandfalls and so on in particular they had this idea that if you take a Lagrangian proteases of somebody manifold so L 0 1 sits inside and 0 minus times and 1 the miners just means that the site but for 1 and 0 then that should give rise to an infinity functor F L 0 1 From food and 0 2 food and Wine and if this looks funny to you I can we still use some motivation for wide the right notion of morphism from an 0 2 M 1 should be a Lagrangian like this which is Weinstein's simply to create everything is the Groningen and therefore morphism students about the benefits should be the conscience I'm all tell you more about these Lagrangian correspondences in the next section but for now the on it in mind you can think of a graph was about to workers from and certain ones clearly this was real the former king of what you have more houses houses during the resolution "quotation mark the war was 1 of those love directed so Was the institution fellow writers world the closely related sewing good circumstances there's a duality between those things "quotation mark and now it turned out when they try to carry out the school that this is really a nontrivial thing to do on the object level it's not so hard to understand how this functor should work what's not trivial but is not as difficult as Levison workers level we're counting something still more effective but it's going to be this funny objects singularities that will be the subject of this talk come and the hand Woodward's resolution was to study some related objects for the analysis was not so hard that where the result was not not quite this that the Goals of Catherine and me are 1st of all to actually do the analysis for the singular objects so some of the coldest Goldstar Group and then the 2nd goal is to sort enlarged is all the rage frameworks of Air Algerie pictures the grungy burst 1 incident is an infinity and
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on the bigger algebraic framework is that we actually expect there to be at the Infiniti 2 categories I just think of it as infant aversion to category where the objects also what are some large classes and let the manifolds for morphisms early Lagrangian correspondences and the 2 morphisms are for a coaching so the pictures something like 10 0 and and wonder objects grunting responses your morphisms 1 more offensives anyway and to morphisms are given by Florida which aims to sell state also more about this other officials at the section I handtohand so well is all about if I have time regret recapped the progress it made I'm in anything that police say is all explained a new result for settlement of the civil war of the objects used to define construction before a Section 2 I want to mention the result of a handover were able to prove so like I said they work with some of the Norfolk things which refuses to deal with analytically the Arkansas result which was a little bit different than what they had originally aimed at so if you assume that year and the demand to exact setting than they showed that a Lagrangian correspondence gives rise to an infant funded but not between Cu Chi categories between these things called out extended the category which they invented for this purpose the objects of the extent of high category not being withdrawn genes but by formerly composable sequences of Lagrangian correspondences the head of the house yes thank you but that and OK for 1 final thing is that in the audience no OK while you wait for the other janitors and years of great planning so if you're countering generated of something that may have come to mind is for transforms and so there is this thing that at least formally seems very analogous in algebraic geometry so if I take object in Cole of a product of smooth projected varieties X and Y then it's this totally wellknown construction that you can produce from that functor going from TV called next to the V called y I'm really useful and so the hope is that eventually will be will understand how mere symmetry intertwines the sponges F L 0 1 and 4 in car transforms that's really that's really quite distant but I did all right so before I moved to this section any questions there were no when I started now instead of going to in the case of book token great thank you OK so let me tell you a little bit more about the Grundig responses in some examples and that sort of thing was so that notation from now on I'm going write on "quotation mark Roger correspondents instead of writing as a subset underwriters in error right celebrities are examples if I take a Lagrangian in 0 and Rongen and 1 make informed of product that will be Lagrangian in In 0 minus signs and 1 so it'll be a begrudging correspondents there In particular if an zeros appoint this tells you that would run GM's induced a Roger correspondences from the point whatever problems inside OK so vanished before is that if he a home match from an 0 2 NY let's say to somebody morphism minus is graph is a Lagrangian correspondence from 10 0 2 and In the last example which is the most nontrivial agrees is that Hambletonian group actions by the groups give To the granted correspondences to say that I have and G acting on and in Hamilton in fashion so the Hambletonian this means that comes with the map called New from and to the duel of the Leon others is construction cost but the quotient retake the 0 level set of new and cushion of IG get and
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others about the manifold some offices and branches at the 0 level set is a Lagrangian correspondence from vandalism but to push the answer Over the years directly from now this is important feature of the grungy correspondences which is that you can compose them so let's say that I take L 0 1 101 and L 1 2 then the composition is defined by pretending that the grass of functions and doing what we would do to composers functions so that is to say were 1st form the final product of the skies over and 1 in a project down to 10 0 times and to it and so in the example of this is that the graph of the and graph of society are going to be the draft PCI compliance but until now In order for this to be sensible we have to have some real saying that compositions of erotic responses Oregon Lagrangian inner turns out that that's true under the general hypotheses that this is a pyramid Sternberg I which says that if the fire Protestant transversely which is to say that the intersection of their products with the diagonal and walked in and 1 is conversing then pie 0 2 defines Lagrangian version of the final products into an 0 reminds tents and OK and so important notes but there is no similarly general hypothesis you can write down With the property that you get all around him that so if you're going to be in the business of composing arranging correspondence is you have to do stuff with the 1st crunch focus and all right so before it moved to the next section any questions OK In surrogates as the 2nd half of the 2nd section so I mentioned the Civil gadgets in the introduction in outside with their there called suitable Norfolk quilts and the waveform raises you understand what role the grunting correspondences should play when interacted with all Wharf so time were all used to saying that if you have a Lagrangian the defining a natural boundary condition personal records it and what the theory of quilts tells us is that the Roger correspondences defined seeing conditions sulfur tell U.S. seen condition is and also simultaneously define whether Norfolk I don't think OK so History Month service spirits and genus analysts divided into 2 pieces by dryness circle here OK Alex labeled the chunks that has been divided into 4 the patches ice about the benefits of was labeled boundary components by truncheons and then let's label these 1 onedimensional submanifolds by the granting correspondences so just as when you drop in on quilted normal Ramon service label with ends and Els that represents
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PT you'd better conditions
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this represents the system of couple beauties OK so this represents the following system we require map you which goes from this letter Bachand and 0 you 0 met you want going from the right should and 1 then from all points in boundaries circle here they're supposed to be mapped by Yukon to 1 and finally the new interesting additions that we take any point in this scene but would carry out its image under you 0 in on a U 1 note the global defined at that point can we land in L 0 1 so that's that's what I meant by sinking everyone OK with that you you know the SuperSonics important thing I didn't say which is that you anyone was to be Jabal Moffat of course I mean no I'm not imposing like that and it's good your wife is about where all are you talking about this seems OK yeah there are conditions and done when you mean if you're talking about like announcing the quilt is you don't worry you don't allow any but you see don't allow any intersections but for the purpose of this talk I Mulligan reconsidering really specific well so we don't think we think that the question What's said general welding called I tell more afterward OK great the government will follow so now it's time to talk about and figure it's and which falls so and 1 of the great novelists well it's a new kind of singularity that Woodward discovered when they were studying strip shrinking so the situation that they're looking at Is there considering sequences of Napster's domain looks like this I said so this is this is a defining a "quotation mark problems and notice that well if you have a family of 4 quilts with this is the domain India them with the Middle Street shrinking then this funky thing can happen so get out like usual you can have bubbling wherever you like on this quilt and in particular should happen in the middle of this shrinking strip slimmer tried draw what's going to happen at intermediate step so puffing out above all but but we better be carrying the seems along with us when we will this bubble up some of quelled the seems now looks something like that but the bubbles out and if the rate bubbling is proportional to the rate of the Delta Queen 0 you're going to get something interesting happening in the limit so in the limit you'd expect To see a quilted sphere sitting on top of double strip where the scenes on the sphere look like this is and so the spirit has to circles scenes this 1 here this 1 here and they meet at the South Pole that makes sense everyone foreign funds OK and now it's time for the velocity version of the stock so it's in the late eighties for was studying for a strips that asbestos column strips and and done he noticed that even have this funky thing happen where bubbles can occur in procuring can disk bubbling on boundary and he said Well let's see what hypothesis of hypotheses we can put on the situation so that we can avoid at and prevent from high comes and says well for strips are still worth disks more or less worrying homogeneous ones with 1 input 1 output can this levels are also disks with 0 inputs in 1 output so let's instead of regarding them as above let's study still Martin disk with arbitrary many inputs in 1 of it added that the others amazing and structure called for by I rights event as it naive registered and I thought let's try to do this with the great bubble but it turned out that an interesting Eldridge structure seems to emerge from that I and by the way the reason that this is called the figure above all is that if you look at it from the South Pole then the 2 scenes look like a victory are great the idea from the philosophy is found to count quilts like the figure above all political and very quilts since there are no longer bubbles really the primary object of study and you know just like the guy didn't wanna we put some marked points on the scene and since the South Pole is what attaches to the base quilts over and the situation where the figure emerged with regard to South holds the output OK it now before we can do anything we have to have some idea of what do the inputs want to eat so what should this be defining a matter between and awaken understand that isn't looking I'm a little neighborhood of when the smart points so it is little disks nothing in and 1 and 0 With these same conditions L 0 1 0 0 1 I OK and then you can seemed like monastery really that if you fall this thing across the center line this is equivalent to a little have this
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mapping into M 0 minus times and 1 with honest Lagrangian boundary conditions in L 0 1 and 0 0 1 4 the only thing difference is that you want to know the status on the world according to the report 0 appear know I should probably should adversities and it doesn't matter market otherwise with OK and then we know that this sort of thing but mark .period and should be chain in CFC L 0 1 0 0 1 priced OK so now we know what sort of input this guy should produce and well In the removal of scenery fearing that I might not talk about today tells us that we'd expected this output mark .period would produce something in well I better give things names but since this is a laser gun crime this is not 1 to try and tell 1 from 11 to how's your 1 so this that something out in Seattle L 0 1 composed of 1 to tell 0 1 prime composer L 1 2 different I'm in this is the 3rd coaching group at the end and 0 minus times and time we have to find out who was behind way an I understand 1 way or the other side of the road the program I think it is decide a group like that because you can prove that that's where the limit will it in that part of coaching his afternoon this has a yes that's right yeah and maybe I should give zoomed in view I near the South Pole salami cut out a little disk South and then look at it from below I'm not sure what looks like some of the of yeah so anyway and tendency is what makes the result the the coaching here found and I'll come back to this but also say that the the reason that the analysis of these figure equals as difficult as exactly because of this local picture here so if the scenes were coming into the upper .period like that then you have no trouble at all and that's pretty much what the with words and mounted late so I what kind of algebraic structures should we get 1 2 so there's some heuristic heuristics for removing that tell us what we should expect I don't think I'll get into the heuristics felt telling you the results right so the someone as saying over here under counting these very quotes on the precise type narrowed down to mark which lesson and 1 on the right that we expect to define a map from that C C S L 1 2 crimes and want to climb crime tends to city have elements you want to cry tenses the at 12 0 1 Karlsruhe on price 2 CFL fell 0 1 composer 1 2 the 1 composed of 1 to 1 and just make it really clear what I mean when I say that yeah counting these quilts should give you this map what I mean is that defied calls Rhapsody to invite the I by the dinner I want to Y 1 into the left side of X 1 into the Red Sea and that defined the output to meet his son overall 1 z the intersection of the composer confidence the count over the 0 torrential strongarm of figuring Colts passing through the specified coaching chains inadequacies infancy so putting all of the different maps that you get out of this process you Nova however many market analysts even however many on the right what you expect to get is something called C to which has been put these 2 4 categories food and online extends into the food and 0 minus tanks and 1 masterful I category and 0 minus times and to so what I mean by this well what I mean is on the object level it and so on apparently Runyon's to a single around in which when L 0 1 and I want to have good composition is just the composition of them that if they don't then you are going to have do some work on and on the morphism level this year also is useful for I'll try to remedy that thinking on the workers levels getting put however however many more in each of the 4 pi categories and spit out a single more define exactly by this process right and I said that we have some expected relations with these maps and in the case of C 2 where relation is that C 2 is expected to be in by phone which is essentially the same thing as saying that it is expected to be an infinitely functor and attends a product of these 2 4 categories To this trip I can agree with those who and let me drive the picture that gives us that expectation the Soviet guy was merciless
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year we were confident that you have to know of piracy of guys who make the mission following the July tells you you don't need to know any higher so it is To being invented by functor that's defined just in terms of the operations on the 3 new categories involved and that the eyes are not the same thing as a symphony operations is significant that the Infiniti operations are the same thing as C 1 right OK so that his example of that going here isn't that I mentioned let's consider a onedimensional on space of figuring quilts with To mark .period on semen and 0 1 points on rights is so think about how this thing can degenerate II let's think about what the value that should be so what they can happen is on these 2 mark .period can come together so when these tumor was going together and will get a twomatch spirit sitting on a figure equal to so the 2 fear is divided into 2 by this the circle here I think I know others something happened for instance and the 2 scenes can can come together and collide In depending on where the mark .period sovereign that collision happened certainly get different ones different bottles the resulting sofa insensitive the tumor points do not come together as the seems collide the new get too the figure equaled seated on sitting on a day to catch sphere Woods I found or they could come together as this as a circles ,comma gather in which case you'll have a single figure of sitting on the bench sphere OK on and missed anything only I've missed 2 things which is that you can have a floor breaking out at either of these 2 more points so if that is the saving of energy concentration in disguise at the end of the book but at also an elevator ride down the held the break expressions that correspond to these steps and so if you if you count quotes like this 1 I didn't write the Merc points so this brings so this fear with 2 patches if you fall across the scene you see that this is just the same thing as a suitable Norfolk disk mapping the M 1 minus stands and to solve this corresponds to during this seem to operation To deprived of 2 more presence in and 1 medicines and to the act of in this this Barger divides the things that feeding into the lapsing of interfering with the rights now let's invest so this is going to be a new 2 of pursuant to whites to see to that I 1 this 1 here is new to seek to have Y 1 . to act this is cq of life to the 1 why 1 and this 1 is to have new 1 what to so anyway I the fact that these guys rises the boundary of this onedimensional office space for the expected to anyway tells us that these are the expression should send a 0 and this is 1 of the infinitely many relations that must be satisfied in order for seated in OK so I I put which malls entitled somewhat which fall all the answers figure a quilts have 2 circles as seems the structure maps in the category of product you can represent as counts of spheres with 1 circle seemed so why don't we study circles with arbitrarily many circles seems so what we call those quilts which indeed patch which falls so intention is going to give rise to some operation called C D going from 2 energy minus 1 like this times and the through through ends reminder scientists and 1 to 10 0 minus time depending on how I should say that it's only 4 D is equal to 2 that this should be out Aventine mostly from the which is exactly analogous to the fact that the only in the operation which is a chain map is new too I'm great now for every indeed we have were from 1st some we have predicted relations and we
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expect predicted relations for all the just working on the algebra so the 1st relation is that the sum of always of sticking 1 into cells Michigan 0 now see 1 goes from food and 0 1 minus times and 1 to itself and as it turns out basically what I said before is is that C 1 is exactly defined using the infrastructure maps and provide adequate so this is exactly equivalent to saying that food and 0 minus times and 1 user name in the category of this year's slowly taking the series 48 was this also yes on the 2nd relations is equivalent to saying that seem to is an infinitely by functor so that you get by sticking C 1 seems to be there on the left or the right path and also adding and the result of sticking C 2 into see 1 however many times as you like so the dot dot dot here means Philip C 1 with citizens so we know we are 1 hour to hour and we have an idea of of what all the other we expected relationship be sacked the
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algebra is what should be working out and the last thing I want to say for a new 1 Russian Is that while sources this thing I mentioned the title is expected Infiniti category is just we're going to get by using the CDs as a structure maps I'll tell you 1 special case of this whole construction which is when DEC too and died at sea and 0 equal to a point In this case the fact that situation been invented by complements that if you fix fell 1 2 on the Net as a formal corollary of expectations gives rise to an infant from from 1 of 2 camps to defined by counting "quotation mark disks that look like that the reason that I see disks and not spheres is that after your it's defined using these these figure equals these 3 perhaps 4 years except 1 of the patches as mapping to points even just delete that patch flatten out to a disk and this is what you get and and splitting Nova now that what now the imam Woodward did is they straightened out this seems a neighborhood of this singular .period someday studied "quotation mark the disks where there the this teardrop thing so the seems come into the output .period the ministers transverse fashion for it and then really lessons before and we wanted them to the new result is dead and for by category you get the Infineon operations because the spaces geomorphic curves live over the operator association dropped on any of India operations reflects algebraic structure of the opera so we expect that the "quotation mark curves live over another operator in these relations exactly come from that operated structure specifically the operand where typical space the modular space d batch which falls so example so what are these bases called P 3 0 0 is on the space of Thrace quilted disks with 3 boundary marker points the example is an example of this thing not Spears because they're easier drop I'm against it turns out that way you get is well so take this picture cut it out along the edges on there certain edge identifications they have to make that I haven't indicated in this figure but the point is that it'll grew up to you a politics inside and don't what that point Hope represents ears this compact Lumumba's space of "quotation mark disks on and the different faces represent "quotation mark mention 1 2 generations on which are indicated here so the edges are convention to 2 generations and so forth and it's now time to play a game was someone named 2 adjacent faces overrated it traced the fact that the adjacent should mean that alpha and beta have a common code mention 1 degeneration so let's see if I can figure out what that is I promise I haven't practiced the of work perhaps the most celebrated the common degeneration is on yet this the the in this component here gets larger and larger Chile it's the boundary so in a forced to To patch disks and here what's happening is the 2 Attachment Points of these guys are going to come together so what you get is but actually but so I hope that that's any questions about the his owes you could hear was the only I'm sorry OK so what's happening is on the degeneration of Alpha is that this seems here so the border of this green disk is expanding intelligence the boundary of the desk the mark points in this generation have not come together simultaneously so by definition of the space when that happens when a collision of the scene with the Markwoods occurs here forced above loss a call disk so that's what these guys are and after this generation happens in this formerly called the disk becomes an uncle to this disk the way to the hands of Beihai is that these 2 fashion points have moved together and your forced bubble offer disk and OK under the listings on so I don't approve but I expect that it will always be a politics which owns it comics .period and I can say that in particular you can specialize in 2 different ways the associate Dry and you can also specialize it to them multiplicity dry and furthermore to the by Associated drain by multiple future that seems to be a pretty rich offer is obligated to categorical version of this procedure OK so any more questions for while so if you can hear the views of some of the points with a 3 freedom everywhere and this is there is all of them will be the will only use the diseases was viewed as well as the lottery's supposed here is because of example corner which more than 3 days previously earlier there is a similar civil but after exulting up too much of the time but that even happens with the multiple he drama claim that these are generalizations and also just so you're forced into that I have a question usually number about the number of those who generously index so what do we have the will of Bonnaroo right OK suggested questions on anything yet so this has to do with these blew a serious sticks that I alluded to earlier on and the answer is that a funny thing that happens is figuring troubling so
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suppose that we have this picture so a disk with 2 patches and let's look at the generation when this scene gets larger larger till it's the boundary so get to quilted disks stepped onto an uncle to this I would not unseat ask What's the Code a mention of that generation so and that is to say How many gluing parameters are there for this situation and you might think that there to win premieres there's 1 for this note and 1 for this note and therefore this problem should not show up in the algebra messages out it's wrong and there is a strong is that there is no way to only include this snow and based on the scene structure and so far to the right way to think about it is in order to move up this picture something smooth 1st going to have to sort of glue this uncle to disk something that's quilted with the very the in the outer patch so to something like that and then you're gonna have to go the bottles and and turns out that there's a relation between those 3 grueling parameters the 2 NEC lengths at the nodes and the with this very thin passionate introduced in fact that which determines what the net lights must be 1 it's a satisfactory answer inches of grit None the coalition is the son of a the number of things going on to the time that those who were killed because of all the it said the exactly right yet so yeah each conversation yeah that's right and is so it's sort of like the usual situation to is that it gives some noble curve than the different Ozden about each other you can do whatever you want to anyone of locally but here if you have a bunch of a bubble sitting on the scene the same scene then they'll know about each other on which you know that seems simple enough at least in this example will we have many different things in the mark .period turns out to you know you have to think carefully to figure out which bubbles know about which other bubbles and the questions all right so it's time to move on to the next section and which I had intended as a sort of known no I wanted I thought I was wasn't going in and on the other the conferences on a fun especially educational so I wanted to do a little bit of a summary of a standard technique and the knowhow it gets adapted to this situation so I will see if we have the 500 you all of that right so this section is about Fred homeless so 1st of all let me mention that the general strategy for proving Fred wholeness totalling arise Deltoro Toro operator and I'd say that this cleaner is operator grows from each 1 teach and you can do you can create the same threat front results for lots of different functions bases but this is the easiest situation so the procedures 3 steps the 1st step is to prove that do you send I've had on the park which is by definition saying that the colonel this final dimensional and the Rangers closed talking in the next step is to you identified the coca all of the With a neat solution 1 of the new stars Glanzer so this is the formal entry into the 1st mentioned in the final thing is yet to prove regularity result to show that you can identify weak solutions of the start with stances on so here's example Of number 1 so let's say that signals a close Ramon service then the standard Olympic estimates tell you that you have inequality of the form that each 1 moment CD is bounded up to a constant the 8 0 Normandy Utsey plus aged 0 normally seen itself OK now should say this where observed that his appearance saying that some iPhone this is equivalent to condition that there exists an operator can going from it's 0 TB which is the contents of such that you can bounce see actually why the 0 1 1 plus the normal K. C sorry for the font size so anyway but this is exactly satisfies the condition because of the embedding of each 1 into 0 on Sigma disclosed its contacts so that's great so that's proved that almost on closely monitored no I'm skits some stuff because they don't have so much time but let me just say that another example of of proving so my friend almost as is the case that signifies not close the sequel to the cylinder and the Socialists and the sea then there's this problem and the promise still against amidst have that each 1 of city's bounded by the exterior Norman D. C at the close the base of exceed so that means you know these functions visitor on C but the problem is that the relics there no longer lies because were not on the contact remnants of us therefore H 1 embedding 2 8 0 is not contact so we don't yet know that this operators estimate that there is not an act against insurers because you us think of a sequence of bond functions adjusted opted for right so let me just say that the resolution is to study the asymptotic operator which is the limit as disposed to plus or minus infinity of the Doha operator and the gas and ordered operators of course s invariant and so you can argue that there's an injectivity estimate for it's
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actually nice amorphous and that's what allows you to fix this problem you can therefore replace this age 0 1 seat with an 8 0 norm on a Compaq some cylinder on
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rights around and say anything more about that about what you do about that resolution in quilted case it turns out you can you can make it work but it's you can make it work actually do some work instead of doing that I think I'll just say what the analog of elliptic estimates are in the Cold case so this is what I mean by the Baltic I apologize the Ministry going about 3 minutes over so abilities let's say we won understand the figuring "quotation mark while we puts a mark .period on there no and we would we would like to know that this defines event on problems of the lira's Dover operator in this case is for home but in order for it to be true we certainly need to have some kind of changes aleady conditions on the correspondences but let me not thinking about that it's exactly what you'd expect and so on to get the argument started we need a lift against us which you can get locally so far away from this bad .period you can get these slipped against it's just from the standard elliptic estimates but then you have is bad .period down here so let's figure out what happens there so let's cut out a disincentive .period and that's going to cylindrical coordinates but on that and so then when you get is a soda With these non street scene rights are how the heck are you going to get an elliptic estimates on this thing given this weird structure of the scenes and the answer is no well look at my thesis let's look at this juncture Cerveza constant with constant height rectangles are looking at the translates and we can note that you can see you can try straightening out the scenes in each of these rectangles so what I mean by that is choosing diffeomorphisms of these rectangles with pictures that look like that so what you see is a sequence of quilted squares With 3 patches where the width of the middle age's drinking 2 0 they have done so the hand Woodward came up with an estimate for Colts with domains like that and on ii upgraded that in my thesis to in particular include the case when the public structures not standard which will happen here because you're using his diffeomorphisms which are going to be tweaking the complex structure and all right so that's all say about Fred homeless on goal just finish by saying that we we now know as their own that In this case DU friend home as an operator from each 1 yeah I know I'm not a hundred days you were serious about the right side D'Alessio ideas is 1 of the 1 mentioned that 1 list of analysis both of these started with and I'll tell you which ones checked off it so the 1st announces goal was a removal of singularities Figure Eight quilts and more generally which falls on the console that's done in the next thing is compactness theorem Vermont licenses of such things OK the next thing is owners and working in a political context is is going home that's the goal but we haven't achieved that yet when I said Fretilin and classically work on this Sunday we would really like it's 0 interest really like something like HK intersect WK minus 1 for to the analog 1 and level differentiation down and I think we can get that and then there's still some gluing staff will have to do in order to prevent the political context and so this is sort of part way down you know just because of what I was talking producer would like this in the Polly folds context we like it and with slightly different functions bases and final thing that's really not done it is going to undo strip shrinking now so this is the 1st of the whole world will know I'd say that we need to figure out for in order to be able to put into appalling folds yeah once we have forgotten I think we're really ready to turn the belittled so anyway for is nothing except on the docket thanks retention In the 1st usually last fall plus a little of the use of force is the home of 1 of the last 2 years has always tried to live close to what you can do all of the leaders in calling something like that of the Woodward did classical gluing 1 situation I don't know if you did on generalize that classical willing to sort of which falls and I don't know how general you can do that but yeah I think we need to do is polypropylene which we have not so much of idea the elimination of the employment during the actual we wanted to if you use the teardrop and what you get is maps between extended categories so these guy categories where objects are formal sequences of ranching correspondences which is on formerly somewhat easier to deal with but is I quite a bit further removed from geometry so that's why we want these things amongst high categories for it for furthermore even if you're the teardrop if you have a bubbling the ball is not inhabited up structure it's going to have his which fall structure she would only be able to 20 visited often you exclude all kinds of every kind of bubbling that happens when structure can also happen to be used your I wasn't actually proposals I do resent you have probably but it's a little bit of a perverse thing to do because in order to prove something like that you have to completely understand the which falls and therefore wiry thinking positive you here questions about that and I think because they know that some of you have probably close to the level of independent of the influence on then there will be coordination to so you're talking about the stigma is still there ,comma as their home .period see yeah like What's the symbols example of this wealth of it yeah let's look at Rose so here we have a whole bunch of nodes and they're not all notes at the same time and I can't answer that very well in that time I have but the idea is that that limited Cicero down was not legally offered space that you'd write down on your 1st try the thing you radon 1st try would have some structure which are it's not the had a cast images of algebra so we've stuck on some additional cells in order to make every stratum correspond to algebraic expression and the ones that your views of exactly the new faces that you get In a pleasing thing about this picture so the Sheraton is exactly the point of tells you that c 3 will define a homotopy between the fungus corresponding to different Lagrangian correspondences the bonds captioning Chris Woodward they did and about proved such homotopy statements although they had to do some funny analysis with DeLay functions because they hadn't done this resolution so a pleasing thing about this resolution is that you won't have to do such funny mucking around in functions if you have an idea of clearer you must have patience that this was fueled by a lot of capital the supposed usually what 1 of this but I think it's interesting goal would be wise take police were sponsors coming from syntactic quotient so going from end to end G and and then this is expected to be a functor between food of those 2 manifolds so how can you relate through good and too frugal and Margie witches my understanding is that's that's an interesting question Eldoret geometries think could be interesting here too the more generally have some ideas but rather not speculate about the reasons for this year and has to go through all of this is that you have here is not the only way you yeah there's a job to do here is what you need to have a word congress group is 1 of the world and where is the evidence that you have to worry about and I do I do not have anything says will say that we should talk afterward respectable slightly worse than this I don't it was the use of the mn