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2/3 Knot contact homology and string topology
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Title  2/3 Knot contact homology and string topology 
Title of Series  2015 Summer School on Moduli Problems in Symplectic Geometry 
Number of Parts  36 
Author 
Ekholm, Tobias

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  10.5446/16319 
Publisher  Institut des Hautes Études Scientifiques (IHÉS) 
Release Date  2015 
Language  English 
Content Metadata
Subject Area  Mathematics 
Abstract  We show how to relate knot contact homology to a version of string topology. More precisely we express knot contact homology in terms of strings in the singular space which is the union of the three sphere and the Lagrangian conormal of the knot that split in a certain way when they hit the knot. 
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00:03
so John nothing thank you so we all of Florida's responsible drinking game for those the afternoon looking so so so today I want to talk about how he would compute this knock on the commodity in terms of low trees and then describes some consequences of these calculations also so let me just recall what a dating service and we have never knocked inside bathroom and then we we take its call normal you know ,comma normally used which is the set of units facilitated the points in In its it's full them every year members of pulsating words it sits in the unit "quotation mark about of arthritis so and that's it is settled instead lie over the Norton and with which have vectors that perpendicular to tenants they said publicly Taurus and insulation in its initial tolerance and we want to computes what I call the the Sonoran contact commodity of of the slander case events algebra Over the group brain on the 2nd commodities I'll tell you when real right go firstly this and then generated by but let's perhaps L see what these coefficients ring actually is so there are basically 3 variables in this 2nd commodities also so what is there to is that knowledge X here's the median P in the Taurus so you think of the Taurus as a boundary of neighborhood of the not and then there is also general this ozone topologically this is just 4 3 times is still so there is a menace to Lefors most so this is actually a threedimensional the space and so also the coefficient rating there there's nowhere to run victory for the fear is actually equal to that of the polynomial on the 3 variables into the eggs to the peace and when I arrived last want Cuba in right thinker with To take but this so Q is related to the Estill factor and it simply related to the to the the longitudinal region on the common will of the people who told you what was going early should thank you look so now In the theory that we are going to compute was Justin's just beginning line they're sold the if we compute the differential of a rare quarterly accounts homomorphic cars in the stations for the following shapes and has a positive Pinter day and then several negative by interested B1 B2 the 3 and basically the homology variables in cold relative declares all these these capped off suitably so 1 can talk too much about this caps but if you close golf then you get the beast with a boundary on this Atlantic a inside years it has a 2nd home right so
04:23
and and and then basically so this is a horrible thing "quotation mark within these good punches .period positives are related as basically the 1st goal today is to tell you how 1 could actually counts in this set up a such these so the 1st observation is that these could these unit contended bundling factors contact morph into the wanted space so as to the map is just so if it takes a it's in the base year why the fibers of wise unit vector that they extend their map why in the 1st minute component of its witches perpendicular to why unless you take this extort 1 this is somehow this notice of Q P C to his conduct morphism sewing factory can we can compute by calculating on 1 of the most on a contract awarded wanted space surface and so let's this kind of study visited so the idea was to use float trays and let me know tried to describe what they actually war so if we given so this is just this is the story His to and if we're given said is J1 promised so we can projected into the 0 jets banks which is called the disk or the front projection and what is it while so that the front than the phones all members has sold generically 2 types of singularities cast educated they have the worst month .period where it looks like a breath function and this is as to terms and then there's the following singularities only so that is because the edges where it looks like that's so here's summaries this 0 section in his or direction so it's because bench on the other singularities this 1 looking like that is that captain cascade is coming together and the the Cardinals article lured into this Cast direction so in other words if if you have a and then you can draw weights in threedimensional while also can nested I'm sorry would draw some while graphs and and that determines for you than a strong and so again remember that locally and locally years 0 section years of your Europe Piotr then the fire firemen and the cement Man and the issue locally have z also said and Q 1 judo and then you can sulfur for the peak toward in terms of all of the function and the duke orders by taking derivatives so so the from determines for you there the finish on look at so and but in particular if you're at peace in general of course you may have more than 1 such session and they may even make singular but let's forget about the singularity of so we stay outside the Single Locus men over this point you you have locally the number the 3 of functions grants of functions went so Eurydice Aundrea fouryear locally defines some finite number of functions unless you can get point because bench recitative it's difficult to say what his functions but for for generic points in the bases over the open their sleep outside coalition won something they have these functions and so now I wanted to tell you what is afloat tree then the float and some trees 1st I should say Wyoming but there were the focus of lotteries have nonsensical dimension 1 in the race outside you know outside the image of this cusp entrance defense onedimensional rights so once you're outside that you really have sheets when you're at the custody of the couple's shoes and then this "quotation mark hottest so that's fine it doesn't matter if they cause isn't just any difference but it's so so so I am going to explain to you what are the this Mooresville trees that we need to count and they they will exactly they 1st of all the other finite dimensions of prominent Aurelia can actually find them which is good but then that they would it would turn out that the correspond exactly to what occurs it gives you a calculation of this differential interst Mauceri data so the faltering is a man the you from trade quoted got most of his career structure internal in this case as into the base and if I do it so can and we require that along each edge there the floor trading the image of the trainee agrees with if thinks also Riemannian metric on this system so we can talk about and so so it follows the gradient this is the cost of doing something like that it follows the gradient of although these local function differences and the paper and and so society is a this so gradients statements and then at the vertices no certain matching conditions and you always have like his sickly water what all of these things and and you can see if you look at these things you can actually he fits naturally to tell the To the here is spread I minus if J but the bond that there already is the sheets right so there's a lot of is if I and J and accosted and just take these lines and lifted twice you can lift it to somehow to it's uh uh 2 it's too To the sheep played belongs to enter this can you just lifted straight up and there is a orientation religion local body with but that it's supposed to look like automorphic occur so that when the only 1 of them I I think you always where you a minus sprayed it with a this is the last thing you you I couldn't itself so here so this this thing here lives in in the basis of this it's Ernesto and the an over as to why this is so so I have to draw on many more
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axis and in this this is a big axes containing the fiber the fibers of the story tho an already I can think of it as a relief to all the way to the Lashonda and rights of the strong so this is a chief of this is another sheets as just here projects down an ice release them up and as long as I'm outside singularity loggers and I'm fine so but I haven't sort told you about this not quite there yet so similar but limit draw that join won the men's lives of region there will illustrates something so so suggest just the the measure 1 this is easy to make precisely theory has always get XX and then there is the accessory of the missing miners it's as old as I would have maybe 1 line like that on 1 line like this this and this is the front This is my land and if I go over to the 6 White Plains then expected to do that this event and and I have wife is equal to the CDX so someone these 0 sheep 1 sheets so seriously lying here and 1 shaved his line the bomb and my phone line just kind of a following of course is a stupid thing is just in onedimensional these lying in the base but on this thing in the base I could lift them to either these these she or that she practice because they're just just above here tonight if they're more dimensions there's nobody it's just a local differ right and I now require that when I come to these these the vertices I want believed to match rights also somehow if I look so now I'm drawing 1 of these sheets into into which I believe so I I have these incoming thing from that I hear that I should continue on the same street that I should be able to to to patch them here so that the top the list is a continous curve so To make basic it would look like vulnerable these was so but instead of instead of so it's a bit basically is said so at least this injunction there is 1 thing looking like that then there is 1 other Street where it looks something like that and and a 3rd Street where it also looks like the 3 local pictures such as summer trying to do this the chosen 1 more thing this is onedimensional picture where I work in 1 so there's no In the Russian 1 it's going to straight in dimension to it's making intermittently at this 1 is this :colon and less than To make these onetwothree sheets where the 3 boundary components leave for this off in the sense of it's what it's supposed to look at but now for our calculation this isn't some sort of many years ago sigh I saw some of the 1st people to explore this tree thing it was there as he forgot all wrote a paper about these uh where you have a year would correspond to having the wrong and without any singularities what what I did was I extended to the case when there are singularities not by now I'm just he in the paper there wasn't any restrictions singularities but certainly it was find to do this in demand for Ford to dimensions where the only these so let me tell you what or their counterparts these region this that weekend so rigid I think that's a great deal you to the alliance of the Republic I felt that you can of exactly where where where vertices everything so so so so anemic that they actually the vertices not stay away from the bad part so so they can be there but some of that the treaty always begins and ends either of these the cusp bench critical points about about so sewerage injuries there's only the following the purchases so they can have want and there it could be a Air raid according some sense of soldiers on Friday at 101 and 1 like this so so here you see this is a from picture you see a kind of a critical critical points that this this would be a record where at the point and the owner of a small I'm can certainly started just like Mauceri so this is sort of a remorse Morris Morris vertex Moss critical .period relations physical they can be of 2 flavors just like before so I when when you warranties filtering is the either you go up up the rape downgrade could operate Portis posted bunted down the airport is negative .period just like for the Home Office curves and we look at film trees with only 1 positive punch the political vertices and then there is another type vortex where the changes goes right into the 2 them To the that that's an end to user Of course at level like a flow line coming his way and then there and this is a positive function difference just shrinking 2 sureties so it's a kick if it can you go there and so I should 1 more picture made according to the UN based more the floor this semester and this is amount picture on the front although awareness as to pictures and if I draw the picture in these these other exwife thing then this is just so grim commodities like these In this it will not increased like that the fact that it still want this is 1 way and then I have listened to the intricacies so a game I have to to mainland critical points so so that's just in some sense annually to meet the special case of this but that's when I want to have for example a positive by nature that they I mean it To then I would try to do it to flow out of this minimum but I cannot there's no it's just constant flow rate doesn't love but if I have some other shaping the between years then what I can do as I can take this thing and I can speak it and then the thesis of flow in different directions that looks like that's resort the general diversion of some combination of Ortiz is about what it means to be used it happens only if these Is lacks the maximum in you will hear this year here the but with the with the the political implications
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of the deal was reached after another test you not so right so so here's our runners adherence as he goes between floor like so and 1 in here goes between flown a nursery 1 answer this England the gunmen here In this woman marries them between coming from 1 of the reasons that this moment is used for the last half of the cost of area placing want the annual flower show I saw the light in not all of this socially in these Patriots are looks like something like these that they had just passed by this in a very flat thing just passed by this Susie In the fall just goes you find ways of missing the limited the extremely flat to the floor line that just goes as buying that there would be this wonder just continues but that can decide to stop sheets here this kind of how I wanted you hear that I haven't taken the limit yet but there what do what I'm doing some scaling the fibers down so in that in that so I have a strong sitting somewhere and I just scale the fibers coming close closing closed 0 section before sufficiently small when is sufficiently close there will be 1 1 1 correspondence between trees in these limits In dimension wonders maybe always but dimensions it's OK so so there there this this is the east into violent and then there is the difficult environment which is called the switch and so the switch is a that Castrejon quality Month is a floor line between national name labels are Lebanese this sheet with so this is in the bottom of a mail was here this is list other sheets from where it is difficult and also to here I haven't the fuel line said that going between the 2 2 is the 3rd Street no this this top sheet and and another 1 and it starts out creeping up along the lower 1 and then can go up and the lives on the Upper once in Kansas State by either bench so it's a phone line is tendency To the gas here and indeed it starts out series on the and the lower 1 and it's moving up to the upper 1 this sense of busily unfortunately invisible in the Russian while so this is the the only true highdimensional inputs men OK and finally there are the the 3 3 Maryland vertices 1 In the very seriously and so did so on the basis of broadbased more promptly as the base this is still in there somewhere is the image of this cost so this is seen as a hero for these seeds to shape his amendment multiplicity is 0 here and to irritate you because he's accustomed to and then you have the floor line which actually is a welldefined only up to this point because then it stopped but the continuing the following his tendency to 2 2 to the to the fold and that this tendency it switches from 1 to the other so sometimes you think you have a family of lines then this is peaking specific 1 that gets stuck onto the scene but the and then things there is a little it is the 1st something called White 0 where where there just isn't so that there was the 1 that I developed as a just 3 smooth streets and ended the comes in between 2 and decides the speed incident it can happen anywhere something yeah so I think we could do something like this was a basically when you lift it it will look like this associated 0 this function to this century what this is like there is a similar 1 which which I call it why 1 which involves also the cast pensioners somehow the picture is the same but the fear is missing loved and speeds and right right pull over the custody this is why 1 trait and in the picture indeed is the following in terms of forms that they have a floor line becomes scene then decides to speak institutionalize between them between the 2 she speeds and that those are 2 ways to go that's why I 1 and then there and analysis is a sign of trees they are in some sense so that they can a complete as they started critical point and the close up either critical points at this and so it's a very it's as if it will be occurred exactly as boundary 1 watercolors the ghosts From endpoints of cost too points of portraits some exactly looking and memory loss because the and the main ceremony reason for saying this so why don't you agree yes and because I look at region trees so you can certainly imagine something like these things but that would mean 1 Perantre family so so so but but when I look at him for generic data which is someone I care about is they they would have only driving licences driving tortoises and if you would look at 5 AM families than they would have to look at some deeper tendencies as such as such as answer to this is the calculation 4 4 4 0 greeted trees here the alliance the man the head the of the Texas cost they could cause they could cost us find the that that sausages and it's a matter of right locally looking at today they can cross the certainly is I need not be embedded this is the political you have to do that the very possibility this is not honestly the rest of the year yeah and 1 and 1 half of the year ,comma is history so they they're playing different roles that fight but but I have I have year always like Candace Iquique holes so I know which 1 comes after and I need to you know how much much that all of the I think he's the importance of yeah you could say that whether that's also is that I just have a local numbering and and when I leave these events up in some she tried and and and actually I have this orientation conventions a 1 1 of this lease is oriented toward the punch and other 1 is oriented away from trying to win a comical Towards a puncher I know which if they should continue on and I require
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that it's oriented so that it matches the I think the oriented coming in on the options for Amendment the company also so that it might make sense oriented ,comma "quotation mark that's that's why I can lifted the names of the best so so this is a some kind of theory and it's a lot of work to prove that they were actually correspond with because of years of work and the House no no .period Moss function is the the functions such as opposed to think about whether functions locally defined by the strong in itself so that strong gives you locally some number of functions the gradient line sort of the differences of those functions so so it is only In order to understand is that theory 1 need only to the 1 exercising his so to these exercises the followers in the end the White Paper on associated taking October this is this is this 0 section and then you take some of the sheep which is sloped saved in this way and then you will get flow lines going from left to right and then you leave that up up up a sheet but on the large changes in value isotopic lead to meet so that interviews arrange all singularities I'm actually resides member rational and trying to with the slice here is these these are the 1st randomized controlled In this funding of the research to date is ring and I adjusted check whatever vendors you would find they can all these all these singularities playing Roland and maybe 4 polish phone of people isn't centenary interesting exercise because it shows that these trees so it remained in trees or regions of the sitting there but but but here you will find trees some of them in some way that have the property that they believe in the 1 printer family but the speed of the tree completely fixed but that this is moving cynicism of these floaters and sums as desperate and normally take so this is a message I'm not saying this will take up all my time following the lead to discussions on his behalf of what the justices Jennifer reminder of anything serious is some the solitary leaving as they wanna highdimensional family can be completely fixed some places another place so this is allowed and the they want so I think a bit this squeezing me saying that they have looked at state and went through much of that but the it's doing something you love and it's the killings he said about the limits of the Navy's unsurprising that so the winter dismisses them he stated that now so did the Lashonda Inc the game differential can be so that this that there is actually stronger but let me just stated latest to can computers by counting float trees he that is so the 91 1 this 1 extension of this the remaining also smaller 1 extension of this theory which is the following so it is all of this and I also know that I saw in the recent local mineral course when you try to calculate the role but but the reason is basically these that the trees that really looking at of doing this we look that the region and onedimensional boundary and this guy has called mentioned sedition never passed this point an individual so was they but when it tried to calculate the will and world so they say some kind of extension of these which we will use so so now imagine that that you have your some the Shawnee Indian scientists 1 witness to this will be at the ,comma only the unlocked in our case assumed that you have so this is shown in looking at the line here than any other some other historian which lying land which lies in a small neighborhood of Islam the 0 to land the Ciro has enabled which looks like the 1 just based Landis 0 there's going to the neighborhood in this business so so we take longer inside the oneyear space 11 0 inside To understand minister no we're interested in counting the on land at sea at Orlando and then what we do again we take Lebanon and be that these pinging thing which allows me to that of relate beasts and entries and now Stalinist stating this can be very sloppily but the theory is that what you should look for this you should look at these land 0 0 wind felled trees growing out on the boundary so so basically you can never be released shows us that you can have sort the been the skin somewhere this narrow region most of these narrow pieces of the 1 more thing so so basically if you pretend that you know everything for some reason above all morphing Beeson and the cereal and you know the flow trees between London the 0 then you know all that on more from the zoo Newfoundland as well and then there's the fear and says that in order to counter the solemnity land then you can count instead with a quote naturally quantum "quotation mark control trees than they appear in many other is "quotation mark cascades and what not to do soak it up so now let the actually that's actually tried to apply these machinery so calculated not contact would yield on so so so the plan should say that the plan is the plan is actually to can't not contact the motive for the on off and then it will actually quite easy to calculate for ending so what's the what's the strategies strategies as holds yes it's about this is the last thing I don't knows we because we have just 1 positive punches to such a guy would have to have to post a punch in this case but in general if you count everything and mean if you look at 1 or 2 or more vigorous such things will happen but it also much easier in some sense the action in only field of the universe but if that's the right so so that the plan for calculation it is as follows so if we think that I'm and that defines the producers
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says this to between normal down which is the Taurus and now if you
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take any other note you can
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Brady around them and also came and that that gives you this land Kate but then in fact if you think about it Islam the case seats now inside someone gets small neighborhood London you so in order to calculate
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the thing we need to understand all the homomorphic these homeland you and then all the trees have been undertaken under you and this is actually kind of
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1 condemned and let's start with the 1st needs a draw the thing and so was so and we take this we take that used to be just exactly like round circles in the in the X 1 accident claims as a way of trying to draw from certain Member what was somehow this year's 0 strange what happens
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there is a from what was his last the last so there was about the same or 1 specialist 1 0 at the expense history there was a see what the position adults the goal some now we have in the Cold War and we need to understand what's happening to these these little circle of collectors and then somehow we would be done so wanted to wait for you it's very easy so here is the answer Is is this section and ii would draw them or accidents to be just like it in starting at the order going to infinity so the lift of the circles are here when I'm here the wine doll to the position victories positive so I would end and that over at the policy it is 0 so get some kind of circle like these so so some it's asserted that I see and hear the function is positive in here it is negative and that's it but now we have symmetries so we can just
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wing it around so the actually leave the from too little normally yes news Strange Torres met winter singularities which replaced Cone cones over the balls and where are the red courts where the rent costs so this is not an unfortunate lead with genuine genetics I would perturbed in a 2nd but that the court's 1st they actually coming in an S 1 family right over the equator a this 1 family accords which corresponds directly by normal geodesic 1 should have said something so serious courts and if you Petrovic believed be so you can make it Short on 1 side you get the costly and long the other side and then remembered his grading formula have in the grading for says that the grading of seeing 1 integrating of these 2 so In order to calculate the differential we only have to care about what happens to the sea and it is stated in its bid clear what happens right we have to go flow lines going out from C 1 goes up of tours Poland 1 goes down to a simple not for the poll is a little bit the Jenin so we need to perturb its amendment to see what's really going on in terms of treaty 2 floors of munitions used Polish so this so that they general the yellow thing here is the 0 section the white thing or graphs of the that's the front so they define the functions graphs of function so here if I'm standing at this point they have exactly 1 function difference the difference between this year and that she and if I form the gradient flow that decreases function that just flow straight up to the people so there's a little fellas that it's their terms such realignment going up 1 going down here it's lead with unknown what happens because it's too nongeneric that I would Petrovic so that becomes it is also my successor is what happens when the London calling for political questions on many things but in summary then in the middle is the Middle States certainly version of that anyway see if I Sears the opponent if I do it from the tall it somehow looks like these words have to Turkish 8 cynicism about adult now I'm going to try to draw Austria what it looks like when I went every resulted in Ireland the projection because it's simpler there will be forecasts and to double lines and outside here is of course looks exactly the same this may be kind cited difficult to see the 1st I'm hearing and think about it but let me draw you that if you have been effectively associated with drug use Donald yes yes you can so you'd have to roll it up a little bit along sitting instead of having it in the so you would find for inflection points which are these these guys here but these you cannot yet but this can you can never get to direct cost to vinyl records you always have because they can't read orientations so you know itself or if you do it think this is a thing of the past which have not only the 2 top places him in the back of the head of the class this amendment passed along he is thinking of moving Europe this in turn will you have to to was a very decent levels but that's where I had in my life this year 2 men don't have to go along with it was time the people in whatever wherever will to so so so anyway What is that if you look at the bases in the front and the Vermont and of this this is just a cylinder like these and some other middle circle is complete acute and the projection so it now we need to figure out what's going on with this fellow trees illegal appeared in fact 1 of them looks like these I should do with hints of 1 looks like this and the other 1 looks exactly the same in the beginning but I won't go and and then at speeds over here and in this picture it corresponds to the coming from here and coming from here then it can go either this way or that way this among the these 2 options 1 the eye shape these kids maybe as this 1 was there this cylinder is premature these English Fondren itself saying that in this is this is almost a isomorphism excited crashes the minister and when I draw the boundary of this believed so these trees basically looks like he's 1 those down 1 way and the other 1 bill goes down the back today it is still occurs and when you lift them or not topic the can different ones so in fact we have for days and we find that is the is equal to 1 and there is some science region was discussed but no minors into the X minus the the people ask you only to the X into the cities so there and I'm I'm also not because some sort of running out of time so I'm not going to tell you go to Q right now it's
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somehow basically no I I'm going to go to Q so basically you have to count intersection numbers of these these to introduce lying above the floor lines with some fiber and and if you take this vibrantly to life here then you get exactly this calculation supplement North this a choice again camping so what is that that therefore descending living in 4 different junta because you have to say that I'm not telling you there to flow lines that we see starting from here but then what what's going to happen up here there's no clear from this picture we have to see how they continues on I have to I am excited to formalize going good 1 phone line going up up to this point but then they're different histories 1 going straight and won going and again kind of even illustrating this not so so prolific analysts have predicted the minute that of actually agreed here USA what's to the north agree the cells so this is our calculation and and also does pretty much behind time behind schedule so let me try to move to there's something this calculation so now this 1 key thing that will appear In the talk tomorrow as was "quotation mark augmentation vital to say what is the real thing the only is the social isn't to say that is the only is just differences in singling thinks of city so so I wanted to talk about the augmentation presentation the right Chris probably the most important there invariant derived from his contract was so so we have this we have our algebra is a Orlando case and then we can look at the locus where you have the chain into integers windows which some point to some coefficient but I like to think that the authorities and the media and he's still see a lot of children and he said and I don't know that they different degrees is building where will those think that this is not part of the of the of the use of limited means changed civilians have room for any more these people go while the other half half time I have 8 8 has agreed to release and see appears in his boundaries but do really want to the differential decreases degree but 1 of these medical and the cost and the constant although degrees 0 goes into the ring which is decreasing please L so no intention writing a instead the locusts the so where should say the locus all of the following in the in the space candidate to the exit to the P the and Q where varies chain wrapped like that J right so 11 let's take a look at this our 1 and only examples well what this is all sort of way change when the says the treated differently and it lies in degrees 0 with so so here for the unlock this is the missing because the only thing that it requires using this requires is that there the image of the French and the gray 1 is killed by this left some because so this this and this we went right to try to define the Epsilon In the image and the French Resnick so we see the equations of the equation for the augmentation rioting please equal to the twilight of the twoday excursions into the paint color scheme and to the extent but the now In the last few minutes let me somehow just very fast explain what happens with the general not so I am going to have follow this we will keep the qualifications the fall strategy and value for the house In the end of last year when I had the right to use the right the budget so that right is 0 set this thing which is used ornamentation Poland thank you but it's so at at this if you have a general not that as indicated a little toward women rated along on and in fact so here I'm going strangely the of the straight lines and then the Braves we consists of some strands here but in fact you what you can do is you can have the strands wall kind of increase along not accept you have to close them up when you're here said so this is somehow because you know what is a great turning much is oddly keep on going upwards interns and just have that outwards motion dominate that means that there will be no my normal courts at all here rather 1 of them will be here said so so so if you haven't Brandon instructions and then you have scored sort of AIG here then the idea and they they go both ways for so you have something to and squared from here and also has the highest in the world so that when you know the
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delegates who produce straight years of the twisting does not really interact with the courts at all levels were then it if you look at it from the top is the knowledge of what is being extremely close to the region 1 Susi that there will also be these these costs that correspond to the courts on their own or operate so she will have some kind of course see J underscore the other ones not even close call the of agency had the authority to go public with the text the accord in With this week to buttons services so there are remnants of the long course on them and that that goes between different trees and could now what we need to do we need to understand what are the flow trees on the on this Torres and in fact so these lambda K inside J. 111 0 is given by by the functions so let me so introduced some more corn year I have my circle of the cornered s along the circle then I have some I have some corn size along the the along that makes me In the end this is the closest neighbor others as 1 and that this 1 is given by the sewer and and if I have a like of Brandea said that defines for you a vector jail onstage a strand increase the so a summary as a function arrest .period is moving in that these physical around and the function of these guys just if FDA Tom S. adult son to the summaries that the function that generates this this fall hopes and it makes it not so hard to define the flow lines in central so let me just tell you that we feel about so here and now drawing this is the front this is a front of the final demand 0 it's a Taurus and I'm doing it like you like a square and the yellow lines so far what's going to the polls this summer is the prematurely the column .period the then when I leave my stresses in my life like ideas and the adjacent to be here the yeah and this is actually a JI find beating doesn't emotionally to ease my my stance just go completely straight then the relevant stable unstable metaphors here just go completely straight flight is 1 of this is the only the only stabilize their mindful of what the law where the function differences stay positive so if find there also flow lines can of falling down here but then very rapidly the function the physical negative and I cannot find any trees so trees have to go to follow these things so for an unblinking eye care technicians and now what happens when I introduce a twist when I introduced a twist I change the phone line like these it's going instead like that and this 1 is falling down here and coming back to win so so now In order to counter trees just need to keep track of how many times did it intersects something here is so it's not together terribly difficult to do it's something Sokol notorious gives submitted to organize things but and basically and you can find 1 that raised by composing the states so soft serve 1 tree we look like that and then you think this to the next thing in the remaining another train like these and then eventually according to collect them up here so this is 1 piece of the Berlin Wall as say something for 2 more minutes is 1 piece of the calculation but not that in these Peachtree defied orders can carefully I can also draw the images of what other being a little more strictly they look something like this so I need to make sense of of being these would trees can of attached on Omdurman and this can also be read all from this picture so basically this picture tells you Everything and gives you a similar expressions let me just very schematically finished saying will tell you the little intention :colon on itself so what you want you there is always then I will be so so users would have to stay in the green indeed we want you have CIG and the idea of having the greatest hero if you have any ideas and basically is so you find something like this that the on CID and constants I know their release shortages these these libraries the the HLA ideas the so that there would would have been any idea what motivated the head of the greatest I don't have to cancel some sense of the Indian DCI Jake and 8 8 good the news of the BIG that's in if you think about this this is just as some kind of polynomials he's the address in the AID and this is the mother polynomial PB and J so there are no what
1:00:04
we want to we will would work with sources saying that these augmentation variety of polonium and how to find it when we need to find values where all these polynomial have common roots so basically you can find this orientation polynomial by elimination series so so so you have to take your multitudes of about like this and all this picture finding there is actually a form of from break presentation take this thing and then it did you do you do elimination you get the pollen and this put enormous called augmentation polynomial and in this case the animation the extra simple because this gun wasn't eliminated the last month in general is more complicated so it maybe that this finished by by stating what I wanted to say and that and maybe 1 exercise of the ideal exercise has been the scene was clicks please say calculates for and so the Q is the queue is somehow related to having something on this From this these cities to get into phone if you wish this religion is more difficult to get control plane can find disorientation :colon for orientation variety is more complicated locate so what they intended to to say and I think most of the was the following the news the is just lending and the also some other than myself and see what mean much of this is the real person and the other process some of the think apologies but the best showing that that fact that woman patient so their 1 considered an commodity and nose mostly on so that 1 can prove In directly using some kind of relation to fundamental group actually groping of fundamental broke and that's how we go about it in this paper that in the making 108 but many proved something and maybe a stronger so he proved that it was not so hard to see from this spring topological things April that this each of the 2 minus the times at a polynomial the North which is a polynomial into the city to the page divides now the augmentation polynomial at the and into the key to P and Q is equal to 1 so and a polynomial by workers :colon Rosca nobody work of other people using condoms recognizes Donaldson sum of if you're not on lot to do have some interesting representation from 1 group complement interested to which gives you some so a point or something derived from S O 2 representation in World Group and so services in fact is a limitation but only restricted contains the polynomial and the starting point from them and for tomorrow's talk is that from this this augmentation therefore gives you a kind of deformation of they put on there was also found by other means from from physics and then the the 2 work is too can related to and to talk about coming after relations 1st operatives if I all 3 of the women 1 look at rest what point he the service the you know and then the answer is that you don't but it's it's so it requires proof because the even taken on a more and uncertainty can save the bids original courage does not pass through any of these bad things but now you're taking a rather badly needed right it is squeezing the the stronger and more on the ground if you wish down to toss section where basically everything you like about will morph occur theory like this estimate hitting the internal metric an external metric everything is going up in the wrong way so someone has to prove that was in the lead you compressor preserve producer preserve this this this this property sites it is true about we wrote something that we never really finished but but but but it's an unfair statement because it's very hard control distance of the Indian bowlers the house so in the paper that I wrote I basically know exactly these 2 types in but but in general you can do it generally and you don't have to know anything about this other singularities accepted the article image so Texas the a ceremony in the writing but not above it's there is something to prove you cannot just stated because the region they won't go there because in in the limit we don't really have your favorite lived experience it's not always as to no necessary in fact what turns out that the the 2 these in the divine but the awesome additional factors here it is not is not the quality is just these additional factors ,comma actually from from the other such representation at GE allowances and so these knows his some about what these flight connections on local and this is the 1 the really good looks like he looks at you in the front of the highway system that competition this you have to represented somewhere to get to know what it is if I had known I think Russians were used with this year and which will improve correspond to takings of apparel copies of this cartridges associated so no head 1 the after center few
00:00
Polynomial
Homologie
Divisor
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Oval
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04:20
Logical constant
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Kontraktion <Mathematik>
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Finitismus
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20:44
Greatest element
Group action
Differential (mechanical device)
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Orientation (vector space)
Correspondence (mathematics)
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Sheaf (mathematics)
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Sign (mathematics)
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3 (number)
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37:09
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38:19
Point (geometry)
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1 (number)
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45:47
Point (geometry)
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Kontraktion <Mathematik>
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53:27
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1 (number)
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Process (computing)
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Category of being
Proof theory
Arithmetic mean
Vector space
Order (biology)
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Directed graph
Point (geometry)
Polynomial
Trail
Presentation of a group
Divisor
Variety (linguistics)
Distance
Theory
Wave packet
Frequency
Gaussian elimination
Root
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Subtraction
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Expression
Line (geometry)
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Estimator
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Spring (hydrology)
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Local ring