Here's the definition of recursion from the book. We're going to take a problem and break it down into smaller and smaller sub-problems until we get one small enough that we can solve it trivially and then put all the results together. And this usually involves a method that we'll call itself. That's sort of abstract. Let's look at some examples of recursion.  Okay, clever visual trick. Let's consider another example that might be more close to home and that would be a 16-person competition bracket. You can think of this 16-person bracket as two 8-person brackets. Each of those 8-person brackets is considered to be two 4-person brackets and the 4-person bracket can be considered to be two 2-person brackets until we finally get down to a single person and that's the lowest level. Getting on to what we're doing in this course let's consider a linked list. If I were to remove the head of the list what would I have? I'd have another smaller list. If I got rid of the head of that list I'd have yet a shorter list and I could keep doing this until I finally got down to the empty list and that's where I'd have to stop because there's no place else to go. When we have data structures that are defined in terms of themselves like the tournament bracket or the linked list recursion is a good way to deal with these structures. Now, here's the key of this. Usually recursion involves a method calling itself. So first we need to investigate what happens when a method calls another method and this is going to be a review of something that we talked about in computer science 75. So here I have a main method that is going to... excuse me... I have a main method that's going to call the calc1 method and the calc1 method will call the calc2 method and each of them is going to return some value. Let's see what happens when we put this into action. When we call calc1 in this line we have to go on hold because we can't decide to result until we know what calc1 returns to us. This puts another... what's called a stack frame on the Java's internal stack and now we know what a stack is so it's beginning to make sense when we call it a stack frame and is now 10 and now we get to this statement and now in order to figure out what the product is we have to call calc2. We have to put this statement on hold and call calc2 and still being 10. The sum becomes 15 and now we're going to return the sum. When we return the sum, two things happen. First, the value that we're returning replaces the function call and I'm calling them functions instead of method but it's the same thing. And the second thing that happens is the stack frame for the method we're returning from goes away. Now I have 15 times 6 which is 90. I can do the assignment and I can return product. Again, two things happen. The 90 that I'm returning replaces the method call. The stack frame disappears and I'm ready to proceed with my main method which is the only one that's left on the stack. So that's a review of how method calls work every time we call a method. We have to go on hold open up a new stack frame and then when we return we return the value and we pop off the stack frame to return to where we left off. So now let's see how this works when a method calls itself. What we're going to do is we're going to give the sum of this list. The sum of the list 5, 3, 7 that's 5 plus the sum of the remaining list 3 and 7. Well what's the sum of 3 and 7? That's 3 plus the sum of whatever 7 is and that's 7 plus the sum of the empty list and that's 0. So we have 0 plus 7 plus 3 plus 5 that will give us the sum of the whole thing. So that's the idea of what we want to do. And here's the Java that does that. Here's our sum method and it's going to be given the head of a list. If the head of the list is null we return 0 because an empty list has a sum of 0, total of 0 for those elements. Otherwise we'll get the data from the head node and then add that to the sum of everything else, the rest of the list which we access by saying head.getNext. So let's look at main where we're going to say find the sum of the starting at the head node of this list 5, 3, 7. I can't figure out what total is until I evaluate the right hand side here, which means I have to call the sum method. Head will be 5 because that's what getHead gave us. That's not null. So I'm going to return 5 plus whatever the sum is of the rest of the list. That means I have to go on hold because I can't return the whole thing until I find out what this second operand works out to. That means I'm going to have to call sum again. This time with head.getNext and so now the head is going to be 3. The sum of this list is going to be 3 plus the sum of whatever comes after 3. That means I have to go on hold again because I can't return it until I know what this second operand works out to. I call sum one more time and this time 7 is the head and I'm going to return 7 plus the sum of whatever the next element is. Again I have to go on hold because I have to figure out this second part. I call one more time. This time the head is the null element and finally I have something I can return. I can return as 0 because I know what the sum of an empty list is. Again return does two things. First thing it does is it replaces the method call with the return value which is 0 and it pops off the stack frame. Now that I have the 7 plus 0 I can return that to the person who called me. This 7 plus 0 is going to fill in this thing that I went on hold for. The 7 plus 0 gets returned replaces the function call and now I'm back in this call where I had 3 plus 7. Now I have a result that I can return. I return that to the person who called me which is the 10. Now I'm back here. I have 5 plus 10 which is 15 and finally I can return that to main and the 15 replaces all of that and that becomes my total. That's the idea of what happens when a function calls itself. We go on hold and we keep calling the function over and over again until we get down to a case where we know what the answer is and then we climb back down the stack replacing the values one at a time until we get to the bottom level where we started.


Last time we started the topic of partition affinity on CW complexes, we proved the major theorem there namely existence of partition affinity subordinate to any open covering assuming one of the propositions. Let me recall that proposition first here. So, it is an elaborate proposition which says that given a partition affinity, suppose X is obtained by attaching k cells to Y, then given a partition affinity on Y, you can extend it to a partition affinity on X. Moreover, the local finite ensured in the covering which ensures local finiteness for Y will get extended to the covering which ensures local finiteness for X in such a way that the sets themselves, you know extended themselves intersecting with Y will be the corresponding open subset. The open subsets themselves are also getting extended here. Before proving this one, we will need another step namely one at a time. Notice here what we have got here is X is obtained by attaching Y obtained by attaching k cells to Y. There may be infinitely many cells. So, what we do want to do is first do it for one cell at a time and that is same thing as saying that start with a open cover for DN or DAK and a partition affinity subordinate to the restricted cover on SK minus 1 on the boundary. So, you must be able to do the same thing here whatever is done here for that particular case. So, that is our first lemma we want to prove. So, let us go to that lemma now. So, this is the lemma we want to prove first now. So, you be a open covering for DN theta, theta alpha, alpha ground lambda be a partition affinity on the boundary SN minus 1 subordinate to the restricted covering U restricted to SN minus 1. Then there is an extension of theta to a partition affinity theta hat on DN which is subordinate to the given covering U. The proposition is same thing with infinitely many cells at a time the same time. And then just it is not SN minus 1 it is happening inside Y and so on. So, let us first prove this one. So, this is the major thing that we have to do.  So, this is the open covering of SN minus 1 ok. Well compactness of SN minus 1 ok and because the entire family is locally finite SN minus 1 finite is compact the whole thing will be what actually finite that will be only finitely many nonzero functions here. So, it follows that we can find a uniform epsilon 0, epsilon, epsilon, SN 1 ok and opens up sets a alpha in SN minus 1 ok such that support of eta alpha intersection SN minus 1 is contained inside the these are viscous remember what this definition n epsilon of a alpha ok and then disclosure and that is contains a U beta alpha. U beta alpha is a given open subsets from the open cover.  If needed you can take even smaller there is no need actually that itself will work. So, but instead of saying that I have said it exists as such a thing that is all. So, you take the closure of that that will contain this one that closure is contained in such one for ensuring that this closure is here you may have to take a smaller thing that is coming. Here we are using the notation of lemma 1.2 which remember this is these are collars of length epsilon of width epsilon all point coming from rays coming from the center to a point inside SN minus 1 ok. So, take a point on SN minus 1 and then take a portion of the layer of length epsilon. So, these are union of all those things as a varies over as the point varies over a alpha that will be as n epsilon. So, take the closure of that way ok. Now, let R from n epsilon SN minus 1 bar to SN minus 1 with a retraction. What is the retraction? These are points of d n remember x divided by x divided by norm x. So, that will push back push the vector back to the boundary ok. So, that is a retraction. Put eta prime of alpha as eta alpha composite with retraction. So, eta alpha is defined on SN minus 1 ok. So, now, you composite with this retraction it will defined on n epsilon SN minus 1 closure of this one ok. So, it is just eta alpha composite R. R of that will be inside SN minus 1. So, eta alpha makes sense. So, that is the definition of eta alpha prime. So, I have extended all eta alpha in a small neighborhood of SN minus 1 ok. These neighborhood I have very specifically said what it is. It follows that support of this eta alpha prime ok. This will be all these points wherein they go inside the support of a support of eta alpha here ok. So, support of eta alpha prime is also contained in the u beta. So, whole thing is contains u beta here all right. So, eta alpha primes are extended very carefully. Now, by T's extension theorem there exist continuous functions eta alpha second eta alpha people will double prime also eta alpha second prime is 1 double to 2 dashes prime second eta alpha second dn 2 I will be I mean these are extensions of eta alpha. See eta alpha prime is defined only on this closed set is a closed subset of dn. So, I extended to the whole of dn that is all. How I get extension I do not know this T section theorem that is an extension ok. And this is not good I want to control this one. So, I will do that later. So, by Eurion's lemma there exist a map G from dn to I which is 0 on the complement of u beta it is a closed set ok. u beta is an open subset of dn. So, take the complement inside dn. So, there it is 1, there it is 0 and 1 on the support of eta alpha prime. It may be eta alpha prime is contained here u beta. So, these two are disjoint closed subsets. So, I can do this one ok.  So, this is the modification I am on. So, what is the property of this one? This will be 1 on the complement ok wherever it was defined, but on on the 0 on the complement. So, it has again it is the support will be inside u beta and 1 on the support of eta alpha prime on eta alpha prime it will not change it ok. So, all that I want is eta alpha prime with all these properties to be extended to the whole of it, but its support must be inside u beta only that is all ok. So, these are eta alpha bars ok. Next let us look at all those points where in sum of all these eta alphas is 0. Remember sum of these eta alphas is a finite sum here now ok all indexing I am writing that with that way, but the number of things which we have started with non-zero things they are only finite remaining all other things are 0 we extend them by 0 for the whole of dn no problem ok. So, this summation it is definitely equal to 1 on the boundary Sn minus 1 that is a partition of infinity ok. So, suppose this is 0 if it is not 0 this f will be empty I have no problem if it is 0 let us take all those points that is the close subset f of dn. Now, you choose a partition of infinity f a a belong to a on dn which is subordinate to this open covering some other partition of infinity ok. So, we put them these two together in a nice way so that the old thing eta alphas these these alphas they are not covering the whole thing you see they are partly 0 on f if they are if this is f is empty then I could have taken this itself divided by the sum then I have got a partition of infinity. Since, I cannot do that I have to take another partition of infinity f a ok on dn which is subordinate to you and look at all those a prime a belonging to a such that support of p a intersection f is non-empty or those are the ones which are important for me. Amongst all these so this a is actually finite set ok I do not mean I do not need that. So, look at all those points all those functions which are non-zero on f support of p alpha intersection f is at least closer to that support should not be should not should intersect f does not mean that f is actually non-zero but support of p alpha intersection f is non-empty.  So, it contains some u alpha because I have started with a partition which is subordinate to you right here partition of infinity subordinate to you for all a inside a prime I have a refinement function there put lambda prime equal to lambda union this extra a prime ok and beta prime equal to lambda on beta on lambda prime and this beta prime on a prime gamma on a prime ok extend beta prime extend beta to beta prime in this way ok. So, I have taken then this beta prime will be a refinement function for the new family along with it alphas it alphas. The only thing now left out is to make them into partition of infinity they are some smooth function their supports cover the whole of d n that much I have ensured. So, this is what it so this was some my f so there is some u 1 here ok for each of them there is something so I have put the whole of f inside u 1 does not matter.  You can study this picture as a model this n epsilon are this is the color of the boundary here ok. Finally, notice that on the boundary not only on the boundary on this color itself because we have defined this as attraction theta alpha bar by using this attraction some total of eta alpha bar is 1 because that is the case on this on the sphere. So, it is true on the neighborhood also ok in this this part it is identically 1 eta alpha bar sensors are 1 and therefore n epsilon s n minus 1 closure intersection f is empty because f is where some total is 0. So, these two are disjoint closed sets. So, now I take one more horizontal lemma here that phi from d n to I b a continuous function such that phi of f is 1 and phi of this n epsilon s n minus 1 bar is 0 ok. So, that the extra functions are multiplied by this function phi. So, they are not disturbed, but they do not enter n epsilon. So, there is a killing this this function is killing. So, that is what I have made phi a is all of them are multiplied by phi. So, they do not enter this color at all ok. So, that is the that is the point and in the in the inside this whatever cover for f there they are identically 1. So, their values on f that has not changed. So, that is the whole idea. Therefore, what happens is away from f this eta bar some total of eta bar is non-zero on f this summation is not 0. Therefore, this capital phi which is sum of all these will never be 0 ok and this being a finite sum of continuous functions it will be continuous on whole of the n. So, I can divide by this function eta alpha hat these are the final modification of my eta alpha we started with. First I take eta alpha bar then eta alpha prime and eta alpha double prime and eta alpha bar and eta alpha hat now ok. So, eta alpha hat is eta alpha bar by capital phi lambda a is a roll phi a into phi divided by phi. So, that now if you take the sum of these it will be equal to 1. So, you put them eta alpha hats on a in lambda and lambda a is from a is from a prime this is my extension. This is extended family ok. It is obviously partial affinity because I have divided by this capital phi which is sum total. Subordinate also we have taken because we have taken this beta prime in this exactly same way. Wherever eta alpha bar is not 0 the same thing as this one because you are dividing by non-zero function and so on. So, subordination function etcetera will not change ok. So, condition 2 and 3 has to be 1 and 4 we have verified condition 2 and 3 we have to get ensure ok. So, that theta hat actually becomes an extension there are 4 conditions remember that. So, start with eta alpha which is equal to eta alpha bar on s k minus 1.  These things are 0. So, this will sum up to 1 here and hence eta alpha hat which is eta alpha bar ok is equal to eta alpha on s k minus 1. So, sum total is 1 I am dividing by that one. So, the sum total on s n minus 1 is this whatever you are dividing is 1 eta alpha hat will be eta alpha bar, but eta alpha bar is eta alpha that we have seen already. Therefore, these are actually extensions ok restricted to s n minus 1 they are old functions. So, this takes care of 3 ok. Finally, let w equal to w x x p not s n minus 1 been open cover for s n minus 1 which ensures local finiteness for theta. We have to get an open cover w x hat for d n which ensures local finiteness of theta hat such that w f of w hat is same thing as f of w. This f is the set of all points all indexing indexes ok wherein theta alphas the eta alphas are non-zero that is what this f is ok. So, notice that because of the compactness of d n local finiteness is obvious that is not the point, but what are the opens of sets which ensure that have you extended them that is the important point here. However, the emphasis is on latter half of the condition which has to be handled correctly. So, that has to be verified ok. So, for that I have to do this work namely define w hat of x to be n epsilon of w x remember what is n epsilon. So, in this picture if this is w x something here like w x n epsilon will be growing it up to this collar like this like that ok. Do not go out of n epsilon put up all that this much of this much of collar for that that is the definition of n epsilon ok. So, w hat of x if x is in s n minus 1 take n epsilon w x intersection with s n minus 1 it is with w x. So, that part is fine in the interior you take whatever you like actually you take the whole of d n one single set that is an open subset of d n anyway all right. So, x is the interior of d n. So, clearly the family is an open cover for d n and as a remark as you have earlier w x hat ensures local finiteness of theta local finiteness because there are no finite elements there. But the indexing sets are also correct is what we have to say we have to verify that f of this n epsilon x these are the whole things right they must be f of w x ok. There are no extra indexes coming here this is what you have to show. For this we observe first of all that if you take lambda a these are new things lambda a intersection n epsilon x n minus 1 is empty and these n epsilon w x are contained inside this one. Therefore, this lambda a support of lambda a intersection this f n epsilon is empty ok. So, w x is contained is n epsilon of w x and r home this one is retraction. Therefore, eta hat of alpha z is eta prime of z it is eta alpha r z where z belongs to n epsilon. For every point in that neighborhood we push it to the boundary and then take the value this r z is the z divided by norm z right. So, that is the condition see there from a and b it follows that both sides of this 5 are contained inside lambda the what is lambda prime it contains of lambda and a prime the members of a prime do not come here this precisely what I am saying. Support of lambda a intersection this is empty that is part a ok. So, these are both the sets are inside lambda they are ok but why they are equal because of this p from b it follows that w f of w x f of w x is what point alpha such that eta alpha is not 0 right on w x it is contained inside f of n w it is a larger thing ok because because it is retraction this is just it is contained in this and that is all it is finally take alpha belong to n epsilon f of n epsilon w f larger one take a point here ok f of that one ok what is the meaning of these there exists a set a inside this is set eta hat of alpha z is not 0 but eta alpha first z it is eta bar of alpha z that is eta bar of r z if this is not 0 but r z is a point of w x. So, I have got a point in w x eta alpha is not 0 that means this alpha is inside w x f of w x. So, every point of f of w x is there which is not actually we need to have that one one way it is enough but they are actually equal is whatever so this comes to the point of the lemma having stated that ok. So, this was the hardest part of the entire story of Paragama of partition affinity for this one ok now we can compare proposing 1.4 very easily ok and this was needed this is very important otherwise when you go to infinitely many things namely at a single vertex for example, you might have attached infinitely many one cells ok then for each of these absorbency to choose they may go keep going smaller and smaller you never know what is happening there. So, you have to control that on the other hand this part was not at all necessary if we had assumed that the CW complex is locally finite we would want to prove it for all CW complexes not just locally finite CW complexes ok. So, the last part is not very difficult at all. So, start with a open cover for you open cover for X ok theta beta alpha lambda of partition affinity on Y and beta be the infinite function and W be the open covering which ensures the local finiteness of theta ok. For each J in J let us have FJ from SK minus 1 to Y with attaching map for the Jth cell DK that is the attaching map ok and FJ from DK to X be the characteristic map ok this is defined from whole of for each each DG or each DK. Now consider theta J ok J is fixed at J is a theta J to be eta alpha composite FJ ok. So, I am pulling back the partition affinity defined on Y I do not want the whole of Y I want it only on Sn minus 1 Sn minus 1 to Y I have map F continuous function you will pull it back on on Sn minus 1. So, FJ eta alpha complete FJ. So, these will be partition affinity on SK minus 1 and support it to each one support network F inverse of FJ inverse of UIs FJ inverse of UIs which is a restriction of an open cover FJ inverse of UIs to take FJ inverse of UIs that will be open cover for DK ok. So, this is a restriction. So, now I apply the previous lemma ok for each of these now I have my entire attention is one single DK one single cell ok. So, by applying above lemma we obtain extension theta J hat let us denote them by psi J alpha where alpha belongs to lambda union lambda J. There I had lambda union lambda prime but that was only one now I have to index it it will depend upon which FJ I have taken. So, I index it by lambda J that is all ok of theta J on DK which is subordinate to FJ inverse of UIs what is the cover? This is the cover it is open cover for each eye ok with the refinement function again beta J from lambda union lambda J to eye which is an extension of beta all these the lemma guarantees me for each J I have an epsilon there but that epsilon I have to index by epsilon J 0 less than epsilon J less than 1 as in the lemma for each W inside W ok and each J inside J look W epsilon J to be take FJ inverse of Ws ok for each W FJ inverse of W take then take N epsilon J of that ok all these if you drop out this J here and F here and so on this is the this is the lemma. So, instead of arbitrary W now F inverse of J FJ inverse of Ws there that is all ok let us take this one. So, this is shorter notation epsilon WJ instead of big this one. We then have an extension of the members of of the open cover FJ inverse of W of Sn minus 1 to opens of set this W epsilon J W belongs to W these are open subsets of dk such that their intersection with Sn minus 1 is precisely FJ inverse of W because it is N epsilon J of that is an extension of this that is the property that for each J the index the indexing sets F of set of all indexes here F of W epsilon J is simply as F of the intersection with Sn minus 1 FJ inverse of W ok and that is precisely equal to FW because I have taken FJ inverse of FJ the function to be precisely what is eta J of FJ ok that is why it is FW ok whatever happens inside Y all those only I have kept there here this is the restriction this is the pullback of the cover of the functions on Y ok. Put lambda prime equal to lambda union all these lambda Js as J varies over the indexing set now I indexing set as J ok that beta prime with extension of these all these on lambda it is beta and lambda J it is beta Js. So, take that one all the beta Js we have extended for each alpha belong to lambda prime define rho alpha prime from the disjoint union of Y and all the dk's on the disjoint union you define first namely on as follows on Y it must be always eta alphas no change there ok on Y let rho prime of alpha eta alpha where alpha is inside lambda for all the new things it must be 0 ok that is what we want anyway we do not want the new things to enter Y right. So, on Y it is this these are the functions ok on dk alphas what are they on dk alphas for each dk J they must be rho prime of X must be the new psi alpha J of X if alpha is in this one for other lambda Js of course they must be 0 because they are disjoint anyway. So, from all other lambda Js must be 0 ok. So, this is now the collection of this all rho alphas ok they are defined on dk the additional dk union Y but when you attach along the boundary what happens to this one they will agree with the corresponding function in Y here ok because this is just eta alpha composite the functions where nothing but what eta alpha composite f alpha f f Js ok. So, they all these functions will now factor down to define a continuous function from X to I for each of them there will be a rho alpha prime I will drop that prime rho alpha from X to I if you look at a quotient map. So, rho alpha of Q X X is some point there is equal to rho alpha rho alpha prime of X that is the way we have defined. So, it is well defined because of on the boundary s n minus 1 f they are f Js given by f Js ok. So, we claim that this family is required partition of infinity which extends the order in a partition of infinity on Y ok that is very clear because that these on Y they agree with this one by the very definition that part is fine ok. So, let us verify one by one if alpha is in lambda then rho alpha restricted twice eta alpha ok on each phi J of the decay of J. So, rho alpha composite phi J is rho alpha prime which is psi J of alpha then support of this one is contained with u beta, but on lambda it is equal to beta prime ok capital lambda it is same thing as this one. Now, suppose alpha is in some lambda J and such a thing is unique right alpha is either in lambda or in one of the lambda Js. Then support of rho alpha is phi J of see support of psi J is with psi J of alpha ok I have to take phi J phi Js are the characteristic functions right phi J of support of that is inside u of beta Js ok. So, because we have pulled back this the inverse image of these things for the original cover ok. So, this u of beta J, but beta Js are by definition beta prime is beta J when alpha is in lambda J. So, it is u of beta J. Therefore, the family theta is subordinate to the open cover u with beta prime as the refinement function ok. Now, let us take w belong to w the old local finiteness steering covering for wire put w epsilon ok I want to extend each of them how equal to q of each w epsilon alpha this is the w epsilon J epsilon w epsilon Js are inside happening inside each disk d J right. So, each disk you take this column take the union of all that and q of that the image of that ok. So, that is my w epsilon automatically this w epsilon becomes open subset in x because intersection with each d J if we if you take the inverse image it will w epsilon J ok. So, each w epsilon is an open subset of x also for each J let us take w J denote the interior of tk phi of that. Remember phi J is a homeomorphism on the interior. So, look at the copy of that that is my w J ok. Now, you take all the w of silence w is coming from capital W or curly w add this extra interior of each cells that is an open cover for the whole of x. And that will ensure local finiteness for theta add as far as all this y's are concerned it will come from w epsilon inside these things inside each d J only things which are non-zero in in this interior of this one are coming from d J that is will be only lambda J. So, it is a local finite this will ensure local finiteness ok. Finally, we have to look at this patching of the index in sets themselves f of w J for each J f of w J set of all alpha and lambda J lambda prime support of rho alpha intersection w is non-empty it is a definition ok. As soon as you write J here this will be happening in lambda union lambda J right for each J this is what it is it is it is inside this particular thing. Now, by the compactness of d K and local finiteness of theta J it follows that only finitely members of theta J's are non-zero here. So, therefore, it follows that this f of w J is finite. Similarly, for w x or w first observe that f of w epsilon is all lambda inside all affines lambda prime support of rho alpha intersection w epsilon is non-empty. So, this may be because w epsilon's are infinitely many w epsilon J's are there ok. But, the whole thing is none of them will contribute anything from acquire anything from these lambda J's this is the whole important thing. So, it is a subset of lambda itself this is this is the key here. So, this is the final thing here. So, we have extra thing if you allow one of them then you do not know how many you have allowed. So, I do not allow any of them. So, whether they are finite any finite does not care. So, no no things come here. So, it is considered lambda. But, then f of w epsilon is same thing as union of all these w J's because that is the definition of w epsilon ok. And, but each of them is f w's here ok. So, finally, this summation is similar to the final theorem also we add this one for any x in x suppose x is inside y then rho x is 0 for its lambda alpha x is not in lambda. All these lambda J's they are all those functions are 0 right. So, only the summation here in lambda prime summation only alphas which are inside lambda will contribute. But, then this is a partition of unity on y therefore it must be x. So, it is eta alpha of x. So, it is 1. Otherwise x is inside one of the phi J of interior of d k alpha d k d k of J ok that a J is unique also ok. Then x equal to phi J of x prime for some x inside that because it is in phi J of this one. In this case rho alpha x is 0 for all alpha not in lambda J prime where J prime is for all of a J inside J prime where J prime is not equal to J only when it is in lambda J ok you will get something. So, this some total will be either it is in lambda J or in lambda ok. So, all this big thing does not come. So, only lambda in lambda J will come summation because all other things are 0 rho alpha of x is 0 right because x is infinite in d k of J that is why. But, these are by definition psi J alpha ok on this one on this d k it is a partition of unity therefore it is equal to 1 ok. So, this completes the proof that theta hat is an extension of theta. So, this part is easy it is more or less the repetition of the final theorem that we have proved in some sense except that this is the key thing here which ensures that the local finiteness is censured here because of this one ok yeah. So, I have told you already that the existence of partition of unity with horse dorseness actually ensures that the space is a paracompact. If you do not know paracompactness you do not have to worry about that right now. Paracompactness is essentially in practice it is essentially to prove the existence of partition of unity. That is the key in all the analysis everywhere in algebraic topology or differential geometry wherever you want ok. But, some basic points of topology has to be done like that that is why Dio Donner thought about this partition of unity ok. So, that was the theorem that I was telling corollary every CW complex is a paracompact. I think this is where we stop today. Thank you.


Okay, maybe I start the last lecture. So first recall, what we have is 4k categories. If you have x omega compact symplectic manifold, I skip the orientation B field, and also the first chain class of tangent bundle is 0, then we get a triangulated category, category 4 omega field, which is 4k category. And object of this category have kind of representatives, which are singular Lagrangian subsets and certain data on them. In particular, if you have object in my category, which has representative supported on some singular Lagrangian subset, sitting in X, then we can associate with this object, it's some class which will be closed chain of L with coefficients in Z, top-regulate chain, defines commolder class, and then, of course, we get a class in n-scommolder of n-band space. This will be image of this class, but now if x is scalar, so we extend to some complex structure and choose n0 homomorphic form omega, then we get a stability structure. And the central charge on k0 goes like this, so we map to n-scommology and then integrate form omega, get a number, so that's n-band integration, okay. And semi-stable object as well, which can be represented by special Lagrangians, a special Lagrangian, and multiplicity is some kind of positive here. And I recall what is a CETA special Lagrangian, a CETA is an R, argument is a following. We have two equations, the restriction of a symplectic form vanishes, of two form vanishes, and restriction of certain real n-form, which is imaginary part of exponent, we have conditional rate to two form and n-form. And what we have in this picture, actually it's something pretty nice, if you can see the set of equivalent isomorphism classes of CETA semi-stable objects in given homologer class, middle homologer of CETA. Here of course we should have this argument of integral of omega theta is equal to theta, we should have this property. To speak about such things, it's certain kind of non-archimedean object, it's this model space of semi-stable object in this given class, it's kind of non-archimedean space, it maps to something more concrete, it maps to sigma special closed chains, closed Lagrangian chains, so we get closed and special. This is defined over QTR, it's already algebraically closed, maybe Q bar, it will be key points of some non-archimedean space. But this is a finite dimensional kind of real space, maybe singular one and so we get something like tropical or real space. This is huge homongo set and here we get some finite dimensional piecewise linear space. And I claim that in this geometric situation it's very similar to stationary skevers, on this non-archimedean space we get scalar metric, we get kind of non-archimedean scalar metric, which means that here we get just remaining metric, it's singular space but at least some part of strata has remaining metric. And what is the origin of this remaining metric? Let's consider this chain, kind of like Q of E or maybe image of Q of E and take a tangent space, let's try to move this single chain, single chain means that you get some kind of union of Lagrangian variety with certain multiplicities, it will be multiplicity and all the boundaries are zero, but all pieces are special Lagrangians and when we move to special chains, then on each most locus we get a section of normal bundle and the normal bundle you can identify, you can substitute form omega, kind of, if you have variation of L, which will be, this is infinitesimal, if this is tied to omega, and you get one form on each stratum and this form will be closed because we preserve this condition. But you can substitute n form, this guy, this imaginary part and we get n form, get n minus one form, just vector field, substitute vector field and it will be again closed and the metric is defined by the following nice formula, suppose you have two variations, infinitesimal variation of my special Lagrangian, then what I do, I take sum over alpha, I apply it to one map, I apply it to another map and in fact it doesn't matter to which order, one can check that it's, you get the same result, actually it's symmetric and positive definite when you apply this thing twice you get integral completely positive density here, two tangent vectors to space of special Lagrangian chains, it's bilinear form, yeah, it's called symmetric and positive definite quadratic form, no, no, no, it's over real numbers, so everything's over real numbers, okay, so you get metric and this one can check that this space has a fine structure because it's identified with first homology of L and it has a fine structure and metric is given by second derivative of some convex function, yes, yeah, so we get, this Foucaille category gives example like a quivers, we get non-incremented model, we get stability, but more than stability, we get Kepler matrix on model space of stable objects and in fact the Kepler class depends only on commolded class of omega, but actual Kepler metric depends on a mutual position of capital omega and small omega and her get infinitely many parameters because one can say that you fix your complex manifold and choose any Kepler metric in a given class, there's a huge functional space of this and as symplectic manifolds will be all simplectomorphic and then you get automatically for each choice you get its individual Kepler metric and even projection from this guy to this guy depends on the choice of mutual position of this two geometric data, so this abstract stability story depends on finally many parameters, second class of small omega and capital omega, but actual, so this thing depends on finally many parameters, but this map and kind of a fine structure and metric depends on the choice of parameters. No, mutual different kinds of facts, you have manifold plus simply two form and plus n form and there are infinitely many parameters for such a guy. Okay, yeah, so that's one story and next it's very, very coarse property, it's something which is again called support property, a long time ago when we start to think about this thing and I will explain it's in this example in a moment, so one can form it for any triangulate category with stability condition, so support property says two things, the first thing is what is written here, this central, this homomorphism from K group to complex numbers goes through some finite rank lattice, it's not some huge thing, it's okay, so it factorizes via certain map, I call it class, to some lattice which is finite rank and some homomorphism from lattice to C and here an example for 4k category, one can take a gamma, it will be middle homology and mod out by torsion because torsion part that's completely reliant from torsion part central church vanishes and the second property is the following one, pick a norm on finite dimensional space, cross R which is Rn, which norm it does matter because they're all up to constant equivalent and then there exists the constant, the pinnacle norm such that for any theta and for any semi-stable object what we have, if we take a class of this object we'll get element in this lattice, so we apply this norm in this space, is bounded by constant multiply by absolute value of the central church which is just complex number, now so it's kind of some property controlling which classes, which class in case theory can have semi-stable object we cannot, so we call support because we get certain subset of gamma when you can expect a trivial object and why it holds for, it holds for 4k category, it's kind of very simple argument because when it has special Lagrangian, it's supported in special Lagrangian or supported in special Lagrangian, then absolute value of the central church is the same as kind of, is the same as integral of, now a real part over, what I denote kf e, and it's the same as up to overall constant as remaining volume of this chain, now if you know remaining volume then you can estimate integral of any closed form like choose some basis of middle coromology of axis coefficients in R and lift to some representatives to some closed forms, closed and forms, I don't know what it's called like beta j whatever, some forms, and those forms because many of them are compact, there are some of which forms, this is some uniform bound and coefficients in the spectral remaining matrix, so kind of you can see zero norms, it's less than some constant and essentially you use this constant, so you see it immediately that all coordinates if you choose basis give you, give you the bound, so it's completely trivial property, but that was a geometric reason and this suggests to keep it in general, it's just for 4k categories and a claim that it's kind of obvious for Queer case, the support property, let me just tell you one way and why it's really obvious, first of all you get category is representations, maybe db of finite dimensional representations of Queer over some field and then how to make some charge, we map to dimension vectors z to the set of vertices, it should be my set gamma and then map to c by dimension vector goes to some of the i d i, we choose some complex number in upper half plane to make this stability, yeah so that's the first property, it's really by definition, it gets a lattice generated by set of vertices and why we get support property, this second story is because semi-stable objects either belong to hard to just ordinary representation Queering Degree 0 and then it's the hard nabillion category, then we get all d i will be non-negative number since some of them will be positive and what you get, you get kind of map positive factant, map to the part of a half plane, it's kind of clear geometrically that lengths of vectors here and here are kind of commensurable or it could be shifted in shift to hard or it could be shifted over the element of hard then depending on parity of shift you get either all the i positive or all the i negative, you get shifted by negative, you get minus electrostatic, yeah so it's kind of completely backwards in Queer situation and what it implies, support property in general implies the following thing, if you consider set of the central charges is c to semi-stable for some theta, you get a subset of complex numbers, it is discrete and it has at most polynomial density at infinity because what happens like imagine station with gamma is let's say z cube, rank of the category z cube, so you get gamma cross r is r cube and by central charge it maps to c which is r squared and what happens you kind of can draw a picture like this, imagine three dimensional space maps to the plane by vertical projection and here you draw some kind of cone around the kernel and now consider integer points in r3 which do not lie in a in a light cone and that's only where if you translate what this means support property, it means that that's where stable object can lie and if you consider the circle of finite radius you kind of do things in a finite direction, you get kind of finitely many integer points in this thing and number goes polynomially depending on rank of the lattice okay so that's why it's gives certain control and with this support property it's kind of much easier to formally improve kind of a version of bridged and basic result than the model space of stability is complex manifold and namely suppose we're given a homomorphism from k group of category two to this lattice and now want to make a stability condition with support property with central charge goes through this lattice that of stability conditions such as Ga goes via lattice and and has support property as it's I think it maps to by forgetting everything from home to gamma to c which is cn and dimension vector space just to get this map Z gamma and and the claim it's locally homomorphism so what it really means that you can move a little bit central charge and then you can make a new stability condition so in particular get a complex manifold because it just some kind of covering of cn maybe sometimes infinitely many but wait so first of all support property will guarantee that you can move this your central charge by certain amount non zero amount and like here can rotate it a little bit it still get projection and everything is well defined so you kind of have a very low bound on how much you can move your central charge and now if you move central charge a little bit what will happen in original suppose fix some direction theta and you get cc semi stable objects and all the central charge lie on a ray and kind of form a discrete set and you're interested don't interest go far away consider kind of finite part of the things so you have object with finitely many central charges now you change your central now you restrict the deformed to some kind of small part of cc sigma and then what happens this central charge will they move a little bit you get things in in in a angle sector but you get to also kind of finitely many things on each finite distance so it looks like a quiver so you can define for this deformed subcategory what is semi stable object or not just looking in this sector and just recalculate defined hardness infiltration by by finite trans properties you get uniques sub-object with maximal slope and so on yeah so that's that's the whole definition but geometrical it's actually very interesting because in general what one can say is what support property says that property says is the following you can make some kind of continuous object for each ray in c we can associate a closed convex cone in the in gamma tensor r which lies over this ray and it will be strict convex cone it will not contains the whole line it will be less than 180 degrees so what what what do you really do you consider not this this way consider small neighborhood we get very narrow angle sector take pullback and take convex hull of the all vector switch represented by a semi stable object and the support property says you get certain kind of convex cone but then you can't limit when this thing goes to zero you get one parameter family of street cones and this one part of family of street cones gives you certain game if you possibility you start to rotate the central charge and then again make convex hull in the fiber because this intersection this set will be no longer convex and it gives certain interesting question convex geometry how far you can go up you get kind of some a priori bound for how far you can go from different things I think it's very attractive question even you have just finitely many rays then you get some kind of like you know analytic continuation up to certain domain it's looks pretty attractive yeah okay what do you mean by this this is not until the ray and the convex are because it's the unique there is a quality no no no the quality no no what what happens through life you get a certain family of convex cone cones but now but now you rotate projection and then the pre-image of the ray will be no longer so intersect with some kind before intersecting with some one parameter family of hyperplanes but we consider this different family of hyperplanes intersection are no longer guaranteed there will be convex so it should take convex hull and so on yeah so yeah yeah so so what we have here we get two stories we get in fukai category we get non-archimedean caliometric we get support property and just want to say says it and support property generalize to fukai category risk coefficients which I explained at the end of last lecture so it was kind of interaction between algebra and geometry this fukai category risk efficient and I recall you what what happened is this service coefficient so at the end of last lecture I have some I discovered some very simple situation of categories with constant coefficients so constant coefficients there was certain category C0 which was equal to category put at each point it will be some triangulated category give some field and my many fault was a torus with some symplectic form but now I want to put also complex structure so it will be elliptic curve and this should be some subtle charge with values not in numbers but in forms which can be said the following suppose you have a central charge from your category we're going to get some stability structure is this central charge and then you define map from zx kind of for each point on my torus I get a map from which one of cotangent bundle the volume element to just one form here and it will be given by the formula is the x will be the zero multiply by certain omega which omega will be one zero form holomorphic one zero form on elliptic curve so it was essentially like product situation and carmologically we get not many parameters but geometrically we get many parameters because in local coordinates you can always find local coordinates x and y so x will be no not point of x will be this on capital X so that form omega will be the x plus e dy so it will be really imaginary part of complex coordinate and symplectic form in just certain two form the row is anything infinity function of x y it should be non-constant story and this will affect the whole geometry of stable object at the end of the day like in previous situation so the whole category here total category it just some kind of tensor product by 4k category of v and by mirror symmetry it's the same 4k category of torus the same as coherent shifts on elliptic curve and this guy will be boundary of category of what's called tate elliptic curve or non-archimedian field over my k and tate elliptic curve is you take kind of analog of c star as people on joint right to take multiplicative group as analytic space okay kind of a fine line minus point as analytic space and defined by powers of q the q is normal less than one and q is element of the field t to the area or the thing which is jingle of positive real number okay and and then we get hypothetical stability condition from k0 of c to z n cross z squared then maps to c the first guy is what because it's from here we get from c0 we get class of or by i assume that now this thing has support property yeah and you think it's also will have support property if k0 of my category c0 map map to some letters it will be first map and this one will be first homology of my torus and the and the map will be product if you get element here and here what we're mapped to it will be product of two complex numbers it will be z0 multiplied by integral of omega yeah so that's that's the picture which one can use for it iterative construction of of this story and what was nice about this story that because special around these two dimensions are so simple it just straight lines in this local coordinate section y so one can try to see what are special Lagrangians and so just remind the stable object that kind of complicates things you get on a torus you get certain graph which is a straight in affine structure related to capital omega the simplex structure you don't have any social structure it has finitely many edges and then then they get certain decorations for each edge in alpha you choose an object of your fiber category the alpha with the properties that c to alpha plus argument of omega restricted to this edge which is constant should be equal to your capital sigma choose and so you get these things then they get more complicated things then they get certain gluing data that vertices which I explained in a minute and the third you should add some boundary chain solution of some marocatan equation it's a solution of certain algebraic problem in form power series and this is really complicated and possible but hard to make explicit and this is and I want to explain that this second part is relatively easy kind of is the goal of my next few minutes will be kind of how to remove these things completely from considerations so what are these gluing data just assume for simplicity the c is equal to zero it's just use of convention when you get a vertex of the graph what we have we have kind of vertex and some n spikes for certain number n and and in fact there was a picture this can be since like a skeleton of disk with n added intervals and infinity and this local category local 4k category is representation of river n minus one it's end of the day and maybe I'll just draw again this my spikes here so we get something like n plus now just draw horizontal line and we get some n plus spikes goes up and minus goes down and then we measure angles we get c to 1 plus sigma 2 plus and so on and get a measure angles in opposite direction so you get just some batch of numbers there between zero and pi I assume for simplicity the snow horizontal direction now we have to put some objects with certain slopes on this I think and how what is the gluing data just to join to the surface square turns out it's it's something very nice gluing data which is kind of part 2 here this gluing data is the following it's just one object in category a0 which is heart of t structure in a which will be a billion category and two two filtrations by numbers from zero to pi exactly by these numbers and these numbers iterations which are not hard on our simulation I call them anti-hard on our simulations but what does it mean it means that each associated gradient is semi-stable with corresponding slope but the order is kind of opposite to what is in definition kind of wrong order and this thing is not canonical hard on our simulation is canonical but this is there are many of them wrong order and what does it mean so you get some sub-object sub-object many times in zero and you get some another chain of sub-objects and when you take associated graded factors you get e something like e1 negative e2 negative up to minus negative and here you get e1 positive e2 positive and plus positive and they belong to certain just to this using this rule up to sine up to shift there belongs to heart of t structure we write even a0 sigma minus plus belong to a0 pi minus sigma i plus so you get the adventures that you get everything is an abelian category it's much more convenient to think you don't uh it's like vector spaces yeah so it's not complex as up to homotoputes now so this is a nice description what uh what is local data but this is really nasty and complicated algebraic beasts and I want to explain away how to avoid it completely it was pretty miracles not in exactly this situation but I will kind of similar situation as up with slightly same simpler situation namely it will be non-compact version and this is this will be related to generalized honeycombs and horn conjecture which you'll see in in a few minutes so now many fault is r2 not not elliptical but just plain omega will be dx plus id y so to just see just usual see but x y now global coordinates and but symplectic form is certain density as before we are always positive and I put some secret which looks very very strange strange constraint which does allow to be constant density of course naively I would like to constant this it will be not good for me namely the two things that if I integrate consider volume respect to symplectic form of any strip in any direction then it will be finite but if I integrate any sector which is bigger than zero degree I get infinite yeah such thing exists for example I take things equal to one over one plus x square plus y square it will be logarithmically divergent for things interconvergent we can rise to some power between one and two I think this yeah so the essentially definitely exists and now for simplicity I assume that my category which I start to play with it I can reduce now to case of a billion categories a billion category will be representations of some cure or some field and and stability is given by kind of cure like data this number's in up a half plane and it's category over for small a and big cat now a will be representations of quiver over big k which is then serious with real exponents it's could be any field in fact any field with continuous evaluation not necessarily serious yeah actually a slightly like it's will be not all representation representations such that one the following condition holds either there exists the collection of norms or matrizations new I in matrizations of these correspond components set it for any arrow TR maps unit ball to unit ball or equivalent one can say is the same as my representation has a model of a ring of integers of okay and in my first lecture if you still remember I considered such category and define some flow here and this choice of there was a kind of theorem in first lecture that if object is semi-stable in this category it's if it only if it's admit some model said that if you have make reduction towards there you will get semi-stable guy and such thing you can call kind of baby a case of harmonic matrix e in of a is c to semi-stable if there exists a matrization such thing I called matrization of e collection matrization such it's it's resolvable said that corresponding representation over residue filtered c to semi-stable it's called harmonic metric okay now what is this generalized Koini-Kohm correspondence it's it's kind of conjectural and one-to-one correspondence between between two things yeah first you fix object of a so it means that representation of q or over residue filter which admits a model but I don't choose a model so we get some representation and first things which I do I consider two anti-harmor-axial filterations of e over this both things are over k over k and of course they are labeled by my interval 0 pi and as I explained this formulae it is a picture like this which I used to be a vertex where I use capital K as kind of new abstract feel forgetting that it's non-archimedeon and the next thing is choice of hermit of harmonic matrix which I explained just one minute ago what is harmonic metric matrix on all associated graded which are semi-stable so they have associated grades so it's a graded plus f plus minus of certain sigma of e yeah that's one object and second is and second object is a pass in the space of matrix infinite from infinite time from minus into infinity which produce finite non-compact spectral network in in in r so it will be certain in r square so it will be in natural network will be kind of l what goes on so the picture will be not this picture it will be something similar but here you get some interesting picture in between and maybe some race will be separated by two or three so with the complex structure can you say spectral network means r2 c then no in r2 is equal to x oh c whatever in c yeah in my manifold x yeah so what what does it what does it mean so you get map from t to new t and it maps should be is kind of c infinity in certain sense outside except finite many number of of bad times yeah kind of piecewise continuous piecewise smooth and for smooth for not bad d it has a speed and speed will be a r-filtration in as explained speed in all the situation is the same as r-filtration on a representation over the residue field these things over small field and there are two axioms when it gets r-filtration get some real numbers some finite collection of real numbers which labels this filtration kind of our labels of these filtrations which are numbers lambda i maybe depending on t you get finite collection of real valid function are in certain one-to-one correspondence with the following guy you intersect your think with horizontal line y y equal to t and get finitely many points so the bit moment of time when you intersect vertex of the thing so it will be x coordinates the correspond to x coordinates can this line intersect the vertices the bad thing bad things intersect vertices cos vertices and the good are just in between when you get different branches yeah now so this real numbers and this is real numbers and how i identify them here use my symplectic form in this complicated function r the identification depend on r on raw and namely it will be in my plane get point zero zero yeah it kind of will be like zero zero and it uses some integrals in certain way namely lambda i of t of t is equal to integral of from x i of t to plus infinity of x dx it will be convergent integrals maybe i should put infinitesimal also the interval over any rays finite minus the integral from zero to plus infinity dx so it can integrate from zero to x i integrate or finite interval and get certain one-to-one correspondence between between real numbers and with some open interval r yeah so it's some this x 80 never is not zero anyway this could be zero of course yeah it doesn't matter this could be position with respect to coordinate axis is irrelevant zero and and this one x in the second condition so we draw some picture get some kind of multivalued function and then there's this picture that should be straight intervals intervals and rays and then because it's guess filtration we get some associated graded guys and associated graded are semi-stable ct semi-stable the ct is pi minus slope of my ray yeah so it's will be some speech produces pitch which i want kind of on torus but on plane but now i don't have to solve any this boundary chain it's just familiar of metrics give me this picture yeah that's actually it was fabian hide and he proposes about your to this point of view which kind of seems to simplify enormously this is boundary chain one can completely remove and now what is what is the correspondence the correspondence is given by it's only there's a map in one direction if i have this pass which i call harmonic pass uh because at the end of the day it's kind of harmonic object yeah from harmonic pass two iterations over two anti-harmory filtration over the big field and some norms on associated graded and harmonic metrics graded quotients um so for example how to reconstruct f i'll just go to one direction reconstruct f minus f plus will be similar formulas um so we get metrics on my representational cure like keep on one vertex we get vector spaces yeah metrics on vector space parameter and it's kind of moves its uh shrinks in some direction explains another direction and it has some certain behavior it goes to minus infinity and as i explained in a situation when you get this family of metrics usually get kind of flags the limit the growth in some direction faster than in other the limit could be not unique but here then the situation it's unique limit uh and uh namely if you move to kind of negative direction what you see you get uh certain rays go with some some one slope another with another slope and another slope so get finite collection of slopes that kind of y y goes to minus infinity here to the interesting and story and you get certain slopes which gives me first some numbers theta c to i you get slopes slopes stabilize for negative time and you get these numbers sigma one sigma n i think it will be sigma one minus sigma one two two minus these angles and this appears with a certain multiplicity and if meant will be filtration by by by growth and different uh when considered two different directions the the the the growth rush of norms will go to infinity for different similar eyes uh grow kind of norm considered vector in one terms filtration last term filtration the ratio of norms go to zero infinity depending on order and why it's so because uh what happens when I calculate what happens with norms you calculate the area of the things in between and this goes to infinity take expanse so so it gets diverged but if if they stay the same the it's will not go to infinity yeah so exactly this convergent of integrals and arrays say that they have the same speed of growth in in branch of parallel lines been completely different if they're separated and harmonic matrix on associated graded can be defined as a following query on associated graded you define it as a following query uh just rescale take a vector sitting in some sort of terms of filtration and now have metric depending on parameter t and then rescale omega t by some number by let's say t to to certain number and number will be the following thing it's will be kind of integral which calculate you pretend you start with reference point zero zero through the picture draw this angle sector c to i then go to this y equal to t goes to very close to minus infinity take this triangle and integrate over omega omega volume omega area of this guy yeah so it's certain things which goes to infinity but if you risk after after rescaling you you will have no divergence okay and similar get a plus in your plus and now i'll just finish before the break with two examples of this yeah so uh story first example make your adjust point category just vector spaces and stability this given by number minus one so nothing to speak any metric is harmonic nothing to speak about and if you return this definition there's only one anti-hardness infiltration because only one direction and you see uh what happens we have two anti-hardness infiltration don't do any sequence of extra space but then to two harmonica just two metrics get vector space with two norms place over key these two norms and what pictures you can draw the only thing you draw just a bunch of parallel lines nothing else how one is related to another yeah when you have two norms when you have two norms on vector space on k to power r that's r dimensional space then we can simultaneously diagonalize them and then we get a singular values and consider logarithm of singular values work of singular values get just bunch of real numbers some of them coincide some don't coincide and in fact each of these lines will have certain multiplicity to be coinciding numbers and here when i get this lines you draw kind of x some certain numbers x i in r multiply by multiply by y belongs to r this will be collection of my lines and um what the correspondence it's again based on this reference point you get somewhere point zero zero and uh because all these identifications was accurately not wasn't translation variant at all and my nothing was this translation variant so it's given by certain integrals lambda i is the integral from if i if x i is positive and minus integral from x i zero times r if it's negative you get again this axioms this volume is finite it's extremely crucial for for for the right yeah so that's it's uh yeah so just in this case it's series of simultaneous deganization of two Hermitian norms and in next case that's really a few cases which you can start in next cases here is this one and we pick non-trivial stability condition when you get three types of semi-stable objects dimension one zero one one and zero one and consider e which will vector space maps identically to itself and we get two anti-heir anti-heir anti-heir signal filtrations one is one step so you just get this object it's already semi-stable okay or two steps namely you consider a representation by zero map is this representation it's a sub-representation of my add things and the quotient is this representation yeah so this arrows means different things it's a morphism of representation it's this arrow and quiver yes it's sorry to use arrows two meanings in the same picture uh so you get short exact sequence it's also filtration and now when you consider so this 20 hard but now we have to choose harmonic matrix on associated which are three guys and harmonic matrix on three associated graded things and the same as three matrix on v three norms on v and there is a kind of famous question in complex linear algebra what you can say about eigenvalues of three Hermitian matrices with sum equal to zero there are some horn inequalities and the same answer but for different questions functions suppose we got three Hermitian norms in the complex vector space for each two of them we get logarithm of singular values and this satisfies some again some inequalities what is it and this similar question non-archimedean case you can see the three norms on non-archimedean it's just the same inequalities all all together and the they correspond to this honeycomb picture which how to draw it yeah i do you do picture like this that's a typical diagram when you draw this diagram you get only three possible direction for edges and general expression will be exactly this honeycomb picture this horn conjecture uh this uh kind of you can intersect this far away with these three particular lines and get collection of whatever r three r real numbers this total sum equal to zero it's like this logarithm of singular values for different pairs of Hermitian norms or non-archimedean norms and the horn conjecture for many times says that things exist it's only such picture exists yeah and um there are many proofs none of them is very canonical and from this abstract picture the abstract will follow because what happens here i get kind of this nasty function row but imagine that you start uh with uh kind of three three times the same norm you get three times the same norm you draw picture like this nothing this will be this uh by this conjecture it will be this unique pass it will be exactly this guy now start to move it a little bit and even a little bit you put some parameter epsilon and suppose this guy will be of size epsilon and this area will be of size of epsilon square and for kind of all measure for in leading terms you'll just really see this displacement so this vertical lines so if you kind of rescale your uh valuation then for small epsilon you get kind of laxation flat space this this function which decays into infinity will be irrelevant because it's and then you get as the leading term you get this horn conjecture but yeah yeah i have to say that we still didn't prove this whole story how we can want to prove it it looks like uh this two anti-hearing field filtration is the norm so kind of like boundary conditions the plus and minus infinity for your pass and then you get your pass connect uh in space of matrix you get certain constraints that you get this finally many things and this pass is one can write a solution of certain Euler Lagrange equation uh you write certain action functional of certain Lagrange and depending on time mu and mu dot which is this filtration and in certain things this guy's convex in the last variable and then it's like mechanics it's negative curvature space you should have unique geodesic but it's like fincillarian metric on very singular space yeah so it's a bit of nightmare uh this equation why is this solution it's kind of really regular why you don't have why you can have infinitely many things and there is some very beautiful upper rebound for the following reason when you get original hardener you see that for this picture you get two anti-hearing field filtration for your object and then for each vertex also get some pairs of anti-hearing field filtration so this object we still we still anti-hearing filtration gives you you get some stable they have the center charges and now you consider central charges of one part of the filtration and they ordered and take the sum of complex vectors you get some pass in a complex line which maps to the center charge of your object or you get another part of hardener filtration you get some certain polygon and if you make kind of Legendre transform to this picture it turns out this is polygon is decomposed it can form the same picture but it still be finite polygon decompose in these lengths which are only discrete set of numbers by support properties decomposing some other polygons which are convex one and of course because they have bounded area bounded below area you get only finitely many possible stations so there's some kind of tiling these two anti-hearing anti-hearing syndrome polygons there's some kind of game kind of finitely many tiles of different size of various size bounded size and you just try to get the game so it will be like kids kids toy all those stability structures so this is the this this solution extremizes this yeah no no no when when you get any such picture you decompose a priori given polygon by tiles of which belongs to some finite set so there are only finitely many possibilities for all this game coming from this if you extremize the action of action no no you can see the Legendre transform of this picture you get some convex function take Legendre transform you get kind of dual picture yeah for each vertex which vertex you replace by two-dimensional cell no but the dots are coming from this action this action you consider as no action it's no no it's it has nothing to do it's a solution of earlier Lagrange equation it's just reformulate all those properties it's like shortest geodesics and stuff for something slayer and metric but as then there's also a way to draw this picture certain it's it's also straight lines it's just kind of the same the vectors which are parallel to z but it's a really a central charger for some objects yeah okay so now we make some small break no yeah so it's kind of really neat way to reformulate things this bounded chain kind of completely disappeared so question about norms some geometry of these buildings and so on now let's try to apply to periodic case apply to x which is r mod z was r mod z how we apply so if you want to reformulate notion of this bounded chain on a torus go to universal cover universal cover and on universal cover you get kind of two periodic pictures something like something of this nature now what we do we again start to intersect with horizontal line y is equal to time now get infinitely many points of intersections yeah yeah universal covers if intersect with horizontal line we get infinitely many point of intersection infinite number of intersection points so it looks that you should get infinite dimensional representation of a quiver but we have still shifts x goes to x plus integers and should should act on this story action and there's kind of obvious guess what do here you should have norm on infinite dimensional presentation which in invariant under shift norms norm of t is invariant under the action and now one can make this kind of exercise what what is all about what does in infinite dimension space the action and norm invariant under the action yeah it looks kind of complicated things just think about individual vertex and get some vector space infinity will be some infinite dimensional space of a non-archimedean field although it looks rightening or the queen feels already something infinite it's doubly infinite and the action means that it's module over loran polynomials in one variable and it's kind of finitely generated models because up to shift you get finitely many orbits and let's kind of assume it's not over all polynomials it's it's kind of module over a ring of functions on tube domain on annulus instead of kind of analytic function annulus and annulus is given by inequalities for some epsilon yeah so it'll be serious in loran series instead of loran polynomials loran series which converges in this domain whatever it means yeah and yeah so to and it's kind of result I assume that it's three module over this guy so it's the infinity is module over the six means it's a section of analytic analytic sections of some analytic bundle on annulus and now let's make you a definition a bounded metric on this thing is metric restricted to circle say is a pre-norm on this v infinity satisfied to to constraints such that it's z invariant it means that multiplication by z for the infinity to infinity is isometry and this word bounded I say it's a some pretty weak property for boundness modulus you do the following I assume actually this guy it will be three module choose trivialization is module over or trivialization of your bundle then there exists a constant depending on trivialization said that for any element of the infinity which you can imagine like section of my bundle and suppose it's non-zero what I have I have if you can see the norm of this section it's up to up a low bound is the same as a supremum of this s of z over unit circle it will be in coordinate space we took maximum of coordinates on circle now so that's some condition why it's a reasonable why called it's a metric on the bundle it's metric it's a Hermitian metric or a non-committal metric on the space but not on the fibers it's kind of different notion but one can make an exercise one can apply a situation because all words make sense of a complex numbers instead of non-argumenting field for k equals c you can see the ring of analytic function holomorphic bundle sections you got infinite dimensional space and I want to heal but norm and with the same boundness condition and the same this Hilbert norms are the same as Hermitian norms on on the circle which are measurable functions yeah it's kind of a pretty funny map says this bounded matrix yeah let's trivialize bundle we will bundle and this we'll consider this kind of something a bit larger than around polynomials is efficient cr and this is norms and take this sort of reference there's a claim that the bounded matrix are the same as measurable maps from s1 to Hermitian r by r matrices and this thing is bounded above and below and less than c to some identity like between sandwich between uniformly between two things so Libeck measure yeah yeah I don't know what is Libeck measure on this non-argumenting circles it's hard yeah but our complex numbers it works completely perfect it's it's a really funny way and the idea is that it's kind of a good notion of Hermitian metric on analog of a non-argumenting geometry this formula is kind of globally it's a bit strange way yet now when you consider pass let's mean now it changes horizontal line you can see the Hermitian metric but you restrict your bundle not to unit circle but circle of larger larger radius so in complex case it means that you just make Hermitian metric everywhere yeah yeah so this whole story it's kind of a formulation is a whole pass it has a following in this station has a following meaning you just construct Hermitian metric on vector bundle on whole annulus not on circles but everywhere for each t of r a bounded norm uh bounded metric on very restricted on very restricted to to the circle epsilon 12 of z complex point of t yeah yeah that's it will be story yeah and here yeah it's kind of a start analysis something is kind of wrong we want it seems to be continuous but it's measurable yeah so there's some question of regularity yeah and so my philosophy is it's all kind of analytic questions is non-argumented and should be reduced to question in real geometry when we understand that and yeah but the basic thing that's all this object of 4k category will be by mere symmetry transformed to bundles with norms yeah so there'll be no more holomorphic disks it will be purely kind of a kind of replacement of 4k categories by some kind of most mainstream mathematics i have to say that's just just no holomorphic these foundations and so on yeah all this picture generalized to several variables if you get kind of n variables in c star or k star and you can see the product of circles and you want you can do the function consider vector bundle defined in small neighborhood and define a metric as a norm on the space of sections which is in the very for which multiplication by each coordinates are unitary operators and to turn us out is the same as differential geometric notion complex case it's kind of abstract linear algebra infinite dimensional stories reduced to usual geometry and all this fits to a mere symmetry mere symmetry is the following station i recall it when we have model space or whatever super conform field theory if you go to one direction to get some manifold with scalar metric you get another manifold with scalar metric but if you go simultaneously you get some sickles you get kind of tropical base on which you get vibration by Torah and will be dual torus vibration for different picture and at this moment you identify 4k category on one side and coherence shifts on another side yeah so i'll just assume that i have kind of torus vibration i suppose b is a manifold with z-affine structure you can imagine again torus or rn then you get x the cotangent bundle v you get torus vibration and it will be symplectic manifold uh fibered by lagrangian tori on on the base or it could be something else could be some a non-archimedeant variety let's say m which also projects to the base and the notion of torus vibration means the following it's base it's locally embedded to rn open domain and the map is given the following it's and you get kind of like invertible coordinates they go to a logarithm of norms so complex numbers it's completely clear and kind of formula is the same in non-archimedeant case so you get kind of generalized tube domain yeah yeah yeah yeah you know i mean kind of both varieties if i interpret a model on one variety and it will be the symplectic manifold and b model will be on dual variety it will be yeah yeah this is this is a mirror symmetry situation yeah yes yes yes yeah yeah so so uh a usual homological mirror symmetry uh says the following that uh when you consider something like fukai category of x omega is equivalent to uh bounded category of coherent shields but actually better to write something else it's called perfect complexes on m and perfect complexes is um what is this is perfect complex i remind you if you get it's kind of can be defined complex geometry the same as non-archimedean geometries you cover your things by open domains you can in good situation finite covering is enough it's really like like a drawn picture on each your alpha you draw finite complex of finite dimensional vector bundles not trivial maybe locally trivial or can you even make it small can you make it's even trivial of vector bundles and on intersections you get two complexes and you should establish quasi isomorphism some map which induces isomorphism homology and then on triple intersection you should make some homotopy and so on yeah it's kind of concrete way to present the object of this perfect complex because of some abstract stuff but that's that's a picture yeah so that's in this category over both over k and what all the story seems to suggest that there is a kind of equivalence of categories of a ring of integers not kind of extended to the following thing i remind you this 4k category to construct this the following way we have this various singular Lagrangian subset we get categories defined of ring of integers take inductive limit and then invert small parameters let's do not invert small parameters this kind of limit or singular Lagrangian space of this 4k category 6 which is over o of k should be equal to perfect complexes with norms yeah that's yeah so this question what is perfect complex with norm yeah usually in great geometry when people do things that they have kind of very simple picture class of norms they consider they use things like models make reduction to the ring of integers of errors because you have a variety of bundles and so on it's will be kind of norms but but norms are very special it's like p2s linear with integer slopes and that's definitely we need something more as you will see so it one should do it kind of by hand because in general norms is i don't know what it's this very huge infinite objects but there are some kind of last class good norms which we can which are concrete objects given by finite amount of data just for individual vector bundle i want to say what is a good norm how suppose i get kind of point on your space i'm analytic you want to describe in neighborhood of point what is a bundle with a good norm so each bundle has one norm no no many more of course normal like in bundle can use commission norms arbitrary to each point yeah it's it's a similar story yeah yeah to make certain formula one want to write some kind of formal producing norms for all neighborhood and what came to my mind is the following you first to choose finite collection a and one finite collection of sections near point of dual bundle and such that they span all dual bundle so so we can see the linear combination of section get at each point they form a basis as they contain the basis then we get also some collection of germs of analytic functions yeah both these things are still non-archimidiant objects i'm serious with some residue field and then there will be some third guy will be a collection of maps fi are functions let's say continuous functions on set positive rate of power and two real valued functions for i from one to n one yeah this will be linked with real analysis these functions and when you have such a data we can define for each point y closed to x and for any vector in a fiber i want to define norm and the definition will be the following it should be maximum over all i from one to n one you consider s i of v at point y you get some element of your field or maybe extension take enormous this so and at least one of these guys will be positive strictly positive because this thing spent everything for the even v is non-zero and multiply by some zero number which you calculate completely formally exponent of fi and applied to real numbers which are norms of this f one of y f and two of y yeah from this non-archimidiant guys produce some non-zero non-negative numbers up get arbitrary way if you like some real number and yeah so it's some kind of norms which from which you can get formula and can manipulate and i think this is a kind of good class of norms which is efficient for stability issues now so the question is why one thing is related to another for example here's some Lagrangian subspace subset why when you get norm how you get Lagrangian subs subset in real symplectic manifold yeah it is kind of completely different story yeah so let's hear some open question how to do it and i have certain guess it's how to get single Lagrangian subset l and also on smooth spot to get finite dimensional complexes and so on so the proposal is the following there is something in shift theory called micro local support for shifts it's by Kashiwara and Shapira you can see the shift of anything it could be more or less whatever you want of anything on a smooth manifold in fact maybe c1 i think it's enough manifold then i get b say manifold y then i get closed coisotropic conical subset it's micro local support of shift sitting in cotangent bundle and yeah first of all it has some part on zero section zero section it sits on zero section where the shift is non-zero it's already zero covector will be part of micro support but what to do with non-zero covector so get kind of like x zero belongs belongs to this micro local support if and only if it's f is trivial near x doesn't belong to micro local support if it's on this trivial but point xp for p non-zero cotangent vector doesn't belong to micro local support it means that you can do the following you can see the kind of half choose local coordinates make a half space and consider sections or shift on this half space it's can extend a little bit outside like solution of like heat equation you can extend to positive time but cannot do it negative time so the shift of solution of heat equation has some interesting micro local support because they can move only forward and time but not backwards and this is micro support it's exactly about such properties for arbitrary shifts there's this possibility to move things and it's pretty not trivial is that you get something coisotropic and in a good situation it's lagrangian and i hope that in this situation we get lagrangian subset but this is conical and where is the shift now to do shift the idea is the following guy we get shift e whatever shift e perfect complex with norms it's called e with some norms and now i will construct a shift on base of my torus vibration multiplied by real line with some coordinate tau and what will be germ of sections at point b tau say it will be sections of on neighborhood of on the torus kind of with norm less than tau plus epsilon for any epsilon for small epsilon you make small and small i think it's certain definitely shift and conjecture that it's for for good metrics micro local support is lagrangian but it's lagrangian in what in cotangent bundle b cross rt and it has two properties is conical because the whole story is conical conical and also invariant under shift by tau because we can multiply by constant things it doesn't really change and if you try to think what this doesn't mean it means you to get non-conical lagrangian cotangent and b but then we have also multiplication by you have kind of automorphism of my situation i can multiply by coordinates this invertible coordinates in the i and it means that the whole story is invariant under shift and then in projects it has kind of have infinitely many branches but then projects it gives certain object on a torus vibration not on a cotangent bundle it will be have infinitely many things and when you shift you get yeah even in the simplest case you get kind of like link infinitely many where all curves things like this in one variable and you shift to get things on it on a s1 vibration and then one can make a simple exercise for example if you get certain domain in b some domain and you get real function from this domain w2r arbitrary real function like sinfinity function and then consider trivial bundle kind of o or m kind of trivial rank one bundle but it put kind of trivial metric multiplied by exponent of f of ring of projection of point you multiply by function which is positive function which is pulled back my positive function here and then if you make calculation with this definition with the shifts and so on you get image of graph of differential which is Lagrangian manifold in cotangent bundle and project to a torus yeah so that's by whatever intuition it's really what should what should one correspond to another and then it seems that one have a now try to and with this language one doesn't use the word for category at all no holomorphic disk nothing can just some game with this filtration song and the proposal is the following for algebraic geometry how to construct stability condition you make this non-archimede in a specific operation construct formally dual guy should be something should be of course in real life you get not some fibers are singular should be done in singularities but even without singularities it's a big question now on this symplectic manifold which construct completely formally just torus vibration you choose some complex structure choose some holomorphic form and now start to solve a real monchampere equation that this micro supports a special Lagrangian guy yeah so it's yeah so it's eventually kind of top told you I start this for category is the same question but now the story that I don't have this algebraic horror which related how to go to this boundary chain it's purely the kind of usual mathematics and of course it's in high dimensions looks horribly complicated question if analysis geometric analysis was single Lagrangian subspace what is going on but the advantage that we can do things in arbitrary dimension and in last lecture I explain things which are much simpler get only straight kind of graphs step by step when it iterate this construction with torus and two-dimensional torus and so on because in complex one-dimensional special Lagrangian just graphs it's not a big deal but so analysis it's kind of only questions troubles disappear but then the price to the pay is that you get not kind of iterated a serious ring kind of very iterated limit and not not a single limit here we get single limit just one norm and not norms of norms or norms and so on but complicated analysis so something should be paid here so the question is can one do without non-recommended fields at all yeah those non-recommended fields it's just kind of like the generic families that's the next level of complexity but what I can try to guess here just doing on this case with one limit that one should really give a description of your stable objects as complex of vector bundles and the length of this complex should be something in one domain something in another domain and this people in differential haven't done they usually still locked in question about what what happens if it's just one vector bundle or it's the stories could be not even complex of vector balls should be described as complex of different lengths and different parts and why it's so because in this svz picture we just take projection by torus by some base and you're interested in special Lagrangians and special Lagrangians if you consider how look it's projection here it could be smooth but when project here it has some folds and somewhere it has some like one pre-image somewhere three pre-images and then you intersect you're represented like rank one bundle or maybe rank three bundles in certain degrees the muscle Phoenix gives the degrees so it means it should divide at least your base on several domains with some real boundaries and then to domain describe the complexes of its own lengths and this suggests maybe it's a complex station also should divide by open domains but that's kind of pure fantasy there's no real reasons here this is a perfect context with the norm and in the construction how the norm comes here norm because I put this norm new of section here I use this norm here here yeah yeah and and also the kind of the last remark this when I construct stability I always say I say I have complex structure and holomorphic volume form and that's actually not good because the stability structure depends on its abstract object that depends on only on co-homology class of this volume form and it's cannot run through all co-homology of your manifold so this kind of basic trouble is that if consider some let's say Calabi or Variety X first we start with symplectic manifold and then choose all complex structures G and then it will be holomorphic forms and when considered homology classes you get certain elements and middle homology with complex coefficients and they run certain subspace and it's not all not a domain open domain image is not an open domain in a modular space in in homology and by bridging ceremony should kind of variety stability freely in some neighborhood yeah from three-dimensional Calabi varieties you get kind of holomorphic Lagrangian cone here but not the whole domain and the idea that's maybe we can what we really need from this form for this form we need the properties that omega restricted to any Lagrangian subspace is non-zero and it looks to be that's the only condition so there's a question of linear algebra you take space r to n with symplectic form you can see the ends power of dual space complex numbers and you have considered elements here which non-vanish on each Lagrangian Gay actually there are two components of the spaces some kind of right oriented and wrong oriented you can see the only right oriented part and for all of them it looks that all this analysis should go through like for holomorphic form so what don't don't really need complex structure at all we define flow given by argument of the restriction and so on and then the total area decreases and maximal value of arguments also decreases yeah so there are a bunch of interesting inequalities which goes in right direction and it looks that's that's a right class of geometric data for stability condition more general just as complex forms okay and yeah so it's will be kind of eventually marriage of analysis and algebra yeah yeah okay thank you