Good morning folks. Steve Rhodes coming to you live from the shores of a pleasant Delray Beach, Florida. This year 11 am update and we've got the mixed bag out here. You've got the Dow, which is trading to the downside. It's off 90 points. The other US indices are trading to the upside. That's including the spot ball of Chillingix S&Ps up 24 points a half percent, a little over one and two tenths percent for the NASDAQ. That's 192 points, eight tenths for the Russell, 15 points, two and seven tenths for the Semis, 107 point move there. Trendy's are flat. Gold's off eight bucks. Silver's up a nickel. Light's recruit is back $3. Natural gas's up 11 cents. The third of your Treasury, she's printed out at $1.2221. That's up 20 ticks. Let's figure out what all that means by looking at that nine panel market update chart. We begin with the ESMini upper left hand side. So the ESMini is attempting to form a new profile. It is above price. That is a bearish message out there. It's price target on a further rally. Then a further rally would be an indication with a close above Friday's high. Friday's high out there is up at the 4760-25 level. We've gotten up to 4760-250. That's the first day in the last five or so where we've at least gotten above the prior day's high. So let's see if we close above it. If we do, odds favor a further rally. In that further rally, I would target that 4799 level. Now a further rally is likely to unfold if the spot volatility continues to close below the 50-day exponential moving average. It closed below it on Friday. It's trading below it right now. It being 1386. The NQ is also attempting to form a new profile. Now the interesting thing here, it's up into the sell zone. The sell zone is between 16583 and 16749. So during the Trader Zen Show, let's go take a look at those intraday charts, see if there's any kind of signals there. U.S. dollar index struggled to get above descending trend line resistance, profile resistance out there. It's just a consolidation. It has a profile support level between 100.51 and 101.09 and resistance up at that 102.26 level. Gold traded lower this morning. It has found support at the bottom of its daily profile. You can see it's trading in between a couple of trend lines, both rising and falling trend lines out there. Not much else going on inside of Goldilocks World. Silver is also attempting to form a new profile. It's got support potentially at 2336 resistance up at the 20, I'm sorry, 2306, I think I said 36, and 2362 would be your resistance level. Lights recruit, testing support, bottom of its daily profile in 7014. Folks, stay tuned for the Trader Zen Show. But if you all have to start your mind to have a magnificent one, thanks much for joining us. We'll look forward to speaking with you again soon. Take care now.


Shipmates, C&O Gilday here with my wife, Linda, to wish you happy holidays and to express her thanks for what you do day in and day out for our Navy. While this year is different than years past, it's especially important for us right now to take time and focus on what really matters families, friends and loved ones. We could not be more proud of how each and every one of you, active and reserve, uniform and civilian, have so resolutely stood the watches past year. There is no doubt our success is due to the personal dedication and the sacrifice of each and every one of you and the support of your families has been absolutely critical. During this holiday season, please know that the Navy is grateful for you and your family's service and let's take time to remember those deployed around the world, far away from their loved ones. To all of our sailors, civilians and to your families, happy holidays.


Okay. Well, thanks everybody for making it out. Today we've got Alex Rice and Osky from Iowa State University, and he will be talking to us about the edit distance function of random graphs. Go ahead and take us away Alex. Thanks for coming. Yeah, thank you for the invitation to your virtual seminar. Today I'll be talking about the edit distance function and related to random graphs. This is ongoing and kind of dissertation work. So the motivating question we have is, given a fixed graph F, how hard is it to remove all induced copies of F from a graph of density p. The cases that we'll take at first are F being a complete graph on three vertices also a cycle graph on three vertices. And the next graph we'll consider is K22 also known as C4. And the main problem is to consider what happens when F is a random graph. And I also claim that this problem is is connected to other sort of famous extremal questions one one being the Turan, the Turan number of a graph will we'll see we'll see that come up in this case. So what do we mean by hard work. And for this problem we're going to measure it in terms of what's known as the edit distance. So given two graphs on the same vertex set. We're going to record their edit distance as the cardinality of the symmetric difference of their edge set normalized by the number of possible edges. In this case we can kind of intuitively think of an edit as a unit of work. So how much work does it take to transform this graph to this graph. If the rule is that you have to add and delete edges with weight one over entries to on on each of those operations. So let's see what edges they don't have in common. So here are the edges in the first graph that do not lie in the second graph so we have to add those we have to delete those. And here are the edges that are in the second graph that are not in the first graph so blue edges we have to add red edges we have to delete. And if my arithmetic right we've deleted for added six. So that should be 10. And the right number to normalize by is 36. So their edit distance between these two graphs, at least as far as label graphs goes is point to seven repeating. Are there any questions up to this point. So let's let's make our first attempt. How hard is it to remove a k three from a graph. So one natural thing you could do to remove k threes from a graph is to make the graph by part that would be you would delete some number of edges in order to make a new graph we could call it G prime, where G prime is part eight. If you do this efficiently. Then in general the best you could do is by deleting about half of the edges. So if if your graph had density. P, then you have to have deleted something like P over two proportion of the edges in the worst case. That worst case would be attained by several things a complete graph with a bunch of isolated vertices that could also be deleted by a random graph of density P. And we could ask, like, in what sense is this type. If, for example, P equals one. What are we asking well we're taking some graph, we're taking a complete graph. Oops. And we want to know how many edges we need to delete from it to make it to make it triangle free. And this is in some sense the same as saying, how far is the graph from being triangle free. So you could think of the graph. Okay and my, okay and minus the edges in the deleted graph is triangle free. And we could call this graph maybe G prime. And how many edges could G prime and possibly have well this is this is Tehran's theorem that the number of edges in G prime is at most floor and squared over four. And so at the very least for people's one this should be tight and there are arguments to make to ensure that this is tight for other values of P. So we're going to graph what we've shown so far. So on the y axis will say distance. And on the x axis will say density. And what we've sort of argued here is that our, our function should take on the value, this being one, that a complete graph takes about half of its edges to be deleted. So this should be one over two, one, one over two. And at any other value of P. We've argued that the best you could do is P over two. Okay, so that that's sort of a functional way of displaying what information we've communicated so far. Are there questions about this. Feel free to turn your mic on and interrupt me there it's there's always a chance I miss chats in the comment. I really don't mind interruptions for these kinds of talks it's it's very easy to go fast when you don't want to. Okay, yes and I've already I've already explained this point Tehran theorem says that we're tight at that value. And we've already graphed what what this looks like for different values of P. Let's try a harder example, because this one, this one you could do in a first course in graph theory. Let's try something a little harder. So let's suppose that G is a graph of density P, and ask how hard is it to remove cycles of length four from C from G. Every time I say remove a graph. This necessarily has to be induced. So I'm always removing an induced graph. So if we translate immediately into the following. What if we want, we're, we can instead ask, what is the distance of your given graph G to the set of all graphs without induced C for that's just a direct translation. And this maximum being taken overall and vertex graphs of density P in both cases. And we can tighten this notation a little more describing the graphs that have no induced C for us just for the C for some of you in the room may notice that this is an example of a hereditary graph property. And we'll talk more about that on future science. So how can we remove all C fours efficiently but also indirectly. And then, okay, this is where I will pause and ask questions. So does anyone have a really basic way to remove C for and to do this we're using additions and deletions of edges. I think we have a mic on for something else. Leave all the edges, you can add all the edges. Okay, delete all the edges at all the edges. This is this is a softball. So the two constructions we're going to consider first is remove all the edges from G and add all the non edges. And the graph has density P. We know already about how many edits we've made. In the first case we've taken out all the edges. So the distance to this constructed graph is P, because we're always normalizing by entries to. And in the other case, our constructed graph is the complete graph and since we're adding all the missing edges. Well we're just adding in one minus P proportion of the edges in the complete graph. And we want to know how optimal this is. One thing I will say, before we move on is that if you use these two constructions, again, distance being on the y axis density on the x axis, then our distance to see for that we've measured is at most this red line. And most this blue line, where at the middle we have one house. And so when we ask is this efficient. What we're essentially asking is, is there another way is there another algorithm we could have used to do this in general where the curve is below the curve that we have at any point. Okay, that's what I've already said that the distance is at most this minimum of these two functions. And it turns out yes, there are there are ways to do this in a more efficient manner. So consider the following sort of structure on the leftover here we have something that looks like a complete graph, something that looks like an independent set. And in between I put gray gray here means it could be anything. The claim is that if your new graph G prime can be split in a fashion that is an independent or a clique on one side and independent set on the other with anything in between. Then I claim that that new graph has new C for. So why would this be true. Well, this is true sense. Where would you put a C for if you could embed it in here. So C for has four vertices. It's clique numbers to its independence number is to. So you would have to put two of its, one of its cliques here, and one of its independent sets here. But that's impossible just by just by inspection. This is a clique. And this is an independent set of cliques of size to an independent sets of size to intersect. And this is non trivially in this case so this kind of embedding is impossible. So if the graph that you produce G prime looks like this in some sense, then it naturally has no C for. So our new editing procedure is just going to mimic is just going to make a graph G prime that looks as much like our original graph is possible. The new graph has this decomposition. What we'll do is we'll randomly map about X one proportion of the vertices to the black region. Map the remaining vertices to the white ring region. And we'll do this ID for every single vertex. We're going to make no distinction about the structure of our graph G. The reason why we're we're almost entirely forgetting the structure of G and just remembering its density. Because when we do this. Okay, again, I should say that the vertices that we map into the black set. We're going to complete that graph that induce sub graph. And on the vertices that map into the white set. We will empty that graph and then that means that the new graph we created does in fact resemble the structure. So now we have to ask, how many edits did we make. Well, the only edits we made involved edges where the two vertices fell into this pile and edges or none. Edges that fell into this pile and non edges that fell into this pile. So those would be the two edits that we made. And that would just be represented by this kind of quadratic form that the expected number of edits that we made under this random mapping is about one minus P. X one squared plus P X two squared. And it's pretty easy to see how to minimize this. You just set X one and X two to be one minus P and P. And that's what we did in expectation something like P times one minus P. And if you compare the function. If you compare the two functions that we have. Yeah. So overall the distance that we have here from G to the form of C4 is at most this function. So we went from something like, I think I did read first, and we did much better. This has a peak of one half with height, a quarter. So again with distance on the Y axis density on the X axis. This construction is a sizable improvement, especially when he is equal to a half. Sometimes the structure can be quite complicated. So this you could kind of look at it and say it looks like a ball of rubber bands. But it's actually a template that's made from a strongly regular graph. It has the name q44 in some in some contexts. And this was actually entirely necessary for understanding distance to form of K to four. Complete bipartite graphs have a lot are end up being quite complicated for this problem. So let's take what we have so far and move it into a little more general setting and try to make things a little more formal as well. So a hereditary property of graphs is a family closed under relabeling vertices as well as deleting vertices. So far we've only considered hereditary properties generated by forbidding some small graph. And let's think about why that's actually a hereditary property. So if you take a graph that forbids some induced F, and you delete a vertex will clearly it still forbids F. And likewise, if you relabel the graph that forbids F, then you haven't done anything to the structure of the graph and so it should also forbid F. And in fact, any hereditary property can be written as forb of a family of graphs, which is to say, any graph that forbids all elements of the set F. Why do we study hereditary properties. Some would say that most interesting graph properties are hereditary. And that would be a bit bold for me a grad student say. So I will quote someone else. This is from a ball of Washington wine rights paper on basically quantifying a little bit more of Sherman and Zito's argument about the speed of hereditary properties. They go into a little more structural detail in this paper. And let's also do proof by authority. Olga alone and his student Yuri stuff also said that most interesting graph properties are hereditary. So we have, we have, we have authority figures who are making this argument. And maybe we should also try to believe it in our own right. You've seen lots of hereditary properties. So in an intro graph theory course you would probably see all of these properties being bipartite is hereditary interval graphs planar graphs are all hereditary. Not all of them are interesting for this problem planar graphs are sparse. So you don't really have to think about removing how to edit in order to make a graph planar you just delete all of the edges except a few. There's also money in hereditary properties, the focus in prize plus a $10,000 bounty was awarded to these four authors for their 178 page proof, which characterizes a perfect graphs. And there's a whole program of studying similar kinds of characterizations of graphs such as graphs that embed on certain surfaces. Does anyone have any questions up to this point I've been kind of speeding through. I'd like to think of this problem involving metric spaces. I suppose these lines could be a little bolder here. So if we take all the graphs on four vertices. So the same. If we take all graphs on four vertices. And we circle those up to isomorphism. In this case it's just circling them up to edge count. We get a smaller metric space. And the edit distance between different graphs is really just the number of edits needed the number of edges needed to take to get between the graphs. If you reduce it up to up to isomorphism. More interesting picture would be all the graphs on four vertices already reduced up to isomorphism. And we have to remember what is our problem. So we're considering the distance from a hereditary property. In this case we're we've circled all of the bipartite graphs. And we want to know what is the graph that is furthest from this of given densities. If we just wanted to know the one that's furthest we would look no further than over here on the left a complete graph is this from being bipartite, at least of density one, maybe this is the graph that's furthest of density, five, six and so on. Yeah, so that would be our optimum. And another picture that I have quite fondness for, because I like graph limits is the following picture. So in blue, I'm going to draw this as. HN. By that I mean this is the each of these is a each of these pancakes is sort of a slice of all the graphs of the given order. The whole picture is by the way GN if you wanted to give a notation for the graphs of order and in this sense what we're doing is, we want to know the graph, or the graph sort of in the middle of this, the graphs of density P that have the furthest distance to this outside disk. And for sort of convexity reasons that make more sense with graph limits this is this is sort of the right picture for this. So what is our original problem. The remaining question, originally asked by Alan and stuff was the following. Let H be hereditary. And I should mention every time I say hereditary I mean hereditary and non trivial in two senses. The one sense being that the hereditary property actually is infinite that it contains graphs of every order. And the other trivial case would be that it contains the set of all graphs or all but finitely many graphs. What they showed is what they asked us as follows. What if you want to know just the graph that is furthest from that hereditary property. It's actually a small lemma to actually prove that this limit is well defined that it's not just a limb soup. And for each and we are asking among the graphs of order and what is the furthest. This is a more restricted version of the problem we were originally asking, but it's in fact the motivating problem. What they showed is that for each hereditary property. There's a parameter P, governing the randomness of a random graph. This is the air to shiny random graph of density P star. And it really attains this maximum distance EDH star. Another thing that's worth mentioning is that anytime one of these results is shown it usually involves a couple applications of the regularity lemma. But there are also fairly slick proofs of these results using graph limits. And some of those are due to Bologna, Lovasa and Segati. The next sort of step that gets us back to our original general question is if you have a hereditary property, pick any value of P between zero and one and define EDH P as the limit of the maximum distance from the hereditary property that you could get if you're looking only among graphs that have density P. So let's put some sort of epsilon error term in here and it would give you the same quantity. We just chose to round to floor of P and choose to, but you could you could round within a small window of that to and still be considered density P. So what Bologna and Martin showed is that for all values of P. amount, this maximum is in fact detained by a random graph of density P, which is a bit strange because we're called that we're going to be studying sort of a stochastic version of the question, which was the distance from four of a random graph. And these theorems are all saying that the furthest graph from forbidding a random graph is another random graph, which is a little annoying to have to come to terms with philosophically. The other main tool that Bologna and Martin found was that this function, if you take as your argument P on the set 01 to 01 is continuous and concave down. So why is that useful for Alan Stovs question. Well, what it means is that if you were to graph and edit distance function. That's maybe not starting and ending at one, but concave down functions. Graphically speaking have a fairly easy to identify maximum. There are cases where it's flat. So it could have multiple maxima. But that's the whole benefit of noting this convexity or this concavity. So our question is, what are the maximum edit distance, the value P star and the edit distance function in general, if H is defined as forbidding a random graph. Okay, so this is a good moment to pause and and ask for any questions. Because there are a lot of things going on and we're going to go in a couple different directions. So the big question is how do we actually compute the edit distance function, presumably once we get the edit distance we can compute the maximum value as well as the place where that maximum is attained just by looking at the graph. So what we're going to define here is a notion called a colored regularity graph. We've already encountered one in fact we we took a template like this. In the CRG language, this is a black vertex. This is a white vertex and this is a gray edge. CRGs are a more diverse generalization of this.   In this decoration, anytime we edit a graph to make it resemble one of these CRGs, what we're doing is completing everything that's black, emptying everything that's white and doing nothing to things that are great. And there's a good reason why we don't use gray vertices, more or less because of counting lemon if you wanted that. With these CRGs, you're pairing it with an optimization problem, much like the one we had before. We measure the distance from a generic graph G to a CRG like this along the path that we took when we completed and emptied accordingly. In this case, we recall that we had the formula one minus px one squared plus px two squared. We also talked about how we maximize that we're generally, well, you can unpack this expression as in terms of the matrix governing this quadratic form. And it's important to stress that we're optimizing over the set of x, where x is a non negative vector something to one, which you could also say as a vector in the standard simplex with two coordinates. Let's see what would happen here. So if if in fact this edge was black, you would have a one minus p here. If this edge were white, you would have a p here and here as well, because it's naturally a symmetric matrix. So for every CRG you have some sort of matrix here in later slides I might call this mk of p. So let's go to this example. And what I'm going to emphasize on this slide is that not all vertices of a CRG are actually useful.  It's a fairly depressing case, but it does happen. So in this case, the matrix corresponding to the CRG at the value p does have a one minus p here, one minus p here because of this black vertex and a p here because of the white vertex. And again these numbers all correspond to the amount of work that you did, deleting edges and adding edges. So here the quadratic form has a different kind of optimization. In this case, it's possible the optimum in in any case is obtained by setting one of the variables equal to zero, which that tells you that this is not a good structure that it was actually these two structures doing all of the work. So we need a notation, we need a good notion. We need a good definition to capture this. It seems to be a couple slides ahead. So we call a CRG p core. If the minimum value attained by any of these different weightings this x is attained by some vector where all of the weights are strictly positive. And another another way to say this is that it's the value you get this kind of quantity that's governing the distance. You should say here, the quantity that's covering the distance from your generic graph G to your new graph G prime edited according to K actually uses all of the vertices of K, which is to say you shouldn't have used any kind of sub CRG of K that you strictly speaking got more had to do less work if you included an extra vertex. So why are we talking about all this technical stuff involving CRGs. We need to remember that this problem is related to hereditary properties. So, in this, in these two examples, recall that C four doesn't embed into this CRG, which means it's a good. This is a good kind of CRG to associate to a property that forbids C four. And likewise, we had briefly discussed that K to four doesn't embed into this wild CRG. And for that reason, this is a good CRG to associate to the hereditary property that forbids K to four. For this reason, we find for any hereditary property the set of all CRGs which embed only graphs that satisfy H. If you only forbidden one graph, then KH is just all see all CRGs that don't embed your forbidden graph. Does that make sense. By the way, how much time do I have left about 24 minutes. Okay. Thank you. So, all of the CRG business is actually useful because of this theorem, which is a kind of joint theorem of two authors so in the same paper that Bella and Martin had their previous results. They also showed that CRGs are useful to get the edit distance function. But there could be an infinite number of CRGs and they didn't quite get that this in FEMA is always attained. This in did in fact show that for all P, this, this infimum isn't is attained. Why is it a minimum. Well, it's a minimum because we're always wanting to find the best CRG that is useful for editing a generic graph of density P. The minimum is always attained by a PCOR CRG because well if a CRG is associated to a hereditary property so is any sub CRG. And well if you had a CRG that had some vertices that weren't doing any work you might as well just delete them. So this minimum is always attained by a PCOR CRG. I have to define that here, but we will come back to this notion. Now we're going to talk a little bit more about our original problem. What are the PCOR CRGs or just generally what are the CRGs that don't embed our random graph. I did not define the random graph. So this is this is this can be thought of as a as a strange kind of coloring question. Let's go back to a little bit of basic graph coloring. So if, if you take K to be the CRG with W white vertices and all other edges gray, something oops, something like this. You use that CRG with all edges gray. Now the picture goes away. So we're mapping a graph into this CRG is the same thing as a W coloring of the graph, because what are we doing we have a we're mapping the vertex set of your small graph F into the vertex set of some CRG K but all of the vertices are white, and there's no edges in between. The gray area means that we don't have to worry about whether there are edges in between or not. And since all of these are white vertices, this is just a proper coloring of your small graph F. If you're using one fewer vertex than the chromatic number, then your small graph does not embed into that associated CRG. Moreover, we can compute the G function of this CRG. If there are, if there are W white vertices and no, no edges other than gray edges, then your quadratic form is pretty boring. It's only considering the edge deletions that occur on your your W many white vertices. And on those vertices, you're editing by deleting so so you have to be so this is proportional to P. And that's our G function there. And similarly, you could use the chromatic number of the complement and a CRG with only black vertices and gray edges. And you get this sort of general upper bound of the edit distance function concerning one forbidden graph F. And so based on this, Martin in 2013 conjectured that if you are forbidding a random graph. Then the edit distance function is no more than what you get from taking a minimum of these two chromatic bounds. So why are these the two quantities you see here. Well, because this celebrated theorem of Bologbash says that chromatic number of a random graph with P fixed is concentrated around its mean, and its mean is and not over to log one over one minus P event not. And again, this is the quantity that you're plugging into here. And when you work it out, you get this quantity. And similarly, you get a another quantity over here that you would plug in for B for the co chromatic number just by replacing one over one minus P with one over P should have a P not here, I suppose. So basically this conjecture of Martin is that you can't really improve this bound for a random graph. But yes, there should be a tilde here. Are there any questions about that. One thing I should also mention is what are the optimal values that you get out of this. So we have, we have this edit distance function graphed with density on x axis distance on the y axis. So we have two lines, red and blue. And the goal from Ellen and stuff was to compute this peak point, which you could call P star. P H star. If you wanted all that. And if you had this, if you had that this formula was asymptotically tight, then you would get these values for the, for the value of P star and the value of EDH star. So we have a more general setup that we could use instead. And this is a good benchmark to test whether there are ways to improve this bound.  But there's something else we could do. We could take any CRGK and use a bunch of a multiple number of copies of it. So we could, for example, take CRG like this and clone it a bunch of times. So we would take that, make another clone over here, make another clone over here, and exclusively put gray in between. That would be this quantity we have called M times K, M times K. And so for any CRG, we could define that as a kind of chromatic number. So we let K sub K of G be the minimum M such that G actually does embed into M copies of the CRGK. And Bolo-Bosch and Thomason, this is a special case of their theorem, but it's the most useful version for what we're talking about. Bolo-Bosch and Thomason showed that they showed that for fixed K, that you can compute this K chromatic number of a random graph of any density that's fixed. Asymptotically, using quantities we already know, such as GK and, yeah, such as just GK. And you'll note that this is proportional to what it should be for the chromatic number, for the ordinary chromatic number. So this shouldn't come as too much of a surprise since this is an example of it. If you just plugged in K being a single white cortex, then this would work out exactly to be the quantity that it is for the chromatic number. So this is in some sense a generalization of that theorem. One important obstacle to this is this only explains constructions that use multiples of some fixed CRG. There's also a sort of pathological possibility that the CRG that you have has a tremendous number of vertices, in which case it wouldn't be fixed relative to N. It could possibly grow huge. And so it would be very hard to characterize when a CRG embeds into, or whether a random graph embeds into a CRG. If that CRG, if you have no control over any of its structure or even its size. So we introduce a sort of trimming. For this, we have to introduce a definition, which is reminiscent of things we've done already. If you had a CRG like this, and you put gray in between, then it's natural to kind of consider these two pieces as components. Let's say that the components of a CRG are the subgraphs that are held together by non-gray edges.  Lemma of myself and Martin is that this is useful for finding peak or sub-CRGs, as long as P lies in between one third and two thirds. So what the lemma says is that if you specify a tolerance and you start with any CRG, then you could shrink to a peak or sub-CRG so that two useful properties holds. One, the G function didn't change all that much, which is to say they're almost just as useful for embedding a random graph in terms of that chromatic lemma. The components are actually bounded order. And that's useful because we could have possibly been dealing with a CRG with an unbounded number of components in terms of N. So overall our argument for proving the result for the edit distance function associated to a random graph, first we trim any CRG that could possibly break the bounds that we already have. Again, the bounds that we're trying to break are this, so our pathological CRG might somehow have something below the peak point. If it has something below the peak point, we're not happy. Right, so if it looks like this, then obviously our original construction, one of these two constructions was not optimal. And by concavity, it's equivalent to ask if this function lies below the curve anywhere, then it also lies below the curve here. That's just a sort of concavity argument. So our trimming lemma, what it does is it shrinks the CRG into something more manageable. So now each of the components lie in a finite set. Since it's a finite set, all of them can be considered fixed in terms of our tolerance and our value of P between one third and two thirds. And we're just going to apply Bologbash and Thomason everywhere. We're going to cut our big graph, our big forbidden graph F into a huge number of pieces. And use Bologbash, Thomason to basically color each of those pieces using the elements of this graph, of this set of CRG's B, in the most efficient way possible. And by that I mean basically up to a tolerance the chromatic number of that little piece, the B chromatic number of each of those pieces. And we just assemble them all together. I mean, this is, this seems like a pretty straightforward argument once you get your hand around Bologbash's theorem, Bologbash and Thomason's theorem. The tricky part is actually this lemma, this piece. The trimming lemma took a lot more work than it might seem on the surface. The general algorithm for proving this trimming lemma is to take to shift immediately to a PCOR sub CRG by deleting vertices of weight zero, and then delete a vertex of lowest weight repeatedly until you reach a certain threshold for the value of the weight vector, the optimal weight vector. And then stop once you're at that point. I argue that, as far as the underlying graph is concerned, the degree is bounded. That takes an argument. But how do you show that a bounded degree graph has bounded components. Well, you would need to show that the diameter of each of those components is small. And that's what we end up doing. At this point we do have to go back and investigate some questions about PCOR CRG's. In Marshall and Thomason's paper, what they found is that PCOR CRG's need to have a very restricted color pattern. If K is PCOR and P is between zero and a half, then no edges are black, and no white vertex is incident to a white edge. So what that basically means is that the white vertices are isolated islands, and the other interesting components have only black vertices and white and gray edges. So this CRG would have five components. That's under this first case. So we're kind of assuming that this P is between zero and a half. And so it's possible for this thing to be PCOR. We're not saying it's PCOR for every P. It's just that this is a necessary condition, necessarily would have to look like something like this. And for P between zero and a half, you just swap the colors in this sentence. So actually it can't be PCOR and P equals one half because of this color restriction. The result classifies all one half core CRG's in general, and it also classifies all two vertex CRG's on any interval. What we introduce now is a new notion of a type of CRG that we'll call a Dalmatian CRG. We recall that we have components defined in terms of non-gray edges. The result states that on a certain interval, the only PCOR CRG's have these specific, have a very specific definition. We're calling this a Dalmatian CRG. Each of these is a Dalmatian CRG. So a Dalmatian CRG is a CRG that, the white edge relation is essentially transitive.  The interval where this holds that this is the entire classification of PCOR CRG's is the interval one minus phi in verse one half where phi is the golden ratio. We don't really have a clear reason why the golden ratio is coming up here other than it solves the quadratic equation. And if you were to negate sort of take the photo negative of the same question, we're swapping black and white. And so the compliment of Dalmatian CRG's are the unique PCOR CRG's on the complimentary interval one half to phi inverse. What it does is it enables computation of edit distance function on this interval of positive length previous results only had characterization on the interval, or at the point at the point P equals one half. So now we can actually compute an edit distance function on a positive length interval without having to do any CRG magic and optimization. So the finalization conjecture, I think in a 2013 paper that asks whether this is possible, and at least on this interval it is possible. The way that the trimming lemma was actually proved was with what we'll call spectral prohibition. So we formed the underlying graph of a CRG by deleting all gray edges and forgetting all of the colors. So the CRG that we've seen a lot is should be 3k one plus all this. That's already there. Yeah, so the underlying graph of the CRG has these five components, which are just found by forgetting the colors, except for gray gray is a non non edge. And we say that a graph is P prohibited. If it cannot be an induced subgraph of any PCOR CRG. So I guess I should say an induced subgraph of the underlying graph of a PCOR CRG. In our paper we define this only in terms of CRG so we said a CRG is P prohibited. If it can't appear as a sub CRG of a PCOR CRG. So what we showed is that if your graph G has minimum eigenvalue lambda, then then the graph G is P prohibited on the interval that this is defined in terms of that minimum eigenvalue lambda. And the proof of this is kind of fun. You take that optimization problem and you infer that necessarily the matrix in the middle has to be positive definite, because you can cheat, you can decrease by exploiting a minimum eigenvalue. If it's negative or zero system future work. This is involved in the Iowa State discrete math RTG. The class involving undergrads, no no undergrads graduate students, a postdoc, and my advisor, and one of our mini goals is to widen the classification. It's tight if you only use Dalmatian CRG to classify, but we want to widen that interval a little more to see whether we could understand when wider intervals where we know all of the PCOR CRGs. The other result would be to widen the interval for which for which the approximation. The CRG trimming lemma also holds, and I am personally interested in understanding this question of hereditary property defined by forbidding a inhomogeneous random graph. So if you've seen other random graph models, you may even, you may have encountered those that are associated to to graph limits. There's a way that you can define this using a graph limit, and I am interested in what happens to this hereditary property if you're forbidding one of those. In particular, do you still get for some values that the graph looks like this. And this is a sort of Wild West territory because we just barely know things about inhomogeneous random graphs, we barely know that their clique number is concentrated for a generic inhomogeneous random graphs we don't even know where it's concentrated. So we have no right to also know where the chromatic number is concentrated. So we have no idea what this line and this line should even look like in the inhomogeneous setting. And those are sort of precursors that are necessary to start thinking about this problem properly. So this is my last slide. So I'll just say thank you for your attention. Thank you for having me at your seminar. Thank you, Alex. If we could all get some thank yous. Thank yous for Alex in the chat or turn your mic on and say thanks that work as well. Yeah, maybe some some claps in the camera. Give us those to put in. Okay, and oh, there you go. Like a little little emoji there. Okay. If anybody has any questions go ahead and ask away. Sorry. Can we do anything if the target graph is sparse instead of instead of dense. That's a great question. So the question is, whether we know what the edit distance function is if you forbid in a sparse random graph is that that's your question. Yeah, also, also you start with also a sparse graph. So you say start with, are you talking about a forbidden graph, or are you talking about if you want to know the distance of a sparse random graph to a hereditary property. So I guess there's a you have a starting family of graphs right then you want to reach another another family of graphs. So here I guess, both of the starting and and the target family are both dense. So those are stars. Are there any theory on that. So, I should say, the setting is sort of, you take any graph. So these are all the graphs of order n. So you can take some graph and then you highlight in blue are for hereditary property. So what you're proposing is that what if instead you had a some other subset and you wanted to know the distance from I don't know all these I. Yeah, so so we're talking about generically what is the furthest graph or the furthest graph of density P to H. But yeah, that's that's a totally natural question. What about for this triangle free graph to see for free graph. That's that's an interesting question and I don't have an answer for you. Yeah, I guess if we don't restrict the starting family, then when if we go from a dense to a sparse one then the edit function is always is it one or something like that. If you went from a. So again, your hereditary property is forbidding a random graph. Or no. Just like whatever we forbid just the family we want to go into is stars is all stars. Oh, you mean this family sparse or. Yeah, after we forbid that all these forbidden graphs and then whether this is sparse. Yeah, let's see this. No, no, I was saying like the other all the elements in HR sparse. Let's say the set of planar graph or something like that. Yeah, yeah, yeah, where H is a sparse or small. Yeah. So my understanding from this, I'm going to misspell this but China men. I'll blur the leather letters together so I don't make a typo and Zito. So they have a theorem that essentially classifies the growth rate of a hereditary property into different classes. And so there's also a later result of see Balog. I think I talked about this Balog Bolog and wine right, which gives a little bit more structural information in the various growth types. My understanding I might be wrong about this is that in the sparse setting, or when your hereditary property is small, then these graphs are either nearly empty, or nearly complete. So my understanding of that case and I think some of it some of this can be inferred from colored regularity graphs. So in that case, yeah, your intuition is right that it would basically be either a zero or one one that the distance would either be P or one minus P all the time. Because you would be moving to essentially a nearly complete or nearly empty set. You could have a different normalization to account for the sparse case or something. Josh, it's pretty hard to hear you. Oh, you could have a different normal is it can you hear me okay. Yeah, that's better. You could have a different normalization than juice to sparse case right. Yeah, so we have distance G prime edits, divided by interest to so you're saying like if you. If it was a graph of linear density, what would, what would that look like. Yeah, so you would only be considering the edit distance function on a short snippet, or maybe the density P is something like one over N. And yeah, I don't know. I don't know I think all of this is in the language of dense graphs, which is why the regularity lemma, and it's most useful forms actually kick in, but I don't know. I think these are good questions. I'm also interested very much in this other question involving what if you're forbidden graph is a sparse random graph, because we know it's chromatic number we know it's co chromatic number. It's just that the arguments haven't been made in this case and some of them might go wrong. If you've, if you've actually had something sparse. Do we have any other questions. Just a quick one. The, well, maybe quick. What is this finalization conjecture. Yeah, that's a good question. Do I get to make a new page. Lots of blank space there. Yeah, maybe I grab it and move it. No, I can't. So the finalization question would be. It comes in a couple of different forms. So basically the most general one would be if a territory. Then, does there exist. That's backwards even. There exists a subset of the forbidden CRGs. And I guess I should also say, can you read this and a is between zero and half. Does there exist an M such that the edit distance function is equal to the minimum overall elements of M. Or GKP. On the sense, let's say P belonging to a one minus a. So our proof works for a equal to one over golden ratio. Yeah, that's something we're working on right now. Do you have any other questions. Okay. Well, thanks again, Alex. And yeah, I guess Josh is there anything that you you want to say about the YouTube channel. To those of us that are here, or can subscribe. Like comment and subscribe. All right. Thank you, Alex. Yeah, thank you. I think this was not disastrous. So go ahead and upload it. All right. Thank you.


ok. This is physical vapour deposition and today we shall discuss on this topic what we know as sputtering. Now, sputtering has a classical definition and this is a process which is basically dislodgement and ejection of material from a solid or liquid surface in atomic or molecular scale due to momentum exchange with an incident energetic projectile. Now, this sputtering is actually a physical vapour deposition process which can be used for depositing various types of elemental metal it can be a compound or it can be an alloy. So, deposition of sputtered material how does it work we can have a very close look. So, deposition of sputtered material actually what happens sputtering is ejection of the material dislodgement of the material from a surface which we call in the language of sputtering as the target this is not actually the deposition process. In fact, the material or the surface which we need to coat or cover with a layer that has to intercept this flux of material which will be ejected by this impingement of the energetic projectile and this is called actually the deposition process and sputtering is the ejection of the material and intercepting that by the substrate material which will condense over the surface of the substrate that is actually known as sputter deposition process. Now, here comes very interesting thing that principle to practice of sputtering that means, how to put this principle or theory of sputtering into a practice and a process of practical interest. So, first and foremost thing one has to do is generation of this energetic particle. Now, this sputtering is conducted in a closed vessel or a chamber which is evacuated and inside that chamber one has to generate energetic particle and which will be the ions of heavy inert gas like argon. Now, to get this ionization again one has to ignite one electric discharge and in that electric discharge ionization of the argon that will take place in the region adjacent to the surface of the substrate and this ionized gas of argon that is also known as plasma and in the process obviously, within that chamber one has to keep one cathode and anode to initiate that electric discharge and in this whole process the target material that means, the source material from which the material will be dislodged and it will emerge out like a flux like a stream of sputter material that is actually kept as cathode. So, this target which will serve as cathode that is actually negatively biased. Now, scheme of sputtering apparatus, now let us have a quick look. So, here schematically we can show like that this is a chamber which has to be evacuated. So, it is connected with the vacuum system vacuum system and obviously, there will be one target which is actually the source material. So, this will be the target and that will be isolated from this chamber wall and it will be negatively polarized. So, this side polarized. So, this will be grounded and at the same time the body of the chamber can be also grounded. So, what is going to happen in this process here we can keep the substrate that means, the material or the object which need to be coated. Now, here this substrate this is actually the substrate material which can be grounded or which can be also biased. So, this can be in the ground state or it can be also polarized. So, what is going to happen say for example, this is titanium target that means, basically it is a titanium disk or a titanium plate having a cylindrical surface or a rectangular surface and this is the negatively polarized target with this power supply. So, this is evacuated down to at least 1 into 10 to the power minus 4 to 10 to the power minus 6 star. So, that is the order of at least one has to reach that and then actually what we have to have one entry point for argon which will be high purity argon and this high purity argon will be admitted in a metered quantity into the chamber. So, that its pressure can build up may be from 1 millitor to 10 millitor and in that case it is highly polarized and in that case there will be one electric discharge because of the electron which are available on the surface of this target, this floating electron that will be attracted and that will heat this neutral argon in this zone in this zone and here we have splitting of argon ion plus one electron and this splitting will occur because of this electron will be flown in this direction because of this negative positive polarity on this side. So, there will be one cathode and one anode even this wall of this chamber can be anode or we can also put one auxiliary anode and during this course of movement in the space between the substrate and the target what is going to happen this electron will strike and have a collision splitting this neutral argon to positive argon and as a result this positive argon now will be accelerated towards this negatively biased surface and that will strike on this surface and it will transfer the energy and as a result the material will get dislodged from this surface and at the same time it will also emerge it will be ejected from the surface and it will flow in this direction like a flux of spattered material and if the substrate is placed in front of this target in that case this substrate surface will intercept this stream of titanium vapor and that will deposit over the surface on this and it will condensed over. So, atom by atom titanium will build up and it will grow and it will finally, appear as a coating of metal. So, this way one can look into the basic process of sputtering and obviously, what has to be done that control of the pressure this control of the pressure on one side here of course, we have a metering valve like a throttling valve to control the pressure inside the chamber and then we have metered quantity of argon which will be flown through one MFC. So, that is the quantity which will be admitted the pressure is controlled by this throttle valve on the downstream side. So, there should be one pressure sensor continuously monitoring the pressure and we have to have some polarization of this target and the power supply. So, with all this one would expect that sputtering from this target surface and it is the deposition on the substrate. So, this is actually called a DC planar diode because it is working like a DC mode and it is like a diode and it is the plane surface. So, it is called a DC planar diode mode very simple form of sputtering apparatus, but even with this simple form of this apparatus one can see how this process can work and at the same time how this material can be deposited from this target and there will be continuous gradual erosion of this material from the target and this will be depositing over this surface. So, here one has to obviously, operate the process parameter that means, there must be some process variable which has to be handled with delicacy and so that the desired outcome we can have the desired outcome. Desired outcome means at least we are interested in the growth rate of this coating this is point number one that means, how many micron of this coating is building up on this substrate surface. Number two uniformity of the coating over this entire surface of the substrate this is also another issue very important issue considering various engineering application and then number 3 is the structure of the coating smoothness of the coating density of the coating is there be any contamination any poisoning of this coating all these issues are to be also properly addressed and then comes the density obviously, then adhesion of the coating with the substrate that means, how this interface is build up at the very initial stretch. So, all these questions are to be properly addressed and this is exactly the sputtering process. So, one has to look for this cathode current density that means, when this voltage is applied. So, once there is ionization so, we can also find or we can determine straight forward the cathode current that means, ion that is collecting on this surface and from this surface area we can find out the cathode current density or even the power density then also the discharge voltage that means, the voltage which will be also recorded on the power supply unit. So, both cathode current and the DC discharge both will be record displayed and then argon pressure which will be monitored here by this throttle valve on the downstream side and then this SOD, SOD means the standoff between this target and one anode plate. So, actually this distance of the substrate it is in between so, substrate can be one anode or substrate can be also grounded and we can also have a separate anode. So, this distance standoff distance between this surface or substrate surface or a separate anode that is also one important parameter because, the pressure inside the chamber and the mean free path that is 1.1 has to consider and also the flight of this mass of spattered material which will follow this path and it will finally, arrive here. So, whether there is collision or it is moving freely all these are issues are also taken into consideration. Now, in the sputtering process one cannot just ignore this point that is called sputtering yield. Sputtering yield is that the ions which are the projectile in this case which are going to heat or strike the target surface. Now, one ion say for example, one ion is going to strike this surface and as a result of that how many atoms will be ejected. So, that is actually the capability of a single ion which can cause dislodgement followed by ejection and then this flow of the spattered material like a stream. So, all these series of processes are actually at the result of this heat by this argon ion which is serving the like a projectile. So, this is one thing. So, that is actually sputtering yield number of target atoms which are ejected per ion per single ion then it depends upon the target spaces that means, it is the atomic mass of the target and then also it is impinging spaces that means, it is also the atomic mass of the argon in this case that is also one thing then also the incident energy that means, it is striking with a velocity. So, it is mass and velocity basically it is a kinetic energy with which it is going to heat the target surface resulting in ejection and dislodgement of the atom and also the angle of incidence at which angle it is going to strike. So, all these things taken together they are actually going to finally, determine what will be the number of atoms which are being dislodged and ejected per unit ion. Now, we go for sputtering yield and sputtering rate. So, sputtering yield we understand that how many atoms are being ejected per strike of one ion per unit strike. Now, here comes the sputtering rate, sputtering rate means here if we consider this as the substrate surface how it is growing that means, it will be say in nanometer scale or micrometer scale how it is building up on this. So, a relation is already available that means, that sputtering rate is given by 62.3 and it is J into S into m t and this is divided by rho. So, many ions term per minute that is a relation one can look at and this is a very handy relationship. So, what are these J is that is actually ion current density and which is given ion current density milli ampere per square centimeter of the target surface. What is S that is actually atom per ion that means, so many number of atoms of the target material per ion of argon this is actually atomic mass of the target material atomic mass and this is density. So, with appropriate coefficient of coefficient we arrive at this. So, this is an expression for the growth of coating which is continuously being build up on the substrate surface. So, this is one very important relation from which one can assess the growth rate sputtering yield and angle of incidence of the projectile. Now, this is also important in that that sputtering yield this sputtering yield that sputtering from this relation we see this is the sputtering yield it has a determining it is a determining factor determining this growth rate. Now, here this angle of incidence also it does coming picture in this that this is actually the surface and if we consider this is a normal then it can have this is the angle theta at which this incoming argon ion is going to heat or strike this target surface. Now, there is a distribution by which one can see that this is sputter yield and this is the theta and it is something like that it goes like that and then it almost have a drooping. So, this is almost 90 degree and here it is somewhere little below 90 degree we get this value critical where it is corresponding to theta max. Now, this angle of incidence normally usually we consider normal angle of incidence. So, it is actually argon ion which are going to strike under the action of this electric field normal to the surface, but here what happens this surface what we are showing that is a theoretical one that means, it is optically flat and horizontal, but in practice this is not the case we have unevenness on this target surface. So, even if we have a normal incidence of the target on the target this argon ion depending upon the undulation on this surface depending upon this undulation on this surface if we consider this is striking at this point at this point which is normal with respect to the axis of the target. However, with respect to this surface this is not actually straightforward. So, here one has to draw one tangent and normal. So, if we draw at this point one normal and one tangent then we get this angle theta and which is somewhere in between this 0 to 90 degree and as a result we can see the angle of obliquity of this incidence and which is not always 0 degree. Whatever may be the case we see a distribution of sputtering yield with this angle of incidence. Now, comes this momentum exchange it is actually a well known principle or theory that it is a momentum exchange between this incoming argon ion and the atom of this target particle. Now, here if we try to if we show this surface and here we consider the spheres as one arrangement of atom. So, this is one arrangement of atom and what we can see say for example, one argon ion is heating and it is intermediates say it is actually heating this surface and it is in between this two atom on the surface of the target. So, this is actually the target surface. So, it is actually said in the theory that because of this strike what is going to happen this two, this two will be immediately affected this two atoms. These two atoms are going to be affected immediately that means there are actually it is a knock on. So, it is called knock on process. So, one is called low energy knock on and the other one is called the primary knock on primary knock on. So, as a result of this strike from this side this atom is going to move in this direction and that we call low energy knock on low energy knock on and during this process what is going to happen as it is try to move in this direction it will try to displace this one this atom and as a result it has a tendency to eject in this direction. So, this is going to be ejection of atom because of this low energy knock on. Now, comes the primary knock on mechanism primary knock on means this one this is going to be pushed inside this lattice. So, this is actually the primary knock on that will be pushed inside this lattice and it will it may come up to this point and then it will be it will move in this direction just like a reflected atom. So, it will move in this direction up to this point and then it will be reflected. So, this will be the movement and then this will be the reflected movement and during this reflected movement it this atom which is actually representing this primary knock on this is going to hit this atom from the back. So, it is from the rear it is going to hit it strike it and as a result of that it is also has every chance to get released liberated and this is an ejection of the atom. So, these are the two well known principle of momentum exchange by which this low energy knock on and primary knock on are take place. So, this is just a lateral movement and this is a side way. So, it is a push from this side and this is because of the reflection. So, this one is released and this atom is also released. So, this way this ejection of the atoms from the surface of the target material that can take place and here one can also understand that this energy which is transmitted to this target atom. So, energy which is transmitted to this target atom this will be actually if we consider this is the transfer. So, this will be proportional to m t into m i divided by m t plus m i whole square this fraction of the input energy that means, the energy with which this argon ion that strikes it will be that part of the energy will be transferred to this target atom and the energy with which this will be reflected from this surface that will be given by m t minus m i by m t plus m i that is whole square into this input energy of the incoming spaces that means, this incident ion into the energy of the incident ion that part will be the reflected energy it is rebounding. So, this is actually transferred with this incoming energy and this part actually reflected this is the coefficient along with this one. So, this is basically the momentum exchange process by which this ejection of this atom from the target surface that may take place. Now, all these things are done actually this velocity energy whatever may be that will be gained by this incoming ion as a result of the electric field and that is actually done created by polarizing this target and which is serving like a cathode. So, here one graph is very important not only from academic point of view, but also for designing of the sputtering apparatus or the sputtering device and this diagram can be shown illustrated in this form and this is in this way. So, this side we show the voltage that is the breakdown voltage that means, inside the chamber we have the cathode and the anode and a discharge is going to occur and this discharge is initiated at that point of breakdown voltage and that is the threshold value, but this is not just independent of anything it has a relation or it depends upon one parameter it is not just a parameter it is a combination of two. So, on this y axis we show breakdown voltage and on this we show it is actually p into d and that is called p into d tau centimeter. So, on this side breakdown voltage that is in volts and this side we have tau centimeter and with that what we can find here this curve look like this. So, one can find out immediately recognize a point which gives the minimum value of this breakdown voltage this is v minimum and that is corresponding to a particular value of p into d and on both the sides to excite this discharge to initiate this discharge one has to one will require a higher value of breakdown voltage. Now, this is one thing we can look here if we work with low pressure this is the low pressure site or a shorter SOD shorter target anode that means, inter electrode space. So, this inter electrode space if it is short if the pressure is low then p and d both will be low. So, as a result what we find that the threshold voltage to cause breakdown of this break cause this discharge initiation of the discharge of this plasma that also require a high voltage. While on the other side other end we also see there is a requirement of high voltage the requirement is rising steadily and this side what we find if we work with a high pressure or if the distance SOD standoff standoff between the two electrodes if it is too high then also the required voltage the demand will be quite high. So, this is one very important information and a data which will be required by the designer of the sputtering machine and depending upon the working range of this machine one can design its operational range that means, at what pressure it can work what will be the standoff distance or size of the vessel and at the same time what will be the available voltage as per the power supply. So, that the machine can become an useful source of deposition process. So, this way one has to look for this minimum voltage for a given p into d that value. Now, this is voltage current characteristics of a DC discharge. So, this voltage current characteristics of a DC discharge it is very relevant to sputtering. So, one can consider this one. So, here what we find? So, this is actually the voltage and this is the current V i V i characteristics it goes like this and this is the peak point and then it drops down then it becomes steady over a wide range and then again it rises and finally, it has a fall and then it is also rising. So, let us identify various section of this graph. Now, this part is called from this part that is called ohmic conduction, this side that is called down send discharge, this is this drop fall of voltage without any change in current this is actually called breakdown process of breakdown. After this what is interesting that without any change in further rise of voltage the current keep on increasing and this we can see here this is a typical characteristics of a DC discharge in a vacuum tube. So, this goes and this is called actually normal glow it is actually discharge with glow. And this part from this point what we see that if we like to increase current that will cause an increase in voltage. So, this is called abnormal glow and it is exactly where we carry out or conduct sputtering. So, this is the zone where sputtering is conducted and after that if one try to increase current then there is sudden fall of voltage and what we see it is almost going to be an arcing. So, we can see various stages variation of voltage with current and in that process when we find the abnormal glow that is the zone where this sputtering process occurs and that is conducted. Now, this is actually current voltage characteristics for planar diode sputtering. Now, what we have seen the voltage breakdown voltage versus P into D, this P into D means process pressure and the target you know distance that means, S O D. Next what we are looking at that is the zone where we have abnormal glow that means, here along with the current voltage is rising this is the zone where sputtering takes place. And now comes in the sputtering what we see that this is actually V i V i characteristics. Now, here one has to particularly the designer of the sputtering device and the operation for that this V i characteristics is so important. Now, say we can draw one curve like that and this curve it is true for a sputtering device having a fixed dimension that means, fixed S O D and only what we can change say it is just for one target material one this projectile material which is going to strike the target all these things are fixed. So, what we are changing here it is only voltage and current. Now, what is our output it is actually current is what we like to have because larger the current that means, more will be the ionization and more will be the ion current and that will have a immediate impact or effect on the deposition rate. So, one can immediately look into this fact and take necessary step how to change the slope of this curve. Now, this is say for a particular pressure. So, this is actually process pressure. So, with this process pressure we can have a V i characteristics. However, if we change this process pressure this characteristics may change. Now, this change means for all practical purpose from designed and operation point of view from engineering point of view one should look at the current and the voltage that means, with what voltage we can achieve that cathode current. Obviously, lower the voltage benefit more will be the benefit. So, the point to be considered here is with what is we can have this large magnitude of ion current and this is one of the subject in the design of the cathode. Now, it is a planar it is a planar DC diode very simple. So, here what we find that this curve is very its rise is quite steep that means, physically it means for a small value of current we have to go with a very large value of this volts it can be few thousands of volt few kilo volt and we cannot even get a high value of current. Now for all sputtering operation one may expect to have certain cathode current means certain cathode current density power density with certain pressure and they reasonably applied voltage which is not too high. So, for this special consideration has to be made in the design of the cathode that means, with the small value of voltage with a pressure range which is very very favorable to get one of the very best structure of the sputter coating one can get a high value of cathode current. So, this is one of the requirement in the design of the sputtering system. Now comes planar diode sputtering one can look into this planar diode sputtering that means, here actually if we have one target or cathode which is this cathode and on this side we have the anode and the whole thing is within a chamber. So, this is polarized you have argon entry all are all polarized obviously, these are all polarized if this is a situation then what we can find out here we have three distinct zone this is the main plasma volume this area which is shown by this area and this space this space this is called actually cathode dark space and this zone anode sheath. So, what happens the electrons which are available very close to the target surface these are known as primary electron that means, electrons which are available here these are called primary electron and these have to be brought in within this main plasma volume and for that we have one accelerating voltage and during this process these are known as that means, the electrons which are in close vicinity close never of this target or the cathode this is target or cathode. So, these are called secondary electron. So, these are actually secondary electron now when they are brought in by this accelerating force inside they are called here this E that becomes the primary electron. Now, they have gained certain energy here they are very weak. So, now, they are within this field they are the primary electron. Now, this primary electron they are going to strike make collision with the argon neutral and here it is very very important that this strike or collision must result in argon ion one argon ion plus one electron and with that this argon ion is going to now it will be accelerated on this surface and with that it is going to strike this surface causing that means, what has been already mentioned that means, this ejection and dislodgement of the target, but at the same time this argon ion can also cause emission of what we call secondary electron. So, one argon ion that is going to eject the atom from the target, but at the same time to generate secondary electron and which once it enters this plasma volume that becomes the primary primary electron and this way the process repeats itself. So, it is very important that this primary electron must generate argon ion and this argon ion will be accelerated towards the surrogate surface. So, this is a just a cyclic process and this way the process will keep on going and in this what is important that partial pressure of argon this target voltage, this breakdown voltage and this cathode anode this distance those are put together they have a combined effect and accordingly we can see as the output what is the cathode current and also the deposition rate. So, this is what we know as very simple planar diode sputtering. Now, in planar diode sputtering we can have lot of variations. So, though it is simple it can have lot of variation that means, it can be a single target machine and just mean that means, one cathode it can be an elemental cathode say any metal it can be titanium, aluminium, copper and so on it can be one alloy, it can be even compound it can be a alloy of say silver copper it can be nickel, chromium alloy it can be titanium, aluminium alloy it can be a compound say for example, aluminium oxide, molybdenum disulphide, tungsten disulphide something like that, but instead of single target we can also have a multi target machine. Multi target machine means we have this kind of vacuum chamber and the target may be positioned like this and here we have the substrate. So, the deposition this is intercepted. So, there are two materials and which can have simultaneous deposition instead of two we can also have four targets. So, here it is an intermixing that is also possible, but by this multi targeting what we can have by this multi targeting we can have either instead of a mono phase we can have a multi phase, multi phase layer that means both are ejecting material. So, both will form an alloy. So, it is not a mono phase it becomes a multi phase layer, but it can be mono phase multi layer it can be just. So, it is a multi phase single layer, but it can be a mono phase multi layer that means it can be interrupted. So, say this is target A and this is target B. So, it can be just like an alternate layer A and after that we can put another layer of B and this way the process will repeat. Of course, in this case we need a precise I mean precisely control automation to run the system to have a perfect blending of this multi layer having the required thickness uniformity of the thickness then structure of this each layer. So, those issues should be there, but the advantage is that by this alternating by just sandwiching the mechanical property for example, can be remarkably improved by this way, but it can be also a graded layer. Graded layer means we can start up this way say this is just the substrate and with this substrate what we can have A it is A here 100 percent A, but then what we can do we can have a grading over this this may be a trick layer. So, this is A and here what we have this is actually A and B together say this is A x B y in that combination we can have a buffering with this buffering the value of x and y can keep on changing continuously. So, from A to we get A x B y and as we go in this direction that means as the coating keep on building and thickness keep on increasing what we have finally, at this end we can have just pure B. So, we start with 100 percent A in this position and then it is gradually converted into A x B y value of x is at the very beginning very high y is low, but as the coating thickness keeps on increasing value of x will be decreasing and y will be increasing and finally, it will become 100 percent B. So, this A is compatible to the substrate however, B has the required property functional property. So, B to substrate that can be handled by this sputtering and in this case obviously, we can change we must know the sputtering rate of this cathode A and the sputtering rate of this cathode that means, target and that one has to look from this curve and this is actually what we know that this sputtering yield which if we say this is S and here we have the this energy which is given in electron volt E v that is the energy and with that we can have available energy we can have a sputtering rate like this for a material we can have a sputtering rate like this say for example, we can give this illustration say it starts with say 0.5. So, 0.5 means S is what is S? S is actually target atom liberated by the strike of one single argon ion. So, it is so many atoms of the target material for one single stripe of one argon ion. So, this can be as high as say 3.5 to 4 and here it is as low as 0.5. So, it depends upon material. So, it can be on this side high end it can be silver and on this side it can be titanium, but this graph should be available and one can find out from this graph and doing some experiment and trial run he can find out what should be the actually the sputtering rate sputtering rate and accordingly one can one can find out what should be the cathode current for A and what should be the cathode current of B and gradually that has to be adjusted to see that it is an absolutely a graded layer which can give you that super quality of the composite coating. So, this is multi target now comes substrate biasing. Now, this is also one important issue that what we have so far considered it is just the target it is just the target and here we have the substrate. So, target will be negatively biased and this is grounded and this is actually the chamber. So, it should be the substrate and here one can look into this substrate and so here we can have some kind of biasing and with this also we know that with this we can have substrate biasing that means, it can be during sputtering it can be before sputtering and that can be used for cleaning the substrate and also changing the structure of the substrate during sputtering. So, it is just not that argon is going to strike this surface for getting this ejection of the material, but it can be also very closely precisely controlled impingement on this surface to have proper structuring of this built up layer. Similarly, what we can have? We can also have admittance of the reactive gas in addition to argon. So, when we have target material falling on this we can also see that if we have one reactive gas, then this reactive gas and this material which is getting condensed over that surface they can also form a coating of a product which is the reaction say titanium from this target and nitrogen. So, we can have also TIN. So, this is called reactive sputtering. So, with this what we can see that in summary sputtering is basically ejection of material from a solid surface, because of impingement of the same by highly energetic projectile. The argon ion strikes the source or the target surface which is acting as the cathode in the sputtering chamber. The flux of the sputtered material is intercepted by the receptor surface of the substrate. The cathode current discharge voltage, process pressure and the standoff distance are the main process variables. A mono phase or multi phase coating can be deposited by sputtering. It can also be extended to deposit multi layer or graded layer coating. Substrate bias and substrate temperature may have strong influence on coating properties.


We're about to see a model lesson from the Deep Dive Workshop. As a reminder, the intent of this model lesson is to really give teachers space to see and feel what lessons can look like in their own classrooms. That's why we do a role play activity. Instead of having folks stay in their teacher role, we ask them to be in the learner role. The lesson you're about to see is really going to leverage the pair programming feature. So look out for that. All right, everybody, before we get into the curriculum and start actually doing a lesson as learners, we're going to take ourselves into the position of your students by looking at a few student profiles. So you see here we have five student profiles, five descriptions of students. I'm going to give you a moment to read these and as you read, I want you to reflect on whether or not you see any of your own students in these profiles. Further, I'd like you to think about what your role in responsibility is as a teacher in supporting these student needs. Take a few minutes to read and reflect and then we'll come back together. I definitely see student number four in my classroom. We focus a lot on collaboration and so it's nice having students who are always eager to work with other students and support them in their learning. Yeah, I think the challenge though is when you have those students who just want to blurt out the answer to everyone because they're so proud that they know it and they want to share their knowledge. Pacing them when you have other students who are still trying to work through things is definitely a challenge. Absolutely. So in the past couple of years of teaching this course, I certainly have seen a lot of student number threes in my classrooms. They're certainly very engaged early on, but as the lessons progress, a little bit more difficult. They get really discouraged and aren't as persistent as I'd like them to be and they work through all the hints that are provided in each puzzle. But after they run out of hints and they give up entirely, I'm trying to get past that. I see student number one a lot. We have a really big disparity in my class. Some kids are doing a ton of this at home and then they come in and they already know everything and they really are excited to share, but then they get ahead of everyone else and they can sometimes get forward when I'm trying to pace it appropriately for the rest of the kids. And it makes the kids who aren't doing at home, who maybe don't even have access to computers at home, makes them feel hind and then it creates a bad dynamic in the class. So I'm still figuring out how to deal with like engaging those students who are really ahead and excited because I don't want them to feel like they're doing something wrong, but also making sure that they're not kind of like taking all my attention. That was some really great discussion about the students that you're seeing in your classroom and how that relates to the support that we need to provide as teachers. I want you to keep those profiles in your mind and think about how you can channel the needs of those students as we move into our model lesson. So this is a role-playing activity where I am going to be acting as the teacher and you are going to be acting as learners. You are putting on what we call your learner hat. I'm going to put on my teacher hat and you're going to notice that I sort of slip into a mode of talking to you like you're my students. It's not because I don't respect you. It's because we're doing a role-play activity where you are going to act as learners in a classroom. Any questions about that? Okay, great. Let me give you some context for the lesson that we're about to do. This lesson falls in course D that's meant for third-grade students and it's pretty far into the course. It's lesson number 12. It introduces the concepts of while loops and it uses the farmer. In this lesson, we are going to be working with one another. So you are going to need one computer per two people. I'm going to give you a moment to get that together. And when you see my next slide, we are all going to be in our role, each of you as learners and me as a teacher. Okay, hey everybody. Welcome back to technology class. Last time you were here, we used a new block in Code Studio. Can you raise your hand for me if you remember using this block? All right, a couple of you remember? A couple of you don't? That's totally okay. We're going to talk about it a little bit.  Raise your hands please, remember. Yeah, right here.  Yeah, so it helps you do something multiple times. So this block, you might remember, we called this a repeat block. And repeat means we're doing something more than once. And who remembers where the question marks are right now? What can we put in there? Numbers, right? Yeah, we can put numbers in here. So we use this block when we want to do something a certain number of times. Make sense? All right, today we are going to start off using that block again, and we're going to build to the next kind of block that lets us repeat. In order to do our task, we're going to be working on two of our computational practices. The first one we're going to work on is creativity, coming up with unique solutions all of your own. And the second is collaboration. So that's working with another person to solve problems. That's what computational practices we'll be using today. In order to do these tasks, we're going to pair program. How many of you remember pair programming? Most of you, awesome, awesome, just a quick recap because you've been away from technology class for a little bit. When we pair program, we each take a role and we take turns in our role. So someone is the driver. That means somebody is using the keyboard and mouse. And someone is the navigator. So that's the person who's keeping track of everything we're doing and giving instructions to the person in the driver seat. Today we're going to switch roles every two minutes. And I've got a timer up here that's going to go off when it's time to switch. Does that sound good? All right, so let's go ahead and get started. I need one computer per two people. I know you grabbed them from the laptop cart on your way in. So open up your computer, one per two people. Go to code.org, the website. And in the top right corner where you see your name, I want you to click and then select pair programming. Once you're there, you're going to pick my class and then choose your partner, whoever you're working with. Pick their name from the list. Can you give me a big thumbs up when you're connected to your partner? Awesome. Okay, your task for right now is to go to course D. That's the course on the code.org website that we've been working through. Today we're on lesson 12 and you are going to get started by practicing with the first three bubbles. All I want you to do is the first three bubbles and you're going to take turns pair programming.  So we need to use the repeat block to take advantage of just one. But not five, right? Oh yeah, four. Once we're there at the repeat block. At the end, open dirt file, close term move. Do this, bring this repeat out. Sorry. All right. Change it to two. Bring another. Yeah. So six. So change to ten. Okay, so perfect to move to the key. Yes. So move forward, move forward. Shall we run it? Okay. Awesome job everybody with the first three bubbles here in this lesson. I want to take a minute and recap what you just did. In these activities, you always knew how much dirt you needed to move. How did you know how much dirt needed to be moved? Yeah, it showed you the number, right? You were either taking away or adding a known amount of dirt every time. I have a question for everybody. What would you need to be able to do in order to do the same exercise if you didn't know how much dirt needed to move? How would you solve that problem? Could you use a repeat block? Yeah, so that's a super different way of using it, right? Up to now when we use the repeat block, we've done it a certain number of times. But instead of saying, dig 10 times, we want to say shovel until there isn't any dirt left, right? Dig until it's done. So this is introducing a new type of loop, a new block. We're going to watch this video to learn more about how this block works. And then you're going to get back on your computer and you're going to use it for yourself. All right. Now that we know more about how while loops work, you're going to go back on your computer still with your partner. You're going to pair program to use this block. Sound good? Awesome job today, everybody. Mrs. Smith is here now, so you're going to head back to her class. But I'll see you back here next week where we're going to keep going with while loops. All right, everybody hats are off. Time to talk pair programming. Show of hands. How many of you have used pair programming the tool in your classroom with your students? All right, a couple, but not everybody. We're going to take some time right now to look at how pair programming shakes out in your teacher dashboard. So you're going to notice on the screen here that when we pair program, the icon changes just from a single person or number to this cluster of students. This is how I as the teacher know that students completed this level with the pair programming feature turned on. I want you to take a moment now to go to your computers. If you weren't programming on your machine, open your machine up as well. And I want you to look at how your dashboard looks now that you've completed this lesson with a partner via pair programming. We're going to take a couple minutes to do that and then come back and talk questions. So that was the model lesson from the deep dive workshop. We hope that seeing that gives you some context for how it actually looks to model a lesson and to talk to teachers as other students in that role playing context and gives you a little bit more detail about how we think the pair programming feature comes out in the lesson.