For the past 11 seasons, the 58th District Baseball Championship has been decided between Allen Central and Pressensburg. And for the last five consecutive years, the Black Cats have went home with the trophy. Picking up in the sixth inning where the Rebels had already had a two-run lead, Seth Hall up to the plate sends one to center field, but there was a tag at third and Rico Dominguez slides in safe at home giving the Rebels a three-run lead. But with that saying, defense wins games was a true one tonight. Trying to get the double play, but they could only just get one out. And then the very next batter, Chase Solar, snags a line drive at second base and to seal the deal, Pressensburg's Scott Stapleton pops one up to Kane Taylor in right field as he catches the last out of the game and for the first time in five years. The Rebels are the 58th District Champions. Final score, Allen Central six, Pressensburg three. I also had the chance to speak with head coach BJ Aldrich after a long anticipated win and he mentioned a 10th man on the field, his late son, Trey Aldrich, who passed away in August of 2014 and would have been a senior this year at Allen Central. They fought hard all day long. They made a mistake, they turned around and fixed it. We made another mistake, we fixed it. This group of seniors right here that we have, this is something we talked about before the season even started during pre-season. We said, hey, we're not going to get in that district tournament and choke again. My baby boys up there in heaven right now looking down on us and that's what allowed this team to be so great is because of him. I just want to say thank you to everybody, thanks to these boys for allowing them to be a part of my life and my baby's life. Daddy loves you, Trey.


So I'm going to be talking about symplectic embeddings of products. This is all joint work with Richard Hind. Let me start by giving you an introduction to the kind of questions that I want to talk about. So if I'm given two symplectic manifolds, M1 and M2, then a symplectic embedding, so I'll give you a definition. A symplectic embedding is a map of psi going from M1 into M2 where psi is an embedding of smooth manifolds. And psi pulls back the symplectic form on the target to the symplectic form on the domain. So one can then ask the following question, which is if I'm given two symplectic manifolds, I'd like to know if I can find a symplectic embedding of one into another. So given M1 omega 1, M2 omega 2, does there exist a symplectic embedding? So I'm going to move over to here. And I'm going to write my symplectic embeddings like this. So the S here just stands for symplectic. So this is the basic kind of question that we want to know the answer to. It turns out that this is quite subtle. I'll give you an example to illustrate why it's subtle in a moment. But anyway, so that's sort of the basic background. Could the image of psi be in the interior of M2? At the moment, no. In the examples that I'm going to give, that generally will happen. But I don't require a condition like that. I should say that this is particularly subtle when the dimensions of M1 and M2 are the same. And that'll certainly be the case in this talk. If the dimensions are off by, say, 4, there's an H principle and things like that. So you should really think of these as having the same dimension. But when I give an example, you'll see what I mean. OK, so any other questions about any of that? Should be pretty straightforward. OK, so let's go to dimension 4 now. And like I said, this problem can be fairly subtle. And the following example, due to McDuff and Schlank, is particularly illustrative of this. So some of you have no doubt seen this before. But I'm sure some of you have not. And it'll also be quite relevant to what else we're talking about in the talk. So what McDuff and Schlank wanted to know is they wanted to know how this symplectic embedding problem worked for some very special shapes. In particular, they defined the four dimensional symplectic ellipsoid. So I want to think of this as a subset of C2, which I identify with R4 with its standard symplectic form. And I also want to study the four dimensional ball. OK, I'll write a 4 here. Which is just an ellipsoid where A and B are the same. And they wanted to know when an ellipsoid admits a symplectic embedding into a ball. So it turns out that we can encode this in the following function. So we can define a function C of A, which is given by the infimum over real numbers lambda such that there exists a symplectic embedding from E1A into a four ball of size lambda. So this is a function from R to R. And it turns out that C of A for A greater than or equal to 1 completely answers. I guess I'm going to call this Q1, because there's going to be other questions that come up later in the talk. So C of A for A greater than or equal to 1 answers Q1. Well, that might not be immediately apparent if you're not used to thinking about these things before. But the point is these symplectic embeddings, well, I can scale them. And they're also symmetric. EAB is the same as BA. So all I have to know is this function for A greater than or equal to 1. And that determines exactly when an ellipsoid embeds into a ball. So McDuffin's just like we're able to actually just compute this function. And I'll show you the answer. And if you haven't seen it before, I think you'll find it pretty cool. So, well, let me graph the function for you. So like I said, our function eats a real number and spits out a real number. A basic consideration is that C of A is always greater than or equal to the square root of A. This represents the fact that symplectic embeddings have to preserve volume. Well, so now, actually I'm going to move this down a little bit, what the function looks like. So I'll draw the square root of A here. So here's root A. And we only need to know what's going on for A greater than or equal to 1. So at 1, the function is 1. And then what the function does is I go via a line for a little while. So if you extended this line, it would go through the origin. And then at 2, well, the line veers off to the right. So let me say a little bit about geometrically what this is saying. So this occurs at 2. And this occurs at 4. So from 1 to 2, you should try to visualize what's going on, especially if you haven't seen this kind of thing before. For 1 to 2, this is saying that this embedding problem satisfies extremely strong rigidity. So here I have an ellipsoid. Maybe I have an e1, 1.5. And what McDuffin-Schleig's computation is saying is that the optimal embedding into a ball comes from including it into a ball. It's saying that from A between 1 and 2, we can't do any better than inclusion. Actually, that's an old result of Hamilton and Jacob on that. OK, I guess this was known before. It was known up to 2 and 1. But I mean, if you're thinking about these embedding problems for the first time, you should think of this as an expression of quite strong rigidity. However, after 2, the function suddenly veers to the right. And this is a horizontal line until it can't go anymore. So here you should imagine an ellipsoid that's four times as long as it is tall. And this evidently can fully fill a ball of size 2. I mean, it can occupy 100% of the volume. So well then, of course, the function can't continue anymore. So what does it do? Well, it does the same sort of pattern. It continues by a line that if you extended it, it went through the origin. And then it veers off to the right. And it does this again, and it does it again, and it does it again, and again, and again. And it does it infinitely many times until it terminates at, well, this number here is the golden mean to the fourth power. OK, so this is the golden mean to the fourth power. OK. So now, what does it do after the golden mean to the fourth power? Well, this is about 6.8. And actually, it's also extremely interesting after 17 over 6 squared, which, yeah, it should be obvious to all of you once you've seen the golden mean is appearing. So this is about, well, this is exactly 8 and 136. So after 17 over 6 squared, the function just is the volume. So here, c of a is just equal to the square root of a. And so now in between, well, it's kind of a mixture of this behavior. It's mostly the volume, but there's a few times where it rebels and it's obstructed. So actually, right after tau to the fourth, there's a rather long ish step. And then I think there's eight more steps that get rather small. And other than that, it's just the volume. So this is actually the piece that we're going to be most concerned with in this talk. But it's a beautiful result. So I'll give you a second. I'll pause and see if you have any questions. But I guess the way you should think about it is that if the ellipsoid is stretched enough, then all obstructions vanish, except for the volume obstruction. But if the ellipsoid is close to round, then there's this very delicate, of course, there's something else I want to say about this. Maybe you were going to ask. There's some very delicate behavior. So of course, there's something else I wanted to say about this behavior. So like I said, this number is 1. This number is 2. This number is 4. Well, you could ask, what are these numbers here?  So if I have a step like this, while the pattern, I'm going to call the x-coordinates.  I'm going to call this Bn. And then this would be An plus 1. Well, so what the An's and the Bn's are is it turns out they're determined by the odd index Fibonacci numbers. So they're both ratios of odd index Fibonacci numbers. The An's are squares of ratios of consecutive odd index Fibonacci numbers. And the Bn's look like this. And here the Gn's are just, and these are the Fibonacci numbers. So because there's this staircase determined by the Fibonacci numbers, this is called the Fibonacci staircase. So apparently, if the ellipsoid is closed around, there's some delicate combinatorial behavior. So are there any questions about that? Where on the graph is the infimum actually at T? Like for which values of A is there a? Is it actually a minimum? Well, I think it's rarely a minimum. I think it's rarely a minimum. It depends what you mean by a minimum. If you ask for when the interior of the ellipsoid is bent into the closed port, then it is a 10.10. Everywhere where the graph, like at all these numbers, but for example, at 4, technically, this is saying that you can't find an embedding of a closed E14 into a closed B2. You can find an embedding of an open E14 into an open B2. But this is saying that you can find an embedding of a closed E14 into a B2 plus epsilon. Yeah, but then you can prove that that means that the open one, the open one, embeds. You could also just write this function. You could write this function with open ellipsoids and open balls, and then it just would always attain the minimum. Any more questions? Is this map for a lolliport ellipsoid?  The actual maps. Yeah, that's a very good question. So there are maybe a few of these that you can write down, but in general, this uses a technique called some plectic inflation, which is certainly rather non-constructive. I want to specifically go ask for it. What? It's the one at four. No, I don't think you can write down the one at four without some plectic inflation. But oh, maybe with another curve or something. Oppstein had a very nice way of seeing that, which I could explain to you. Using the algebraic geometry of Cp2, and you look at the quadric in Cp2, I can explain. And I guess you can probably do also E1n squared or something. Can you do E1n squared into B of n? Can you do or explicitly? OK, so that's part of the four-dimensional story. And well, another part of the story that I want to say, well, there are various ways of thinking about why this is true, but the result, which is going to be of most relevance to this talk, is a result from slightly later, also due to do so. So a little while later, Dusa found a purely combinatorial criterion for explaining when one ellipsoid embeds into another. So Dusa showed that, so now I'm going to go to open ellipsoids to get the cleanest statement, that there is a symplectic embedding of one open ellipsoid into another if and only if, well, the sequence NAB is less than or equal to the sequence NCD. In a minute, I'll tell you what these sequences are, but this is just some sequence of real numbers. And here, less than or equal to means term by term inequality. So this is a purely combinatorial sequence determined by A and B. In fact, what it is is, so NAB is the sequence whose kth term is the k plus first element in the matrix MA plus NB, where M and N are non-negative integers. And sorry, here I mean the k plus first smallest element. And this is index starting at k equals 0. So what this is saying is given positive real numbers, A and B, you take all possible non-negative integer linear combinations, and then you list them in order, including repetitions. You get a combinatorial sequence. And using this combinatorial sequence, you can completely determine when one four-dimensional ellipsoid embeds into another. So just to give you an example of how this N thing works, actually, we're not going to do too much with the combinatorics of the N sequence in this talk. But to orient yourself, for example, if I wanted to do N11, well, the first term would be 0, because I can take 0 times 1 plus 0 times 1. And then I get 2 1s, because I can take 1 times 1, 1 times 1. Then I get 3 2s. And well, so you can associate these combinatorial sequences. And indeed, you can see why this result is true, at least the Fibonacci staircase part, by using this. Although that wasn't how McDuffin Slank did that. What is relevant to our story is that this sequence NAB, well, it looks kind of somewhat beautiful, but somewhat random. It has an interpretation for us that will be relevant. This is the sequence of ECH capacities of the ellipsoid EAB. So here, actually, I guess I'm going to call this embedded contact tomology capacities. I'll spell it out for you. This ended up being messily written. So these are sometimes called ECH capacities. So we'll return to ECH actually quite soon. What embedded contact tomology is, it's a variant of the kind of some plectic field theory ideas that have been discussed here. It only exists in dimensions for contact three manifolds. And you can use it to get four-dimensional, symplectic embedding obstructions called ECH capacities. And from that point of view, McDuff's result says that this obstruction coming from embedded contact tomology is sharp for embeddings of ellipsoids. So you don't have to understand too much about this, because we'll talk about ECH again rather soon. So any questions about any of this? Sorry, could you say that again? Or, clearly, was that how this estimate, half of this estimate is preferred? No, well, I mean, of course, Deuce would be the person to ask. But my impression is that I think I first proved a sort of necessary and sufficient condition for embedding. But it was in 10 terms of NAB, NAB sequence. It was a different sequence. And then Helmut would say, well, you should consider these things, which have been done. And then I realized that the sequence of the obstruction, the way I described it, was the same. So also, Michael showed that there were obstructions in developing the sequence. OK. So this is a very nice story in dimension four. And I guess I'll briefly remark that, of course, ellipsoids are rather simple shapes. You can push these ideas a little bit further. So for example, with Terra Home and Henri de Presse and Alessia Mendini, we've been seeing how this story would work for embeddings of symplectic toric four-manifolds and things like that, toric domains, too. So you can push this story somewhat further. I mean, it's certainly not the case that we have a good understanding of when one symplectic four-manifold embeds into another. But some of this does generalize in a nice way. However, what I want to tell you about today is the story in higher dimensions. So in higher dimensions, the situation is much less poorly understood, even for ellipsoids. Sorry, more poorly understood, even for ellipsoids. So, well, so for example, you could ask, when does a ellipsoid, e, a, b, c embeds symplectically into an ellipsoid, d, e, f? Well, so you could maybe say I should look at the obstruction coming from embedded contact tomology. But like I just said, there is no obvious obstruction coming from embedded contact tomology because there is no known analog of embedded contact tomology in higher dimensions. So there's no known analog of e, c, h. Of course, there is an obvious analog of this NAB sequence. And you could maybe guess, you could still try to guess that this combinatorial sequence still understands the embedding problem. So you could think that perhaps I should put interiors here. You should think that perhaps a result like this should be true. You could think that perhaps a result like this could be true. But unfortunately, this just isn't true. So even this is not true. So we'll see actually a little bit later why this is not true. But some of you maybe have heard of the work of Gooth on the non-existence of intermediate symplectic capacities. And that can be used to show this, for example. So even this combinatorial sequence is just not the right thing. So I think it's fair to say that even for these ellipsoids, we're pretty confused about how the embedding story works. So with Richard Hint, we've been wanting to understand a more modest goal. So I want to understand the following more modest goal. So we want to understand, well, we'd like to understand the following function that's an analog of the function studied by McDuffin's length. So we're going to define C twiddle of A to be, well, the imp over lambda such that an ellipsoid crossed with an R2N, symplectically embeds into a four ball of size lambda crossed with an R2N. So sure, we can fix an N and we can try to understand this embedding function. So if you like this, so of course the talk is called symplectic embeddings of products. Maybe you can see why the talk is called that now. But if you like this ellipsoid paradigm, you should think of this as some ellipsoid where most of the factors are just set equal to infinity. And similarly here. So this is just some question about extremely stretched ellipsoids. So well, now a basic observation is that this function C twiddle, it's always less than or equal to the function studied by McDuffin's length. That's because if I have an embedding of my four dimensional ellipsoid into a ball, I can always just take its product with the identity and get an embedding of the product. So in a way, this is a question about how much flexibility one has for embeddings of products. Can one actually do better than the product embedding? Well, so you might think, now I'm slightly running out of chalk. So you might think that in fact C twiddle of A is equal to C McDuff's length of A. And in particular, then it wouldn't depend on N. However, that's not true. So for example, there's the following theorem due to Richard Heind, which is a kind of refinement of Goose construction. So OK. So Heind has shown that there is a symplectic embedding from well, N, E1A cross R2N into a four ball of size 3A over A plus 1 cross R2N. So well, now I'm actually going to draw a picture for you. So at this point, maybe some of you see why this says that this can't be equal to the McDuff's length function. But I still want to draw a picture because it's going to be very important for understanding what's going on. So note that the McDuff's length function is always greater than or equal to the square root of A while 3A over A plus 1 converges as A goes to infinity to 3. OK. So again, I'll put this in pictorial form. But this says that eventually one must be able to do better than what McDuff and Slank did because there is no volume obstruction to this problem, for example. So by the way, if you're not used to thinking of this. Is it 0 or what's the state of your hand? Yeah, A is bigger than 1. And N is bigger than 1. So again, if you're learning about this for the first time, you should think of A as extremely large, maybe even infinity. And then it's kind of an amazing fact that such a thing exists because we're squeezing an entire R2 factor. We have essentially an extra factor of R2 here. So it's kind of amazing. By the way, to answer something that Vivek said, this actually can be done explicitly. So this is constructed explicitly using a kind of catalyzed folding, which is probably meaningless for maybe even 98% of the audience. But there's a construction due to Dusa, Felix, Slank, and others called symplectic folding. And you can't quite use symplectic folding to prove this. But you can use a kind of catalyzed version, which is very similar.  At least for some of us, this is kind of a lot easier to understand this catalyzed folding picture. So this is a very explicit embedding. OK, so anyways, now I want to draw you a picture. Would it be possible to prove any result like this without an explicit construction? Yeah, that's a very good question. So at the moment, most of our only ways of producing embeddings come from some explicit, in higher dimensions, come from some explicit construction like this. I mean, of course, one could take a non-explicit construction of embedding of an ellipsoid into a ball and then cross it with the identity. But yeah, in dimension four, we have these symplectic inflation tools, which are very powerful. And there's not a good analog of that in higher dimensions. So most of our ways of getting non-trivial higher-dimensional symplectic embeddings are pretty hands-on. So you're not saying that Sharp is just constructed one for each end? Well, we'll get back to the Sharpness question in a moment. But yeah, at the moment, I'm just saying he constructed one. And if you think about it, that shows that the McDuff's length result cannot possibly be the answer here. So let me draw you a picture to underscore that. This picture is just supposed to emphasize. So I'm now going to try to graph C twiddle of A. Well, the graph of 3A over A plus 1, well, I just have to start it at 1. It's asymptotic to 3. So here's 3A over A plus 1. Now, this square root function, well, it is a square root function. And in particular, it crosses the graph 3A over A plus 1 at some value. And remember, C of A, the McDuff's length function, is always above this white curve. Whereas what we're saying is, so the hind result says that whatever our function C twiddle is, it's always less than or equal to 3A over A plus 1. So evidently, once we get past this point, this product function cannot be equal to the McDuff's length function. So now a natural question is, what is this point? Well, this is kind of a fun exercise. So is anyone good at mental math? What is the point where 3A over A? What is the A value for which 3A over A plus 1 and square root of A intersect? Well, it's nothing other than the golden mean to the fourth. Yeah, exactly. So this intersection point is the golden mean to the fourth. And in fact, if you draw the Fibonacci staircase, what you'll find is it actually bounces between these two graphs. So the Fibonacci staircase looks something like this. In other words, the upper graph here does clip the top of all of the steps. So based on this, Heine conjectured in an earlier paper that C twiddle of A is actually equal to the McDuff's length function for A between 1 and 10 to the fourth. And Richard and I proved that. So the theorem here, this is due to Richard and I, is that C twiddle of A equals C of A for A from 1 to 10 to the fourth. So again, if you think geometrically in terms of what's going on, it's kind of a cool situation. This is saying that the optimal embedding in the Fibonacci staircase region is just given by lifting the McDuff-Schlank embedding. It is just given by a product embedding. And then that's the only part of the McDuff and Schlank graph that persists for this problem. Once you move past the Fibonacci staircase part, the rest of the graph is not relevant. So somehow this Fibonacci staircase really is a robust feature of the embedding problem. Do you know if the C twiddle depends on n at all? No, that's a great question. So of course, this says that C twiddle doesn't depend on n from 1 to tau to the fourth. Richard and I are currently thinking about what happens after tau to the fourth. And we don't know. I mean, it's actually a fun question if you like trying to figure out which moduli spaces of holomorphic curves are empty or non-empty. So I would guess that it does not depend on n. But I have no way of showing that. I should say, actually, that there are a few more points that are known due to Richard. So actually, also if a is equal to 3d plus 1, where d is an integer, then the graph is also given by the yellow line. So there are a few more points on the yellow line. So the 3a over a plus 1 graph line. So that's when you know that C. So you actually do know if this is a equals 3d plus 1, where d is an integer greater than or equal to 3, greater than or equal to 3, then you actually do know that these are correct values of C twiddle. And these also don't depend on a. So this is an earlier result due to Heind and really Ellie Kerman, too. So this is this Heind-Kerman thing. OK. So I want to say a few words about how the proof goes. It's sort of good news for you, because this is a conference on moduli spaces of holomorphic curves. And the proof involves a lot of tinkering with moduli spaces of holomorphic curves. Also, interestingly, it uses embedded contact tomology quite a lot, which is interesting, because it's a higher dimensional problem. So are there any questions about this, though, before I give some sketch of how the proof goes? Are you going to say why it's a critical number? I don't know why it's a critical number. I mean, especially if you look at this folding construction, you would maybe never guess that it's ever sharp, because it just looks like you kind of do this, you kind of do that. So it's sort of amazing that this is somehow a pretty persistent feature of the graph. But actually, it does seem to be. So again, I was saying with Terra Home and others, we've been thinking about how this story would work for other, say, toric targets. And this kind of part seems to be a pretty persistent feature of what's going on. Actually, there's also some very interesting commonatorics in Tau to the Fourth, involving Earhart polynomials. So you can find triangles with an irrational slope given by Tau to the Fourth that their lattice point enumeration looks like the lattice point enumeration for an integer triangle. That's something very special about Tau to the Fourth. So would you just to describe your argument in general, would you say it's like the common hind argument that's just up in a slightly different setting? That's right. Yeah, in fact, I was going to say, yeah. So it's pretty similar to the Heine-Kurman argument, but with a new input from ECH. Right. Well, I mean, they sort of were using ECH. They sort of do, but it's more explicit. They do some sort of neck stretching kind of thing. And yeah, we don't do that. So OK. But actually, I want to answer Deuce's question and even write something on the board. So what's the idea? And well, some of you are probably not used to thinking about these symplectic embedding problems at all. So I'm going to spell this out in a little bit of detail. So it actually suffices to show that C twiddle of the B ends is the same as the McDuff's Lang function of the B ends. That might not be immediately apparent, but that's not that hard to show. So what I'm saying is, remember, we want to show that the graph is given by the purple thing, purple in honor of Joe, by the way. So we want to show that the graph is given by the purple thing. Well, what I'm saying is it actually just suffices to show that these are the correct points by some fairly simple elementary observations. I mean, it helps to notice that these lines actually go through the origin, which you would never guess from the way that they're drawn. But they should go through the origin. And similarly, these lines are horizontal, which really helps. So that's just a rescaling argument that we used. That's exactly just an argument that was used by McDuff and Schlanck. So we just want to show this. So now we know that C twiddle of Bn is less than or equal to C of Bn, because as I said, you can just lift the McDuff Schlanck embedding. So you should think of what we want to do is we want to find an obstruction to finding a better embedding. OK. So find an obstruction to a better embedding. And well, if you're, so I'm going to make that 0.1. If you're not used to thinking about these symplectic embedding problems, well, I want to make the following point, which is that, generally speaking, that obstruction should come from some sort of holomorphic curve. I mean, of course, there are other obstructions to symplectic embeddings that are a little more variational in nature. There's these Eklund Hofer capacities. But I think it's fair to say that it's reasonable to look for a holomorphic curve that's somehow obstructing something better. OK. So now we just want to find the right curve. And this is where we use the Hein-Kirman argument. So you should think of the curve. It's going to live in the completion of some cubordism of high dimension. I mean, in this case, it can really have any even dimension. So we want to find some holomorphic curve living in a completed symplectic 2n manifold. So the Hein-Kirman idea is to find a curve in a four dimensional cubordism that lifts nicely to higher dimensions. So that's a little vague. I'll fill in some of the details later in the talk. But this is where ECH comes in. So the way you should think about it is go back to the McDuff and Schlank picture. Of course, the McDuff-Schlank statement that these are points on the graph is also an obstructive statement. There's an obstruction component to that that you can't do any better. So the idea is find the holomorphic curve that's giving McDuff and Schlank's four dimensional obstruction. And then, well, this thing is like a product. So roughly speaking, what you want to do is you want to take that holomorphic curve and instead of mapping it into the completion of the ellipsoid cross R2N by, say, sending it to 0. And so then to find that holomorphic curve, we want to use embedded contact tomology. So that's the general idea. And yeah, this is essentially the Hein-Kirman argument with this ECH used somewhat differently. Are there any questions about that? What is the VM in your supply structure, the VM sequence? Oh, this was this discussion about how, as you can see, these lines go through the origin. And so Dusse and Felix have this nice argument where, well, these ellipsoid embeddings, they satisfy some basic axioms. I mean, so for example, they're monotonic. Like, this function has to be a priori monotonically non-decreasing. And it also has to satisfy a subscaling property, meaning once I have some point on the function, if I draw the line through the origin, connecting that point to the origin, the function always has to stay below that line, or it can't go above that line. So using those two points together with the McDuff and Schlank information, you can recover the whole graph. And this is kind of a version of an argument that was used by McDuff and Schlank in their paper. So there's no sort of interesting holomorphic curve theory here. It's just the first principles of the embedding problem. OK. So now let me be a little bit more explicit about what I mean by how I actually want to do this. So we first want to give a brief crash course on embedded contact tomology. So this is going to be a very brief crash course, hopefully more of a course than a crash. And again, the motivation is we want to use embedded contact tomology to find this curve. Now we have our ECH crash course. So you should think of this as kind of the complete opposite of a mini course, where everything is done carefully and in much detail. OK. So for ECH, I have a three manifold. So now y is a three manifold, and lambda is a non-degenerate contact form. So now the embedded contact tomology of this pair, well, it's a variant of, say, some plectic field theory. OK. In other words, well, it's the homology of a chain complex, I need better chalk, which is generated by certain sets of ray orbits. And the differential d, certain, well, embedded, except there may be just mostly embedded curves in r cross y. So I mean, we've been doing so much with contact tomologies and this kind of thing at this conference that I think I just want to draw your picture. So the picture you should have in mind is really the kind of curves that get counted. And while the kind of curves that get counted, you should maybe just have a picture like this, say, by which I mean to illustrate the following. So the curves can have genus. They can have arbitrarily many positive punctures. They can have arbitrarily many negative punctures. That's why I'm saying that the chain complex should be generated by sets of ray orbits. I mean, in cylindrical contact tomology, you just have a cylinders. You have one orbit up and one orbit down. You have a whole set of orbits here. But we don't actually want to count all such curves. We want to single out embedded curves in some clever way. And we end up singling them out by this thing called the ECH index. So specifically, we count ECH index 1 curves. And that somehow forces the curves to be essentially embedded. What I mean when I say mostly embedded is a component like this actually has to be embedded. But if I have a trivial cylinder here, this could be multiply covered. So this ECH index is a great thing. This is all due to Hutchings, of course. And well, we're actually almost done with the crash course. But you might ask, why are we doing this? Well, one reason is that embedded curves are very nice. I mean, certainly the kind of transversality issues for these embedded curves are much more mild because they can't be multiply covered. But another reason, which is a bit cooler, is that this embedded contact tomology, well, it looks very much like a holomorphic curve theory, but it ends up being isomorphic to a gauge theory. So Taubes has shown that ECH is canonically isomorphic to a version of cyber-gwitten fluorochomology. So this star here is just some grading business, which we don't really want to talk about. So this is the cyber-gwitten fluorochomology. And well, I don't want to say much about this, but this is a gauge theory. It also only depends on the three-manifold y and not on the contact form. So somehow embedded contact tomology allows you to encode topological information about the three-manifold coming from gauge theory into information about the contact and symplectic geometry of the three-manifold and associated four-manifolds. So that's sort of the end of our crash course. But for our purposes, actually, what we care about is that we can use this isomorphism to define kabordism maps. We can use this to define kabordism maps, by which I mean, well, if there's one thing that should probably be clear from this summer school, if you want to define a kabordism map on some version of some plectic field theory, normally that gets extremely hairy. However, cyber-witten fluorocomology does admit kabordism maps, and this has worked out in nice detail by Kronheimer and Morovka. And so we can use the fact that cyber-witten theory admits kabordism maps to get a kabordism map on embedded contact tomology. Now, a priori of that kabordism map then would count some kind of cyber-witten monopoles, but Hutchings and Taubes have shown that these kabordism maps satisfy some nice axioms. In particular, they satisfy a holomorphic curve axiom, which will be very useful for us. In particular, a morphic curve axiom. So remember, I said before that we want to use ECH to find some nice four-dimensional curves. And that's the idea. We're going to induce a kabordism. That's going to have a kabordism map. We're going to apply the Hutchings-Taubes holomorphic curve axiom to get a holomorphic curve. And then we're going to pay very close attention to it to make sure that we can lift it correctly. So any questions about that? Are you going to say what an ECH capacity is? No. I will not say what an ECH capacity is. Interestingly, yeah, you can ask me later. I don't refuse to ever talk about that. Interestingly, we don't actually have to know much about ECH capacities to do any of this. And this is maybe an idea that Michael has recently put in writing, which is that you probably should really think about the curves themselves and not so much these capacities, which are kind of user-friendly, but for very delicate applications, they lose some information. So we're much more interested in the curves themselves. But nevertheless, I'm happy to talk about it. OK. By the way, this symplectic caportism, I could write more about what I want it to be. In this setting, I actually want it to be, well, you might think I want it to be an exact symplectic caportism. I actually only want it to be weakly exact. But I think I'm going to skip those details. But you can ask me later. OK. Where does this portism go between what and what? I haven't told you. So here, this is part of our crash course about ECH. I'm saying any time you have an exact symplectic caportism between contact manifolds, you get an induced map on embedded contact demology. Now I have to tell you how to induce that caportism for the purposes of our problem. And I'm saying you would think that you want an exact caportism, but our caportism is actually only going to be what's called weakly exact, which is fine for what we want to do. But anyways, I'm going to say what the hold of it more than code max it is. More or less, yes, at a later moment. So I'm now going to, so this is all just general facts about ECH. I'm now going to induce for you the caportism that will be relevant to us. OK. So I now want to go back to the proof. OK. So I want to produce some sort of symplectic caportism. And well, the general game here, which is certainly familiar if you've done this before, is that the caportism should come from an embedding. So by the McDuffin-Schlank computation, so by McDuffin-Schlank, we know that I can embed, well, an ellipsoid stretched by a factor of bn. So remember these bn's, these were ratios of odd index Fibonacci numbers that are off by 2. So I can embed this bn. I actually want to add a little epsilon for sort of technical reasons that you can disregard at first. And I want to embed this into the interior of, well, I'm just going to write this as an ecc plus epsilon, where this c is equal to c of bn plus epsilon twiddle. OK. So remember that c of bn was itself gn plus 2 over gn plus 1. OK. So there's a lot of epsilons here. And they're put in here because of the convention. I learned this phrase from a paper by Obstein. They're put in here because of the convention that statements should be correct instead of approximately correct. But you should maybe ignore them at first reading. So the reason we're putting these epsilons in here is remember, we wanted a non-degenerate contact form. And if I took a ball here, I'd have a more spot contact form. So you really want to think of this as a kind of ball, but I'm just perturbing it a little bit to be irrational. And similarly, bn is a rational number. So I want to perturb this to be irrational. And also, there was this point that Duce and Nate and I were talking about that I can't actually embed this into the interior for what's coming. So I might have to make c a little bit bigger because it's just an inf. But so if you're learning about this for the first time, you truly can just ignore all the epsilons. And that might help your sanity. So I'll draw you a picture. So this is the kind of picture that you should have in mind. Here's my ecc plus epsilon. And then my ellipsoid is just embedded in here in some strange way. So here's an e1 bn plus epsilon. So now I can write down my cabortism. So I'm now going to remove, well, psi of the interior of e of 1 bn plus epsilon to get a cabortism x. One second. It was close to the epsilon. This is the whole point of the equation. Yes, that's right. So this gives you a weakly exact some plucked cabortism. So you should think of the cabortism as this thing x. So here's my cabortism x. So I just removed the interior of this ellipsoid. It's sitting, the whole ellipsoid is itself sitting in the interior of the ball. And I remove it to get a cabortism x. So now my holomorphic curves are going to sit in the completion of x. So I want to complete this by adding cylindrical ends. Yes, it means that the, see, the issue is that this is a symplectic embedding, but it might not be an exact symplectic embedding. So I want to complete x by adding cylindrical ends. OK, so again, weakly exact is maybe not so important. I'm just saying, well, in an exact symplectic embedding, you would like your symplectic form on here to be d of a 1 form that restricts to the boundary as the relevant contact forms. But we are not given an exact symplectic embedding. So we can't necessarily find that. However, all we actually need is we need the symplectic form to be exact because we want to rule out some kind of bubbling, for example. And we want it to have, we want its restriction to the boundary to be d of the relevant contact form. So that's actually all you need to define cabortism maps. But I assume most people are more used to thinking about exact symplectic cabortism. So you don't lose much. Well, I guess I'll just write to both sides. So this is a cabortism with two boundary components. And I want to add a cylindrical end to both boundary components. So what x bar looks like is it's x. And then I add a cylindrical end to the top. So I add a 0 infinity cross a boundary of a ball.  And I add that on, on the boundary of the ball. And then I also add on a negative end on the boundary of the ellipsoid. Does that make sense? So you take this thing and you add an end to the top and an end to the bottom. This is a pretty standard thing in this story. So the curve lives in x bar. So our curve will live in x bar. In a moment, I'll draw the curve for you. But let me answer Vivek's question and tell you a little bit about what the holomorphic curve axiom says here. So anyways, I've produced this cabortism x and I'm attaching it to get a coarsen x bar. So now remember, ECH admits cabortism maps. Because ECH admits cabortism maps, I get a map, which I'll call psi twiddle, because it's induced by psi, mapping, well, the embedded contactomology of the boundary of this ball to the embedded contactomology of the boundary of the ellipsoid. And it will satisfy the holomorphic curve axiom. But so now first, we want to do a very easy calculation. So an easy calculation, well, so this easy calculation involves the following. I want to tell you a few things about the embedded contactomology of this. Of course, I haven't really defined embedded contactomology in any rigor. But remember, there's some rape orbits involved.   Yeah? I think I'm going to channel my inner Muhammad and take eight. But then I'll wrap it up. OK. That seems to be a popular strategy in this conference. But I will take eight. Well, maybe I should stop pretty soon. I'll take a few more minutes. Yeah. OK. So this takes two. I just want to wrap up at a reasonable place. And I'm not going to be able to wrap up in one minute. But I promise I'll be able to wrap up relatively soon. OK. So this has two embedded orbits. Shh. Let us continue. Yeah, yeah. OK. So this has two embedded orbits, which I'm going to call alpha 1 and alpha 2. Well, the action of alpha 1 you can see is c. And the action of alpha 2 is c plus epsilon. And well, but yeah, I apologize for going over. I really do. I just want to finish this general train of thought. And well, the boundary of the other guy similarly has two orbits, beta 1 and my conventions are the same. So you think of beta 1 as short and beta 2 as the longer one. So anyways, the point is you can show that there exists. You can show that my map phi twiddle, well actually remember, in each stage I'm allowed to take orbit sets. So if I take this orbit alpha 2 and take gn plus 1 copies of it, I can show that that gets mapped to beta 1 with gn plus 2 copies. And so now the holomorphic curve axiom. So the point of all this is that the holomorphic curve axiom now says that there is some broken, some possibly broken, holomorphic curve alpha 2 to the gn plus 1 to beta 1 to the gn plus 2. OK, I promise. I really am almost done. So just to emphasize Vivek's question, so what the holomorphic curve axiom is saying here because I know that the kabordism map sends this element of ECH to this element of ECH. I know that there's some, well the holomorphic curve axiom says that there's some holomorphic curve, but it could have multiple levels. Here you can show, however, that in this particular case, partially because of the nice properties of the Fibonacci numbers, this curve has exactly one level. You can show in this case there's exactly one level. And you can actually write down the curve. So the curve looks like, well, and I'll draw it for you. So the curve looks like this. So there's gn plus 1 positive ends that are simply covered and 1 negative end, which is a gn plus 2 fold cover. So you should think of this as the quote Fibonacci curves. OK, so now I'll just say the point and then we can go eat lunch. But I do want you to have this picture in mind, because these are very nice curves. So now the point of these curves is if you stare at the index formula, so these are index 0. That's sort of just a general thing that happens in this kabordism map story. But the point is if you stare at these curves and apply a Fibonacci identity, you'll see that when I lift these curves, essentially by just crossing with r to the 2n, then they still have index 0. In principle, when I put it in a different modular space, its index could jump. But if I write down the modular space correctly, these have index 0 in all dimensions. And so we can use them to do the Heine-Kirman argument. What is the Heine-Kirman argument? Well, remember, I want to show that I can't do better than the product embedding. So what I do is I take some random embedding, which might not be a product. And then I can isotope it to the product embedding if I allow myself to shrink the ellipsoid. And so by doing that, I can get a one-parameter family of almost complex structures and try to move this curve in the normal way. So then I just have to prove a compactness result, showing that these curves actually persist. And I also have to do some delicate sign counting to make sure that there weren't two of them to start with. You can do all that. And what you find is that even if I had some non-product embedding, this curve is still living there to obstruct it. So anyways, you have to do this compactness calculation and also a sign counting. But the picture that you should have in mind is really this picture, because these are nice curves. OK, so I think I'll stop there. Sorry for going over time. I apologize to the organizers.


Hey everyone, welcome to today's day trading recap for Thursday, May 13th. Nice green day today, plus 1076, but I'm actually a little disappointed in that. And the reason is is because I overstayed my welcome. I was up well over 2000 earlier today and I gave some back because I placed some trades later in the day that I probably shouldn't have. But anyway, let's go through these trades. I'll show you what we did. Starting with the mighty 90s, we didn't have any trades. On the Paris trade had one in gold versus silver, small loser for minus 127. And then on the runners, plus 1203 had a trade in Boeing that was exactly a scratch, zero. Baba plus 565, beyond minus 304, Netflix minus 270, Peton over 1000 bucks, Roblox, first trade in Roblox plus 183, Roku minus 920 square 505, biggest winner Tesla 1095, and then the Nasdaq minus 676. So let's go over these. I will show you what we did here. Starting with Boeing. So Boeing was the was a runner. Let me switch over here. All right. So Boeing, Boeing had this nice initial push. In fact, on this little pullback here, it was trying to get long and never did just move too quick, didn't have a chance to get filled and this thing fired up. So that would have been a really nice winner. But then when it pulled back to yesterday's open, got long here, it bounced, got out of half my position for over 10% profit was looking for a continuation higher, just didn't happen rolled over.  Next trade, Baba. So Baba was a nice winner for 565 on this volume spike here when it bounced just a little bit, got short kind of chopped around for a while and then pushed down, got out of some pushed down to this pivot, got out of the majority of my position. And then when it popped up here, I thought maybe our little downside was over. So I went out cutting out of the last piece. And so booked that one for 565. Now, obviously, it did have some order on the downside, but I did not stick around for that. And then beyond. So this was one of my late day trades. And I, you know, I don't really mind that I took this trade because I really like the setup. This one just didn't, in fact, work. And of course, it's beyond. So beyond has a way of getting us as chat about beyond. It's his favorite. So anyway, I mean, this thing just continued to push lower, push lower, push lower, push lower. And then when it flushed and had this nice volume spike, a lot of times that signifies exhaustion. And so when this thing chopped around and started to push up above that big volume price bar, I was looking for a bounce and that didn't happen. So this thing just came down and cut out right when it broke through the lows here. So manageable loss, but minus 304. That was kind of a late day one that, you know, like I said, I don't mind that I took it. It just didn't work. Netflix, Netflix, I had three different trades in. So let's take a look at Netflix. The first of which had this volume spike here. So when it bounced up to unchanged, I was looking for a continuation, some more downside, but that didn't happen. It kept pushing. And so when it broke above this bar, just cut out, kept my, kept my loss pretty manageable there. And then, and then when it pushed, I looked at that as a, what we call our volume reversal. So I got long here and cut this nice move up. In fact, got out of my last piece right here at this pivot, right at the top, which is a good exit because this thing then fell apart. And then once it fell apart, it was kind of chopping around here and the market was getting weak. So I got short right here, cut this nice move lower, got out of, uh, I think half the position. And then, um, I was looking, I was holding for a break of the lows just never happened. And when it bounced up, I cut out of my last piece. And anyway, net, net on those three, uh, minus two 70 in Netflix. Peton, big winner, over a thousand bucks in Peton. And look at this, the market just closed five minutes ago. Look at Peton. What a rally all the way from all the way back up to highs of the day closed up over six and a half percent. Crazy, crazy move in there. But, uh, what I did in Peton was on this big volume spike here, waited for this little pullback. This was my first trade of the day. And then this thing just ripped higher and just, uh, booked some nice profits, uh, had held on to one last contract, see if this thing was going to continue to rip, did pull back, held on, held on. I was waiting to see if it would get at least up to that pivot. Didn't quite get there. And so when it started to roll over and the market got started getting really weak, I just cut out of my last piece, but booked, uh, 1,025 on that one. And then Roblox, first trade in Roblox, uh, stayed super small in here, but had this, uh, had this big volume spike here. So on this bounce, got short, pushed lower. I got out of half of my position and then it bounced back up again, took a little heat and then rolled over. I think I got out of my last piece right down here. So I was super small in there. First time trading it. So just kind of watching it. I like the options in there. They're, they're liquid, got filled easy, and I like the way it moves. So we'll be trading more of that, but booked 183 on that one. Roku, Roku got me for a 920. So in Roku, uh, had this, it started to consolidate right out of the gate and then made this big push up. Uh, and then when, so when it pulled back to this level, I got long here, started to make a big bounce, got out of half my position for a nice profit. And then when it came back down, I added again, thought we still might have some upside, but then it just kind of continued to flush. So I ended up cutting out of it and just taking a loss. Um, and then, you know, the market was getting weak, so I didn't want to have any long zone at that point. So I ended up just taking a loss and moving on. Uh, so minus 920 on that one. And then Squarespace booked 505 on this one. And this was a, um, let's see what we do here. Oh yeah. So on this volume, was it this one? Oh no, is this one here? Yeah. This volume spike here, uh, is it broke through the highs of the day and then pulled back, got long right here and caught this little push up, uh, got out of almost all of it right here. And then it started to pull back. And so I cut out of the last piece and booked a nice 505 there. Tesla plus 1095 on Tesla, Tesla, me likey Tesla today. It was good, good, uh, good trade. So, uh, Tesla had some initial weakness bounced. Couldn't even get back up to the high of the day. So when it came down here and broke through the lows on this little bounce back up to untrained, uh, unchanged, it tried to go positive. I got short right there, looking for more downside and sure enough, this thing just flushed, got out of my last piece. I think it was right here. I was debating holding one, uh, for some further downside, but kind of talked myself out of it. Uh, cause it's, you know, I was up over a thousand bucks and, uh, I thought we might be getting a bounce here. We're getting some higher volume bars. And so I just cut out, obviously in hindsight, did have a little bit more downside. Uh, but still booked 1095 on that one. And then lastly on the NASDAQ, uh, this was a late day trade. And this is one, you know, this is where I say I kind of overstayed my welcome in the day trading today.  And so when this thing broke down, uh, and then started to bounce here, I was looking for some more downside, you know, to kind of have a late day flush. Uh, so when it bounced here, I got short and then added, uh, so I got short two here, got short another two here at the pivot, looking for a rollover and just never happened. This thing just ripped up. I guess the only thing I probably did not get on this one is I let this thing go a little bit too long.  But so I, I gave it a little bit back, but minus six 76 on that one. So still, still a nice green day plus 1076. I'll take it, especially on top of a big J big day yesterday. So we'll be streaming live tomorrow on Friday and then every day next week. So look forward to seeing everyone in the live stream room. See you then.


Well thank you. The first person I need to thank is my wife. That is entirely appropriate but it's not what you might think. As part of our preparation for this experience this week, she encouraged me to write an acceptance speech just in case. But I saw no need. I really felt it would be probably the most futile exercise since I drafted Tim Tebow in my first round of the Fantasy Football League. But I am very grateful right now that she insisted that I jot some thoughts down on paper because I am so stunned right now. I don't think without this paper that I would have a single sentence to utter. Education is a noble profession and I am proud to have dedicated my life to it.  It's a remarkable reward and now to receive this additional reward is almost more than I can comprehend. I am honored and I am grateful. I want to thank the selection committee for what must have been a very difficult decision. I'm especially humbled by this award because of my interaction over the last few days with the other amazing teachers and especially the other finalists. And it's been exhilarating. It's been a reaffirming experience for me to know that there are so many good teachers out there working hard every day to make a difference. I want to thank all of the sponsors of this event. I want to thank anyone who was associated with this elegant affair this evening. This has been outstanding. I want to thank Linda Howe who nominated me for this award. I want to thank the Pennsylvania State Education Association. They advocate every day for teachers and students and I appreciate that. I do want to thank my wife Joanne not just for this piece of paper but she unselfishly tolerates the many hours I spend correcting papers and preparing lessons. My son Jordan is with me. He is the joy of my life and I'm glad you're here tonight Jordan. My mother is also with me and while she is not a professional educator she was my greatest teacher. My thoughts also naturally turn to my late father who was the finest educator I ever knew. I want to give a special shout out to all my students both past and present. They are the reason I'm here tonight and while I am especially grateful to be recognized at an elegant affair like this I am absolutely confident that my colleagues throughout the country would be equally grateful for a simple gesture like a thank you. So I would encourage everyone to celebrate teachers everywhere. Again I am deeply honored and I shall work very hard to justify the faith that has been placed in me. I thank you with all my heart. Thank you very much.


Everybody forced here ripping origins football hobby. This is eight box number three Pick your team style got Aaron E with the Jags Alvin s Falcons and Dolphins Brett W. Eagles Felix with the Pats Florian Niners Gregory M with the Cardinals Saints and Jets James s Seahawks Jason M a Giants Jeff S Steelers Jesse a Panthers Jim G. Bills Joseph C Vikings Cash has the Packers Keith B Browns Kenny H Broncos Washington Mark L Colts Matthew G Titans Michael B Ravens Mo Emma chiefs Robert R Skip the name. I'm oh I'm chiefs Oren has the Texans Robert R Bears Robert and Raiders Shane R Cowboys Towel in has the Chargers Troy s Lions Tyler W. Bengals Tyler P Bucks and Victor F with the Rams. Good luck everybody Let's go ahead and get started here Teams on sale on the next one. Here we go Rookie Stetson Bennett for the Rams out of 40 or 149 we go Rams 2 1 25 is sake one for the Giants Kyler base and Rookie CJ Stroud for the Texans Chargers Rookie jumbo Jersey to 99 Quentin Johnston Ice card Chargers Colts Proving Ground Auto out of 99 Alec Pierce Colts Alec Pierce there and Chiefs on the Rookie patch auto to 99 Rishi rice with a sick patch here The Rookie Premier there Rishi rice RPA to 99 Good color match Why is this pack not worth it? There it goes Okay, Falcon's Desmond Ritter out of 99 There was a Bijon just based on car auto earlier Shane to 149 Mims rookie for the Broncos Kyler base followed by a rookie Duce Vaughn for the Cowboys Steelers Jack Hamm to 25 on the Hall of Fame patch You must be jacking there you go Steelers Lion Sam Laporta one of one receiving stars Rookie autograph That's the prize you write Troy s with the Lions. Oh Man Spencer and Jace are gonna pass out. Oh, yeah. Oh my god Detroit Detroit one of one rookie auto and there you go And there you go and a redemption for Michael Mayer Raiders Rookie Jumbo Jersey Auto Raiders Jumbo Jersey Auto Michael Mayer That's not gonna help with the delusion Yep, double zeros they sign the cards Like right after they get they get signed the cards before camp and everything before they even put on actual NFL uniforms and pads Rick it a 99 Ty J Spears for the Titans. It's like the autos. You'll see double zeros on Jerry Judith 75 for the Broncos Rookie Bryce Young for the Panthers And a Laporta rookie for the Lions and then a lot of the ones that are printed later that they don't need to get signed in person They'll have their numbers on Rams Jersey car Hall of Fame the 99 Isaac Bruce few scooters Colts 75 Anthony Richardson Oh Mark L with the Colts now big old colts Sometimes you got to wrangle them Colts in the 75 orange auto big congrats to the Colts Monster hit there and a Luke schoonmaker rookie Jersey auto for the Cowboys All right breaks hot breaks hot spots up in the next ones Teams on sale teams on sale. No commanders cards. Yeah, I mean sometimes with with low card counts. You might not see Certain teams to the last especially had to have less cards the product, but I mean it says it's seven cards a box So it's a 149 is Mike Williams for the Chargers Rookie to 125 Don Tavion Wicks Packers So they're running the commanders though for your G tuck got car base Mims rookie for the Broncos Lions rookie patch out of 199 Jameer Gibbs Nice card for the Lions Clayton tune rookie auto for the for the Cardinals All in card Ricky QB auto and a Titans rookie jumbo Jersey auto to 99 Ty J Spears Ty J Spears Titans a 90 on RPA to 25 is Michael Thomas Saints Rookie to 149 is Jonathan mingo Carolina got a fields base and a mingo base rookie for Carolina Will Anderson rookie patch card to 75 on the hometown roots For Hampton, Georgia, that's where Shane's from right Hampton, Georgia Go well Anderson Texans three color patch Sam Howell number to 25 on the proving ground auto for the commanders Second-year commanders auto here There's a Washington card and redemption is Michael Wilson Ricky jumbo Jersey auto for the Cardinals rookie out of 125 Josh Downs for the Colts So 149 Steelers George Pickens Yeah, then Hennan Hooker is gonna have value to for a while George Pickens rookie Sean Clifford for the Packers Jaden Reed rookie jumbo Jersey to 99 for the Packers Rock spring. There's a Rock Springs not far from here Shane Jaden Reed jumbo Jersey Packers rookie Broncos with an on-card rookie auto 49 of Marvin Mims great hit for them They were on sale man Some people hitting for low low price Broncos and a Will Anderson junior rookie Jersey auto for the Texans now Okay, Kyler out of 99 for the Cardinals get the start this week. He's back 75 Garrett Wilson for the Jets Sauce Bay CJ Shroud rookie for the Texans rookie patch guard Bryce young Panthers out of 199 this card Panthers Chargers Quentin Johnston number to 25 rookie on card auto on the receiving stars Very nice hit Chargers now and a Chase Brown rookie patch auto to 49 for the Bengals Well, now you know Shane now you can talk about it with with us Okay, so Next break number four Cardinals Falcons Broncos Patriots Saints and Steelers six to go All teams mark down Patriots gone number four things down to five now Yep, five to go. Yeah, there's there's a lot of adults But there's also like a lot of kids that play for like me for kids Kind of Adults have played now played it when they were kids I didn't start playing till recently because people here play it's 149. It's Mike Evans for the Bucks Ricky Rishi rice is 75 for the Chiefs Shane. You should use your daughter's account. We'll go play Tyree kill base schoonmaker rookie for the Cowboys Rookie patch card Titans Ty J Spears number to 75 three-color patch Victory Royale. She sounds better than me. Let me go Titans Dolphins Devon H and rookie on-card auto Great hit dolphins and a redemption on the back of the break is gonna be Jalen Hyatt Giants Ricky jumbo Jersey auto Hyatt for the Giants Give me a sec here guys. We'll keep going. We'll do a recap Steelers gone.  Maybe she can become pro If they're good at it if you're not good don't play all day, that's bad if you're not excelling then yeah Start a recap here with the relics got a Bryce young Jameer Gibbs Isaac Bruce Jaden Reed Quentin Johnston a Ty J Spears will Anderson jr. And jackham Autos now we had Giants Redemption Jalen Hyatt jumbo Jersey auto Cardinals Michael Wilson jumbo Jersey auto Raiders Jumbo Jersey auto Michael Mayer They're there teams It's teams of it's solo teams of two three or four and those other game modes with it's like 50 versus 50 All right autos now we get the Von A Chan cleaton tune Will Anderson jr. I think initially it was just solo at the very very very beginning and then it went to like Duos in play There's like quads and doze uh schoonmaker Rashi rice nice rpa ty J Spears as well Alec pierce auto to 99 mims on card to 49 RPA to 49 chase brown either quentin johnston to 25 and a howl to 25 second year nice howl Two awesome hits here orange to 75 rookie auto anthony richardson for the cults And a one-of-one sam loporta receiving stars auto for the detroit lions Two monsters right there. Thank you both Thanks everybody for joining congrats to the lions and the cult spots there and to the others that hit moving on now to number four


Give the people what they want. Give the people what they want. Give the people what they want. Your weekly movement news roundup. Today is the 19th of November. You're with us on Give the People What They Want, brought to you from People's Dispatch, your favorite movement-driven news site with Prashant and Zoe, and I'm Vijay from Globetrotter. My mother would like me to say that today is Guru Nanak's birthday, but, well, I suppose the best birthday present the great sage could have received Prashant was the withdrawal by the Indian government of Prime Minister Narendra Modi of the three farm bills, which had occasioned a massive protest running almost a year, incredible protest and a major victory for Indian farmers and farm workers. What's been happening over there, Prashant? Right, Vijay. I mean, it's difficult to really quantify a day like this because, you know, you spend, there are moments when these kinds of things happen. They don't happen very often because what we saw was, let's take a look at the balance of forces, so to speak. What we saw was on the one hand, you have a right-wing government and Narendra Modi, which has basically, you know, taken over almost all livers of the state. And it has control over not only state institutions but also civil society institutions, for instance, like the media. And they passed a series of laws last year, which would have seriously affected the livelihood of farmers. Now, it's important to note that these laws were passed without any consultation with the farmers. They were rushed through parliament. And at that point of time, a lot of people were very unhappy about it, but people were like, what can we do? But on the other hand, farmers' organizations had been preparing for this moment. They knew from many months before the law was passed, the law was passed in September last year. They knew that this was coming. And they had been mobilizing across the country. And it's important to note, across the country, number of small agitations, number of mass mobilizations taking place over the months. And in November, at the same time when an all India workers strike was taking place, the farmers mobilized around Delhi. Now, again, for those of us who watch the visuals, I would encourage people to go back to people's dispatch and watch some of those videos on news click as well. They started this procession on November 26. Farmers from around Delhi marching towards the capital. They were saying, we go into the capital. And the police directed barricades. They tried every single way to stop them. The farmers just kept marching on. And they kept marching on in tractors. And again, this is not just a small bunch of farmers. We're talking about tens of thousands of people, whole families, old people, young people, women, no distinctions, people coming prepared.  Because the farmers then even then knew that this was going to be a very long fight. And they're not allowed to enter the city of Delhi, the main city itself. And they say that we are going to sit down here and we are not going to go back until these laws are withdrawn. Now, even at that point of time, which is two or three days into the strike, there were many people who said that, okay, they've kind of camped around Delhi. How long are they going to sit? It's not possible to sit forever. So at some point, the government is not going to give in. At some point they're going to go back. But that is not what happened. What happened was that the farmers continued to camp around Delhi. There are farmers who stayed in those camps for six to eight months. These encampments became proper many villages almost. There were community kitchens, there were bathrooms, there was an entire social system built around these camps. And the farmers had just one very simple demand. Withdraw the laws, we'll talk. The government tried negotiations in which they played this whole hand of trying to negotiate but not willing to withdraw the laws. The Supreme Court tried to intervene. There were stock of expert committees. But the farmers remained laser focused. That's the most important thing. They expanded their agitations to neighboring states. There were these mass gatherings of tens of thousands of people or farmers in various states who expressed solidarity, some of whom came. In January, for instance, there was an attempt by the government to break down these camps. Again, at that point, there were mobilizations. And this basically after a point it started, the government started realizing that this was not a problem for them that was going to go away. And that I think is one of the most important lessons. The enduring significance of these kind of occupations. And I say occupation very knowingly because we've seen similar occupations in various parts of the world. Where popular movements take over spaces. And they say that we have these demands and we are not going to go back. And this has been probably one of the most longest and powerful occupations that we have probably seen in the world. So what we see is over close to a year passes, by the end of November, the farmers were planning yet another major upsurge in terms of agitations. And there were elections coming up in a couple of states as well. And the best guess is that the government realized that this could prove to be a costly affair for them. Because they had tried everything. They had tried negotiations on their own terms, of course. They had tried various pro-government media outlets had defamed the farmers called them terrorists, called them anti-nationals. They had said that this was just a mal-section of malcontents. They had said that these were old fashioned farmers who were not catching up with the new liberal reforms. They had said that these were anti-growth people. They had tried every trick in the book. And the farmers had stopped protesting. So what probably happened was that the government did a calculation and realized that it did not make sense anymore to continue with these laws. Of course, the government and its allies are now spinning. It is a gift to the farmers as a concession, as a noble gesture. What is important to note that during the course of these protests, close to 750 people have died. These are 750 families whose lives will never be the same again. And I think the farmers have very clearly made some responses. People are celebrating, of course, but the farmers have been cautious. They have said that, one, this is not the end, because we are not going to go back until this promise of withdrawing farm laws is actually put in writing. That is until parliament passes the necessary steps. Two, that the farm laws are not the only demand. The farmers for decades have been asking for proper pricing, which the government can ensure. About 400,000 farmers have apparently committed suicide in the last 20 years. So what about all these families as well? All of these questions that are very much in contention, in the debate right now. So a major victory, yes, but the farmer leaders and the farmers movements also very clear that they want justice. This is not a concession as far as they're concerned. This cannot be written off as just a total gimmick, because this is not just begging for attention or whatever. This is a very organized mass movement with very clear demands. And I think the fact that they have won this victory today is actually an inspiration to popular movements across the world, because this shows that even in the face of such a very powerful establishment, people in large numbers united by a cause, people in large numbers mobilizing in an organized fashion are able to achieve something. So I think very interesting, as they say, vibes from India to the rest of the world today. And while we can, people might debate, what is the long-term political impact? What is going to happen in the elections? All these are maybe questions. But I think the more, especially for our show and especially for, say, an organization like People's Dispatch, really the question is this today shows what movements are, what movements mean, and what movements can do. I think it's important what you say, Prashant, just to underline this, of course, the struggle of the Indian farmers is a frontline in a global battle against agriculturalists. The three farm laws put in pushed through parliament by Mr. Modi's government. They have a majority. They didn't take consultation. They pushed three farm laws into law, which are actually part of a global architecture of uberizing farmers. Getting farmers to basically operate like uber drivers, where they don't have the ability to control the agricultural way of life, and which is why farmers saw this as an existential struggle. This is a life or death matter. Indeed, as you said, 750 people gave their lives for this struggle, a life or death struggle, and it's not going to end. The question of minimum support prices on the table, the question of the actual withdrawal of the laws on the table, they don't want to believe Mr. Modi, because Mr. Modi, in the statement he made this morning, said that, well, we didn't consult the farmers, therefore we should go back for consultation. This is a year, a year, a full year, after the farmers have been asking for the withdrawal. So it's hard to actually take at face value what the government of India is saying, but an incredible story. It should be seen as a victory of agriculturalists around the world, not just the farmers in India. The farmers in India fighting in a way of proxy fight for farmers around the world saying, no, we don't want digital agriculture. We don't want big corporate agriculture. We want to control our farming and we want state support for it. I think that's the key issue. They understand agriculture cannot function without state support. And I think that's important. Well, that's one victory. On this show, Prashant, Zoe, we're not used to celebrating too many good things. We generally celebrate terrible things, coup d'etats. I didn't say celebrate, did I? I meant we generally report on terrible things, coup d'etats, murders and so on and so forth. Here's another good story. I was told, Zoe, that I was told by the United States government that on the 15th of November, the Cuban revolution was going to be ended. Today is the 19th of November. Four days later, what's happening in Cuba today? Well, the U.S. was slightly off in their projections. The Cuban revolution is very much still alive, slightly off. And as we've been reporting on people's dispatch on this show, essentially, you know, the U.S. has been in cahoots with their, you know, far-right opposition groups based in Miami with kind of some new players emerging within Cuba. We saw Junior Garcia, who's a playwright who had founded this organization Archipelago, and they had been, you know, building towards supposedly a huge mobilization, huge civil society protests on November 15th, the day that Cuba opens up to tourism. Also the day that Cuba was reopening schools now that they have vaccinated not only large, almost all of the adult population, but also many children.  Their permit for the protest was not approved by the Cuban government due to the obvious links of this organization with the U.S. State Department.  And in the end, we didn't see anyone on the streets really. There were really isolated protests in a couple cities with maybe a couple dozen people, but these were largely overshadowed by people supporting the revolution and mobilizing in support of the government and above all against, you know, attempts by the U.S. attempts by this, you know, mafia in Miami to subvert the Cuban revolution. And so, you know, for example, there was this really beautiful encampment of young people in Havana, the red scarves, and they were, you know, camped out for 48 hours, you know, reading poetry, creating music, sharing, debating, really engaging and discussing what it means to defend the Cuban revolution today. And they even invited Cuban president Miguel Diaz Canel to participate. He, you know, accepted, we don't, this isn't the kind of thing we see in a country like the United States or much less in India. The president go to this encampment of young people and engage with them, dialogue with them, discuss the pressing questions that Cuba is facing today. And so, and then, of course, across the world, people mobilize in support of Cuba. There were in some cities small, small groups of opposition supporters who tried to kind of make a scene at the embassies. But again, they were overshadowed by people by and large supporting the Cuban revolution and, you know, of course, rejecting U.S. imperialism. So this is another great victory. Of course, as in India, with these victories, we always, you know, it's not the end. Originally, they had scheduled the protests for November 20th and it, you know, they shifted the date earlier because of the reopening of the country. So will they try to mobilize again tomorrow? It's unclear, but of course, you know, it's been 60 years since the Cuban revolution and the U.S. has never been successful, but they've kept trying.  So, of course, this is not the end of U.S. attempts to de-sebelize Cuba, but I think it represents a really important victory, especially in this moment where Cuba is at a very challenging point where it's gone through the COVID-19 pandemic, where its economy has really been impacted by the lack of tourism to the island. So now it's kind of a turning point where now there will be, you know, more revenue coming into the island with opening to tourism, with the high level of vaccination. They really have turned a corner in terms of the COVID-19 pandemic. On November 12th, they registered zero deaths from COVID, which is a milestone, you know, many countries have been unable to reach. So I think it's a really exciting moment. We have to celebrate these victories because if we're only talking about, you know, the challenges and the challenges we're facing in this world, it can get a bit grim. And so I think we have to celebrate this farmer's victory. We have to celebrate once again another victory of the Cuban people and the people of the world against U.S. imperialism. Well, history goes in zig-zag. So as you say, you never know what's coming around the corner. I read a piece at People's Dispatch yesterday very excellent report. I was first struck by this business of end SARS. I thought end SARS. What does this have to do? Is this, you know, the thing that came before COVID, the SARS epidemic? Turns out nothing to do with that. Another victory this time in Nigeria. Prashant, what is this end SARS business and shouldn't people go to People's Dispatch every day and read the kind of reports that I read yesterday? Absolutely. The report by our colleague Pawan actually is a follow-up on a very important issue that kind of was rock Nigeria last year, which is again, drawing threads to some of the many issues we've been talking about. This was a protest. The SARS is a police unit, a paramilitary unit in Nigeria or WAS. We'll come to that later, of course. But which was very notorious for its brutal ways in terms of how it treated people, issues of rapes, beating, kidnapping, all these classic paramilitary tropes that we've talked about in Latin America, for instance, or for that matter in Africa. So last year saw a massive round of protests against this organization. And that is how end SARS came to me. The end SARS became this umbrella term for this WAS movement, which of course began with police brutality and then also went into many socio-economic questions that young Nigerians were talking about. And young Nigeria, of course, facing an issue of unemployment, a lot of youth aspirations not being fulfilled. In many ways, this mirrors the kind of protest that we've covered in so many other parts of the world. For instance, in many other countries in North Africa, in countries in Latin America, where one small incident, for instance, of police violence leads to a larger systemic uprising against the structure of society itself, against the structure of how the government is organized. So these were mass protests that were taking place throughout the country towards the end of last year. And one of the most tragic instances of the police response to these protests or the government response to these protests was what is called the Leaky Tolgate Massacre in Lago State. Now, when the massacre took place at that point, it was believed that somewhere around 10 to 12 people had been killed at that point in Japan. And that itself, there was a huge amount of condemnation. But a year later, a very important report has come out, which is commissioned to find out what actually happened. And the revelations are shocking because it says that close to over 100 people actually, close to 115 people were killed on that day in at Leaky Tolgate. And what happened was a massive cover-up operation after that. Now, this report, the news of this report itself is very interesting because there were very strong suspicions that this report would be buried because the governor of the province who actually appointed another committee to look into this report, so to speak. And it's good that the contents of this report came out because it actually reveals the extent of state involvement in this issue because on the one hand, you had the army and the police who fired on these protesters. But equally shocking was the fact that civic agencies in Lago State, for instance, the organizations involved in clearing the streets, the organization involved in managing the Tolgate, all of them had a role to play as well. For instance, the fact that the day after the massacre, three large cleaners or trucks with cleaning brushes were brought in to remove all traces of the bleeding that had taken place. After about 100 people were reportedly killed, the army and the police basically took a lot of these corpses and dumped them in various parts of Lagos. And to this day, it's not clear about the identity of some of these victims, it's not clear. It is just believed that they are victims of the Ansar's protest. And the panel makes a very important point. Incidentally, for instance, it says that several unidentified bodies were removed by the Lago State Environmental Health Monitoring Unit and they were deposited at various mortuaries in the state. So look at the name of the organization, it's the Environmental Health Monitoring Unit and this is what it was doing last year. So the panel also makes a very important point that the protesters at the Leiki Tolgate were not any kind of threat to organized order. They were actually sitting peacefully. What we talked about about the farmers also, they had their own system for sanitation, they were extremely well organized. Many of them were sitting waving the flag and chanting the national anthem. And these were the people who were fired upon justice in a different way the Indian farmers were called as terrorists and anti-nationals and all that. These were the people considered as threats by the Nigerian state authorities. So there's been a series of recommendations about disciplinary action against military authorities, disciplinary action against police authorities. But the key question is that, you know, whether this report itself will prove vital in revitalizing the end SARS movement because protesters are pointing out, for instance, that SARS has been dissolved, but there's something called SWAT just taking its place, which is again a classic technique, give the, you know, rename the organization, call it something else and it keeps doing the same thing. And SWAT units continue to do pretty much the same thing that the SARS was doing as well. So this is actually in some ways a very powerful vindication of those protesters. The panel has even, you know, called for compensation to be paid. It is called for the toll gate to be renamed as the end SARS toll gate in honor of those victims. But the real key question is that, does this bring a lot more long term justice? Will it give a shot at the arm for the protests and revitalize them further? So definitely something to watch out for. It's an important story. I mean, we have to look at it as a escalating, you know, set of victories of the people because they've revealed, you know, the moment the government changes the name of a unit like that, it's a victory. Even though, well, you can say it's the same, but no, it's a victory. Shifting focus, we're going now deep into the question of trade. Complicated story to share with you friends. U.S. Trade Representative Catherine Tai was in Tokyo. She met the new Japanese foreign minister, Mr. Yoshimasa Hayashi. They met. They had apparently very cordial meeting. They signed or re-upped the U.S.-Japan partnership on trade, a very important initiative. But on the table, Mr. Hayashi apparently put to Catherine Tai is the question of U.S. engagement in the trade deals taking place around East Asia in particular. Mr. Hayashi apparently brought back to the conversation the comprehensive and progressive agreement for trans-Pacific partnership. Remember the TPP, this has a new name now, comprehensive and progressive agreement. Many countries signed up. The question of course is that China, Taiwan and the United Kingdom have applied to join. This has created some political problems. Australia says we're not going to let the Chinese into this new TPP. Japanese want the United States to return to it. They would like to see China blocked also. But friends, let's pause for a minute. The Chinese are already in a major trade deal with these countries. It's called RCEP, Regional Comprehensive Economic Partnership. RCEP has 30% of the world's population inside it. 30% of global GDP. Australia is part of it. It's a very complicated situation here. What the Japanese are asking of the United States is to so-called, quote unquote, re-engage in trade policy. But it's contradictory. The United States says we will re-engage, but China has to be cut out. How can China be cut out? China is the principal trading partner of almost every country in East Asia and of Australia. Impossible to cut China out. These countries are actually dancing on the head of a pin, trying to keep both feet on the top of that pin. Not easy to do, I can tell you. Not that I've ever tried to dance on the top of a pin, but doesn't look like it's easy. And that's essentially what Mr. Hayashi talked with Catherine Tai about just a few hours later. He called Wang Yi of China and they had a telephone conversation where Mr. Wang Yi explained to Mr. Hayashi that the world is a complicated place, friend. And Japan better figure out what it's going to do. The world is a complicated place. Let's go on the other side of Eurasia. The Russians, Nord Stream, the big Russian gas company, Gasprom, had built a pipeline with a bunch of European partners called Nord Stream 2, which was going to to some extent bypass Eastern Europe. Also put it in harm's way from NATO ships to circle those waters north of Germany and Poland. Nord Stream 2 has run into some legal problems, it seems, in Germany. So that's halted. Meanwhile, gas prices or energy prices in Europe have shot up by 15%. What have the Russians done? Again, interesting. The Russians had previously been selling Chinese markets natural gas. Remember, the Chinese made a pledge to get off coal. The way to get off coal they thought was we'll buy Russian natural gas. Well, in Eastern Siberia, the Chinese have been procuring Russian natural gas. Now the Russians are saying, wait a minute, we're going to build another pipeline. It's called Power of Siberia 2. And we're going to bring you natural gas from the Yamal fields. These are the same fields tapped by Nord Stream 2. What the Russians are saying to the Europeans is, if you're going to make economic problems for us, we're selling our gas to the Chinese. Now, what has this story been? What have they taken you through a journey through? It's a story about the fact that this confrontationalist policy, which has infected trade relations, has actually put many of the US partners in a very complicated, difficult situation. Europe is going to have energy shortages for the remainder of the winter because they can't settle the question of Nord Stream 2. This is not a light matter. Many people will get their electricity cut off during this winter. There will be rolling blackouts in Europe. On the other side, in Asia, countries cannot afford not to be trading with China. Got to understand trade policy for this complexity. It's being hampered by politics. Where are the free traders when you want their voices to be heard? Silent, it seems. Meanwhile, of course, in Libya, we hear there's going to be an election. My old friend Saif al-Islam Gaddafi once wanted to make weight on the Mediterranean through his hat in the ring. So did Khalifa Haftar, the man that I'm happy to say I revealed as a CIA agent in 2011, also in the ring, and three or four other people. We're going to come back to the Libyan election because it's a mess, and we need to spend some time talking about it. But Zoe, I know there's a lot of elections again in South America, and you can't do a show without talking about a South American election. What's happening over there? Just not every November, December, and February, March, April. But this Sunday, elections in Chile and Venezuela. The Venezuelan elections are the super elections, the local level, sub-national elections, but hundreds of positions that are going to be voted on. Council members, governors, mayors. Really crucial elections, even, I mean, we might say, okay, but these are local elections. What does this matter? These are actually quite important, quite historic because the far opposition will be participating for the first time since 2015 when they essentially broke with constitutional order. Also, observation delegations with the European Union first time in 15 years. This is also historic, so these are very big advances that what they might lean to is that after these elections there is a possibility that bodies such as the European Union might loosen their sanctions that they have in Venezuela. A lot of the justification for why they impose these sanctions is that Venezuela is undemocratic, that they do not have free and fair elections. Now they're on the ground. They'll see for themselves. They'll be able to see how Venezuela has one of the most... They have the most advanced actually electronic voting system on the continent, and so that will be kind of revealed in those elections and over in Chile, a bit to the south. It is the first round of the presidential elections. This is going to be a close race. It's really a cast who is the far-right candidate who is trailing behind. I think he was in third place just a month ago and a half ago. He's kind of pulled right up and he's right neck and neck with Gabriel Boric. And it's most likely it will go to a second round, which will be held in December. It's likely to be between these two front runners. But it is... I mean, as you've written, Vijay, you were in Chile. This is a very important moment for Chile and democracy. It is a moment where Chile is going to re-rate its constitution that was written during, of course, the dictatorship of Pinochet. It is these elections and this whole process of renewing the democracy in Chile that has been kind of cemented by neoliberalism, by denying the rights to people, exclusion of important communities. As we mentioned a couple of weeks ago, of course, there's also the current militarization of the Mapuche territory. So it's a very important moment in Chile. And between a far-right candidate and a left candidate, it's really... The future could go either way. And of course, as we've seen in neighboring countries in Latin America, of course, once the left candidate wins presidency, it doesn't mean necessarily that they're assured an easy passage. But it really could mark a shift in the waters of Chile, which has really been a stronghold of conservative neoliberal policy in the region. It has always been an ally of regional efforts against Venezuela, against Bolivia. Of course, they denied that Bolivia's access to the sea. So these are really important to watch out for. And in Venezuela, we'll see how the U.S. responds if they're going to say that, yes, this was a free-fair election or continue their campaign of siege and embargo on Venezuela. Where else are you going to watch a show, at least in the English language, that covers stories that we have covered in half an hour? You've been listening to give the people what they want. Your favorite weekly news roundup brought to you by People's Dispatch, one of the best places for regular news coverage. Prashant and Zoe, co-editors of People's Dispatch. Go and bookmark the page. Go and read Pawan Kulkarni's article. Read something, for God's sake. I'm from Klockrota. You don't know what that is, but go and find out. Thanks a lot. See you next week. See you.