WEBVTT 1 00:00:00.530 --> 00:00:04.760 Courant Events Right: But I'm gonna have the last talk of the afternoon. It's also the last talk of this 2 00:00:04.900 --> 00:00:11.620 Courant Events Right: Three-day workshop. It's going to be, Rick Kedian from Yale, who's going to speak on independent covers. 3 00:00:12.310 --> 00:00:13.350 Courant Events Right: Thanks, Greg. 4 00:00:13.640 --> 00:00:20.109 Courant Events Right: I didn't get a mic. Oh, sorry. Oh, you need a mic? I don't know about that. Is this one working? Yeah. 5 00:00:20.210 --> 00:00:20.900 Courant Events Right: Yep. 6 00:00:22.210 --> 00:00:23.390 Courant Events Right: It's turned on. 7 00:00:24.910 --> 00:00:27.130 Courant Events Right: Is it turned on? Yeah. This value. 8 00:00:28.220 --> 00:00:39.240 Courant Events Right: Oh yeah, okay, thanks for the organizers. I didn't intend to give a talk here, but I guess there was a cancellation, and I got shoved into position, so this is the first time I give a talk on this, and it was… 9 00:00:39.780 --> 00:00:49.490 Courant Events Right: somewhat short notice preparation, is it not… is it not on? It is on. 10 00:00:49.680 --> 00:00:50.830 Courant Events Right: Drop another button. 11 00:00:51.010 --> 00:00:53.049 Courant Events Right: Maybe the other side. 12 00:00:55.640 --> 00:01:09.729 Courant Events Right: How about that? Is that… is that better? You have to press a button. It's on! It's on, apparently. I'll try to talk louder. Is that better? Okay, 13 00:01:10.150 --> 00:01:10.960 Courant Events Right: Yep. 14 00:01:11.110 --> 00:01:16.819 Courant Events Right: Thank you. Anyway, this is a joint work with, Catherine Wolfram, who's at Yale and, Zurich. 15 00:01:17.570 --> 00:01:35.570 Courant Events Right: this year. And, you know, just to start off, you know, pretty much everything I learned in math, I learned as an undergrad. So, you know, when I was a grad student at Princeton, they just didn't have grad classes. They had to… at that time, so, you know. 16 00:01:36.030 --> 00:01:55.669 Courant Events Right: All my talk is going to be linear algebra and a little bit of Fourier analysis, and if there's… if I'm using some… something you don't know, please raise your hand. It should be accessible to everyone, all right? I'm… I'm not going to talk to any experts, because there are no experts on this subject, it's a new subject. Well, okay, maybe there's one or two experts. 17 00:01:56.140 --> 00:02:01.270 Courant Events Right: Okay, here, well, what's the… what's the goal, right? There are… there are certain… 18 00:02:01.460 --> 00:02:15.290 Courant Events Right: stat-meg models, right? Dimer models, fantasy model, Ising model, which we know, and the reason we know them is because they're somehow free fermionic. You can use some linear algebra, determinants, faffians, and so on to study them. 19 00:02:15.400 --> 00:02:28.159 Courant Events Right: And there's an interesting history of those… of understanding their scaling limits, what happens when the lattice basically goes to zero, and the, you know, many of us in the room know that, and 20 00:02:29.700 --> 00:02:45.039 Courant Events Right: There are some other models for which we know less about what's still some non-trivial things about, like percolation and the fibertex model and so on, but the first three are free fermionic in the sense that they 21 00:02:45.140 --> 00:02:51.619 Courant Events Right: you can study them with linear algebra. And… and so I want to define, the new 22 00:02:52.110 --> 00:02:57.629 Courant Events Right: Fairly rich family of other measures that make models. 23 00:02:57.750 --> 00:03:12.639 Courant Events Right: which are determinantal, and they're going to be in ZD in general, although most of the pictures you're going to see are two dimensions, and we're just getting started on three dimensions, so I won't go too far into that. And the tools I'm going to use are simply, you know, very simple tools, linear algebra and 24 00:03:12.640 --> 00:03:22.820 Courant Events Right: some Fourier analysis. But, you'll see that the Fourier analysis questions we ask are kind of sophisticated, so if anybody is an expert on that, I would be happy to talk. 25 00:03:23.420 --> 00:03:24.460 Courant Events Right: Later. 26 00:03:27.690 --> 00:03:30.680 Courant Events Right: Okay, so let me start with the definition. 27 00:03:33.320 --> 00:03:39.280 Courant Events Right: Every talk should start with a definition. 28 00:03:39.590 --> 00:03:56.619 Courant Events Right: I've got a graph. It's a set… you can think of it as a binet set, or if you like, a graph, and a collection of subsets, which I'm going to call tiles, although they're going to be overlapping things, so maybe it's better to call them, like, shingles or something else. 29 00:03:56.620 --> 00:04:02.810 Courant Events Right: But, let me just call them tiles, because that's what we have been calling them. And associated to this. 30 00:04:03.160 --> 00:04:04.240 Courant Events Right: his aid. 31 00:04:04.390 --> 00:04:18.170 Courant Events Right: Matrix, a rectangular matrix, where the rows are indexing the vertices or the elements of the set, and the columns are indexing the tiles, and it's just an incidence matrix, so 1 or 0, depending on whether the vertex is in the tile or not. 32 00:04:19.440 --> 00:04:32.700 Courant Events Right: Okay, so if you like, you can just start with a rectangular matrix, forget that stuff. And what's an independent covering? It's a covering of the graph, or the set, with some subset of tiles. 33 00:04:33.100 --> 00:04:36.360 Courant Events Right: And, if the graph has 34 00:04:36.860 --> 00:04:49.040 Courant Events Right: you know, n vertices, it's a covering of size N, so the one tile per vertex, but the… the union of the satisfies two properties. Union of the tiles 35 00:04:49.720 --> 00:04:52.130 Courant Events Right: Here's the whole graph, so you have to cover 36 00:04:52.350 --> 00:04:56.939 Courant Events Right: And it has to be independent. What does independent mean? It means that the corresponding 37 00:04:58.050 --> 00:05:03.899 Courant Events Right: submatrix of the incidence matrix is a full rank. 38 00:05:04.190 --> 00:05:14.210 Courant Events Right: So the… basically what I'm doing is I'm… I have this matrix, I'm taking a collection of columns of the matrix, which spans the corresponding 39 00:05:14.400 --> 00:05:16.070 Courant Events Right: real space, space. 40 00:05:18.460 --> 00:05:19.529 Courant Events Right: Does that make sense? 41 00:05:21.030 --> 00:05:23.289 Courant Events Right: Let's just do an example. 42 00:05:25.180 --> 00:05:31.419 Courant Events Right: This is just the submatrix of D whose columns correspond to the tiles in F. 43 00:05:32.670 --> 00:05:45.300 Courant Events Right: So D, the columns of the matrix are indexed by the tiles, the rows are indexing by the vertices, and I want to pick a collection of tiles such that the corresponding square submatrix has non-zero determinants. 44 00:05:45.750 --> 00:05:50.910 Courant Events Right: Let's see, let's see an example. They play no role at the moment. 45 00:05:51.620 --> 00:06:04.939 Courant Events Right: edges of the graph play no role, but that adds some geometry, and eventually we'll do some Fourier analysis where the geometry plays a role. Okay, here's a very simple example. My graph is just the interval from 1 to n. 46 00:06:05.840 --> 00:06:09.960 Courant Events Right: Okay, 1 through 10, and my tiles are just gonna be… 47 00:06:10.010 --> 00:06:25.769 Courant Events Right: collections of three consecutive integers, so… so, which can slide… which can be slipped anywhere along there, and… and I'm going to allow the tiles to sort of go past the endpoint, so the… the first tile just consists of a single vertex, the second tile is the next 48 00:06:25.770 --> 00:06:33.700 Courant Events Right: first two vertices, and the third tiles, the first three vertices, and then it slides along. So tiles are all subsets of three consecutive vertices. 49 00:06:33.910 --> 00:06:37.370 Courant Events Right: Including the tiles which sort of overlap the boundary. 50 00:06:37.990 --> 00:06:46.070 Courant Events Right: Okay, so there's a… I can just index the tiles by the numbers 0, 3, and plus 1 if I think about the location of the center of the tile. 51 00:06:46.910 --> 00:06:47.660 Courant Events Right: Right? 52 00:06:48.120 --> 00:06:57.919 Courant Events Right: Everybody with me? So that gives… my matrix now is a… in this case, there's 7 vertices, I think, and there's 9 possible tiles. So as a sound, my 9 matrix. 53 00:06:58.250 --> 00:07:09.079 Courant Events Right: And the, the, right? So, for example, the first pile only contains the first vertex, and that's with the matrix, and that's the incidence matrix, and now you can ask. 54 00:07:10.490 --> 00:07:14.620 Courant Events Right: What's an independent cover? It's a collection of 7 tiles. 55 00:07:15.960 --> 00:07:25.590 Courant Events Right: In other words, well, there's 7 vertices, so I need to pick 7 tiles, and then I want the corresponding submatrix, 7 by 7 submatrix, to have non-zero determinants. 56 00:07:25.780 --> 00:07:34.230 Courant Events Right: And you can ask, that's a fun question to work out, if you're bored, which, maximal miners, which 7x7 minors in this matrix have… have full rank. 57 00:07:36.720 --> 00:07:39.660 Courant Events Right: Anybody want to guess, or know the answer? 58 00:07:40.960 --> 00:07:42.150 Courant Events Right: I'll tell you later. 59 00:07:45.650 --> 00:07:56.019 Courant Events Right: Okay, here's another example. This is kind of the even more motivational example, which is called the Uniform Spanning Tree Model. G is a… now a bipartite graph. 60 00:07:57.220 --> 00:08:11.159 Courant Events Right: I pick up one vertex, these are all called the root vertex, and the set of tiles now are just the edges of the graph, and then I'm going to add one more tile, which is just a single root vertex by itself. 61 00:08:12.210 --> 00:08:15.350 Courant Events Right: Then, the… 62 00:08:16.880 --> 00:08:31.400 Courant Events Right: Well, here's the… here's an example for the… when the graph is a square. You get this 5. In this case, there's, 5 possible tiles. There's 4 edges plus a root vertex, so there's 5 columns, and there's 4 vertices. That's the matrix, and the… the, 63 00:08:31.580 --> 00:08:38.610 Courant Events Right: theorem, which is… well, it's in graph there, it's called the Pointer Ray Lemma, which says that your 64 00:08:38.909 --> 00:08:44.159 Courant Events Right: Minor that is, the, the, the… 65 00:08:44.710 --> 00:08:57.020 Courant Events Right: Miner has full rank, if and only if it's a spanning tree. It's an independent cover is the same thing as a set of V minus 1 edges and V0 itself, which make a spanning tree. 66 00:08:58.340 --> 00:09:12.309 Courant Events Right: And, you know, it's not so hard to prove this lemma, you can prove it on your own. Conversely, any spanning tree is an independent cover, so the set of independent covers is exactly the set of spanning trees. As you can see in this example, there's four possible spanning trees. 67 00:09:12.310 --> 00:09:19.070 Courant Events Right: Of this graph, and each of them has 4… 4 parts, 3 edges and 1 root vertex. 68 00:09:19.850 --> 00:09:20.810 Courant Events Right: Make sense? 69 00:09:23.110 --> 00:09:25.109 Courant Events Right: Who knows the point-grade 11? 70 00:09:26.380 --> 00:09:34.929 Courant Events Right: Now I do. Now you do, okay. I don't know why, I don't know, maybe it really was point Craig, but that's what my graph theory book told me this was called. 71 00:09:36.100 --> 00:09:41.780 Courant Events Right: Is there any connection to the usual pancrella, which is that the closed form is exactly… 72 00:09:42.150 --> 00:09:43.810 Courant Events Right: The way you got in? 73 00:09:44.960 --> 00:09:48.550 Courant Events Right: Oh, I don't know. I don't know. Maybe it is, yeah, must be. 74 00:09:49.010 --> 00:09:54.320 Courant Events Right: Anyway, I… As I was making the slides, I was watching this movie, Dune, with my daughter. 75 00:10:03.260 --> 00:10:09.590 Courant Events Right: Okay. Just in case you're… you were asleep. Okay, 76 00:10:11.160 --> 00:10:17.649 Courant Events Right: There's a natural… Sorry? I had to give this talk in Hollywood. That's right. 77 00:10:18.150 --> 00:10:29.640 Courant Events Right: There's a natural probability measure on independent coverage. Now we understand what an independent cover is. Sorry, I'm kind of looking over here, but you guys are also in, you know… 78 00:10:30.390 --> 00:10:36.099 Courant Events Right: Okay, well, some of them aren't pulled back up. There's a natural probability measure. 79 00:10:36.530 --> 00:10:43.410 Courant Events Right: Where the probability of an independent cover is proportional to that determinant 80 00:10:43.540 --> 00:10:54.049 Courant Events Right: Right, the determinant of the corresponding minor, I called it the upper end before, that's a mistake. Well, okay, anyway, typo. You just take the determinant squared. 81 00:10:54.780 --> 00:11:11.659 Courant Events Right: And then you divide by a normalization constant. The partition function is the determinant of DD star. That's, well, there's a nice linear algebra result, which says that for a rectangular matrix, the determinant of dtt star is the sum of the 82 00:11:12.980 --> 00:11:14.930 Courant Events Right: Maximal miners, squared. 83 00:11:15.790 --> 00:11:17.179 Courant Events Right: Coach keeping in theorem 84 00:11:17.580 --> 00:11:25.209 Courant Events Right: So this, this is the probability, natural probability measure to put on a, on a, independent cover. 85 00:11:25.670 --> 00:11:32.610 Courant Events Right: And the theorem is that this measure is a determinantal measure, and it has a kernel, well, this complicated 86 00:11:33.470 --> 00:11:40.249 Courant Events Right: combination of D, so D star… D star is just the transpose of D, DD star inverse, times D. 87 00:11:40.640 --> 00:11:51.259 Courant Events Right: And if you know about spanning trees, this is a generalization of a theorem of Burton, which was originally to Burton and Pimentel from the 90s, maybe 93, for the spanning tree case. 88 00:11:51.260 --> 00:12:01.989 Courant Events Right: And, you know, this theorem is essentially due to either of them, or… or some… some combination of them, and more generalized by, in this paper. 89 00:12:02.140 --> 00:12:06.160 Courant Events Right: Benjamin E Lyons, Paris and Schramm, 2001. That's where I learned it. 90 00:12:08.560 --> 00:12:12.560 Courant Events Right: Let me explain what that means. What's a determinantal measure? 91 00:12:12.740 --> 00:12:23.690 Courant Events Right: Maybe we, for physicists, we should call this a pre-fermionic measure, but in math, it's a determinantal probability measure. What is it? It's just a measure on binary sequences of length n. 92 00:12:24.380 --> 00:12:37.090 Courant Events Right: probability measure. It's determinant, so if you can find a certain matrix, n by n matrix K, which is called the kernel, such that, has this property, that for any subset of indices. 93 00:12:37.450 --> 00:12:40.210 Courant Events Right: the probability that… 94 00:12:40.630 --> 00:12:51.580 Courant Events Right: for every index in the set S, the corresponding variable is 1, is given by the determinant of the S by S principal minor of that matrix K. 95 00:12:52.130 --> 00:12:53.719 Courant Events Right: So, in some sense, all the… 96 00:12:53.830 --> 00:12:59.069 Courant Events Right: Probabilities that you want to compute for this measure turn out to be determinants of matrices. 97 00:12:59.920 --> 00:13:02.489 Courant Events Right: Constructed from the… from the kernel K. 98 00:13:02.730 --> 00:13:15.350 Courant Events Right: So that's where these measures are sort of defined by linear algebra. And the example, of course, the standard example is the spanning tree measure. Edges in a uniform spanning tree form a determinantal measure. 99 00:13:16.840 --> 00:13:27.559 Courant Events Right: Right? If you… if you take a graph and you ask, is this edge in and this… is that edge in? This is a 2x2 determinant of the… of the kernel that you associate to the… that graph. 100 00:13:29.300 --> 00:13:36.100 Courant Events Right: Another example is the edges in the dimer model for a planar bipartite graph. This is a theorem of myself. 101 00:13:36.260 --> 00:13:48.420 Courant Events Right: Maybe it was known earlier. And there are some continuous examples of determinantal measures as well, where instead of probabilities, you get probability densities as determinants. And one of the 102 00:13:48.710 --> 00:13:54.840 Courant Events Right: ones we'll use in this talk is this, example of the Gaussian analytic function. 103 00:13:54.960 --> 00:14:01.590 Courant Events Right: This is due to Yuval Perez and Valen Birag in 2005. What we do is we take a… 104 00:14:02.320 --> 00:14:06.100 Courant Events Right: Power series, just a one variable power series, complex 105 00:14:06.360 --> 00:14:16.650 Courant Events Right: function, complex analytic function, where the coefficients are independent, IID, standard, complex normals, 106 00:14:17.010 --> 00:14:20.979 Courant Events Right: And this is a power series which converges inside the unit disk. 107 00:14:21.010 --> 00:14:37.940 Courant Events Right: And you can ask, where are the zeros of this function? They're random zeros, random locations of zeros, and the… that set of zeros, the locations of those zeros, is a point process which is determinantal, and the density is given by, 108 00:14:39.950 --> 00:14:42.519 Courant Events Right: The determinants of some… of some… 109 00:14:43.370 --> 00:14:47.149 Courant Events Right: Well, currently some, yeah, operator in this case. 110 00:14:48.660 --> 00:14:56.339 Courant Events Right: And there are lots of other examples of determinantal processes and random matrices, like GUE, CUE, Ginibra Ensemble, you've probably heard of those. 111 00:14:56.640 --> 00:14:58.199 Courant Events Right: Are they frickle, German? 112 00:14:59.880 --> 00:15:06.310 Courant Events Right: No, well, nevermind. 113 00:15:07.390 --> 00:15:24.999 Courant Events Right: Not in this talk, for sure. Okay, all my determinants are going to be defined, here. So here's a nice linear algebra fact, right? I said we're going to use some linear algebra. This is maybe something you didn't learn in your linear algebra class, at least. I didn't learn it in mine, but I learned it recently that, if you have a 114 00:15:25.000 --> 00:15:33.999 Courant Events Right: K-dimensional subspace of Rn. W is a K-dimensional subspace of Rn, and suppose they have a basis for W, W1 through WK, 115 00:15:34.560 --> 00:15:48.859 Courant Events Right: Then, you make this rectangular matrix, K by N matrix, whose rows are the vectors W1 through WK. Then this expression here is the matrix of the orthogonal projection to W. 116 00:15:50.210 --> 00:15:58.300 Courant Events Right: Okay? Is it… who knows this result? Who learned this in linear algebra? All the Europeans. 117 00:15:59.170 --> 00:16:10.819 Courant Events Right: I… I don't think so. I double-checked, but I could be wrong. Okay. The rows, the rows of the vector… You're correct, you're correct, right. No problem. 118 00:16:12.000 --> 00:16:31.400 Courant Events Right: Okay, so this is the thing which comes out in the kernel for my independent covers, and so the corollary is that the kernel of the independent cover determinant will measure, right, that matrix which defines the probability measure is, in fact, the matrix of orthogonal projection to some subspace. 119 00:16:31.510 --> 00:16:36.990 Courant Events Right: Right? Remember, our matrix is indexed by the set of all tiles. 120 00:16:37.170 --> 00:16:47.020 Courant Events Right: And the subspace is the… I mean, the corresponding substrate is for… the matrix is the matrix of projection onto the kernel of D, 121 00:16:47.280 --> 00:16:50.029 Courant Events Right: Right, D is the… D is the… 122 00:16:50.200 --> 00:16:53.899 Courant Events Right: Remember, the mapping from the space of tiles to the space of vertices. 123 00:16:57.450 --> 00:17:01.200 Courant Events Right: Oh, okay, so, let's see. 124 00:17:02.130 --> 00:17:05.450 Courant Events Right: Yeah, let… okay, so… 125 00:17:05.790 --> 00:17:12.279 Courant Events Right: What's a comic? A comic is the complement of an ick, an independent cover, right? 126 00:17:13.030 --> 00:17:14.960 Courant Events Right: So, it's this… 127 00:17:15.720 --> 00:17:27.909 Courant Events Right: It's the set of tiles which you don't use in your independent cover, right? If you have an independent cover, you have a certain set of tiles, you take all the tiles which you don't use, that's… that's the complement, that's to call the comic, and… 128 00:17:27.970 --> 00:17:39.019 Courant Events Right: That's a property of determinantal measures, is if you exchange the zeros and the ones, then the new measure is also determinantal, but the kernel just becomes the identity minus the original kernel. 129 00:17:39.930 --> 00:17:43.609 Courant Events Right: So the natural determinantal measure on the 130 00:17:43.980 --> 00:17:49.219 Courant Events Right: for comics, as the kernel i minus K, where K is this previous kernel. 131 00:17:49.540 --> 00:17:55.099 Courant Events Right: And so it represents orthogonal projection to the perpendicular 132 00:17:55.660 --> 00:17:59.090 Courant Events Right: subspace to kernel D. Kernel D perk. 133 00:17:59.720 --> 00:18:13.989 Courant Events Right: And the nice thing about a comic is that in many cases, it's smaller than the original, independent cover. Since the size of the comic is the complement, that's the number of tiles minus the number of vertices of the grid. 134 00:18:16.140 --> 00:18:20.600 Courant Events Right: Okay, so then, if you only care about the comic, then you can… 135 00:18:22.080 --> 00:18:38.290 Courant Events Right: rather than use the matrix D, you can use the matrix which forms the basis for the kernel of D, and then the probabilities all work out the same. The probability for a particular comic S, maybe I should have called it something besides S, but it doesn't matter, the… is this same kind of expression. 136 00:18:38.560 --> 00:18:42.230 Courant Events Right: Okay, so now we can go back to our original example, just to… 137 00:18:43.420 --> 00:18:49.429 Courant Events Right: summarize what I just said, this was our original example, this was our matrix D, 138 00:18:49.630 --> 00:18:55.519 Courant Events Right: This is a 7x9 matrix of full length, so its kernel is only two-dimensional. 139 00:18:56.290 --> 00:19:13.980 Courant Events Right: And the basis for the kernel is just given by the… the… I put them in rows here rather than columns. It's just given by the… these functions such that the sum of any three consecutive entries is zero, and I could choose my basis. Well, there's several ways to make a basis, but just using some complex numbers. 140 00:19:13.980 --> 00:19:19.910 Courant Events Right: cube roots of 1, that's a basis. The 2 by n plus 1 matrix. 141 00:19:20.530 --> 00:19:21.980 Courant Events Right: And, 142 00:19:22.630 --> 00:19:30.560 Courant Events Right: Therefore, you know, my independent cover is going to use 7 tiles, but the complement of that is only going to use 2 tiles. 143 00:19:30.860 --> 00:19:33.730 Courant Events Right: It's got the two, right, so two locations. 144 00:19:35.030 --> 00:19:42.640 Courant Events Right: And, the… you can ask, what's the probability that those two locations are at two points, X1 and X2? It's just that… 145 00:19:42.880 --> 00:19:51.170 Courant Events Right: is just the corresponding minor squared of that matrix B that I wrote on the previous slide, which is just the… 146 00:19:51.440 --> 00:19:54.749 Courant Events Right: You know… What do you call it, Matrix? 147 00:19:56.180 --> 00:20:08.510 Courant Events Right: It's just a 2x2 matrix. You take the… you take the determinant, it just comes out to be 0 if X1 is congruent to X2 mod 3, and it's some constant otherwise. 148 00:20:08.900 --> 00:20:16.080 Courant Events Right: some constant, like, Yeah, anyway, whatever the constant is. So, if you, if… 149 00:20:16.520 --> 00:20:27.609 Courant Events Right: If you're interested in which maximal minors of this matrix have full rank, it's exactly those. Well, you have to remove two columns, but the two columns you remove cannot be a multiple of three apart. 150 00:20:29.110 --> 00:20:37.099 Courant Events Right: So if you're… if the two columns you remove are a multiple of three apart, then the minor is 0. Otherwise, it'll be… it'll be some constant, plus or minus some constant. 151 00:20:40.300 --> 00:20:52.729 Courant Events Right: Okay, no big deal. That was just kind of a baby example. We want to do this in higher dimension, but let me tell you, just to pique your interest, let's try a slightly slight variant. It's still in one dimension. 152 00:20:53.620 --> 00:20:55.590 Courant Events Right: Alright, so this is a… 153 00:20:55.770 --> 00:21:08.770 Courant Events Right: I'm going to take the same graph consisting of endpoints in a row, and now my tile is going to be… I'm going to put some weights in my tile, some integer weights, so that translates of the tile 14641, so it's going to be 5, 154 00:21:08.770 --> 00:21:20.339 Courant Events Right: vertices, consecutive vertices, but they'll have weights 4, 6, 4, 1, and those go into the matrix here. And so my D matrix looks like this, looks very similar to the previous one. 155 00:21:20.340 --> 00:21:28.120 Courant Events Right: just all translates of the 1464.1. And… but now the… the kernel is more interesting, 156 00:21:28.190 --> 00:21:36.000 Courant Events Right: The kernel of this matrix consists of vectors, which are 157 00:21:36.990 --> 00:21:41.000 Courant Events Right: Now, f of x, where X is a polynomial degree at most 3. 158 00:21:41.190 --> 00:21:52.259 Courant Events Right: Except for this sign. Like, for example, if I take, you know, 1 minus 2, 3 minus 4, 5, and so on, that's in the… that's in the kernel of this, and so is the same. Each… each row of this matrix 159 00:21:52.260 --> 00:22:01.449 Courant Events Right: That is in the kernel, and therefore, the kernel, because this is of all polynomials of degree at most 3, except for this, minus… except for the alternating, oscillating 160 00:22:02.090 --> 00:22:06.010 Courant Events Right: index minus… minus 1 to the i. 161 00:22:07.730 --> 00:22:15.550 Courant Events Right: So, the probability that… that… so now my… in this case, my… my comic has 4 tiles. 162 00:22:16.110 --> 00:22:32.400 Courant Events Right: I mean, my independent set is, has size, 5, and the total matrix has size 9, 9 rows, so my independent set has 4, tiles, and you can imagine this being very long, it's the… 163 00:22:33.030 --> 00:22:43.020 Courant Events Right: You know, I can make N here as large as I want, and the comic will still have only 4 points in it. That's the advantage of doing the comic rather than the independent cover. 164 00:22:43.850 --> 00:22:49.489 Courant Events Right: And the probability that I get, that X1, X2, X3, X4 points are… 165 00:22:50.070 --> 00:22:53.000 Courant Events Right: My comment consists of exactly those four points. 166 00:22:53.140 --> 00:23:04.970 Courant Events Right: is proportional to the… so what I do is I take the V matrix, and I look at the column, the X1 column, the X2 column, the X3 column, the X4 column, and I take the determinant, but that's exactly a Vandermont. 167 00:23:05.380 --> 00:23:16.569 Courant Events Right: matrix whose… whose determinant is this, you know, Vandermont squared. I mean, when you square it, the determinant is the Vandermont squared, so the probability proportional to the difference of the 168 00:23:16.770 --> 00:23:20.520 Courant Events Right: Xi's squared. Product over all I's and d's. 169 00:23:21.320 --> 00:23:24.599 Courant Events Right: So, it's not a coincidence. 170 00:23:24.900 --> 00:23:44.020 Courant Events Right: No, that's not a coincidence. What's… what's going on is that I, you know, I took a… I took a tile here whose, right, whose coefficients were binomial, so it's… it's, you know, Fourier transform has multiple roots on the… on the unit circle, and that's what caused this interesting behavior here in the… so… so… 171 00:23:45.240 --> 00:23:57.619 Courant Events Right: So it's an interesting determinantal process on four-point subsets of a… of a big interval with… with, but the density is a very familiar, you know, beta, beta equals 2 density. 172 00:23:57.750 --> 00:23:59.589 Courant Events Right: Does it count? 173 00:24:00.010 --> 00:24:02.220 Courant Events Right: Bigger numbers to find me? 174 00:24:03.480 --> 00:24:04.650 Courant Events Right: I get more points. 175 00:24:04.770 --> 00:24:11.119 Courant Events Right: If I… if I took a, you know, 1, 5, 10, 10, 5, 1, I'd just get 5 points with the same… with the same kind of expression. 176 00:24:17.800 --> 00:24:19.249 Courant Events Right: Any questions so far? 177 00:24:23.190 --> 00:24:26.000 Courant Events Right: Because I'm going to go to a higher dimension now. Yeah. 178 00:24:32.300 --> 00:24:33.100 Courant Events Right: Yep. 179 00:24:34.180 --> 00:24:35.969 Courant Events Right: Dude, just pick up, I can't really hear it. 180 00:24:41.890 --> 00:24:47.080 Courant Events Right: Right, the entire was 5 consecutive versees, 114641. 181 00:24:47.340 --> 00:24:49.370 Courant Events Right: And the… the… 182 00:24:54.410 --> 00:24:55.190 Courant Events Right: Oops. 183 00:24:55.420 --> 00:24:56.700 Courant Events Right: I went back too far. 184 00:24:58.860 --> 00:25:02.670 Courant Events Right: Yeah, so this dot represents the center of one of my tiles. 185 00:25:03.480 --> 00:25:12.159 Courant Events Right: Right? So this dot really represents a tile sitting there in… inside the interval from, you know, well, minus 1 to F plus 2, or something like that. 186 00:25:13.510 --> 00:25:14.810 Courant Events Right: Was that your question? 187 00:25:18.120 --> 00:25:21.130 Courant Events Right: Nothing is quite early. 188 00:25:24.040 --> 00:25:28.569 Courant Events Right: I really can't hear you because… What's the role of the weights? What's the role of the weights? 189 00:25:30.440 --> 00:25:34.609 Courant Events Right: Well, they changed the kernel out of matrix, and they changed everything, right? 190 00:25:35.480 --> 00:25:40.590 Courant Events Right: Right, I guess in principle, you know, you could use, other non-integer weights, but, 191 00:25:41.040 --> 00:25:44.950 Courant Events Right: This problem originally came from a different problem where we needed positive integer both. 192 00:25:45.690 --> 00:25:50.140 Courant Events Right: So… Basically. 193 00:25:51.420 --> 00:25:59.150 Courant Events Right: If you want, you can ignore this example and then ask for all the weights to be 1 or 0. It could just be the incidence matrix. 194 00:25:59.260 --> 00:26:10.259 Courant Events Right: Okay, let's go to two dimensions now, and we see some more phenomena that we didn't see before. So here, I just took the… the… G is now a subset of Z2, 195 00:26:10.490 --> 00:26:30.180 Courant Events Right: And my tile is just going to be the L tile. All translates of the L tile, you know, just translates, I'm not allowed to rotate, so the L tile is just the… the polyomino, or the subset 001001 in Z2. And so you can see a kind of a picture over here. What this is, is I took G to be a big square. 196 00:26:31.050 --> 00:26:33.690 Courant Events Right: And it took a random coming. 197 00:26:33.880 --> 00:26:40.299 Courant Events Right: So that I'm not… I'm not drawing the independent cover, because the independent cover has… has N squared… almost n squared tiles. 198 00:26:40.640 --> 00:26:47.599 Courant Events Right: In fact, it will have exactly N squared times, but the… but the comic, the complement of it only has, like, two n times. 199 00:26:49.220 --> 00:27:05.779 Courant Events Right: And what you see is that the tiles mostly, you know, near the boundary, this is a red… this is a perfect sample. The tiles are mostly near the boundary, there's a few in the middle. What did I do for this picture? I just, you know, I just tilted that picture with a linear map. First of all, I took a triangle. 200 00:27:05.940 --> 00:27:23.520 Courant Events Right: Rather than taking the square, I took a triangle and then tilted it so that you can see a little bit of extra symmetry here. If you tilt the… if you tilt the L title, then it looks like a little equilateral triangle. But then I took a comic for that one. So this one has some nice three-fold symmetry, which is not evident in that previous picture. 201 00:27:24.570 --> 00:27:29.290 Courant Events Right: And so what's… what's happening now, in the scaling limit? 202 00:27:32.010 --> 00:27:48.910 Courant Events Right: to this process. It's a determinantal process, and the scaling limit is that it's looking like the zeros of a Gaussian analytic function. Here's a picture of the actual Gaussian analytic function, right? As I defined before, you take a power series with 203 00:27:48.990 --> 00:27:55.720 Courant Events Right: independent Gaussian entries, you look at the zeros, they look kind of like that. This is an actual simulation of the zeros. 204 00:27:56.000 --> 00:27:59.890 Courant Events Right: And, Yeah, so the… 205 00:28:00.070 --> 00:28:08.250 Courant Events Right: theorem of Paris and Bragg should show that the zeros are determinantal process on the unit disk with a certain kernel, which is called the Bergman kernel. 206 00:28:08.720 --> 00:28:20.559 Courant Events Right: It's a… it's a function of Z and W, which is on the… on the unit disk, it's given by this, but this Bergman kernel exists on any domain in the complex plane. It's the projection kernel from 207 00:28:20.680 --> 00:28:23.180 Courant Events Right: L2 of the domain to the 208 00:28:23.330 --> 00:28:42.439 Courant Events Right: L2 holomorphic functions on the domain. So you take an L2 function on the domain, and there's a… there's a projection operator to… to the… you find the closest analytic function on the domain, which is an L2, and that's the… this is the kernel of that projection operator, and that's the theorem. The theorem is that, 209 00:28:42.540 --> 00:28:44.239 Courant Events Right: You take the L-tyle. 210 00:28:44.540 --> 00:29:01.409 Courant Events Right: Here, on a domain in U, you take a lattice approximation to it in epon-C2, I changed coordinates a little bit. In the limit, in the scaling limit, in the limit as the scaling scale goes to zero, the point process for the comic 211 00:29:01.830 --> 00:29:10.290 Courant Events Right: for this L tile, converges after, after that linear math that I discussed to the zeros of the Gaussian analytic function on the domain U. 212 00:29:12.660 --> 00:29:20.549 Courant Events Right: So, you know, if I… if I… as the scale… as the mesh size goes to zero, the scaling limit of this tonic process 213 00:29:21.160 --> 00:29:31.759 Courant Events Right: will still be on the triangle, but if I apply the Riemann map to the circle, I will get the same process in law as the zeros, the Gaussian LA function. 214 00:29:32.820 --> 00:29:43.200 Courant Events Right: This, this, the nice thing about this Gaussian analytic function process is that it's conformally invariant. If you apply a conformal change of domain, it goes to the Gaussian analytic function on that domain. 215 00:29:43.940 --> 00:29:47.309 Courant Events Right: I spent with Jaime, how many, 216 00:29:48.500 --> 00:29:51.430 Courant Events Right: How many points are there in that triangle? 217 00:29:51.540 --> 00:29:58.529 Courant Events Right: Well, it's linear… the number of points is linear in the size of the triangle. So it's like 2… twice the side length. 218 00:29:59.000 --> 00:30:16.359 Courant Events Right: Roughly. I've forgotten exactly the amount, but it's a linear number outside. Most of them go to the boundary, but in the scaling limit, you know, they'll sort of accumulate on the boundary, but there'll be still some inside, and the density of them inside is given by this… this, the same as the Gaussian analyzed function. 219 00:30:18.110 --> 00:30:26.820 Courant Events Right: So it's kind of a nice, reasonably simple… yeah. What do you mean by up to, you know? Well, the linear map which takes the square grid to the… 220 00:30:26.970 --> 00:30:36.060 Courant Events Right: See, if I did it in the square grid, it wouldn't converse to the Gaussian… the conform… the correct conformal structure is obtained by… by this… by this 221 00:30:36.700 --> 00:30:43.399 Courant Events Right: coordinate rather than the square root coordinate. So I have to take the L tile and then tilt it by the… whatever that linear map is. 222 00:30:49.510 --> 00:30:51.090 Courant Events Right: Any more questions about that? 223 00:30:51.810 --> 00:30:55.560 Courant Events Right: Blow up the triangle to a circle, will they go. 224 00:30:56.600 --> 00:31:02.829 Courant Events Right: Yeah, that's right. In order to identify this process with this process, I have to use a conformal map from the triangle to the disk. 225 00:31:03.720 --> 00:31:04.560 Courant Events Right: Yep. 226 00:31:11.680 --> 00:31:21.160 Courant Events Right: Okay, so, well, corollary, the comics for the L-tyle are conformally invariant, because we know… we know what they're… we… well, we've identified their limits. 227 00:31:23.040 --> 00:31:26.630 Courant Events Right: There's no choice of victims were identified as a piece of land. 228 00:31:27.560 --> 00:31:34.330 Courant Events Right: Yeah, well, see the next slide. He asked, what happens if I change my tile from an L to some other tile? 229 00:31:34.550 --> 00:31:45.379 Courant Events Right: I mean, that's… that's pretty much the next obvious question. Sorry, that… but, okay, sorry, let me tell you about how the proof goes, and then I'll get to your question, Martin, right? 230 00:31:45.600 --> 00:31:48.950 Courant Events Right: This… this proof is kind of is not… not… 231 00:31:49.500 --> 00:32:02.119 Courant Events Right: so easy, right? So, you know, just because I fit it on one slide doesn't mean the whole proof is there. This is the… this is a very rough idea of the proof. well… 232 00:32:02.790 --> 00:32:06.169 Courant Events Right: there's two things you need to do, right? The… the… 233 00:32:07.310 --> 00:32:15.160 Courant Events Right: Remember that the kernel of the project… of the kernel of the terminal process is projection on this particular subspace of functions. 234 00:32:15.350 --> 00:32:18.489 Courant Events Right: And that subspace is, 235 00:32:18.810 --> 00:32:29.509 Courant Events Right: Well, kernel of D. What does that mean? It means that your function on Z2 is in the kernel of D if, when you sum the value at a… 236 00:32:31.780 --> 00:32:37.480 Courant Events Right: I didn't make a… I didn't make a slide, but I have a function on… Right? 237 00:32:37.850 --> 00:32:54.790 Courant Events Right: There's the lattice… there's two lattices. There's a set of files, which is also a Z2 lattice, and there's a set of vertices, which is Z2 lattice. And if I have a function on Z2, it's in the kernel if, when I sum it at these three locations, I get zero, because 238 00:32:54.830 --> 00:33:03.710 Courant Events Right: for each vertex, V, I want this tile, the value of this tile, plus the value of that tile, plus the value of that tile, to add up to 0. 239 00:33:04.000 --> 00:33:10.530 Courant Events Right: So I'm looking for functions on G2 which have this property that this value plus this value plus this value equals 0. 240 00:33:10.930 --> 00:33:12.230 Courant Events Right: That's the kernel. 241 00:33:13.180 --> 00:33:19.309 Courant Events Right: least, at least, you know, and, and then there's the boundary conditions, which, I'm just taking, sort of. 242 00:33:19.780 --> 00:33:32.780 Courant Events Right: three boundary conditions, I'm allowing the tiles to overlap the boundaries, but let's not worry too much about those. So, one example of such a function looks like this, right? If I just take omega to the X minus y. 243 00:33:32.880 --> 00:33:37.370 Courant Events Right: Where omega is this, cube root of… cube root of… Give root of 1. 244 00:33:37.480 --> 00:33:44.230 Courant Events Right: Right, so omega plus omega… well, 1 plus omega squared plus omega. That's one function. And the… and the… 245 00:33:44.360 --> 00:33:52.899 Courant Events Right: Lemma is that any function in the current… any function which satisfies that on a large domain, will… will… 246 00:33:53.310 --> 00:34:01.770 Courant Events Right: look like this function here times a smooth function, unless it, it… any function which stays bound, which, you know, is bounded in L2. 247 00:34:02.580 --> 00:34:06.100 Courant Events Right: Right? If, if, if I, if I… 248 00:34:07.140 --> 00:34:24.170 Courant Events Right: If I… if I… I can define any function in the kernel by just defining the values on a vertical strip and a horizontal script, and then… and then the value here will be the sum of the previous two, and this value here's the sum of the previous two. But all those functions will sort of blow up exponentially fast, unless… unless they are of this type. 249 00:34:24.170 --> 00:34:36.129 Courant Events Right: This particular periodic function tied to smooth function. So that's something you have to prove. And in addition, once you have this form, and you apply the D operator, you see that F has to be actually an analytic function 250 00:34:36.130 --> 00:34:49.679 Courant Events Right: of the… just by Taylor expanding the D of F, you see that F has to be an analytic function of this… the tilted coordinate, X minus omega squared y, the coordinate when you get tilted like that. 251 00:34:49.949 --> 00:34:58.640 Courant Events Right: So that's a little… that requires some… some, 40… a little bit of 40 analysis and the uncertainty principle, to prove that step. 252 00:34:59.190 --> 00:35:13.120 Courant Events Right: But then you see that, after, you know, after, you know, conjugating your function by this oscillating prefactor, this prefactor, then it looks just like a smooth function, and then the projection from this 253 00:35:14.310 --> 00:35:21.099 Courant Events Right: On this smooth function, you know, is, it, it tells you that the, 254 00:35:21.100 --> 00:35:36.960 Courant Events Right: when you predict on this lattice, the projection is a discrete version of the Bergman kernel, and then you can kind of wave your hands a little bit, at least that's what I'm doing now, and show that it converges to the… in the limit, the scaling limit, to the actual 255 00:35:37.260 --> 00:35:41.410 Courant Events Right: Bergman kernel, the projection from L2 to the analytic functions. 256 00:35:42.110 --> 00:35:52.210 Courant Events Right: Okay, that's all I want to say about the proof, but there's some, there's a lot of details there. 257 00:35:52.330 --> 00:35:53.719 Courant Events Right: How do I get what? 258 00:35:54.030 --> 00:35:58.580 Courant Events Right: I have the impression that we have a U for a bow, or going to disabled. 259 00:35:59.340 --> 00:36:03.129 Courant Events Right: Yes. Yes, from your county. 260 00:36:03.670 --> 00:36:06.010 Courant Events Right: Yeah, well, we have a… we have a… 261 00:36:06.340 --> 00:36:12.950 Courant Events Right: Let's talk afterward. That's… it's somewhat technical, and and I want to get to the fun stuff. 262 00:36:14.330 --> 00:36:24.520 Courant Events Right: Okay, so now… now to, Martin's question. What… what happens if… what property of the L tile did we use, and can we use that property for other tiles? 263 00:36:24.630 --> 00:36:26.780 Courant Events Right: Right? Well, 264 00:36:27.280 --> 00:36:47.089 Courant Events Right: If we think about the Fourier transform of the L tile, the L tile has this… has these three vertices, 0, 0, 1, 0, and 0, 1, and if we… if we call, you know, Z the Fourier transform of translation to the right, and W the Fourier transform of translation left, then the Fourier transform of L tile is just 1 plus Z plus W, just multiplication by 1 plus Z plus W. 265 00:36:47.680 --> 00:36:48.950 Courant Events Right: And this… 266 00:36:49.180 --> 00:36:56.370 Courant Events Right: polynomial of two variables, 1 plus E plus W, has… has the property that it only has simple zeros on the unit torus. 267 00:36:56.600 --> 00:36:58.370 Courant Events Right: You know, if I… if I… 268 00:37:01.660 --> 00:37:11.040 Courant Events Right: Well, certainly that's true. I mean, if you look for the zeros of this thing on the Uniforce, they're just omega, omega squared, and omega squared omega, and that's the property that 269 00:37:11.600 --> 00:37:14.629 Courant Events Right: leads to this conformal invariance, leads to this Bergman kernel. 270 00:37:15.010 --> 00:37:17.730 Courant Events Right: And, 271 00:37:18.260 --> 00:37:29.320 Courant Events Right: Basically, because of the same story on the previous slide, if you just have simple zeros, then you can multiply by this oscillating factor and find a smooth function. 272 00:37:30.380 --> 00:37:32.470 Courant Events Right: If you just have a pair of simple zeros. 273 00:37:34.940 --> 00:37:53.189 Courant Events Right: So, similar results are going to hold for any comic, as long as you're… when you have only one tile, and that tile has the property that it's Fourier transform, P, has only a single pair of complex conjugate simple zeros on the unitores. For example, and here's some examples. 274 00:37:54.610 --> 00:38:05.330 Courant Events Right: So, some tiles are nice. They lead to conformally invariant processes. In fact, they always lead to linear scales of this Bergman kernel process. Gaussian analytic function. 275 00:38:06.640 --> 00:38:16.649 Courant Events Right: However, there are plenty of other tiles which don't have that property. For example, this one, this tile has, multiple simple root pairs. 276 00:38:17.850 --> 00:38:24.889 Courant Events Right: And so what we get is actually a linear superposition of these Bergman kernels, and it's not conformally invariant. 277 00:38:25.400 --> 00:38:34.549 Courant Events Right: Or rather, there's two conformally invariant things, but each one is shifted and, by a different linear factor. 278 00:38:37.190 --> 00:38:46.010 Courant Events Right: the sort of a superposition of two different shifted and oscillated Bergman kernels. Sorry, that's… I should have made a picture here, but I didn't. 279 00:38:46.140 --> 00:38:48.199 Courant Events Right: But then, 280 00:38:48.420 --> 00:39:05.569 Courant Events Right: Some tiles, like the square, that was Form's idea, intersect the torus in more than just a tiny set of points, and those have weird behavior, which we don't really know how to classify. So, for example, this square tile, the Fourier transform is 1 plus Z times 1 plus W, 281 00:39:05.920 --> 00:39:11.160 Courant Events Right: It intersects the unitaurus in the union of two curves, Z equals minus 1 and W equals minus 1. 282 00:39:12.280 --> 00:39:31.940 Courant Events Right: here's a random sample of that. It's still a determinantal process, it's pretty easy to get your computer to spit out a random sample for the triangle, but the sort of covariance function between these tiles is kind of complicated. If I… what I did was I asked, you know, what's the… 283 00:39:32.040 --> 00:39:45.919 Courant Events Right: covariance between a tile here and a tile at a different location in the triangle, and you can see that the covariance function has the sort of delta functions along the horizontal and vertical, and then it's got this analytic part, and then it's sort of only piecewise analytic. In fact, the 284 00:39:46.140 --> 00:39:47.230 Courant Events Right: the… 285 00:39:47.350 --> 00:39:56.169 Courant Events Right: The square over here is independent of the square over here. The locations… the density of squares here and over here are independent of each other. 286 00:39:57.190 --> 00:40:03.800 Courant Events Right: for mysterious reasons. But, I mean, we can prove it, but it has something to do with the fact that these… these curves are… 287 00:40:04.280 --> 00:40:06.020 Courant Events Right: This polynomial factors. 288 00:40:07.860 --> 00:40:17.140 Courant Events Right: So we get kind of weirdish behavior in this setting, Here's another example with… 289 00:40:17.270 --> 00:40:26.640 Courant Events Right: also weird behavior of a completely different type. Our tile, we just tweaked the tile just a little bit by adding some weight tubes. This is the new polynomial. 290 00:40:27.260 --> 00:40:33.199 Courant Events Right: Now, the zero cent on the torus, is also not a set of points, it's a curve. 291 00:40:33.760 --> 00:40:35.189 Courant Events Right: But a different kind of curve. 292 00:40:35.840 --> 00:40:54.840 Courant Events Right: And, you know, this… here, this… this… this sample was on a… on a… on a disc again, but… but, you know, apparently the disk's not so important as it's on a disk. The… the… the interesting thing is that the covariance, functions for this, for tiles here, has this kind of, 293 00:40:55.010 --> 00:41:00.930 Courant Events Right: Chronicle behavior, kind of like a… interesting… 294 00:41:01.850 --> 00:41:14.779 Courant Events Right: Minkowski or Werenci in geometry. If you have a tile here and a tile out here, they're sort of exponentially uncorrelated, but tiles which are costly related, have some polynomial correlation. 295 00:41:16.620 --> 00:41:21.910 Courant Events Right: Kind of weird, I don't understand, but this is what the… what you get from the Fourier analysis of this tile. 296 00:41:22.440 --> 00:41:29.030 Courant Events Right: Again, I'm just doing linear algebra, right? It's just, you know, to get this covariance function, I just have to invert the certain matrix. 297 00:41:29.410 --> 00:41:36.850 Courant Events Right: Which is based on the, you know, where the… basically finding the Fourier coefficients of 1 over P. 298 00:41:41.550 --> 00:41:59.730 Courant Events Right: Here's… here's… here's another polynomial, the plus polynomial. It's also got a curve, but it's a closed curve in this case, and I… there, the covariance function is… is pretty weird, sort of oscillates in every location, but it's definitely got some long… long-range structure, which you don't really understand. 299 00:41:59.770 --> 00:42:04.160 Courant Events Right: None of these are, sort of, conformally invariant. 300 00:42:04.290 --> 00:42:06.509 Courant Events Right: But they… but we do, sort of. 301 00:42:06.830 --> 00:42:09.639 Courant Events Right: Can… we can compute their scaling limit. 302 00:42:10.370 --> 00:42:12.469 Courant Events Right: A scaling limit of the kernel. 303 00:42:13.200 --> 00:42:18.299 Courant Events Right: Well… At least that we know how to do some Fourier integrals. 304 00:42:23.710 --> 00:42:31.609 Courant Events Right: Yeah, but what happens in 3D? Oh, I had some… I had a picture for 3D. Well, let me… let me read the slide first, and then I'll show you a picture. 305 00:42:31.950 --> 00:42:44.199 Courant Events Right: If you have a… if you have one tile in three dimensions, typically it's not going to intersect the torus in a finite set of points. It'll… it's, it's a… it'll intersect the torus in a one-dimensional 306 00:42:44.490 --> 00:42:46.719 Courant Events Right: You know, some variety. 307 00:42:46.860 --> 00:42:55.970 Courant Events Right: And, and… that's the case where the Fourier analysis is complicated. We don't have any sort of general 308 00:42:56.350 --> 00:43:09.300 Courant Events Right: description of the… because it depends on sort of a case-by-case basis, what that curve is. I don't have a good, I'm a, you know, baby when it comes to Fourier analysis. I don't understand how to do these integrals yet. 309 00:43:10.040 --> 00:43:13.210 Courant Events Right: Still thinking about it, working on it, or looking for an expert. 310 00:43:15.440 --> 00:43:35.349 Courant Events Right: When… on the other hand, if there's two tile types, then typically you get isolated points just by dimension counting. For example, if I take 1 plus Z plus W in the XY… one tile in the XY plane, and one sort of vertical dimer in the… in the Z plane, then the… then you get a funny set of points. I think this… there's only two points here, U equals 311 00:43:35.630 --> 00:43:40.080 Courant Events Right: minus 1 and Z and W are the Q roots of 1, then you get some nice 312 00:43:40.770 --> 00:43:44.710 Courant Events Right: 3-dimensional point process, which has this 313 00:43:45.070 --> 00:43:54.050 Courant Events Right: Well, I can't say conformal invariance, but I can at least say it's sort of stretched rotational invariance. Once you apply the appropriate linear map, you get a rotational invariant process. 314 00:43:55.880 --> 00:44:02.019 Courant Events Right: And again, going back to what I said before, if… if you take your… 315 00:44:02.980 --> 00:44:06.259 Courant Events Right: graph to be Z cubed, and 316 00:44:06.390 --> 00:44:09.699 Courant Events Right: all… you take all the edges of Z cubed plus one extra 317 00:44:10.000 --> 00:44:21.509 Courant Events Right: a large region in Z cubed and one extra vertex, but along with the edges, then you get… do get the 3D spanning tree as the independent set, independent cover model, and 318 00:44:21.840 --> 00:44:24.410 Courant Events Right: Well, we know to what extent we… 319 00:44:24.670 --> 00:44:34.290 Courant Events Right: That's a sort of big open question. There are questions about the spanning tree for which we know the answer, and there has some rotationally invariance of the 320 00:44:34.680 --> 00:44:43.830 Courant Events Right: edge covariance functions, but the… the other questions, like, you know, what is the… what are the branches of the spanning tree look like? Those are still open questions. 321 00:44:44.430 --> 00:44:57.850 Courant Events Right: So just because we can prove that certain things are conformally invariant, or know the scaling members of certain observables, there are other… there can be other observables for which we don't get information, directly from this, procedure. 322 00:44:58.890 --> 00:45:00.959 Courant Events Right: Let me see if I can, oh. 323 00:45:06.070 --> 00:45:10.950 Courant Events Right: Let me… let me pull up the other, just to show you that the one… oh, where did it go? 324 00:45:11.130 --> 00:45:12.010 Courant Events Right: Here it is. 325 00:45:20.770 --> 00:45:24.770 Courant Events Right: I made a picture of a… Don't see that? 326 00:45:25.770 --> 00:45:36.929 Courant Events Right: This is a 3D point process, and you know, in 3D, you can't really see a whole lot, but this is a 3D process, point process on a cube, which comes from the tile, which is the three-dimensional version of the L tile. 327 00:45:38.620 --> 00:45:51.569 Courant Events Right: And, you know, I don't know what you can really tell from this example, but there it is. It's not so hard to make these simulations. These are all exact samples, because there is an exact sampling algorithm for determinantal processes. 328 00:45:53.620 --> 00:45:59.309 Courant Events Right: Any questions? Sorry, what? What's the analog of the barbacco? 329 00:45:59.640 --> 00:46:01.609 Courant Events Right: What's the analog of the Bergman kernel? 330 00:46:03.220 --> 00:46:08.160 Courant Events Right: Is there? Is there? No, this is not the… 331 00:46:09.050 --> 00:46:23.390 Courant Events Right: This is… there's… yeah. I don't know… I don't know what to… what to say about this. I mean, I can write down a formula, but there's no conformal invariance, conformal invariant object in three dimensions. There is no Bergman kernel in three dimensions, so… 332 00:46:23.580 --> 00:46:27.909 Courant Events Right: Well, you could study some truths of the rock equation. 333 00:46:28.030 --> 00:46:32.100 Courant Events Right: the analog of polymorphic function is a solution to the rock equation, basically. 334 00:46:32.280 --> 00:46:42.659 Courant Events Right: L2 speeders and projectile to another ball, projectile to the… Yeah, yeah. Whether that arises in such a model is not clear to me, but 335 00:46:42.870 --> 00:46:49.450 Courant Events Right: Right, so the question is, what's… what's the analog of the Bergman kernel in 3D? 336 00:46:49.900 --> 00:46:56.920 Courant Events Right: Is there some direct… direct operator which, I don't know. Good question. 337 00:46:58.470 --> 00:47:01.519 Courant Events Right: And that… thank you very much, that's all I had to say. 338 00:47:09.180 --> 00:47:10.860 Courant Events Right: Are there more questions? 339 00:47:12.640 --> 00:47:21.610 Courant Events Right: Yeah, Mark. So, like, most of them do then just fail to basin. You still have all the… 340 00:47:21.810 --> 00:47:35.199 Courant Events Right: This one is kind of a non-isotropic, model. Definitely, definitely the points are accumulating on three of the faces, but not the other. I mean, it's some sort of a… 341 00:47:35.710 --> 00:47:47.189 Courant Events Right: You still have the same model of points on top of that? Yeah, yeah, I think that the density of points inside is some continuous function of the distance. 342 00:47:49.370 --> 00:47:58.869 Courant Events Right: Yeah. Does it make sense to introduce some parameters in your model, like maybe weights, then study how they… Oh, for sure, yeah. How they measure depends on the weights. 343 00:47:58.870 --> 00:48:11.770 Courant Events Right: Yeah, that's right. I… for this talk, I just chose the constant weights, but it's kind of like the spanning tree. You can add conductances to the spanning tree, and then you get slightly different behavior, but the global behavior is roughly the same. That's what I think happens in these models. 344 00:48:11.810 --> 00:48:26.129 Courant Events Right: Basically, you know, the underlying Laplacian, which is DD star, you can add some conductances in there and put parameters. Is that what you're asking? Well, there is an interesting connection in the space of parameters. 345 00:48:26.380 --> 00:48:29.150 Courant Events Right: Which is like a generalization that you can actually be shown 346 00:48:30.370 --> 00:48:32.849 Courant Events Right: engine is written in terms of nose operators, and… 347 00:48:34.270 --> 00:48:37.440 Courant Events Right: One of the examples of that would be the… 348 00:48:37.600 --> 00:48:41.270 Courant Events Right: aviation construction for the… for instant dots, for emergencies. 349 00:48:41.660 --> 00:48:42.490 Courant Events Right: Sure. 350 00:48:42.830 --> 00:48:51.970 Courant Events Right: Okay, maybe we can talk. If you have some good ideas, this is great. It's a new… it's a new subject for us, too, so if you have good ideas, please bring them to me, and we talk afterwards? 351 00:48:53.610 --> 00:48:57.719 Courant Events Right: Can you come tomorrow? Yes, tomorrow, okay. 352 00:48:58.350 --> 00:48:59.670 Courant Events Right: Any other questions? 353 00:49:02.030 --> 00:49:03.699 Courant Events Right: Oh, there's one that's back here. 354 00:49:10.670 --> 00:49:13.679 Courant Events Right: Can you speak out loud? Wilder! 355 00:49:17.150 --> 00:49:17.980 Courant Events Right: Often. 356 00:49:27.660 --> 00:49:30.569 Courant Events Right: Can you repeat the question? Yeah, he's asking, 357 00:49:31.160 --> 00:49:37.979 Courant Events Right: Is there a… can we prove that there's a scaling limit in 3D, even if we can't describe it? 358 00:49:42.780 --> 00:49:58.279 Courant Events Right: Yes. Well, no, okay, I'm 90% sure that I can, but, you know, until things are written down, which they are not at the moment, I'm not gonna, you know, be definitive about that. So, yeah, it's a good question. 359 00:49:58.620 --> 00:50:03.400 Courant Events Right: But I believe that the linear algebra all works out, and there will be a… there will be a scaling limit. 360 00:50:06.730 --> 00:50:13.620 Courant Events Right: If there are no further questions, let's thank, Rick for his comic relief. 361 00:50:18.100 --> 00:50:21.540 Courant Events Right: And this concludes this first part of this week's activity.