WEBVTT 1 00:00:00.020 --> 00:00:04.040 Muze Ren: The math is happening on the middle. 3 boards. 2 00:00:07.824 --> 00:00:18.160 Muze Ren: Are you sure I cannot use this? But not so clear? Okay, you can. But it's not one. 3 00:00:23.100 --> 00:00:26.420 Muze Ren: When I'm obedient I'll start when I'm told to start. 4 00:00:27.240 --> 00:00:30.220 Muze Ren: Yeah, that's a good attitude. 5 00:00:36.450 --> 00:00:38.040 Muze Ren: isn't it bad one, was it? 6 00:00:38.240 --> 00:00:41.400 Muze Ren: If you switch the light off on the right one. 7 00:00:43.700 --> 00:00:55.250 Muze Ren: Do you want me to drop? No, I think we already found the optimal. But the thing you want is the dim lights. 8 00:00:55.840 --> 00:01:03.600 Muze Ren: No, I mean for me, it's fine. But typically for zoom, it's better without relying on the library solution. 9 00:01:03.820 --> 00:01:27.990 Muze Ren: Well, I think you know now it's a good compromise combination actually, without the light at all on the blackboard might be better right? Part of the problem, because before 10 00:01:27.990 --> 00:01:43.397 Muze Ren: and yeah, the title is secondary operations, emergent notes and the government variety by algebra. Okay, thank you. Music. Are you recording? Yeah, good. So 11 00:01:44.100 --> 00:01:50.730 Muze Ren: I have to say, this is my 1st blackboard talk in a long, long time. So I'm a bit nervous. 12 00:01:50.960 --> 00:01:56.489 Muze Ren: Also the talk today apologies. The talk today is mostly language 13 00:01:57.070 --> 00:02:00.980 Muze Ren: and like the the meat. 14 00:02:01.800 --> 00:02:12.159 Muze Ren: Sorry tofu is in my talk on Thursday, which is on a completely different topic, and actually unrelated to this workshop. 15 00:02:13.221 --> 00:02:19.089 Muze Ren: So my goal is mostly to introduce emergent notes. 16 00:02:19.390 --> 00:02:32.889 Muze Ren: and the purpose is to make use case happier for his talk. But I think it's actually closely related to what people from the Geneva group are going to talk about tomorrow. So 17 00:02:33.477 --> 00:02:36.979 Muze Ren: so maybe it's not just that. 18 00:02:37.240 --> 00:02:43.199 Muze Ren: anyway, let's start with our secondary operations. 19 00:02:43.300 --> 00:03:09.800 Muze Ren: So I want to tell you that the Goldman Terry of operations are in some sense secondary, and that's not an insult. It's a technical term. So a secondary operation is, suppose you have 2 primary operations whose symbol, so whose highest level term is the same. 20 00:03:09.920 --> 00:03:12.810 Muze Ren: then the difference between them 21 00:03:12.950 --> 00:03:15.970 Muze Ren: is somehow lower order stuff. 22 00:03:15.990 --> 00:03:18.530 Muze Ren: And that's a secondary operation. 23 00:03:19.150 --> 00:03:20.240 Muze Ren: Okay. 24 00:03:20.340 --> 00:03:40.550 Muze Ren: so let's convert it into a commutative diagram. So suppose you have, 0 goes to A goes to B goes to C, you know. Let's add the 0 here, even though this 0 is not used. 25 00:03:40.580 --> 00:03:44.204 Muze Ren: And this means that 26 00:03:46.040 --> 00:03:49.260 Muze Ren: B is kind of the big space. 27 00:03:49.410 --> 00:03:56.830 Muze Ren: and if you mowed out by A, you get C, so C is somehow the top level stuff 28 00:03:57.040 --> 00:04:03.979 Muze Ren: which, if you ignore lower order stuff then you, you. 29 00:04:04.360 --> 00:04:10.739 Muze Ren: you know, if you ignore the lower order stuff, namely, the A part of B, then you only see C. 30 00:04:11.090 --> 00:04:27.419 Muze Ren: So now suppose you have 2 operations. So let's call them lambda, 0 and lambda, one mapping E into B in some other space. 31 00:04:27.880 --> 00:04:34.050 Muze Ren: And suppose the difference when you map into C was equal to 0. 32 00:04:34.590 --> 00:04:40.930 Muze Ren: So one way to say it is to say that you have a map going this way, such that the 2 33 00:04:41.150 --> 00:04:44.229 Muze Ren: triangles that you see here are commutative. 34 00:04:44.810 --> 00:05:02.780 Muze Ren: Okay? And then obviously, since the difference goes to 0 here, it means that the difference came from here. So you get the new operation, Etta, which is lambda, 0 minus lambda, one going from E to a 35 00:05:04.550 --> 00:05:05.640 Muze Ren: okay. 36 00:05:06.076 --> 00:05:23.030 Muze Ren: And in fact, I didn't really need it to be 2 operation. I could have had just one operation such that the composition into C is equal to 0. But in all the examples it will that this one operation will be the difference of 2. So I write it this way. 37 00:05:23.480 --> 00:05:24.070 Muze Ren: Okay. 38 00:05:24.700 --> 00:05:35.669 Muze Ren: okay, let's fansify it a little bit. So suppose the top layer was itself a quotient situation. 39 00:05:35.890 --> 00:05:39.178 Muze Ren: So suppose we had 40 00:05:41.970 --> 00:05:48.749 Muze Ren: d goes to E goes to f 41 00:05:48.960 --> 00:05:52.580 Muze Ren: goes to 0 as the top layer 42 00:05:52.650 --> 00:06:00.040 Muze Ren: and the bottom layer was A goes to B goes to C, and coming from 0. 43 00:06:01.230 --> 00:06:08.179 Muze Ren: And suppose I had 2 operations. 44 00:06:11.734 --> 00:06:16.509 Muze Ren: lambda, 0 and lambda, one 45 00:06:16.520 --> 00:06:19.390 Muze Ren: going from E to B 46 00:06:21.033 --> 00:06:29.479 Muze Ren: such that you could make a commutative diagram like this. 47 00:06:30.000 --> 00:06:33.929 Muze Ren: And, by the way, these 2 operations 48 00:06:34.837 --> 00:06:44.159 Muze Ren: sometimes would be coming from 2 operations going this way. So lambda, 0 bar, and lambda one bar 49 00:06:45.120 --> 00:06:47.429 Muze Ren: by simply composing. 50 00:06:48.130 --> 00:06:59.089 Muze Ren: So like, my input will either be 2 operations going from E to B, or 2 operations going from F to B, in which case I create the 2 operations going from E to B. 51 00:06:59.220 --> 00:07:05.260 Muze Ren: Sorry the commuting screen as they commute for both. Yes, okay. 52 00:07:06.030 --> 00:07:08.740 Muze Ren: Well, in this situation. 53 00:07:09.498 --> 00:07:13.730 Muze Ren: So again, if you have something in F, 54 00:07:14.620 --> 00:07:18.650 Muze Ren: then it came from something in E, 55 00:07:18.810 --> 00:07:24.079 Muze Ren: you push it down to be in 2 different ways, and take the difference 56 00:07:24.500 --> 00:07:31.519 Muze Ren: it pushes to C into C to be 0. Therefore it came from A, 57 00:07:32.940 --> 00:07:55.489 Muze Ren: and if you recognize it as the construction of the connecting homomorphism in, you know, it's very similar to how the connecting homomorphism is defined, but it's a slightly different context, and I will call this thing. God, I don't know where to draw it. I will call it Etta. 58 00:07:56.210 --> 00:07:57.340 Muze Ren: Okay. 59 00:07:58.150 --> 00:08:02.660 Muze Ren: So eta is the secondary operation going from F to a 60 00:08:02.880 --> 00:08:04.320 Muze Ren: okay. 61 00:08:04.580 --> 00:08:06.356 Muze Ren: Okay. The next thing 62 00:08:06.980 --> 00:08:20.720 Muze Ren: is the next thing that occurs in when you talk about the connecting homomorphism. So the next thing that you you you you you learn about the connecting homomorphism is that it's natural. 63 00:08:21.160 --> 00:08:30.859 Muze Ren: So if you have a map between complexes, the connecting homomorphism are are on the kind of 64 00:08:30.950 --> 00:08:44.109 Muze Ren: commutes with the map between complexes. So yeah, sorry. So the the fact that the difference between lambda, not and power, one maps to 0. And C, that's the condition. Right? Yes, yes. 65 00:08:44.530 --> 00:08:49.859 Muze Ren: Oh, did I call it lambda, 0 and lambda 0. Yes, I am. I meant lambda, 0 and lambda one. 66 00:08:50.570 --> 00:08:51.640 Muze Ren: Okay? 67 00:08:52.245 --> 00:08:55.410 Muze Ren: So now suppose I have the second layer. 68 00:08:55.500 --> 00:09:00.880 Muze Ren: So suppose I had. D prime 69 00:09:01.010 --> 00:09:05.570 Muze Ren: e prime. F prime 70 00:09:06.382 --> 00:09:18.420 Muze Ren: a prime B. Prime and C. Prime, and suppose they made the same picture 71 00:09:19.640 --> 00:09:22.310 Muze Ren: among themselves? 72 00:09:26.690 --> 00:09:53.150 Muze Ren: Did I miss any arrows? Maybe not. And then suppose also I had a vertical arrows. So this going amount from A to a prime, and map from D to d prime and map from E to E, prime, B to B, prime, C to C, prime and F to F. Prime. Sorry it gets complicated. 73 00:09:53.680 --> 00:10:15.940 Muze Ren: Then I claim that if all sorry. Then, 1st of all, since the lower diagram is the same shape as the top diagram. I also have a secondary operation, or if you want a connecting homomorphism, eta prime 74 00:10:15.980 --> 00:10:18.099 Muze Ren: at the bottom diagram. 75 00:10:18.920 --> 00:10:20.839 Muze Ren: and I claim 76 00:10:20.910 --> 00:10:23.040 Muze Ren: that if 77 00:10:23.430 --> 00:10:28.249 Muze Ren: all of the vertical squares commute. 78 00:10:29.070 --> 00:10:40.019 Muze Ren: then the square formed by eta in f eta prime also commutes so going from F via eta to A, and then to a prime 79 00:10:40.080 --> 00:10:46.010 Muze Ren: is the same as going from F to F prime, and then via eta prime to a prime. 80 00:10:46.250 --> 00:10:54.130 Muze Ren: Okay, so let me say. And so this is let me write it. So this is naturality. 81 00:10:55.686 --> 00:10:57.820 Muze Ren: We all 82 00:10:58.656 --> 00:11:05.723 Muze Ren: a priori given given squares. So if all the given 83 00:11:06.650 --> 00:11:07.930 Muze Ren: squares 84 00:11:10.390 --> 00:11:11.960 Muze Ren: commute. 85 00:11:12.400 --> 00:11:15.109 Muze Ren: then so does 86 00:11:15.190 --> 00:11:18.100 Muze Ren: they. Square 87 00:11:19.850 --> 00:11:20.850 Muze Ren: produced 88 00:11:22.270 --> 00:11:26.849 Muze Ren: by the secondary. 89 00:11:28.985 --> 00:11:30.250 Muze Ren: Operations. 90 00:11:31.180 --> 00:11:44.599 Muze Ren: And yeah, sorry. Just one more question. So this you say, it's not quite a connection. But if you say that your map is actually lambda, 0 minus lambda one, because they don't seem to appear separately. 91 00:11:45.080 --> 00:12:01.960 Muze Ren: Yeah. Yeah. And I said that at the beginning, I mean in applications, it's always lambda, 0 and lambda one also, you know, if I if if I wrote it as the difference, then I'd have to say that the condition is that when you go to C, you get 0 92 00:12:02.360 --> 00:12:12.739 Muze Ren: right? And if I make them separate maps, all I have to say is that the square commutes so like, I say, 3 words or so. 93 00:12:13.060 --> 00:12:24.310 Muze Ren: Okay, but sorry. But what was your question? I interrupt. No, no. But then, yeah. 94 00:12:24.360 --> 00:12:26.070 Muze Ren: instruction or or not. 95 00:12:26.571 --> 00:12:34.110 Muze Ren: I actually didn't even fully think it through myself. So it's very similar. I, okay. 96 00:12:34.830 --> 00:12:41.190 Muze Ren: okay, maybe it is. I. I just didn't take it through. Okay. Now, 97 00:12:41.830 --> 00:12:44.309 Muze Ren: in the examples. 98 00:12:45.722 --> 00:12:54.009 Muze Ren: The top layer will always be note theory, or some variant of note theory. 99 00:12:54.800 --> 00:13:04.210 Muze Ren: not theory, is always graded or not graded, filtered. Sorry. There is a filtration, the finite type filtration. 100 00:13:04.790 --> 00:13:22.950 Muze Ren: And then you can take the associated graded so the lower level, so the top level will always be knots. The lower level will be always the associated graded of notes, which is often called a or core diagrams. 101 00:13:23.630 --> 00:13:28.800 Muze Ren: And so this is a front door 102 00:13:29.840 --> 00:13:38.180 Muze Ren: taking the associated grade. That is a factor. So if you have a diagram of the top, you automatically get the diagram at the bottom. 103 00:13:39.390 --> 00:13:48.319 Muze Ren: You do not automatically get morphisms from the top diagram to the bound diagram. That's an expansion 104 00:13:48.350 --> 00:13:54.720 Muze Ren: and expansion is precisely again. So the factor takes the spaces. Kind of 105 00:13:55.130 --> 00:13:58.420 Muze Ren: takes the category of spaces to to the categories. 106 00:13:58.510 --> 00:14:15.500 Muze Ren: to a category of spaces. So you get the diagram becomes the diagram. But there isn't a map between them. But then, if you get an expansion. So a contivich integral. So a map between these 2 diagrams, then you get a a commutative square like commutative 107 00:14:15.550 --> 00:14:17.330 Muze Ren: picture like this. 108 00:14:17.390 --> 00:14:21.249 Muze Ren: and then the connecting homomorphism. Here 109 00:14:21.350 --> 00:14:27.309 Muze Ren: the connecting homomorphisms at the top will be the Goldman terrari of operations. 110 00:14:27.620 --> 00:14:31.120 Muze Ren: The connecting homomorphisms at the bottom 111 00:14:31.230 --> 00:14:38.520 Muze Ren: will be the other algebraic counterparts. They're associated, graded. 112 00:14:38.660 --> 00:14:52.309 Muze Ren: and the the fact that this diagram that the the diagram commute that the diagram produced by enter and enter prime are is commutative 113 00:14:52.380 --> 00:14:57.679 Muze Ren: will be the homomorphicity of the expansion of the government derived expansion. 114 00:14:57.980 --> 00:14:59.259 Muze Ren: That's the plan. 115 00:14:59.790 --> 00:15:00.900 Muze Ren: Okay. 116 00:15:01.180 --> 00:15:04.839 Muze Ren: so let's start defining. 117 00:15:05.300 --> 00:15:09.350 Muze Ren: So I guess this was part one. 118 00:15:09.460 --> 00:15:16.650 Muze Ren: And now part 2. Let me talk about emergent notes. 119 00:15:16.800 --> 00:15:17.650 Muze Ren: So 120 00:15:19.020 --> 00:15:21.030 Muze Ren: emergent. 121 00:15:22.285 --> 00:15:23.230 Muze Ren: North 122 00:15:23.940 --> 00:15:27.270 Muze Ren: in a surface. 123 00:15:27.290 --> 00:15:29.909 Muze Ren: cross a little bow. 124 00:15:30.180 --> 00:15:33.830 Muze Ren: and I should say that often 125 00:15:34.110 --> 00:15:38.350 Muze Ren: the surface will be at this 126 00:15:38.620 --> 00:15:45.920 Muze Ren: minus end points. So this minus N points. 127 00:15:46.360 --> 00:15:54.939 Muze Ren: And then, Sigma, cross, an interval will be basically. 128 00:15:55.010 --> 00:15:58.709 Muze Ren: this thing made 3 dimensional. So a room 129 00:15:59.410 --> 00:16:22.899 Muze Ren: with N vertical lines removed. And this is what, in previous talks in the previous talk that I gave on the topic I gave only once before a talk on the topic. It was 2 years ago in La Viable. So 2 years ago I called this the pole dancing studio. Right? So this is PDSN, 130 00:16:23.130 --> 00:16:24.380 Muze Ren: namely. 131 00:16:24.790 --> 00:16:39.610 Muze Ren: again, a a room with N vertical bars removed, and note theory happens inside this room. So kind of the notes are the dancers in the pole dancing studio. 132 00:16:40.210 --> 00:16:41.280 Muze Ren: Okay. 133 00:16:42.300 --> 00:16:46.280 Muze Ren: so let's make a bunch of definitions. 134 00:16:46.320 --> 00:16:50.759 Muze Ren: So 1st of all, let K of s 135 00:16:51.440 --> 00:16:52.590 Muze Ren: b 136 00:16:56.739 --> 00:16:57.639 Muze Ren: notes 137 00:17:00.840 --> 00:17:13.159 Muze Ren: in Sigma Cross. I I'll fix Sigma once I'll fix Sigma, and later I will furthermore fix it to be this particular sigma. 138 00:17:13.599 --> 00:17:29.720 Muze Ren: Okay, so it will not be a part of the notation. If you insist, you could add here a subscript. I don't know a lower subscript left subscript subscript, Sigma, to be clear which, sigma it is. But I mean for me, Sigma will be fixed. 139 00:17:30.110 --> 00:17:31.180 Muze Ren: Okay. 140 00:17:31.910 --> 00:17:33.165 Muze Ren: so 141 00:17:35.140 --> 00:17:37.540 Muze Ren: sorry I'm running out of chalk. 142 00:17:41.380 --> 00:17:42.380 Muze Ren: Okay? 143 00:17:42.660 --> 00:17:43.870 Muze Ren: So 144 00:17:46.780 --> 00:17:54.940 Muze Ren: so K. Of S. Will be notes in Sigma Cross. I with a skeleton. 145 00:17:56.870 --> 00:17:57.659 Muze Ren: S. 146 00:17:58.120 --> 00:18:07.430 Muze Ren: So let me explain what I mean by skeleton. So the skeleton is basically the thing that gets knotted 147 00:18:08.000 --> 00:18:33.209 Muze Ren: so it could be a line. It could be a circle. It could be a circle and a line, and more precisely, if it's a line I should then indicate where it ends, put where its end points are. So, for example. If the arena. 148 00:18:33.590 --> 00:18:36.139 Muze Ren: if the poll dancing studio 149 00:18:37.030 --> 00:18:41.839 Muze Ren: so these vertical lines stands for the poles. 150 00:18:42.590 --> 00:18:46.490 Muze Ren: then the skeleton could be a loop. 151 00:18:46.520 --> 00:18:49.819 Muze Ren: which means that the look could be noted with anything 152 00:18:49.900 --> 00:19:02.360 Muze Ren: or a line. But if it's a line, then maybe I'll indicate that it starts here and ends here, or maybe it starts at the top and ends at the bottom. 153 00:19:02.580 --> 00:19:07.280 Muze Ren: and then to separate. The poles 154 00:19:07.310 --> 00:19:19.389 Muze Ren: from the thing that actually get noted is I'll indicate these with the letter S. So S stands for strands, and also these are pieces of the skeleton. 155 00:19:19.550 --> 00:19:22.349 Muze Ren: and they are to be distinct from P, 156 00:19:22.690 --> 00:19:28.610 Muze Ren: which is part of the ambient space, or, in fact, they are not part of the ambient space. 157 00:19:28.670 --> 00:19:31.959 Muze Ren: So in in so the red age of black and red 158 00:19:32.220 --> 00:19:37.379 Muze Ren: right in Letia blare. They were black and red, but you know I'm I don't want to. 159 00:19:37.430 --> 00:19:42.680 Muze Ren: Yes, so P. For poles, and they are vertical and fixed, and don't move. 160 00:19:42.760 --> 00:19:46.429 Muze Ren: and S in strands, and they can be not noted. 161 00:19:47.540 --> 00:19:49.359 Muze Ren: Please spell. 162 00:19:49.800 --> 00:19:52.409 Muze Ren: Oh, so they are tangles, really. 163 00:19:52.450 --> 00:19:59.740 Muze Ren: I call them. No, because it's generally noted. Object. Yes, I mean all type of noted object. 164 00:19:59.850 --> 00:20:22.659 Muze Ren: And in fact, you know sometimes, if I if I have the right mood, it's not going to happen today. But but but you know, I could restrict my attention to braids. For example. Okay, so I mean when I say not, I mean in the most general sense noted objects. In fact, they could be graphs. Okay, but things that you could not. 165 00:20:22.950 --> 00:20:30.659 Muze Ren: Okay. But let me make one correction. I'll always take linear combination of things like this. 166 00:20:30.690 --> 00:20:36.920 Muze Ren: So and let's say, the the ground field is overcube 167 00:20:37.450 --> 00:20:43.250 Muze Ren: over some field of characteristic 0 about not sophisticated. Okay? Just cheap. 168 00:20:43.610 --> 00:20:44.730 Muze Ren: Okay. 169 00:20:45.720 --> 00:20:46.340 Muze Ren: okay. 170 00:20:47.070 --> 00:20:49.249 Muze Ren: so this is the 1st definition. 171 00:20:50.610 --> 00:20:53.689 Muze Ren: K, 1 of F 172 00:20:54.300 --> 00:20:57.470 Muze Ren: is equal is the same. 173 00:20:57.920 --> 00:21:01.229 Muze Ren: except well, not the same. But it's the span 174 00:21:01.430 --> 00:21:05.890 Muze Ren: the span within. 175 00:21:06.782 --> 00:21:08.470 Muze Ren: The previous space 176 00:21:08.580 --> 00:21:09.770 Muze Ren: K. 177 00:21:10.381 --> 00:21:14.790 Muze Ren: Of S. So with the span within this. 178 00:21:14.810 --> 00:21:19.509 Muze Ren: or you know, if you wish you could call this K. 0 first, st and then this will be K. 0, 179 00:21:19.690 --> 00:21:23.120 Muze Ren: the span within K. 0, or 180 00:21:24.100 --> 00:21:24.980 Muze Ren: no 181 00:21:26.730 --> 00:21:31.330 Muze Ren: with one SF. 182 00:21:31.740 --> 00:21:33.400 Muze Ren: Double point. 183 00:21:34.070 --> 00:21:39.689 Muze Ren: So an Ss double point is a point in which 2 strands 184 00:21:40.357 --> 00:21:44.119 Muze Ren: cross each other, go through each other. 185 00:21:45.432 --> 00:21:47.490 Muze Ren: But this is not a note. 186 00:21:48.060 --> 00:21:59.150 Muze Ren: Okay, so this really stands for. It. It's really a mnemonic, for it really represents. An overcrossing 187 00:21:59.260 --> 00:22:01.950 Muze Ren: minus an undercrossing. 188 00:22:03.320 --> 00:22:04.380 Muze Ren: Okay. 189 00:22:05.610 --> 00:22:10.550 Muze Ren: so this is kind of like an augmentation ideal within. K, 190 00:22:11.330 --> 00:22:16.980 Muze Ren: okay? And similarly, K, 2 of S 191 00:22:17.090 --> 00:22:28.009 Muze Ren: will be the span. Basically all everything is the same except but that with 2. 192 00:22:28.170 --> 00:22:29.630 Muze Ren: If if 193 00:22:29.880 --> 00:22:40.939 Muze Ren: oh, sorry when I say ss, I should have indicated. Both of these lines are strands, not not poles. Poles cannot cross, poles are fixed. 194 00:22:41.280 --> 00:22:50.169 Muze Ren: But I'm not looking at strand pole interactions only at Strand strand interactions at this point. 195 00:22:51.260 --> 00:22:52.350 Muze Ren: Okay. 196 00:22:53.975 --> 00:23:02.229 Muze Ren: so k, 2 of S will be the same thing. But with 2 ss. Double points. 197 00:23:02.890 --> 00:23:07.840 Muze Ren: And this is to be interpreted as a difference of differences. 198 00:23:08.010 --> 00:23:18.449 Muze Ren: Right? So like this expands to over minus. Under. This expands to over minus under. So this is like a linear combination of 4 things. 199 00:23:19.040 --> 00:23:20.100 Muze Ren: Okay. 200 00:23:21.460 --> 00:23:25.170 Muze Ren: then. K, 201 00:23:27.420 --> 00:23:29.130 Muze Ren: flush one. 202 00:23:29.880 --> 00:23:35.450 Muze Ren: So K slash one is defined to be. And now I'm dropping the S. 203 00:23:35.510 --> 00:23:43.020 Muze Ren: Just because I'm lazy. I'm fix you fix a skeleton or you indicate what the skeleton, but I'm dropping it for them for for this. 204 00:23:43.230 --> 00:23:52.479 Muze Ren: So K. Sub. One is k modulo k sub modulo k. 1, 205 00:23:53.460 --> 00:23:57.729 Muze Ren: and let's give it another name. 206 00:23:57.830 --> 00:24:01.979 Muze Ren: So you see what happens if you mode by one. 207 00:24:02.130 --> 00:24:10.499 Muze Ren: if you MoD by one, then basically I mode by these things so over becomes equal to under. 208 00:24:11.100 --> 00:24:14.279 Muze Ren: So the note got ghosted 209 00:24:14.870 --> 00:24:16.559 Muze Ren: became a ghost. 210 00:24:16.670 --> 00:24:29.198 Muze Ren: not in the sense like on tinder when you get ghosted, but in the sense of ghost can pass through each other without. 211 00:24:29.870 --> 00:24:32.910 Muze Ren: you know, they just walk through each other. 212 00:24:33.310 --> 00:24:37.869 Muze Ren: Okay, so I will call it K ghosted. 213 00:24:39.300 --> 00:24:42.067 Muze Ren: But kay ghosted 214 00:24:43.320 --> 00:24:45.385 Muze Ren: is really 215 00:24:48.137 --> 00:24:53.599 Muze Ren: homotopy classes. So basically, it's homotopy classes of curves 216 00:24:53.800 --> 00:24:56.480 Muze Ren: inside our arena. 217 00:24:56.800 --> 00:25:00.769 Muze Ren: meaning maybe a pole dancing studio, maybe some other surface. 218 00:25:01.337 --> 00:25:12.409 Muze Ren: With where where the endpoints of the classes are are specified, or some some classes indicate stands for curves. 219 00:25:13.058 --> 00:25:23.120 Muze Ren: Closed curves so. For example. so so really this is for more topy classes. 220 00:25:23.570 --> 00:25:26.849 Muze Ren: So really, if you 221 00:25:30.010 --> 00:25:34.810 Muze Ren: look at K ghosted 222 00:25:35.652 --> 00:25:41.840 Muze Ren: with the scholar skeleton a line and a and a circle. 223 00:25:42.090 --> 00:25:44.889 Muze Ren: Then this is the same 224 00:25:44.960 --> 00:25:46.010 Muze Ren: Us. 225 00:25:47.073 --> 00:25:53.789 Muze Ren: the fundamental group. So the line indicates an element of the fundamental group. 226 00:25:54.650 --> 00:26:04.109 Muze Ren: Tensor. So I'm allowing linear combination. So it's really the group ring of the fundamental group. When I write pi one, I always mean the group ring. 227 00:26:05.160 --> 00:26:30.040 Muze Ren: And these are fundamental elements of the fundamental group without a base points, which is the same as cyclic words, or the same as conjugacy classes so 10 tensor. What people here denote with Taiwan with absolute value. And this is that just means conjugacy classes of elements of the fundamental group. 228 00:26:30.040 --> 00:26:36.509 Muze Ren: And maybe I'll also say that there is also a 229 00:26:36.510 --> 00:26:39.490 Muze Ren: framed version. 230 00:26:39.740 --> 00:26:50.150 Muze Ren: so I could have talked about framed notes instead of ordinary notes, and then there will be framed a framed version here. But I I, I'll skip that. 231 00:26:50.990 --> 00:26:55.130 Muze Ren: Okay. So I think that's all I had to say about ghosts. 232 00:26:55.970 --> 00:27:04.859 Muze Ren: Okay? And now, somehow, the central definition, so K mode 2 233 00:27:05.090 --> 00:27:15.129 Muze Ren: is equal to is defined to be k mode k. 2, 234 00:27:16.600 --> 00:27:21.889 Muze Ren: and I will call it k emergent. 235 00:27:21.970 --> 00:27:26.789 Muze Ren: So em stands for emergent. 236 00:27:27.350 --> 00:27:30.349 Muze Ren: So what do I mean by that? 237 00:27:31.020 --> 00:27:37.610 Muze Ren: So like, had I modded out by k. 1, i would kill nothing. 238 00:27:38.130 --> 00:27:39.020 Muze Ren: So 239 00:27:39.550 --> 00:27:46.389 Muze Ren: K. Goes that nothing gets killed here, nothing just nearly gets killed. 240 00:27:46.470 --> 00:27:48.159 Muze Ren: But not quite 241 00:27:48.870 --> 00:27:52.769 Muze Ren: so basically in K emergent. 242 00:27:53.890 --> 00:27:59.349 Muze Ren: If you look at the difference between an overcrossing and an undercrossing. 243 00:27:59.930 --> 00:28:03.850 Muze Ren: then after that, you can take further differences for free. 244 00:28:05.100 --> 00:28:08.330 Muze Ren: Okay. But the 1st difference counts. 245 00:28:09.340 --> 00:28:20.289 Muze Ren: and the origin of the word is like my understanding of the French language. So I have an emergent understanding of the French language. 246 00:28:20.570 --> 00:28:25.180 Muze Ren: which means it's not 0. But it's very, very close to 0. 247 00:28:25.680 --> 00:28:33.310 Muze Ren: Okay, so this is noting. Emergent, nothing, very, very little. Nothing remains. 248 00:28:33.330 --> 00:28:34.380 Muze Ren: Okay. 249 00:28:35.503 --> 00:28:46.079 Muze Ren: So you know, I said before, that, like secondary operation is not an insult. Emergence notes is sort of an insult. It's barely barely notes. 250 00:28:46.460 --> 00:28:48.350 Muze Ren: Okay, good. 251 00:28:50.056 --> 00:28:52.400 Muze Ren: Da da da da 252 00:28:53.070 --> 00:29:04.489 Muze Ren: right? And maybe from now on let's make it official. So Sigma is equal to A disk minus end points. 253 00:29:06.591 --> 00:29:09.600 Muze Ren: So meaning that our arena 254 00:29:11.640 --> 00:29:18.559 Muze Ren: is the tall dancing but dancing studio with in poles. 255 00:29:18.810 --> 00:29:21.240 Muze Ren: And let me give you a few examples. 256 00:29:22.230 --> 00:29:22.940 Muze Ren: Okay. 257 00:29:44.670 --> 00:29:48.420 Muze Ren: so example number one. 258 00:29:53.149 --> 00:30:06.660 Muze Ren: Let me take the skeleton. So you know, I I already specify what's the arena? So I just need to tell you what the skeleton is, so the skeleton will be 2 circles. 259 00:30:10.760 --> 00:30:13.469 Muze Ren: mark with one and 2. 260 00:30:14.200 --> 00:30:18.440 Muze Ren: Oh, I'm sorry, even before I should have said 261 00:30:18.650 --> 00:30:23.248 Muze Ren: in general. So before I had like when describing 262 00:30:25.260 --> 00:30:30.350 Muze Ren: secondary operations, I had A goes to B goes to C, 263 00:30:30.820 --> 00:30:45.310 Muze Ren: okay, so just to be clear. My A goes to B goes to city A, BC, or DEF will always be emergent notes in the middle. 264 00:30:45.820 --> 00:30:49.349 Muze Ren: They map into ghosted notes 265 00:30:50.270 --> 00:30:53.169 Muze Ren: right? And they come from 266 00:30:53.180 --> 00:30:54.186 Muze Ren: so 267 00:30:57.530 --> 00:31:01.180 Muze Ren: everything in k up one 268 00:31:02.280 --> 00:31:08.320 Muze Ren: maps to 0 in K ghosted K. Ghosted is K module k up one. 269 00:31:08.750 --> 00:31:12.730 Muze Ren: So the kernel is k up one 270 00:31:14.310 --> 00:31:18.940 Muze Ren: except things in K up 2 271 00:31:19.640 --> 00:31:22.830 Muze Ren: mapped to 0 already in K emergent 272 00:31:24.300 --> 00:31:28.110 Muze Ren: right k emergent is K. Modulo k up 2, 273 00:31:28.430 --> 00:31:33.660 Muze Ren: so I could mold out by K up. 2. 274 00:31:33.910 --> 00:31:42.169 Muze Ren: So this is the space a. I will short this with play up one more 2 275 00:31:43.780 --> 00:31:50.039 Muze Ren: and my operations. We go from Kgh to 276 00:31:50.930 --> 00:31:53.060 Muze Ren: k, 1 more, 2, 277 00:31:53.110 --> 00:31:56.120 Muze Ren: and these look like completely different spaces. 278 00:31:56.250 --> 00:32:07.450 Muze Ren: but later we will impose a further relation, and then they will become isomorphic, and then my operation. But operations will go from homotopy classes back again to homotopy classes. 279 00:32:08.170 --> 00:32:14.569 Muze Ren: Okay, so let's do the 1st example. So the skeleton is this. 280 00:32:15.160 --> 00:32:18.863 Muze Ren: and then hey 281 00:32:19.800 --> 00:32:21.940 Muze Ren: GH, 282 00:32:22.410 --> 00:32:25.869 Muze Ren: so now I'm writing what used to be F, 283 00:32:26.120 --> 00:32:37.245 Muze Ren: so the domain of the secondary operation. So KGH. Or of 2 circles, is really 284 00:32:37.920 --> 00:32:39.920 Muze Ren: sorry. 2 circles. 285 00:32:41.714 --> 00:32:46.000 Muze Ren: Is really 2 homotopy classes. 286 00:32:46.600 --> 00:32:50.870 Muze Ren: So 5, 1 tenzor 287 00:32:51.410 --> 00:32:52.960 Muze Ren: Taiwan. 288 00:32:53.080 --> 00:32:59.990 Muze Ren: So 2 close homotopy classes. So 2 continuity classes in Taiwan. 289 00:33:01.822 --> 00:33:13.070 Muze Ren: And then let's complete the diagram. So this this gets a map from K em of 2 circles. 290 00:33:14.830 --> 00:33:19.469 Muze Ren: And now I need to construct 2 maps. 291 00:33:20.210 --> 00:33:29.740 Muze Ren: lambda 0, or maybe I'll call it lambda, 1 0 and lambda 1, 1, because this is the the 1st one is, for example, one. 292 00:33:30.620 --> 00:33:31.720 Muze Ren: Okay. 293 00:33:32.160 --> 00:33:33.105 Muze Ren: so 294 00:33:34.670 --> 00:33:42.009 Muze Ren: and they also go into K emergent of 2 spirals. 295 00:33:44.088 --> 00:33:47.280 Muze Ren: But let me tell you what they are. 296 00:33:47.800 --> 00:33:48.895 Muze Ren: So 297 00:33:50.997 --> 00:33:53.209 Muze Ren: my input, oh. 298 00:33:53.530 --> 00:33:55.330 Muze Ren: my, input 299 00:33:55.730 --> 00:33:58.490 Muze Ren: is a note with 2 components. 300 00:33:59.600 --> 00:34:00.610 Muze Ren: Right? 301 00:34:00.860 --> 00:34:03.150 Muze Ren: My output 302 00:34:03.800 --> 00:34:16.780 Muze Ren: for via lambda 0 will be you put the 2 components, you stack them vertically, with with the 1st component above the second 303 00:34:17.120 --> 00:34:22.830 Muze Ren: and lambda, one will stack them vertically in the opposite order. 304 00:34:23.310 --> 00:34:29.665 Muze Ren: Okay. So lambda 1 0 is 305 00:34:31.260 --> 00:34:33.479 Muze Ren: port, one 306 00:34:33.650 --> 00:34:34.780 Muze Ren: above 307 00:34:35.310 --> 00:34:36.440 Muze Ren: 2 308 00:34:36.510 --> 00:34:39.710 Muze Ren: and lambda 1, 1 will be put 309 00:34:40.030 --> 00:34:41.150 Muze Ren: 2 310 00:34:41.179 --> 00:34:43.639 Muze Ren: above one. 311 00:34:44.159 --> 00:34:51.120 Muze Ren: and if you want a picture. So here is the whole dancing studio. 312 00:34:53.159 --> 00:35:12.829 Muze Ren: So if the 2 components are red and green, then in one map red is in is reimbedded on top of sorry green is reimbedded on top of red, and in the other map the the order is flipped. 313 00:35:13.210 --> 00:35:14.410 Muze Ren: Okay. 314 00:35:14.700 --> 00:35:26.150 Muze Ren: now. It's just clear. But maybe just as a matter of definition, if you talk pro, k emergent, right? 315 00:35:26.480 --> 00:35:38.249 Muze Ren: These 2 circles. Yeah, they are in principle. They were already positioned in some way. Right? Yeah. So I reposition them. Oh, okay, so you get out. Yeah. Position. Yes. 316 00:35:39.340 --> 00:35:44.079 Muze Ren: So. Or alternatively again, really, my mountain 317 00:35:44.830 --> 00:35:47.719 Muze Ren: in the end. Yes, yeah, yeah. But yeah. 318 00:35:47.730 --> 00:35:58.349 Muze Ren: And then clearly, well, you can put this, I mean, here you get 10 GH of 2 circles, and and the squares commute. 319 00:35:58.460 --> 00:36:11.280 Muze Ren: because once you because modulo homotopy, it doesn't matter where which one is on top and which one is below. And basically you can slide them through each other. 320 00:36:11.690 --> 00:36:15.379 Muze Ren: Then this gets a mark 321 00:36:15.420 --> 00:36:29.249 Muze Ren: from K. So I'm writing. Not the full diagram, only the part that I care about. So k, 1 mode, 2 of 2 of 2 circles. 322 00:36:29.780 --> 00:36:34.420 Muze Ren: And let me just interpret this. 323 00:36:35.130 --> 00:36:40.610 Muze Ren: Okay, so k, 1. 324 00:36:41.727 --> 00:36:43.970 Muze Ren: Basically is maps 325 00:36:44.330 --> 00:36:45.960 Muze Ren: of a 326 00:36:48.380 --> 00:36:49.653 Muze Ren: did as 327 00:36:51.030 --> 00:36:54.599 Muze Ren: a skeleton that has one double point in it. 328 00:36:55.070 --> 00:37:02.879 Muze Ren: so it's or can be span, is spanned by maps of a skeleton that has one double point in it. 329 00:37:03.180 --> 00:37:04.345 Muze Ren: And 330 00:37:06.110 --> 00:37:11.390 Muze Ren: this. Which really, you can think of as a noted graph. 331 00:37:11.950 --> 00:37:13.050 Muze Ren: Okay? 332 00:37:13.490 --> 00:37:19.300 Muze Ren: And then so if I'm thinking of this 333 00:37:19.380 --> 00:37:23.565 Muze Ren: as being generated by maps of 334 00:37:25.560 --> 00:37:27.889 Muze Ren: a skeleton that looks like. 335 00:37:28.280 --> 00:37:29.230 Muze Ren: did 336 00:37:30.060 --> 00:37:30.700 Muze Ren: they? 337 00:37:32.050 --> 00:37:35.079 Muze Ren: After I've mapped this into 338 00:37:35.170 --> 00:37:38.210 Muze Ren: the arena. The Indian space 339 00:37:39.160 --> 00:37:41.339 Muze Ren: I bought out by having 2 340 00:37:41.850 --> 00:37:50.460 Muze Ren: by 2. So basically, these pieces are now homotopy, homotopyp, homotopy pieces. 341 00:37:50.780 --> 00:37:57.350 Muze Ren: Okay, so this can be interpreted as maps of 342 00:37:57.660 --> 00:38:00.720 Muze Ren: a figure 8 into the space. 343 00:38:00.830 --> 00:38:09.769 Muze Ren: and you can also write it in terms of Taiwan. But but but I won't. It's just a a 344 00:38:10.230 --> 00:38:14.819 Muze Ren: basically. Well, I'll just stay stick with it as it is. 345 00:38:14.920 --> 00:38:29.969 Muze Ren: So this is up to homotopy. So so basically, the space like this, this space is the space of notings of things like that into a pole. Dancing studio modulo homotopy. 346 00:38:30.330 --> 00:38:32.256 Muze Ren: and our 347 00:38:34.000 --> 00:38:42.959 Muze Ren: secondary operation is an operation that goes from pi. One tensor pi one into 348 00:38:44.550 --> 00:38:47.770 Muze Ren: maps are of a figure 8. 349 00:38:48.020 --> 00:38:53.899 Muze Ren: Yes, they mentioned like those double points, then they're only from only between different components. 350 00:38:54.230 --> 00:38:59.409 Muze Ren: Sorry, like the double points that you allow. You only allow them in different components, not one component. 351 00:38:59.520 --> 00:39:06.699 Muze Ren: Otherwise you would also get that. Only one of the one of the loops gives you the figure 8, and you have another one that doesn't. 352 00:39:08.250 --> 00:39:12.457 Muze Ren: So i i i think you are 353 00:39:16.010 --> 00:39:17.070 Muze Ren: okay. 354 00:39:22.830 --> 00:39:25.510 Muze Ren: I I should. So 355 00:39:26.640 --> 00:39:27.890 Muze Ren: would be 356 00:39:28.500 --> 00:39:30.889 Muze Ren: precise. You are right. 357 00:39:32.779 --> 00:39:39.400 Muze Ren: And this will come up again a a few minutes later. But 358 00:39:39.929 --> 00:39:47.570 Muze Ren: within the difference of our lambda one and our lambda 0 and our lambda one 359 00:39:47.980 --> 00:39:49.350 Muze Ren: only 360 00:39:51.890 --> 00:39:55.670 Muze Ren: only double points of the type that you indicate occur. 361 00:39:56.100 --> 00:40:00.449 Muze Ren: So you're right. Strictly speaking, this space is bigger. 362 00:40:00.720 --> 00:40:02.530 Muze Ren: but 363 00:40:02.740 --> 00:40:04.450 Muze Ren: our eta. 364 00:40:04.610 --> 00:40:12.402 Muze Ren: Its target is the subspace in which the double point is between the 2 circles. Okay? 365 00:40:13.180 --> 00:40:16.180 Muze Ren: And this will occur again in a few minutes. Thank you. 366 00:40:16.910 --> 00:40:20.400 Muze Ren: And let's do example number 2. 367 00:40:31.270 --> 00:40:32.365 Muze Ren: So 368 00:40:35.840 --> 00:40:38.929 Muze Ren: example 2. 369 00:40:39.800 --> 00:40:45.010 Muze Ren: So now the skeleton 370 00:40:45.410 --> 00:40:49.510 Muze Ren: will be basically this. 371 00:40:49.800 --> 00:41:06.350 Muze Ren: So I mean, so so what I mean by that is that we will be talking about nottings of intervals in a Pole dancing studio, where the interval begins and ends at the bottom in 2 specified points. 372 00:41:07.240 --> 00:41:27.960 Muze Ren: So the diagram. Sorry. 373 00:41:27.990 --> 00:41:30.860 Muze Ren: So, yeah, so so this all. 374 00:41:31.110 --> 00:41:39.839 Muze Ren: But it should. It should be a person which sends K 2 into K 2. It's it's well defined on the course. 375 00:41:40.080 --> 00:41:45.239 Muze Ren: So yeah, I guess it's totally obvious that it says a 2 to K. 2. But I care to see it. 376 00:41:45.280 --> 00:41:46.500 Muze Ren: So. Okay. 377 00:41:48.540 --> 00:41:57.339 Muze Ren: Gee, I hope I didn't get something wrong. No, no K. 2, 378 00:42:19.830 --> 00:42:21.620 Muze Ren: you know it's 379 00:42:22.720 --> 00:42:26.370 Muze Ren: possible that I was a bit too ambitious. 380 00:42:27.010 --> 00:42:31.709 Muze Ren: Sorry. It's possible that I that I set it wrong. 381 00:42:31.720 --> 00:42:34.889 Muze Ren: so maybe I should have just kept it. 382 00:42:35.100 --> 00:42:39.449 Muze Ren: Or maybe here the space should have simply been 383 00:42:39.850 --> 00:42:42.930 Muze Ren: a emergent 384 00:42:43.500 --> 00:42:44.949 Muze Ren: of a loop. 385 00:42:45.230 --> 00:42:46.720 Muze Ren: pencil or 386 00:42:46.800 --> 00:42:49.539 Muze Ren: K emerging of a loop. 387 00:42:50.920 --> 00:43:03.880 Muze Ren: So I mean, I'm not sure if I I'm I'm not sure if what you said it is is a problem. But this definitely solves it. 388 00:43:03.950 --> 00:43:05.020 Muze Ren: Okay? 389 00:43:07.285 --> 00:43:08.180 Muze Ren: Okay. 390 00:43:09.706 --> 00:43:12.250 Muze Ren: So example number 2, 391 00:43:13.130 --> 00:43:20.299 Muze Ren: the skeleton is this, is this, and then, my spaces are. 392 00:43:20.380 --> 00:43:28.770 Muze Ren: K. Boasted of this, which is really 5, 1 393 00:43:29.797 --> 00:43:38.339 Muze Ren: there is a map into it from K emergent of the same skeleton. 394 00:43:40.310 --> 00:43:41.483 Muze Ren: and 395 00:43:44.030 --> 00:43:51.849 Muze Ren: I need to construct 2 maps into well, here it will be again. K. Emergent 396 00:43:52.110 --> 00:43:54.310 Muze Ren: of the same skeleton. 397 00:43:55.433 --> 00:43:58.399 Muze Ren: And I need to construct 2 maps. 398 00:43:58.570 --> 00:44:08.580 Muze Ren: but, like I indicated before, it's actually easy easier to construct them going this way. And then these 2 maps will simply be the compositions 399 00:44:09.770 --> 00:44:16.629 Muze Ren: and the 2 maps. Let's call them lambda, 2 0 and lambda 2. 1 400 00:44:16.920 --> 00:44:27.409 Muze Ren: will be the offending, embedding, and the disbanding 401 00:44:27.550 --> 00:44:28.920 Muze Ren: embedding. 402 00:44:29.490 --> 00:44:31.020 Muze Ren: namely. 403 00:44:31.310 --> 00:44:36.990 Muze Ren: so if you have an element of the fundamental group. 404 00:44:37.370 --> 00:44:51.870 Muze Ren: You can always so so so maybe I'll just write it. Lambda 2, 0. The ascending embedding will look like this. So if you have an element of the fundamental group 405 00:44:52.170 --> 00:44:54.460 Muze Ren: of the complements. 406 00:44:54.830 --> 00:44:57.289 Muze Ren: So but basically the free group 407 00:44:58.960 --> 00:45:00.500 Muze Ren: you can. 408 00:45:01.410 --> 00:45:07.650 Muze Ren: 1st of all, embedded as offending braid. 409 00:45:08.300 --> 00:45:10.780 Muze Ren: and then drop down to 0, 410 00:45:11.640 --> 00:45:16.669 Muze Ren: or you can embed it. So, lambda 2, 1 will be. 411 00:45:17.806 --> 00:45:19.920 Muze Ren: You are 412 00:45:21.380 --> 00:45:22.520 Muze Ren: 1st 413 00:45:23.020 --> 00:45:25.709 Muze Ren: shoot to to pretty high up. 414 00:45:25.750 --> 00:45:30.819 Muze Ren: and then you embed it as a descending 415 00:45:31.060 --> 00:45:32.170 Muze Ren: friend. 416 00:45:33.200 --> 00:45:34.380 Muze Ren: Okay? 417 00:45:36.093 --> 00:45:43.940 Muze Ren: And then. So so this generates 2 maps going this way. But again, that's automatic. 418 00:45:44.200 --> 00:45:49.019 Muze Ren: And then sorry. The map into here 419 00:45:49.060 --> 00:45:57.030 Muze Ren: from k. 1 over 2 of a a single curve. 420 00:45:57.460 --> 00:46:11.890 Muze Ren: and like before this is really homotopy classes of curves like this that have one singularity so curves that look like this. 421 00:46:12.470 --> 00:46:17.899 Muze Ren: So multiple classes of curves like this into the full dancing studio. 422 00:46:18.570 --> 00:46:22.369 Muze Ren: And again, modulo remote opinion. 423 00:46:22.750 --> 00:46:24.655 Muze Ren: And this is 424 00:46:25.820 --> 00:46:27.640 Muze Ren: no, that's my output. 425 00:46:28.410 --> 00:46:32.469 Muze Ren: Okay, so these are the 2 examples. 426 00:46:33.560 --> 00:46:41.462 Muze Ren: Now, I want to get it even closer to Goldman to Riev. 427 00:46:45.090 --> 00:46:47.919 Muze Ren: so I'll introduce 428 00:46:49.840 --> 00:46:51.850 Muze Ren: some new spaces. 429 00:46:52.780 --> 00:46:56.140 Muze Ren: So now this is part 430 00:46:56.330 --> 00:46:58.450 Muze Ren: 2 and a half 431 00:46:58.540 --> 00:47:00.010 Muze Ren: of the talk. 432 00:47:00.470 --> 00:47:06.030 Muze Ren: so let me indicate, by K. Sub, H. 433 00:47:07.540 --> 00:47:14.019 Muze Ren: So again, this could be in any pole, dancing studio, or any other surface. 434 00:47:14.888 --> 00:47:27.100 Muze Ren: And with an arbitrary skeleton. But I'm writing the general definition. So this is defined to be a power series in a variable B, 435 00:47:27.930 --> 00:47:34.960 Muze Ren: in fact, it's kind of irrelevant that it's power series, I mean polynomials. If you wish, in a variable B 436 00:47:37.050 --> 00:47:38.779 Muze Ren: 10 zor 437 00:47:38.860 --> 00:47:40.870 Muze Ren: the old K. 438 00:47:43.365 --> 00:47:48.690 Muze Ren: Except I take B to have degree one 439 00:47:49.610 --> 00:47:53.309 Muze Ren: so, or filtration degree one. 440 00:47:53.380 --> 00:48:06.350 Muze Ren: So if I will be writing K. Up h modulus 2 or up 2, it means I either have 2 B's and an arbitrary element of K 441 00:48:06.360 --> 00:48:14.029 Muze Ren: or one B, and an element of k, 1 or 0 B's, and an element of K 2. 442 00:48:14.120 --> 00:48:16.469 Muze Ren: So B has degree one. 443 00:48:16.990 --> 00:48:19.800 Muze Ren: and I mowed out 444 00:48:20.280 --> 00:48:31.280 Muze Ren: by a relation, and the relation says that a double point and a double point really stands for over minus under. 445 00:48:31.700 --> 00:48:40.259 Muze Ren: So if you have an over minus and under, and of course, everything here is strands. 446 00:48:41.365 --> 00:48:48.160 Muze Ren: Then this is equal to b times. You replace it by the smoothing. 447 00:48:48.770 --> 00:49:00.150 Muze Ren: and the H. Stands for this is like the relation in the Humphrey polynomial, though it's also the relation in the common polynomial. So maybe it should have been sub. C. I'm not sure. 448 00:49:00.690 --> 00:49:03.820 Muze Ren: Okay, that's a bit of choice what to call it. 449 00:49:04.410 --> 00:49:06.210 Muze Ren: So now, 450 00:49:10.078 --> 00:49:15.949 Muze Ren: so by the way, can 451 00:49:16.010 --> 00:49:17.670 Muze Ren: stop age 452 00:49:17.720 --> 00:49:19.580 Muze Ren: ghosted. 453 00:49:20.310 --> 00:49:23.720 Muze Ren: I mean here. 454 00:49:23.920 --> 00:49:28.209 Muze Ren: basically you kill by the image of the image of B. 455 00:49:28.600 --> 00:49:34.289 Muze Ren: So nothing gets to you. You've changed nothing. This is just K. Ghosted. 456 00:49:34.950 --> 00:49:44.749 Muze Ren: Okay, but there is a a, an, a nicer theorem or proposition. So proposition. 457 00:49:45.881 --> 00:49:50.209 Muze Ren: if you look at the map, multiplication by B, 458 00:49:50.440 --> 00:49:59.420 Muze Ren: so B hat like in quantum mechanics. This means multiplication, sorry multiplication. By B. 459 00:50:00.090 --> 00:50:04.009 Muze Ren: So if you look at the multiplication by B map. 460 00:50:04.660 --> 00:50:11.039 Muze Ren: so this is a map from K. Ghosted. 461 00:50:11.570 --> 00:50:17.970 Muze Ren: you know, you can MoD out by age. You don't have to MoD out by age. It's the same as I just indicated 462 00:50:18.020 --> 00:50:25.369 Muze Ren: into K, so basically, if I start with 463 00:50:25.790 --> 00:50:37.490 Muze Ren: K. 0 mode one, and I multiply by BI land in k. 1 mode 2. So this is a map into KH. 1 mode 2 464 00:50:38.980 --> 00:50:51.150 Muze Ren: and even before stating the proposition, I'll tell you why I care. I care, because this is where my target space 465 00:50:51.200 --> 00:51:05.769 Muze Ren: for the connecting homomorphism. Sorry the secondary operations lands. So the plan is to take the previous examples, the the previous 2 examples, and move them by age 466 00:51:06.280 --> 00:51:12.180 Muze Ren: and and the point is that now I will have an identification of KH. 467 00:51:12.530 --> 00:51:18.439 Muze Ren: So the theorem is that this map is an isomorphism. 468 00:51:21.810 --> 00:51:28.010 Muze Ren: So basically KH. 1, 1, 2, which is my target spaces 469 00:51:28.180 --> 00:51:34.070 Muze Ren: is simply homotopy is ghosted notes, which really means homotopy classes. 470 00:51:34.320 --> 00:51:38.249 Muze Ren: So I get maps from home. Homotopy classes to homotopy classes. 471 00:51:38.570 --> 00:51:39.670 Muze Ren: Okay. 472 00:51:39.970 --> 00:51:50.679 Muze Ren: you know. I sorry if you're totally bored. I mean, some of you have seen this, and it's just the same as what I said 2 years ago, I admit it's the same as what I said 2 years ago. 473 00:51:51.310 --> 00:51:54.418 Muze Ren: Okay, anyway. 474 00:51:56.970 --> 00:51:59.569 Muze Ren: So let me prove this. 475 00:52:00.360 --> 00:52:01.740 Muze Ren: So proof. 476 00:52:02.695 --> 00:52:06.020 Muze Ren: So basically, surjectivity is obvious. 477 00:52:06.420 --> 00:52:10.880 Muze Ren: So element in k, 1 are differences. 478 00:52:11.781 --> 00:52:23.559 Muze Ren: Right? K up one. There are differences. Anything which isn't a difference is an image of multiplying something, but by B. So Sir Jed. 479 00:52:23.700 --> 00:52:25.220 Muze Ren: pity 480 00:52:25.280 --> 00:52:26.330 Muze Ren: if 481 00:52:26.450 --> 00:52:27.640 Muze Ren: obvious. 482 00:52:28.729 --> 00:52:49.220 Muze Ren: So all I need to do is to construct them up, going the other way, so I need to construct them up sign going the other way, which will be a 1 sided, inverse. So I need sign such that if you 1st multiply by B, so do this 483 00:52:49.530 --> 00:52:54.599 Muze Ren: and then applied. Psi. So then applied. Psi, then you get that identity. 484 00:52:55.830 --> 00:52:56.950 Muze Ren: Okay? 485 00:52:57.859 --> 00:53:03.790 Muze Ren: So I basically need to define time on. 486 00:53:03.800 --> 00:53:07.789 Muze Ren: Stay! Well, I need to define psi on this space. 487 00:53:07.800 --> 00:53:15.099 Muze Ren: This space is a quotient space, so I'll define it on the generators and check that it satisfies the relations. 488 00:53:15.280 --> 00:53:17.942 Muze Ren: So psi 489 00:53:19.360 --> 00:53:23.909 Muze Ren: I need to define it on things which look like B times something. 490 00:53:24.310 --> 00:53:29.519 Muze Ren: So if I have b times the note, I will simply map it to the note. 491 00:53:30.090 --> 00:53:38.509 Muze Ren: Right? Basically, B hat is multiplication by BI need to divide by B, like the inverse should be division by B 492 00:53:38.700 --> 00:53:45.209 Muze Ren: so, or like division by B. So if you have B times the note, I'll map it to the note itself. 493 00:53:45.940 --> 00:54:02.519 Muze Ren: and if I have. So B, this is B to the power one, if I have B to the 0. So if if if I have a generator in which the coefficient is not a power of B is not B. 494 00:54:02.620 --> 00:54:09.129 Muze Ren: Then I will do something. Fancy I will map it to one half 495 00:54:09.160 --> 00:54:17.819 Muze Ren: times the sum over all the crossings in K. So x in K means crossings of K. 496 00:54:17.920 --> 00:54:24.490 Muze Ren: Now each crossing in ken has a sign, according to ordinary note theory. 497 00:54:24.600 --> 00:54:35.579 Muze Ren: so I'll take minus one to that crossing and then take that, and then take this K. But with the crossing replaced by a smoothing. 498 00:54:38.880 --> 00:54:40.100 Muze Ren: and 499 00:54:40.160 --> 00:54:42.569 Muze Ren: then you have to check things. 500 00:54:42.610 --> 00:54:55.649 Muze Ren: 1st of all, you have to check that. This map respects Rydermeister 2 and Rydermeister 3. And these are routine checks. You. Basically, you know you write Rydermeister 3, and you write with crossings. This is. 501 00:54:55.850 --> 00:55:08.130 Muze Ren: I wrote it unsigned. You you write write away 3, and you just apply. It becomes of sum of 3 terms on the left, the sum of 3 terms on the right. It checks 502 00:55:08.340 --> 00:55:10.899 Muze Ren: likewise for Rydermeister 2. 503 00:55:11.270 --> 00:55:14.769 Muze Ren: And finally, you have to check that it satisfies H, 504 00:55:15.520 --> 00:55:17.459 Muze Ren: but basically 505 00:55:18.480 --> 00:55:22.599 Muze Ren: the right hand side gets mapped to erase the B, 506 00:55:22.880 --> 00:55:24.848 Muze Ren: the left hand side. 507 00:55:25.390 --> 00:55:32.809 Muze Ren: this becomes one half of a smoothing, this becomes minus minus one half of a smoothing. And it's equal. 508 00:55:33.630 --> 00:55:36.290 Muze Ren: So so this map is an isomorphism. 509 00:55:36.990 --> 00:55:59.860 Muze Ren: So yeah, in in fact, yeah, this is exactly the so. So the fact that the Goldman derive operations are well defined follows 510 00:55:59.880 --> 00:56:01.360 Muze Ren: from from this. 511 00:56:02.340 --> 00:56:05.650 Muze Ren: Okay, but sorry. I'm a little lost. 512 00:56:05.740 --> 00:56:10.129 Muze Ren: Whether this is a 2 dimension. You need to check it on 2 dimension or 3 dimension 513 00:56:10.190 --> 00:56:20.999 Muze Ren: Israel, I mean. So this is 3 dimensional. I was just lazy. All of these are crossings. No, no, but but for the government wave is actually on only on the surface. Right? 514 00:56:21.040 --> 00:56:26.190 Muze Ren: You check the cutoffs where the means moves only on surfaces right? 515 00:56:26.300 --> 00:56:27.269 Muze Ren: But this. 516 00:56:27.580 --> 00:56:39.539 Muze Ren: this. This implies the fact that the road Goldman thrive is well defined. This is something else, but it implies. 517 00:56:39.740 --> 00:56:44.720 Muze Ren: but ghost is is not very 3 dimensional. Yeah, yeah, so 518 00:56:44.840 --> 00:56:46.229 Muze Ren: yeah, I had. 519 00:56:46.370 --> 00:56:49.470 Muze Ren: right? Yeah, so it it is actually 2 dimensional. 520 00:56:50.505 --> 00:56:55.860 Muze Ren: The target is is really 2 dimensional. Yes, I agree. 521 00:56:56.990 --> 00:57:17.159 Muze Ren: Okay. So anyway, I I'm I'm kind of running out of time. I'm not as kind. I'm I'm running out of time. But let me say so so let me just let me quickly complete the story and then move on. So this is isomorphic. 522 00:57:17.370 --> 00:57:33.570 Muze Ren: VR multiplication by B with K. Gh of. Okay. So here I was a bit loose here. I didn't indicate the skeleton. 523 00:57:35.850 --> 00:57:40.999 Muze Ren: and and in fact, the map side changes the skeleton. 524 00:57:41.090 --> 00:57:48.460 Muze Ren: Also the relation changes the skeleton, the relation does not fix the skeleton. 525 00:57:50.501 --> 00:57:57.109 Muze Ren: So you know. Here I I it's unclear. What skeleton to write? 526 00:57:57.864 --> 00:58:10.979 Muze Ren: But in the particular case that we have lambda, 1 0 minus so lambda, 0 minus lambda, one ends in the image of one specific skeleton. 527 00:58:11.690 --> 00:58:17.609 Muze Ren: so I can restrict my attention to this one specific skeleton, and it's a single circle. 528 00:58:18.000 --> 00:58:28.759 Muze Ren: and the single circle here is, and K goes of one circle, even if you MoD sorry, so I should mow down everything by km, H. 529 00:58:29.991 --> 00:58:41.340 Muze Ren: And the ghost stuff doesn't change. The emerging stuff does change. But the ghost stuff doesn't change, and this is just pi one. And this is the Goldman Bracket. 530 00:58:41.800 --> 00:58:54.900 Muze Ren: and exactly the same over here. So this is the image via B of the map from K goes 531 00:58:55.080 --> 00:58:56.080 Muze Ren: all 532 00:58:56.950 --> 00:59:00.149 Muze Ren: one line in the circle. 533 00:59:01.280 --> 00:59:03.710 Muze Ren: and this is 534 00:59:04.197 --> 00:59:07.859 Muze Ren: sorry, and I have to add age everywhere. 535 00:59:10.450 --> 00:59:13.659 Muze Ren: and this is Taiwan tenzor 536 00:59:13.740 --> 00:59:15.573 Muze Ren: on one 537 00:59:16.740 --> 00:59:24.820 Muze Ren: absolute value. And this is the so-called what is it called the core coaction. 538 00:59:24.950 --> 00:59:26.450 Muze Ren: the cool 539 00:59:26.460 --> 00:59:27.990 Muze Ren: action 540 00:59:28.090 --> 00:59:39.970 Muze Ren: ma, which is the precursor of the 2 riot core bracket. If you want to get the 2 right core bracket, you have to close this as well and anti-symmetrize. 541 00:59:41.100 --> 00:59:42.200 Muze Ren: Okay? 542 00:59:42.950 --> 00:59:47.870 Muze Ren: So I'm oh, my God, I I have 0 time left. 543 00:59:48.728 --> 00:59:51.680 Muze Ren: But now I should be going to part 4. 544 00:59:51.830 --> 00:59:56.371 Muze Ren: It's a little bit negotiable like how much time we need so 545 00:59:56.880 --> 01:00:04.929 Muze Ren: no, too much. So I mean, so it's so. So I'll just say 546 01:00:05.465 --> 01:00:12.040 Muze Ren: in principle. All you have to do is to apply the Ger factor. 547 01:00:13.890 --> 01:00:18.939 Muze Ren: right? So if you apply the Ger factor 548 01:00:20.150 --> 01:00:21.430 Muze Ren: you get. 549 01:00:22.890 --> 01:00:23.980 Muze Ren: I mean. 550 01:00:24.220 --> 01:00:28.099 Muze Ren: you get that. Everything becomes algebraic. 551 01:00:28.310 --> 01:00:33.829 Muze Ren: And maybe I'll just tell you what is 552 01:00:33.890 --> 01:00:43.439 Muze Ren: the output of the girlfunctor? So basically K of a skeleton becomes so this is becomes not maps into 553 01:00:43.510 --> 01:00:50.719 Muze Ren: becomes a of a skeleton. So, for example, if the skeleton was this, then a of this 554 01:00:51.120 --> 01:00:52.540 Muze Ren: will be 555 01:00:53.770 --> 01:00:58.570 Muze Ren: the space of so-called core diagrams. 556 01:00:58.670 --> 01:01:07.939 Muze Ren: So the core diagrams will have a number of poles because we're still in a pole, dancing studio and 557 01:01:08.500 --> 01:01:13.869 Muze Ren: a skeleton, a copy of the skeleton, and then the cards. 558 01:01:14.200 --> 01:01:16.010 Muze Ren: We'll be going 559 01:01:16.290 --> 01:01:24.563 Muze Ren: well. No cords between the pole, because the Poles are stationary. But you have cords going from 560 01:01:26.890 --> 01:01:28.159 Muze Ren: this to this. 561 01:01:28.280 --> 01:01:35.699 Muze Ren: okay, from from skeleton to skeleton, from from strand to strand and from strand to poles. 562 01:01:36.204 --> 01:01:56.330 Muze Ren: And then you have a few relations. So basically, if you want to ghost something, the ghosting relation is, if you have a chord between 2 fronts, you're equal to 0, and that also the same as saying, or also B is equal to 0. If you do the b extension. 563 01:01:57.050 --> 01:01:59.600 Muze Ren: Then there is the emergent 564 01:02:00.156 --> 01:02:06.060 Muze Ren: relation. So the emergent relation is, if you have 2 chords 565 01:02:06.100 --> 01:02:08.550 Muze Ren: between strands 566 01:02:08.670 --> 01:02:24.539 Muze Ren: anywhere like I wrote them in a certain way, but I mean anywhere going from any place to any other place between 2 strands, then this is equal to 0. Likewise, B squared is equal to 0 also. B. Times, one chord is equal to 0 567 01:02:26.470 --> 01:02:53.320 Muze Ren: and finally, there is the H relation and the H. Sorry. I should have said even before that you have to model by the usual 4 T. Relations, and then there is the H relation, and the H. Relation says that if you have a chord between 2 strands, and it's more fun to draw them with opposite orientations. Then this is equal to B times the strands replaced by a bridge. 568 01:02:53.690 --> 01:02:54.830 Muze Ren: And 569 01:02:55.470 --> 01:03:09.409 Muze Ren: so this is the emergent relation that Yusuke will be talking about in a completely different language. But still it's the same relation in his talk, and I believe it's also very similar to what 570 01:03:10.122 --> 01:03:12.589 Muze Ren: people from Geneva have been playing with. 571 01:03:12.780 --> 01:03:16.580 Muze Ren: So they maybe will be telling us about it tomorrow 572 01:03:16.620 --> 01:03:28.206 Muze Ren: and and I think what I don't have time to do, and I genuinely don't have time to do is to tell you how. 573 01:03:29.830 --> 01:03:31.480 Muze Ren: if you repeat 574 01:03:32.220 --> 01:03:33.210 Muze Ren: these 575 01:03:33.550 --> 01:03:35.370 Muze Ren: stories. 576 01:03:36.550 --> 01:03:45.499 Muze Ren: using A instead of K, you get the Goldman Turayev algebraic. 577 01:03:45.660 --> 01:03:52.340 Muze Ren: count the the Daljibray counterpart of the Goldman derived operations. 578 01:03:52.860 --> 01:03:57.839 Muze Ren: and furthermore, if you use the Kultsevic integral. 579 01:03:58.290 --> 01:04:02.390 Muze Ren: then? Well, for the Goldman picture 580 01:04:03.168 --> 01:04:07.482 Muze Ren: all the vertical arrows are 581 01:04:10.310 --> 01:04:38.820 Muze Ren: make commutative squares. So you get the homomorphicity of the secondary operation, meaning the Goldman Bracket. And for here not quite all the vertical lines are homomorphic, are not quite all the vertical squares are commutative. But when they're not commutative they're not commutative in a controlled way. 582 01:04:38.920 --> 01:04:51.390 Muze Ren: and then you can check that if you map this further to pi one transfer pi one, so it will close the remaining open component. 583 01:04:51.869 --> 01:04:58.759 Muze Ren: Then it becomes well defined, then then sorry. Then all the commuties do hold. 584 01:04:58.960 --> 01:05:00.819 Muze Ren: So everything works 585 01:05:01.170 --> 01:05:02.520 Muze Ren: and 586 01:05:03.001 --> 01:05:11.690 Muze Ren: and I have to say I have to make 2 comments before I quit. So I'm already 5 min over time. So one is, I'm not happy. 587 01:05:12.020 --> 01:05:31.609 Muze Ren: so I really think I I feel that I mean I don't know. I I hope this was clean, but I at least the last bit here can be cleaned even further. I haven't. We haven't cleaned it, but my feeling is that it should be possible to clean it even further. 588 01:05:32.360 --> 01:05:38.369 Muze Ren: and the second comment is that it's not a comment. It's a homework assignment. 589 01:05:38.540 --> 01:05:44.039 Muze Ren: So homework do the same. 590 01:05:44.520 --> 01:05:45.820 Muze Ren: or 591 01:05:48.280 --> 01:06:00.622 Muze Ren: W. Notes and W. Chord diagrams. And the point is that this should like the expansion in this case, would be 592 01:06:01.310 --> 01:06:04.789 Muze Ren: coming from Kashibara Verne. 593 01:06:05.440 --> 01:06:11.995 Muze Ren: and and so this will relate cash, or this will re-relate cashier to 594 01:06:14.262 --> 01:06:16.100 Muze Ren: Goldman to Riev. 595 01:06:16.697 --> 01:06:34.069 Muze Ren: And it's not trivial, I mean, it's not just do the same. I mean, you have to think more carefully. What are the relations? What is H. What replaces H. How to replace this? What? To replace the spaces by? So it it's not trivial. It's a it's a serious assignment. I mean. 596 01:06:34.590 --> 01:06:38.300 Muze Ren: like, it's a it's a term project, not just a. 597 01:06:38.400 --> 01:06:40.439 Muze Ren: you know, next week assignment. 598 01:06:41.220 --> 01:06:42.908 Muze Ren: Okay? And I'll stop. 599 01:06:43.470 --> 01:06:44.480 Muze Ren: Thank you. 600 01:06:48.480 --> 01:06:50.740 Muze Ren: Any questions or comments? 601 01:06:51.200 --> 01:07:06.230 Muze Ren: Yeah, 2 small questions. I mean, suppose you only have one punctures? You have one line. Yeah, you have a circle. Yeah, I remember in your old paper, you could explicitly compute the conjurity integral of this right? Yeah. 602 01:07:06.939 --> 01:07:20.650 Muze Ren: But but I feel for the other generators. Could you complete computer explicitly, I feel the generator is relatively easy for this kind of puncture. The So. Plane 603 01:07:23.920 --> 01:07:24.600 Muze Ren: oops. 604 01:07:26.026 --> 01:07:32.479 Muze Ren: I have not. I've done some explicit computation, but not enough. 605 01:07:32.670 --> 01:07:34.170 Muze Ren: And maybe 606 01:07:34.180 --> 01:07:46.919 Muze Ren: possibly Yusuke will mention something I don't know. But bear in mind that all of this is happening in emergent notes and in emergent notes. 607 01:07:48.750 --> 01:07:50.729 Muze Ren: Those are barely noted. 608 01:07:51.310 --> 01:08:06.549 Muze Ren: So you kill quite a lot. Basically, you model by this relation and modulo, this relation, the the computations are easier. So you you expect to be able to compute more than just the concierge integral 609 01:08:07.082 --> 01:08:16.099 Muze Ren: but do you think you can kind of explicit computer contributing, or of the generators of this scheme. Modules of this punctured plant? 610 01:08:16.580 --> 01:08:17.560 Muze Ren: No. 611 01:08:19.445 --> 01:08:24.660 Muze Ren: it will remain unexplicit at the end, or, you know, written in terms of integral. So 612 01:08:24.790 --> 01:08:25.649 Muze Ren: okay. 613 01:08:26.050 --> 01:08:26.870 Muze Ren: thank you. 614 01:08:27.050 --> 01:08:29.060 Muze Ren: Thank you more more questions. 615 01:08:30.540 --> 01:08:34.869 Muze Ren: Yeah, I I. So since you mentioned your talk 2 years ago. 616 01:08:34.930 --> 01:08:46.200 Muze Ren: right? So in that talk there was also a little bit of a discussion that maybe one can continue. It's like you're looking at. k, 1 or 2. What about like K, 2 or 3, 617 01:08:46.439 --> 01:08:55.240 Muze Ren: and so on. That would the story, repeat in some way right like I have not. I have not. What would be your speculation? 618 01:08:59.260 --> 01:09:05.294 Muze Ren: I have no sorry. I'm really sorry. I have no useful speculations 619 01:09:06.420 --> 01:09:26.420 Muze Ren: but my speculation. But another speculation is so. You know I I was a bit ambivalent about whether I applied the age, relation, or not like at the end. I did, but in the beginning. But but before that I had pictures in which I didn't, and I think there is real content here. 620 01:09:26.630 --> 01:09:30.519 Muze Ren: So I think the Goldman to rival operations 621 01:09:30.540 --> 01:09:32.639 Muze Ren: really do have 622 01:09:37.689 --> 01:09:44.589 Muze Ren: counterpart parts in which you end up talking about singular nodes, embedding of of singular nodes 623 01:09:45.140 --> 01:09:46.470 Muze Ren: and 624 01:09:46.996 --> 01:09:51.630 Muze Ren: and and that there will be interesting things to say about us. 625 01:09:51.830 --> 01:09:56.020 Muze Ren: So like, this is the direction I'm more happy to to speculate about. 626 01:09:58.750 --> 01:10:01.910 Muze Ren: Okay, any more questions or comments 627 01:10:02.530 --> 01:10:10.410 Muze Ren: now. So I'm little curious. Because in the beginning you explain this secondary operations with sound, like exact sequence. 628 01:10:10.500 --> 01:10:16.960 Muze Ren: Yeah, I I feel this is very similar to in physics is they always do the semi-classical limit. 629 01:10:17.490 --> 01:10:21.359 Muze Ren: And you take some content stuff. You take semi classical image. 630 01:10:21.390 --> 01:10:25.450 Muze Ren: I don't know whether your exact sequence can also explain that 631 01:10:25.800 --> 01:10:27.669 Muze Ren: all the semicritic. 632 01:10:28.050 --> 01:10:30.190 Muze Ren: No, I don't know what to say. 633 01:10:33.280 --> 01:10:38.410 Muze Ren: Maybe one more word. My talk on Thursday is totally underrated. 634 01:10:40.286 --> 01:10:44.590 Muze Ren: Okay? Well, thanks. Again. 635 01:10:48.678 --> 01:10:54.870 Muze Ren: the key to schedule, especially for user people like 636 01:10:54.910 --> 01:10:59.590 Muze Ren: before you'll be speaking. Then, in 20 min. Is it okay for you people. 637 01:10:59.790 --> 01:11:03.370 Muze Ren: Okay? So then we do a schedule, then 638 01:11:03.440 --> 01:11:09.270 Muze Ren: 1130 for you to find. So in 20 min. 639 01:11:12.170 --> 01:11:19.010 Muze Ren: Oh, Yusukee, are you there? Oh, okay. 640 01:11:20.100 --> 01:11:30.339 Yusuke Kuno: I I'm sorry I I muted myself sorry I I hear dro. Well, but sorry I don't hear. Well, what did you say? 641 01:11:31.740 --> 01:11:33.740 Muze Ren: Continue in 20 min. 642 01:11:35.520 --> 01:11:36.950 Yusuke Kuno: Okay. As scheduled. 643 01:11:37.340 --> 01:11:39.245 Muze Ren: Yeah, that's schedule. 644 01:11:40.260 --> 01:11:41.280 Yusuke Kuno: Okay. Yeah. 645 01:11:42.290 --> 01:11:51.829 Muze Ren: Music. Will you send me the file? Or you know, a link to the file, or whatever? 646 01:11:52.700 --> 01:11:55.630 Muze Ren: Okay, percentage of the 647 01:11:55.720 --> 01:11:57.190 Muze Ren: methods? 648 01:11:59.340 --> 01:12:01.370 Muze Ren: Yeah.