1 00:00:00,420 --> 00:00:01,253 Hi, class. 2 00:00:01,253 --> 00:00:03,423 Welcome back to modeling complex system. 3 00:00:04,590 --> 00:00:07,950 I've said it before, what I like about modeling, 4 00:00:07,950 --> 00:00:09,990 and I called it a romantic tool, 5 00:00:09,990 --> 00:00:12,720 is that it allows you to do science 6 00:00:12,720 --> 00:00:13,980 in a different way, right? 7 00:00:13,980 --> 00:00:18,720 I think I said you can do it by the fire on a beach 8 00:00:18,720 --> 00:00:21,150 of Lake Champlain or walking down a railroad track. 9 00:00:21,150 --> 00:00:22,380 Doesn't matter where you are, 10 00:00:22,380 --> 00:00:23,820 the lab benches are in your head, 11 00:00:23,820 --> 00:00:25,620 and you can follow your curiosity, 12 00:00:25,620 --> 00:00:28,260 and answer almost any questions 13 00:00:28,260 --> 00:00:30,930 or try to answer almost any questions that you care about, 14 00:00:30,930 --> 00:00:33,300 that you're interested in and follow your own curiosity. 15 00:00:33,300 --> 00:00:35,640 So what I'd like to do in this video 16 00:00:35,640 --> 00:00:38,400 is just give you a few examples of the kind of systems 17 00:00:38,400 --> 00:00:39,297 we're gonna be looking at, 18 00:00:39,297 --> 00:00:42,600 and what are the keywords you should be thinking about 19 00:00:42,600 --> 00:00:45,720 when wondering what systems you can try 20 00:00:45,720 --> 00:00:48,513 and study with the modeling tools we're gonna learn. 21 00:00:49,530 --> 00:00:53,520 So the first keywords I'll use is collective. 22 00:00:53,520 --> 00:00:55,680 And I'm using it as a first example, 23 00:00:55,680 --> 00:00:58,200 because it's behind, this idea of collective behavior 24 00:00:58,200 --> 00:01:01,050 is behind probably the most classic example 25 00:01:01,050 --> 00:01:03,199 of complex systems. 26 00:01:03,199 --> 00:01:06,032 (birds whooshing) 27 00:01:10,907 --> 00:01:13,793 The sound of the birds, it felt... 28 00:01:13,793 --> 00:01:16,020 So this collective behavior is amazing 29 00:01:16,020 --> 00:01:21,020 and it's (indistinct) incredible that (indistinct) 30 00:01:22,890 --> 00:01:24,833 emerge from birds (indistinct) individually, 31 00:01:25,686 --> 00:01:27,210 (indistinct), right? 32 00:01:27,210 --> 00:01:28,350 So it begs the question, 33 00:01:28,350 --> 00:01:33,350 do we need to threshold with self-awareness or intelligent? 34 00:01:33,449 --> 00:01:36,124 (birds whooshing drowns out speaker) 35 00:01:36,124 --> 00:01:37,811 (birds whooshing) 36 00:01:37,811 --> 00:01:42,228 (birds whooshing drowns out speaker) 37 00:01:44,196 --> 00:01:46,946 (birds chirping) 38 00:01:48,719 --> 00:01:51,552 (birds whooshing) 39 00:01:55,410 --> 00:01:57,490 This was louder than I remembered 40 00:01:58,710 --> 00:02:01,320 and I've used the word before. 41 00:02:01,320 --> 00:02:02,760 But as you model these things, 42 00:02:02,760 --> 00:02:04,590 you put a lot of these birds together, 43 00:02:04,590 --> 00:02:08,370 the behavior you tend to care about is emergent, right? 44 00:02:08,370 --> 00:02:11,940 It's something that is not forced top down by a leader 45 00:02:11,940 --> 00:02:15,570 or by the modeler within the rules of the system. 46 00:02:15,570 --> 00:02:19,140 But that emerges because of the many interacting parts, 47 00:02:19,140 --> 00:02:20,670 interacting parts. 48 00:02:20,670 --> 00:02:23,070 The example I'll use for this, I muted this video 49 00:02:23,070 --> 00:02:26,790 because there's some pretty unsavory language being used, 50 00:02:26,790 --> 00:02:29,460 but quickly what's going on here is that the band, 51 00:02:29,460 --> 00:02:33,030 Lamb Of God is on stage on the left side of the screen 52 00:02:33,030 --> 00:02:36,690 and the singer is using some colorful language 53 00:02:36,690 --> 00:02:39,450 to ask the audience to start a circle pit. 54 00:02:39,450 --> 00:02:41,790 And what a circle pit is essentially just 55 00:02:41,790 --> 00:02:45,093 members of the audience running in circles, jumping around. 56 00:02:46,440 --> 00:02:47,760 It's a lot of fun. 57 00:02:47,760 --> 00:02:50,640 But then the singer is not telling them where to do it, 58 00:02:50,640 --> 00:02:52,410 which direction to turn 59 00:02:52,410 --> 00:02:54,810 and it's really through this collective behavior 60 00:02:54,810 --> 00:02:56,430 is that a direction emerge, right? 61 00:02:56,430 --> 00:02:58,470 So you have one circle that you're probably looking at, 62 00:02:58,470 --> 00:03:00,390 in the middle, there's one in the far right, 63 00:03:00,390 --> 00:03:02,550 one in the bottom right of the screen. 64 00:03:02,550 --> 00:03:04,650 They don't have to go in the same direction. 65 00:03:04,650 --> 00:03:06,330 These clusters can merge. 66 00:03:06,330 --> 00:03:08,100 It's almost like fluid dynamics. 67 00:03:08,100 --> 00:03:10,560 But here, it's mostly a human example 68 00:03:10,560 --> 00:03:13,683 of what we saw in the birds just now, right? 69 00:03:15,450 --> 00:03:20,450 So co-evolution is one of the theme of most of my research. 70 00:03:21,360 --> 00:03:23,610 In the previous example, it could have been co-evolution 71 00:03:23,610 --> 00:03:25,950 of different groups, different circle pits. 72 00:03:25,950 --> 00:03:29,610 Often what I care about, what's fascinating to me 73 00:03:29,610 --> 00:03:34,292 is the co-evolution of a dynamics on a system. 74 00:03:34,292 --> 00:03:37,560 If you want, so the idea of a pandemic, 75 00:03:37,560 --> 00:03:41,970 let's say, in a human society with the dynamics 76 00:03:41,970 --> 00:03:44,760 of the system, which can be like because of a pandemic, 77 00:03:44,760 --> 00:03:48,390 there's social distancing, physical distancing going on, 78 00:03:48,390 --> 00:03:51,450 there's the creation of new institutions, right? 79 00:03:51,450 --> 00:03:55,200 So the society is co-evolving with this pandemic 80 00:03:55,200 --> 00:03:57,150 and this interaction of structure and dynamics 81 00:03:57,150 --> 00:03:59,280 is something that I find fascinating. 82 00:03:59,280 --> 00:04:01,650 I chose this random video. 83 00:04:01,650 --> 00:04:03,780 The reference is there, I don't know where it's from. 84 00:04:03,780 --> 00:04:05,670 I picked it because it's beautiful 85 00:04:05,670 --> 00:04:09,060 and it claims to show the growth of a fungus and a bacteria. 86 00:04:09,060 --> 00:04:11,340 I don't know if that's true. That doesn't really matter. 87 00:04:11,340 --> 00:04:14,880 I just love the fact that often when you grow bacteria 88 00:04:14,880 --> 00:04:16,290 or whatever it is, 89 00:04:16,290 --> 00:04:19,020 you tend to get these beautifully complex structures. 90 00:04:19,020 --> 00:04:21,993 Like doesn't this almost look like a city, right? 91 00:04:22,920 --> 00:04:25,560 And then it begs the question of how, again, 92 00:04:25,560 --> 00:04:27,570 something that locally is so simple, 93 00:04:27,570 --> 00:04:30,600 cells and bacteria can grow in these complex ways, 94 00:04:30,600 --> 00:04:33,120 can learn to trade within a colonies. 95 00:04:33,120 --> 00:04:34,380 And those are just beautiful 96 00:04:34,380 --> 00:04:36,273 or interesting questions in my mind. 97 00:04:37,500 --> 00:04:39,660 And at the end of the video here, we saw, you know, 98 00:04:39,660 --> 00:04:42,690 the structure start to decay. 99 00:04:42,690 --> 00:04:44,970 And it brings us to this idea of instabilities. 100 00:04:44,970 --> 00:04:47,460 A lot of the systems we're gonna care about 101 00:04:47,460 --> 00:04:50,070 have multiple stable states. 102 00:04:51,558 --> 00:04:53,970 Going back to the COVID-19 pandemic, 103 00:04:53,970 --> 00:04:56,280 you could take the same novel Coronavirus, 104 00:04:56,280 --> 00:04:59,880 the same virus behind COVID-19, put it in the same person, 105 00:04:59,880 --> 00:05:02,070 in the same place at the same time. 106 00:05:02,070 --> 00:05:05,463 And just because of random noise or you know, 107 00:05:06,420 --> 00:05:09,990 or sensitivity to initial conditions, chaos. 108 00:05:09,990 --> 00:05:13,620 9 out of 10 times would probably be okay now. 109 00:05:13,620 --> 00:05:15,810 There's multiple different steady states 110 00:05:15,810 --> 00:05:18,240 or outcomes that we can get 111 00:05:18,240 --> 00:05:21,540 and it's interesting to know how likely one is 112 00:05:21,540 --> 00:05:22,560 versus the other. 113 00:05:22,560 --> 00:05:26,700 And when do you push the system beyond this unstable point 114 00:05:26,700 --> 00:05:28,770 and towards a different state? 115 00:05:28,770 --> 00:05:30,960 And I studied that a lot in a system 116 00:05:30,960 --> 00:05:35,100 that I care a lot about, which is the Amazon rainforest. 117 00:05:35,100 --> 00:05:38,610 So the Amazon rainforest exists 118 00:05:38,610 --> 00:05:41,340 right next to savannah, right? 119 00:05:41,340 --> 00:05:46,340 Dry grassland, and it's not just because of rainfalls, 120 00:05:47,280 --> 00:05:49,710 essentially is that the rainforest 121 00:05:49,710 --> 00:05:51,960 and these dry grasslands 122 00:05:51,960 --> 00:05:55,173 are two stable states of the same system. 123 00:05:56,700 --> 00:05:58,080 And once you understand that 124 00:05:58,080 --> 00:05:59,940 from a dynamical system perspective 125 00:05:59,940 --> 00:06:02,100 or a modeling perspective, it begs the question, 126 00:06:02,100 --> 00:06:04,560 how much damage can this beautiful system, 127 00:06:04,560 --> 00:06:08,430 the Amazon rainforest take before it collapses 128 00:06:08,430 --> 00:06:10,173 to savannah dry grassland? 129 00:06:11,790 --> 00:06:13,830 And as climate change increases, 130 00:06:13,830 --> 00:06:17,583 as deforestation increase, as forest fire increase, 131 00:06:19,140 --> 00:06:22,470 it is important to be able without burning the forest 132 00:06:22,470 --> 00:06:23,760 to try and answer these questions, 133 00:06:23,760 --> 00:06:25,770 how much damage can it take? 134 00:06:25,770 --> 00:06:28,023 And modeling can help us do exactly that. 135 00:06:28,950 --> 00:06:32,160 So it goes back to this idea now of feedback 136 00:06:32,160 --> 00:06:35,280 that as we do damage either through deforestation 137 00:06:35,280 --> 00:06:39,300 or forest fires, we reduce the forest area 138 00:06:39,300 --> 00:06:41,730 and therefore, reduce its ability to grow back. 139 00:06:41,730 --> 00:06:45,210 So there's what we call a positive feedback loop 140 00:06:45,210 --> 00:06:47,730 in the loss of forest. 141 00:06:47,730 --> 00:06:50,700 But in terms of, you know, biodiversity, 142 00:06:50,700 --> 00:06:53,370 it's just a terrible, terrible negative things. 143 00:06:53,370 --> 00:06:56,463 And this feedback show up in a lot of this of systems. 144 00:06:57,570 --> 00:06:59,910 There's a beautiful example in the next video. 145 00:06:59,910 --> 00:07:01,470 I'm just gonna shut up and let it play 146 00:07:01,470 --> 00:07:04,472 because I love the sound of ice and snow. 147 00:07:04,472 --> 00:07:07,055 (ice rumbling) 148 00:07:10,920 --> 00:07:12,780 [Narrator 1] The only way that you can really try 149 00:07:12,780 --> 00:07:15,180 to put it into scale with human reference is 150 00:07:15,180 --> 00:07:18,780 if you imagine Manhattan, and all of a sudden, 151 00:07:18,780 --> 00:07:22,230 all of those buildings just start to rumble and quake 152 00:07:22,230 --> 00:07:24,780 and peel off and just fall over and fall over 153 00:07:24,780 --> 00:07:25,713 and roll around. 154 00:07:27,570 --> 00:07:31,530 This whole massive city just breaking apart 155 00:07:31,530 --> 00:07:32,630 in front of your eyes. 156 00:07:37,462 --> 00:07:40,045 (ice rumbling) 157 00:07:42,840 --> 00:07:44,430 We're just observers. 158 00:07:44,430 --> 00:07:47,070 These two little dots on the side of the mountain. 159 00:07:47,070 --> 00:07:51,000 And we watched and recorded the largest witness 160 00:07:51,000 --> 00:07:53,373 calving event ever caught on tape. 161 00:07:55,890 --> 00:07:58,560 So this calving event that they're talking about 162 00:07:58,560 --> 00:08:01,110 is essentially just separation of the glacier. 163 00:08:01,110 --> 00:08:03,063 It's a beautiful, beautiful event. 164 00:08:05,400 --> 00:08:08,280 But similar feedbacks happen when you just look 165 00:08:08,280 --> 00:08:10,830 at an iceberg melting or whatever it is, 166 00:08:10,830 --> 00:08:14,370 there's a loss of snow cover 167 00:08:14,370 --> 00:08:17,220 and you get less reflection of sun rays 168 00:08:17,220 --> 00:08:19,200 and an increase in climate change. 169 00:08:19,200 --> 00:08:20,700 And then you have these events, 170 00:08:20,700 --> 00:08:23,820 like this incredible calving event was 75 minutes 171 00:08:23,820 --> 00:08:24,783 or so, right? 172 00:08:25,860 --> 00:08:27,750 But that interacts with climate change, 173 00:08:27,750 --> 00:08:30,870 which occurs on a much longer time scale. 174 00:08:30,870 --> 00:08:34,950 And so, you have different systems, some local, global 175 00:08:34,950 --> 00:08:37,743 or systems of different nature that are interdependent. 176 00:08:38,910 --> 00:08:42,180 And the video that I use to illustrate interdependence 177 00:08:42,180 --> 00:08:46,410 or interconnectedness of systems is a video of bats 178 00:08:46,410 --> 00:08:47,640 in Sierra Leone. 179 00:08:47,640 --> 00:08:50,370 So a lot of my work deals with the Ebola outbreak 180 00:08:50,370 --> 00:08:54,030 on 2014 to 2016 in Western Africa. 181 00:08:54,030 --> 00:08:56,880 I chose this video to talk about it in part 182 00:08:56,880 --> 00:08:58,860 because the video shows that Sierra Leone 183 00:08:58,860 --> 00:09:00,153 is a beautiful country. 184 00:09:01,560 --> 00:09:04,410 And also, it shows, you know, this behavior of bats 185 00:09:04,410 --> 00:09:06,240 and it's an interesting system 186 00:09:06,240 --> 00:09:08,430 because as as these countries grow 187 00:09:08,430 --> 00:09:11,400 and push back against the forest for economic growth 188 00:09:11,400 --> 00:09:14,100 and urban development, 189 00:09:14,100 --> 00:09:16,770 you increase the frequency of context 190 00:09:16,770 --> 00:09:19,380 between the human population and the bats population. 191 00:09:19,380 --> 00:09:23,400 And here, the bats are the vector for a lot of diseases, 192 00:09:23,400 --> 00:09:24,663 including Ebola. 193 00:09:26,010 --> 00:09:29,070 So economic growth is increasing this risk 194 00:09:29,070 --> 00:09:32,040 of Ebola coming back in the human population. 195 00:09:32,040 --> 00:09:33,690 And anytime there's an Ebola outbreak, 196 00:09:33,690 --> 00:09:36,630 it comes as great cost of human lives 197 00:09:36,630 --> 00:09:38,790 and economic development. 198 00:09:38,790 --> 00:09:42,570 In economic modeling, that's a classic poverty trap. 199 00:09:42,570 --> 00:09:45,360 The idea that you're either sick and poor 200 00:09:45,360 --> 00:09:47,070 or rich and healthy. 201 00:09:47,070 --> 00:09:49,590 It's a cartoon but it helps us think about 202 00:09:49,590 --> 00:09:51,660 how positive things like economic development 203 00:09:51,660 --> 00:09:54,600 can have terrible outcomes 204 00:09:54,600 --> 00:09:57,240 because of the interdependence of systems, 205 00:09:57,240 --> 00:09:59,193 in this case through animals. 206 00:10:00,960 --> 00:10:03,300 So the last example of behavior that I have 207 00:10:03,300 --> 00:10:04,530 is just this idea that 208 00:10:04,530 --> 00:10:06,900 a lot of complex systems are adaptive. 209 00:10:06,900 --> 00:10:09,363 The parts of complex systems is adaptive. 210 00:10:10,230 --> 00:10:12,840 The models we're gonna develop are not easy to control. 211 00:10:12,840 --> 00:10:16,470 If you try and intervene, the parts, 212 00:10:16,470 --> 00:10:19,410 the elements of your system might change their behavior. 213 00:10:19,410 --> 00:10:21,870 And I chose a ridiculous video to show that. 214 00:10:21,870 --> 00:10:24,360 So this is a video of a man, a lawyer in Florida 215 00:10:24,360 --> 00:10:28,950 dressed as the grim reaper protesting the reopening 216 00:10:28,950 --> 00:10:30,120 of beaches in Florida. 217 00:10:30,120 --> 00:10:33,783 And I chose the video because I love most protests, 218 00:10:34,980 --> 00:10:37,710 but also because it was a good representation 219 00:10:37,710 --> 00:10:42,390 of behavior that I'm really fascinated with right now. 220 00:10:42,390 --> 00:10:44,220 This idea that if you have different states, 221 00:10:44,220 --> 00:10:45,780 different counties, different towns 222 00:10:45,780 --> 00:10:47,100 with different regulation, 223 00:10:47,100 --> 00:10:50,940 so one closes their beach, the other doesn't. 224 00:10:50,940 --> 00:10:54,330 Well then, resident of the town that did close their beach 225 00:10:54,330 --> 00:10:56,250 can move over to the other town, 226 00:10:56,250 --> 00:10:59,880 and then you end up with even more people than you expected 227 00:10:59,880 --> 00:11:00,930 on the beach, right? 228 00:11:00,930 --> 00:11:05,310 So this is a negative consequence of your control measure 229 00:11:05,310 --> 00:11:08,283 brought on by the adaptiveness of the system. 230 00:11:09,510 --> 00:11:11,940 It's a classic, you know, unexpected 231 00:11:11,940 --> 00:11:16,940 or unintended consequence in public health intervention. 232 00:11:17,250 --> 00:11:19,500 This picture, I've been using for a few year. 233 00:11:19,500 --> 00:11:23,673 When I started studying Ebola outbreak way back in 2014. 234 00:11:24,810 --> 00:11:26,700 It was sad to see, you know, 235 00:11:26,700 --> 00:11:29,100 this was one of the first measure implemented. 236 00:11:29,100 --> 00:11:32,250 So Ebola is transmitted through bodily fluids, 237 00:11:32,250 --> 00:11:36,720 and that can be, you know, blood, sweat, anything. 238 00:11:36,720 --> 00:11:40,770 And the infectious person doesn't have to be alive 239 00:11:40,770 --> 00:11:43,110 to still be infectious through their fluid. 240 00:11:43,110 --> 00:11:46,833 So even after a death has been reported due to Ebola, 241 00:11:48,300 --> 00:11:51,840 health officials would come in and burn the belongings, 242 00:11:51,840 --> 00:11:55,743 sheets, mattress, anything that could contain bodily fluids. 243 00:11:56,730 --> 00:11:59,490 Well, in a very naive mathematical model, 244 00:11:59,490 --> 00:12:00,840 that makes a lot of sense, right? 245 00:12:00,840 --> 00:12:03,750 You're burning the belongings, you're removing a vector, 246 00:12:03,750 --> 00:12:06,663 you're reducing probability of transmission. 247 00:12:07,590 --> 00:12:10,350 With adaptive behavior, what's likely to happen is that, 248 00:12:10,350 --> 00:12:12,390 you know, these are poor communities, 249 00:12:12,390 --> 00:12:15,840 burning everything that someone owned is a terrible thing. 250 00:12:15,840 --> 00:12:18,420 So it's likely that they just stopped reporting cases. 251 00:12:18,420 --> 00:12:20,700 And that's one of these unexpected consequences, 252 00:12:20,700 --> 00:12:24,750 again, brought on by adaptive behavior. 253 00:12:24,750 --> 00:12:27,570 But we sit all over right now, very locally, 254 00:12:27,570 --> 00:12:30,513 even around COVID-19. 255 00:12:31,410 --> 00:12:36,410 So I'll finish with one last example of a system, 256 00:12:36,600 --> 00:12:39,330 something a little less sad, a little more majestic. 257 00:12:39,330 --> 00:12:42,690 And really, the idea is that in a lot of systems, 258 00:12:42,690 --> 00:12:47,340 we're gonna be studying, it's not one or one property, 259 00:12:47,340 --> 00:12:49,890 or those properties are not even easy to define. 260 00:12:49,890 --> 00:12:52,050 It's really all together that they create this idea 261 00:12:52,050 --> 00:12:53,223 of complex system. 262 00:12:54,300 --> 00:12:58,620 So the last system I'll show and I'll let the narrator 263 00:12:58,620 --> 00:12:59,700 describe the system. 264 00:12:59,700 --> 00:13:03,693 Really brings them all together in a majestic way, I think. 265 00:13:04,724 --> 00:13:07,224 (tense music) 266 00:13:09,690 --> 00:13:11,250 [Narrator 2] At sometimes of the year, 267 00:13:11,250 --> 00:13:13,050 seasonal changes make the currents 268 00:13:13,050 --> 00:13:14,940 especially rich in nutrients. 269 00:13:14,940 --> 00:13:16,650 And then the ocean around the sea mount 270 00:13:16,650 --> 00:13:19,509 becomes a virtual soup of plankton. 271 00:13:19,509 --> 00:13:22,009 (tense music) 272 00:13:27,210 --> 00:13:30,963 At such times, hunters gather in astonishing numbers. 273 00:13:31,881 --> 00:13:34,631 (water sloshing) 274 00:13:40,860 --> 00:13:43,860 Bonito, smaller relatives of the tuna, 275 00:13:43,860 --> 00:13:46,620 they're searching for still smaller plankton feeders 276 00:13:46,620 --> 00:13:48,543 that have been attracted by the bloom. 277 00:13:51,750 --> 00:13:56,013 So are these jacks and their prey is nearby. 278 00:14:01,050 --> 00:14:04,230 A school of anchoveta has strayed up near the surface, 279 00:14:04,230 --> 00:14:05,760 even though it's broad daylight 280 00:14:05,760 --> 00:14:07,210 and hunters are on the prowl. 281 00:14:08,744 --> 00:14:11,494 (water sloshing) 282 00:14:12,990 --> 00:14:15,690 These small fish can already feel the vibrations 283 00:14:15,690 --> 00:14:17,250 of the approaching predators. 284 00:14:17,250 --> 00:14:19,920 Swimming at speed, they formed into a ball 285 00:14:19,920 --> 00:14:22,070 and now, they must wait for whatever comes. 286 00:14:24,855 --> 00:14:27,180 I just love this example. 287 00:14:27,180 --> 00:14:29,700 There's everything that I just mentioned is in there. 288 00:14:29,700 --> 00:14:33,210 The seasonal changes that bring us interconnected, 289 00:14:33,210 --> 00:14:35,610 bring interconnectedness with different systems, 290 00:14:35,610 --> 00:14:39,330 collective behavior, nonlinear dynamics and instabilities. 291 00:14:39,330 --> 00:14:41,430 We're actually gonna go back to this example 292 00:14:41,430 --> 00:14:42,933 throughout the semester. 293 00:14:44,160 --> 00:14:46,890 In two weeks or so, we're gonna be looking at why 294 00:14:46,890 --> 00:14:50,340 during World War I, there was more sharks 295 00:14:50,340 --> 00:14:52,950 and less fish than you'd expect in Italian water, 296 00:14:52,950 --> 00:14:56,463 despite the fact that that fishermen were at war. 297 00:14:58,080 --> 00:15:00,750 Much later, we're gonna look at what kind of rules 298 00:15:00,750 --> 00:15:03,390 do you need to be able to simulate these 299 00:15:03,390 --> 00:15:07,140 like collective behavior for fish balls, right? 300 00:15:07,140 --> 00:15:09,000 It's just a beautiful system. 301 00:15:09,000 --> 00:15:10,800 So I don't want to take too much time right now. 302 00:15:10,800 --> 00:15:14,610 I think it's time that we get to work. 303 00:15:14,610 --> 00:15:18,750 So soon, we're gonna be starting to see the different rules 304 00:15:18,750 --> 00:15:21,960 for how we should even start modeling such systems 305 00:15:21,960 --> 00:15:25,710 in a way that embrace their complexity and their richness. 306 00:15:25,710 --> 00:15:27,494 So it's gonna be a really exciting semester. 307 00:15:27,494 --> 00:15:29,250 I'll see you in the next one. 308 00:15:29,250 --> 00:15:30,083 Thank you.