1 00:00:07,958 --> 00:00:15,451 It is a field of mathematics with very ancient roots, 2 00:00:15,667 --> 00:00:21,603 whose objective, put simply, 3 00:00:22,503 --> 00:00:26,003 is to study the evolution in the very long term... 4 00:00:26,012 --> 00:00:31,516 of a system whose evolution in the short term may be given by a simple rule, 5 00:00:32,188 --> 00:00:41,754 but in which even simple short-term rules can cause big issues in the long term. 6 00:00:42,058 --> 00:00:47,290 One very commonly cited example (because people are very familiar with it), 7 00:00:48,653 --> 00:00:53,852 and which was initially defined in the 17th century, 8 00:00:55,535 --> 00:01:01,418 is a planetary system ruled by Newton's gravitational rules. 9 00:01:01,947 --> 00:01:05,473 It doesn't need much more complication to become interesting. 10 00:01:07,351 --> 00:01:12,328 If you have two masses interacting, 11 00:01:12,503 --> 00:01:18,436 you will have what people know as the behavior of elliptical orbits, 12 00:01:18,585 --> 00:01:22,865 which are well known, are regular and persist in time. 13 00:01:23,959 --> 00:01:27,519 But there is a question that has been asked since then, 14 00:01:27,536 --> 00:01:29,785 and which is still not well understood: 15 00:01:29,786 --> 00:01:38,229 If you have 3 systems interacting through the same gravitation laws for the short-term, 16 00:01:38,363 --> 00:01:42,193 which are well defined and understood for a long time now, 17 00:01:44,559 --> 00:01:48,457 what can you say about behavior in the very long term? 18 00:01:49,799 --> 00:01:54,557 This is what interests Mathematicians looking at this system: 19 00:01:54,558 --> 00:01:59,797 If you ignore other limitations, 20 00:02:02,398 --> 00:02:07,506 like a sun exploding or imploding, or something similar, 21 00:02:10,835 --> 00:02:14,196 or other things that may happen in the timescales we can consider, 22 00:02:14,197 --> 00:02:19,771 then, in a potentially limitless timescale ruled only by gravitational rules, 23 00:02:20,131 --> 00:02:24,362 one would like to be able to make interesting conclusions. 24 00:02:24,363 --> 00:02:29,161 Would these planets, which seem to propagate regularly, 25 00:02:29,281 --> 00:02:35,821 start interacting in more complicated ways and escape far beyond where they began? 26 00:02:36,294 --> 00:02:37,763 Thing like this. 27 00:02:37,764 --> 00:02:42,936 That is one example of a question that can be asked, 28 00:02:44,052 --> 00:02:45,967 and there are many other systems. 29 00:02:46,127 --> 00:02:48,057 We can start from modeling, 30 00:02:48,169 --> 00:02:52,139 such as a model with population interactions, 31 00:02:53,651 --> 00:02:57,755 with several populations competing over simple rules, 32 00:02:58,724 --> 00:03:01,955 and try to predict what could happen with the system. 33 00:03:02,253 --> 00:03:07,626 Maybe you want to know what would happen if you made a small intervention in it. 34 00:03:07,994 --> 00:03:12,141 Would it be catastrophic, or potentially catastrophic? 35 00:03:15,462 --> 00:03:18,579 Another thing that is understood… 36 00:03:19,036 --> 00:03:22,312 There are also systems based on purely theoretical models, 37 00:03:22,440 --> 00:03:27,480 used only to explore abstract approaches to dynamic systems, 38 00:03:27,688 --> 00:03:33,798 or sometimes interactions with other fields of Math, like numbers theory... 39 00:03:35,412 --> 00:03:40,502 and a few abstract dynamic systems that don't model physical phenomena, 40 00:03:40,503 --> 00:03:44,915 but only mathematical ones that answer interesting questions... 41 00:03:44,916 --> 00:03:47,986 that don't initially seem to be about dynamic systems. 42 00:03:48,483 --> 00:03:55,276 But what interests people when we talk about... 43 00:03:55,297 --> 00:04:00,383 the potential to say something interesting on long-term behavior, 44 00:04:00,928 --> 00:04:06,427 one major finding from discoveries made along the last century... 45 00:04:10,432 --> 00:04:18,153 is the discovery of the frequency and inevitability of chaotic behaviors. 46 00:04:19,343 --> 00:04:25,766 These are systems in which chaos appears - which may happen even with simple rules - 47 00:04:27,157 --> 00:04:29,607 despite these simple rules, and in certain scenarios, 48 00:04:29,608 --> 00:04:37,629 it is often theoretically impossible to make very... 49 00:04:39,063 --> 00:04:42,275 accurate predictions of what behavior will be at any given time. 50 00:04:42,335 --> 00:04:45,560 In this case, some questions would be hard to answer: 51 00:04:45,561 --> 00:04:52,575 if I have this system, and, say, look at it a million years from now, 52 00:04:53,577 --> 00:04:57,483 at that moment, what would the state of that system be? 53 00:04:57,852 --> 00:05:03,181 Imagine we are measuring something in a physical system, 54 00:05:03,271 --> 00:05:06,862 like the temperature of something, which is evolving, and ask: 55 00:05:06,958 --> 00:05:10,432 What will temperature be at a very precise moment in the very distant future? 56 00:05:10,561 --> 00:05:14,172 That is often a question that one, when faced with chaos, 57 00:05:14,173 --> 00:05:18,120 can't reasonably give intelligent answers to. 58 00:05:18,894 --> 00:05:24,440 On the other hand, you can ask more statistical questions about the system, 59 00:05:24,441 --> 00:05:29,403 which is also something that has been developed. 60 00:05:29,404 --> 00:05:33,265 The idea is, even faced with this unpredictability, 61 00:05:33,266 --> 00:05:35,563 which indeed arises... 62 00:05:37,403 --> 00:05:42,126 in things like the so-called 'butterfly effect', where even small changes... 63 00:05:42,127 --> 00:05:45,783 create changes so large they prevent predictions, 64 00:05:46,084 --> 00:05:54,251 sometimes it is possible to robustly describe not the exact, precise behavior at point X, 65 00:05:54,252 --> 00:05:56,530 but rather 'average behaviors'. 66 00:05:56,531 --> 00:06:01,621 Something like being able to say 'in period X, 67 00:06:01,752 --> 00:06:04,857 average temperature will likely be about Y'. 68 00:06:05,553 --> 00:06:12,014 This is somewhat related to the robustness of statistical laws. 69 00:06:12,961 --> 00:06:15,475 It can be compared to... 70 00:06:15,952 --> 00:06:19,502 a situation in which you are playing a game of heads and tails. 71 00:06:21,724 --> 00:06:26,315 The question of what the next toss will be... 72 00:06:26,316 --> 00:06:31,059 is not one you can say much about. 73 17:09:18,123 --> 00:06:33,975 It's either heads or tails, 50/50. 74 00:06:34,063 --> 00:06:37,111 So you can't predict accurately. 75 00:06:37,112 --> 00:06:39,453 But you can say, quite confidently 76 00:06:39,454 --> 00:06:46,703 that if you do 1,000 tosses, you will have approximately 500 heads and 500 tails, 77 00:06:46,704 --> 00:06:50,586 with error margins that can be well estimated. 78 00:06:50,587 --> 00:06:53,827 You'll have said something useful about the behavior as a whole, 79 00:06:53,828 --> 00:06:55,627 but not about a single toss. 80 00:06:55,628 --> 00:06:59,998 What happens with chaotic systems is interesting... 81 00:07:00,120 --> 00:07:06,629 Heads/tails tosses have inherent randomness. 82 00:07:06,847 --> 00:07:09,633 But sometimes you start with a system deterministic in principle, 83 00:07:09,634 --> 00:07:17,695 but whose intrinsic complications create behavior close to randomness. 84 00:07:17,696 --> 00:07:22,656 And, in fact, the best way to model this behavior, even though it is deterministic, 85 00:07:22,657 --> 00:07:26,659 is through a law that assumes the system is truly random. 86 00:07:26,660 --> 00:07:31,213 It can be approximated quite well by a system very similar to coin tossing. 87 00:07:32,573 --> 00:07:34,997 That allows you to say interesting things in the same way... 88 00:07:34,998 --> 00:07:38,591 about this system's average behavior. 89 00:07:39,001 --> 00:07:43,841 That's an example of a question, and answers, and type of answer... 90 00:07:43,894 --> 00:07:49,568 modern dynamic systems theory can offer when chaos is involved. 91 00:08:01,336 --> 00:08:05,237 I have worked on several aspects of dynamic systems. 92 00:08:12,093 --> 00:08:18,452 I didn't work only on chaotic systems, that was more a general example. 93 00:08:18,453 --> 00:08:27,510 I often work with systems that have a little less chaos in them, 94 00:08:28,226 --> 00:08:33,786 and one of the methods I used the most often... 95 00:08:33,866 --> 00:08:38,293 in different scenarios is the one known as renormalization. 96 00:08:38,294 --> 00:08:41,938 The concept has a certain history, 97 00:08:45,079 --> 00:08:48,409 and represents different things in different areas. 98 00:08:48,410 --> 00:08:54,081 It means one thing in Physics, another in Dynamic Systems, 99 00:08:54,082 --> 00:08:56,200 and maybe others for other fields of Math. 100 00:08:56,562 --> 00:09:02,853 For Dynamic Systems specifically, it is used to try to understand the behavior... 101 00:09:02,854 --> 00:09:07,581 of a system in a certain 'subarea'. 102 00:09:07,582 --> 00:09:10,745 The system acts over a certain space of configurations, 103 00:09:10,746 --> 00:09:15,922 and you would like to focus attention on behavior in a small subset of that space, 104 00:09:15,923 --> 00:09:19,660 but, as usual, over a very long time horizon. 105 00:09:19,661 --> 00:09:26,234 So you obtain a certain system with this new time and space scale, 106 00:09:26,416 --> 00:09:31,155 which you then re-scale - you renormalize - 107 00:09:31,190 --> 00:09:36,259 and imagine your whole time horizon is in fact just one time beat, 108 00:09:36,260 --> 00:09:42,237 and that subarea is in fact your whole space, that it corresponds to the whole. 109 00:09:42,272 --> 00:09:48,199 In certain situations, the system you obtain through this process... 110 00:09:48,200 --> 00:09:53,310 is qualitatively similar to the one you started with, 111 00:09:56,195 --> 00:09:58,228 even though it is not exactly the same. 112 00:09:58,229 --> 00:10:04,617 In these cases, you will find a sort of operation in the space of dynamic systems... 113 00:10:04,618 --> 00:10:07,753 that has certain specific features. 114 00:10:08,938 --> 00:10:13,811 This change of scale on how you observe systems, 115 00:10:13,812 --> 00:10:16,919 going to a smaller space and longer time scale, 116 00:10:16,920 --> 00:10:21,859 leads from one dynamic system to another, but within the same 'class'. 117 00:10:21,860 --> 00:10:30,477 This idea, surprisingly, works quite well... 118 00:10:31,184 --> 00:10:35,361 in certain classes of systems that are not chaotic. 119 00:10:35,362 --> 00:10:39,961 When you consider a chaotic system and change its time scale, 120 00:10:39,962 --> 00:10:43,284 you produce something with much more chaos. 121 00:10:44,972 --> 00:10:48,434 Information loss becomes much faster with a longer time scale. 122 00:10:51,698 --> 00:10:56,047 But for other systems that are more regular than a completely chaotic one, 123 00:10:56,048 --> 00:11:00,698 you have this potential - not always, but in some situations - 124 00:11:00,699 --> 00:11:06,528 of arriving at a system not so different, qualitatively, from what you had before. 125 00:11:06,529 --> 00:11:13,949 Then, as you try to study and achieve longer and deeper time scales for your system, 126 00:11:13,950 --> 00:11:18,919 You begin trying to iterate this process, 127 00:11:18,920 --> 00:11:21,793 and making that operation in the dynamic systems space. 128 00:11:25,472 --> 00:11:30,195 In general, you consider this acting over systems in small dimensional scales. 129 00:11:30,758 --> 00:11:36,130 These are 1- or 2-dimensional spaces - we live in a 3-dimensional one, 130 00:11:41,035 --> 00:11:44,745 but many interesting dynamic systems happen in low-dimensional spaces, 131 00:11:44,746 --> 00:11:48,927 on a plane or on a straight line. 132 00:11:50,180 --> 00:11:52,785 So if you consider these systems, 133 00:11:52,976 --> 00:11:56,473 and look at this action in the space of the systems, 134 00:11:56,641 --> 00:11:59,681 you're looking at them in space that is often of infinite dimensions 135 00:12:00,102 --> 00:12:02,782 What we are studying here, then, 136 00:12:02,783 --> 00:12:07,500 is this operation in a class of systems that live in an infinite-dimension space. 137 00:12:07,501 --> 00:12:11,001 And, interestingly, when a system operates over itself, 138 00:12:11,002 --> 00:12:14,037 you can repeat it - and you are interested in repeating it. 139 00:12:14,038 --> 00:12:17,647 You get a new dynamic system, with a new set of features, 140 00:12:17,648 --> 00:12:22,043 that now acts, that 'lives', in a much bigger space. 141 00:12:24,993 --> 00:12:29,379 What is surprising is that now you're in something much more complicated, 142 00:12:29,380 --> 00:12:32,221 - from, say, one dimension to infinite dimensions - 143 00:12:32,222 --> 00:12:37,334 but it is frequently possible to say very interesting things... 144 00:12:37,482 --> 00:12:39,514 about this infinite-dimension system. 145 00:12:40,122 --> 00:12:43,811 And this repeats often, for several situations... 146 00:12:45,075 --> 00:12:48,861 when we discover this infinite-dimension system now has chaotic features. 147 00:12:51,547 --> 00:12:56,266 And then the methods that are efficient to analyze certain chaotic systems, 148 00:12:56,267 --> 00:12:59,075 including this probabilistic analysis I mentioned, 149 00:12:59,076 --> 00:13:02,135 can be used to later say interesting things... 150 00:13:02,274 --> 00:13:05,384 about the low-dimension system we were interested in at first. 151 00:13:06,382 --> 00:13:10,270 This sort of idea emerged in the 1970s, 152 00:13:10,419 --> 00:13:15,090 and was well developed in several directions in the next decades. 153 00:13:18,184 --> 00:13:22,306 I applied them several times in my work, 154 00:13:22,307 --> 00:13:25,246 on studies of one-dimensional systems, for example, 155 00:13:25,247 --> 00:13:29,059 or certain scenarios on the behavior of billiards, 156 00:13:29,726 --> 00:13:33,152 which is also something I worked on... 157 00:13:33,153 --> 00:13:39,349 and also on certain aspects of certain Schrodinger operators, 158 00:13:39,753 --> 00:13:44,309 which are also modeled on dynamic systems that frequently can also be renormalized. 159 00:13:57,010 --> 00:14:01,713 It's an idea that has been developing for a while. 160 00:14:02,149 --> 00:14:08,681 I applied and developed certain techniques, and incorporated... 161 00:14:08,716 --> 00:14:12,390 For every scenario in which you use this group of ideas, 162 00:14:12,391 --> 00:14:14,899 some features differ, 163 00:14:14,900 --> 00:14:18,741 and you have to integrate them to understand things. 164 00:14:19,672 --> 00:14:24,320 I began working in the low-dimension dynamics of the interval. 165 00:14:24,321 --> 00:14:29,613 These are very well developed applications, associated with some well-known fractals, 166 00:14:29,614 --> 00:14:33,267 such as the Mandelbrot Set and the like. 167 00:14:37,811 --> 00:14:42,827 But then, over time, I became interested in other systems... 168 00:14:42,828 --> 00:14:45,998 that could also be subject to renormalization theories. 169 00:14:49,604 --> 00:14:52,316 You have to understand how these ideas can be applied, 170 00:14:52,317 --> 00:14:55,528 because the questions that it can help answer are different. 171 00:14:55,529 --> 00:14:59,307 For instance, when I began my work on Schrodinger operators, 172 00:15:00,389 --> 00:15:05,226 it was interesting to me how the questions were asked differently. 173 00:15:05,227 --> 00:15:09,688 The ultimate goal wasn't even to understand a given dynamic system. 174 00:15:10,171 --> 00:15:18,737 The system was the means to analyze the associated equation's behavior... 175 00:15:18,738 --> 00:15:21,576 which will correspond to a dynamic, now a non-classical one. 176 00:15:21,577 --> 00:15:26,538 Before, I was talking of classical dynamics, with deterministic laws. 177 00:15:29,882 --> 00:15:32,853 But when I was interested in Schrödinger operators, 178 00:15:32,854 --> 00:15:35,535 the goal was to understand a quantum dynamics phenomenon, 179 00:15:35,639 --> 00:15:40,382 but before arriving at the equation that will say interesting things about it, 180 00:15:40,383 --> 00:15:47,634 you have the option to use this classical dynamic system model... 181 00:15:47,635 --> 00:15:48,875 to understand a piece of the equation. 182 00:15:48,974 --> 00:15:51,665 Actually you have to understand a family of dynamic systems... 183 00:15:54,118 --> 00:15:59,751 associated with the various possible energies for your quantum system. 184 00:15:59,752 --> 00:16:05,032 For each energy there is a different classical system that you can look at... 185 00:16:05,033 --> 00:16:09,813 and will yield a good behavior of that system's functions. 186 00:16:12,539 --> 00:16:14,447 The issue is you need... 187 00:16:14,448 --> 00:16:20,124 I had a long learning process to discover what interested people. 188 00:16:20,125 --> 00:16:21,496 I began with dynamic systems, 189 00:16:21,497 --> 00:16:26,858 but people had specific questions to ask about these operators, 190 00:16:29,223 --> 00:16:33,991 and on what understanding of the dynamics would allow you... 191 00:16:34,246 --> 00:16:38,942 to reach any conclusions on the aspects of that equation at the end. 192 00:16:40,770 --> 00:16:46,466 Renormalization, then, was one part of a theory, but I branched out later. 193 00:16:46,859 --> 00:16:52,271 I started by using it, it was sort of the way I started. 194 00:16:52,272 --> 00:16:58,347 When I began, this was in 2002, 2003, and I realized... 195 00:16:58,348 --> 00:17:05,933 renormalization could be used to better understand some specific problems. 196 00:17:07,704 --> 00:17:13,067 Later, this started yielding interesting results, 197 00:17:13,444 --> 00:17:19,591 such as conclusions about the fractal known as the Hofstadter's butterfly, 198 00:17:23,259 --> 00:17:31,134 merely by using renormalization slightly differently from how it was used before. 199 00:17:33,183 --> 00:17:37,495 Then I went on to other aspects less related to normalization, 200 00:17:37,496 --> 00:17:40,962 which could, after a long time, 201 00:17:42,460 --> 00:17:47,862 be reunified with renormalization for a deeper understanding of the area. 202 00:17:47,863 --> 00:17:50,135 But this took many years. 203 00:17:50,136 --> 00:17:55,076 I still work on it, but there was slow progress for many years… 204 00:17:55,077 --> 00:17:59,469 Then in 2008-09, 205 00:18:00,727 --> 00:18:04,216 I had a better understanding of how to unify these aspects, 206 00:18:05,022 --> 00:18:09,456 including by looking at another area that is also very present in my work: 207 00:18:10,348 --> 00:18:11,878 the Lyapunov exponents. 208 00:18:13,472 --> 00:18:18,429 Specifically, understanding the appearance of 0 or positive Lyapunov exponents. 209 00:18:18,916 --> 00:18:24,728 Lyapunov exponents are certain numbers associated to a dynamic system... 210 00:18:24,729 --> 00:18:28,625 which detect a bit of the instability of the system. 211 00:18:28,626 --> 00:18:33,084 What is the speed at which orbits separate, for instance? 212 00:18:35,350 --> 00:18:42,323 You begin with two configurations, two orbits that are quite close, 213 00:18:42,324 --> 00:18:45,553 and they begin separating at a certain exponential speed, 214 00:18:45,554 --> 00:18:49,612 which is the rapid loss associated with chaotic systems. 215 00:18:58,384 --> 00:19:03,108 I usually work with many potential problems. 216 00:19:03,109 --> 00:19:05,402 I work on one at a given moment, 217 00:19:05,403 --> 00:19:09,299 but I oscillate a lot over a few days or weeks... 218 00:19:09,300 --> 00:19:12,295 thinking in different directions. 219 00:19:12,296 --> 00:19:14,525 What often happens in this work... 220 00:19:14,526 --> 00:19:18,377 is that you arrive at an obstacle where you lack... 221 00:19:20,988 --> 00:19:26,256 ideas on how you can move forward or what the strategy is. 222 00:19:27,828 --> 00:19:29,990 In that scenario... 223 00:19:30,414 --> 00:19:35,544 where will the idea come from that will allow you to 'unblock', 224 00:19:35,545 --> 00:19:39,154 and continue the work on a more… 225 00:19:41,325 --> 00:19:43,527 objective, actually... 226 00:19:45,431 --> 00:19:47,189 in a more comprehensible way? 227 00:19:47,190 --> 00:19:48,611 Once you have an idea to work with, 228 00:19:48,612 --> 00:19:52,238 there are several techniques that can be used, 229 00:19:52,239 --> 00:19:59,164 and many ways to find associations and move ahead towards a solution. 230 00:19:59,165 --> 00:20:03,645 But without a starting point, that gets very hard. 231 00:20:03,646 --> 00:20:05,077 There isn't much to do. 232 00:20:06,652 --> 00:20:09,338 You know that, with current ideas and techniques, 233 00:20:09,339 --> 00:20:11,742 you can't get past a certain point. 234 00:20:12,556 --> 00:20:15,985 When you get to this 'blockage', 235 00:20:15,986 --> 00:20:19,955 there is no formula on how to get out. 236 00:20:23,200 --> 00:20:28,450 There are theories on how discovery happens, 237 00:20:29,895 --> 00:20:32,242 and some say it happens in the subconscious, 238 00:20:32,243 --> 00:20:37,059 and it is not something that can be 'willed' by the person, 239 00:20:37,060 --> 00:20:42,394 but regardless of those, there are interesting possibilities to consider… 240 00:20:42,971 --> 00:20:46,545 You get to a block, you try, and after a while it becomes clear... 241 00:20:47,602 --> 00:20:50,008 you don't have what you need at that time. 242 00:20:50,835 --> 00:20:53,758 Then maybe it is interesting to look at something else, 243 00:20:53,759 --> 00:20:57,711 first psychologically, so you don't feel you're banging your head against a wall, 244 00:20:57,712 --> 00:21:01,205 and also because, if you look into another direction, 245 00:21:01,206 --> 00:21:06,043 you may end up having a new idea, or... 246 00:21:06,070 --> 00:21:09,890 getting in touch with a new idea, without looking directly at the former direction, 247 00:21:09,891 --> 00:21:12,111 and then a new idea can arise, 248 00:21:12,112 --> 00:21:17,959 that you can see as useful to the problem you were working on before. 249 00:21:18,848 --> 00:21:21,039 This happened with me sometimes. 250 00:21:21,340 --> 00:21:23,084 I was working with multiple problems, 251 00:21:23,275 --> 00:21:26,307 and realized what I was working on in a certain context... 252 00:21:28,060 --> 00:21:33,077 helped open the path for progress in another context. 253 00:21:33,078 --> 00:21:35,142 It could give me some… 254 00:21:38,382 --> 00:21:42,296 Sometimes it's just a different perspective on something. 255 00:21:42,297 --> 00:21:44,391 You basically had all elements, 256 00:21:44,392 --> 00:21:52,770 but you needed the right angle to pick the right way to move forward. 257 00:22:23,350 --> 00:22:25,659 That depends a lot on the situation. 258 00:22:27,409 --> 00:22:29,479 Each project has its own traits. 259 00:22:29,480 --> 00:22:38,754 The choice of working entirely without using paper or the like... 260 00:22:38,755 --> 00:22:42,739 is a bit deliberate in some situations, 261 00:22:42,740 --> 00:22:51,253 to try to avoid the temptation of going for a more 'brute force' approach, 262 00:22:52,940 --> 00:22:55,763 more calculation-heavy approach to systems. 263 00:22:55,764 --> 00:23:00,873 I avoid calculating very complicated things, 264 00:23:00,874 --> 00:23:04,583 and if you don't have paper around you, 265 00:23:04,584 --> 00:23:06,867 it limits what you can calculate. 266 00:23:06,868 --> 00:23:07,995 So, instead of calculating, 267 00:23:07,996 --> 00:23:11,566 you need to understand what you are studying better... 268 00:23:12,231 --> 00:23:17,923 until the moment the calculation you need becomes evident. 269 00:23:18,057 --> 00:23:20,804 It becomes so simple that you can do it in your head. 270 00:23:21,315 --> 00:23:25,032 You try to increase your understanding in a more abstract way, 271 00:23:27,319 --> 00:23:31,141 and see how this can be interpreted differently. 272 00:23:31,142 --> 00:23:35,565 More obvious things, like expressions etc., 273 00:23:35,566 --> 00:23:40,709 can be manipulated so you can look at how they can interact differently. 274 00:23:40,710 --> 00:23:44,712 It is a choice I make, 275 00:23:44,713 --> 00:23:49,284 because I think it tends to be very fruitful. 276 00:23:51,448 --> 00:23:55,633 I think there is a lot that can be done in terms of calculations, 277 00:23:56,537 --> 00:23:58,216 but I usually work in collaboration, 278 00:23:58,217 --> 00:24:02,694 and other people can do the calculations, and sometimes they don't get there. 279 00:24:02,695 --> 00:24:05,756 So I prefer to think a bit of what can be done without calculations, 280 00:24:05,757 --> 00:24:12,674 and sometimes it can merge with the calculation work some of my colleagues did. 281 00:24:27,915 --> 00:24:30,317 I don't really look… 282 00:24:31,117 --> 00:24:37,954 In the situations in which I am working out of the office, 283 00:24:38,019 --> 00:24:39,347 and indeed I do, 284 00:24:39,451 --> 00:24:42,524 either walking, or on the beach… 285 00:24:43,262 --> 00:24:48,164 The goal is not for me to be inspired by my environment. 286 00:24:48,165 --> 00:24:51,819 What is important is that it is a calm environment, 287 00:24:51,820 --> 00:24:57,152 and you can be without a clear distraction. 288 00:24:57,589 --> 00:24:58,766 Interestingly, for instance, 289 00:24:58,950 --> 00:25:01,307 the sound of the ocean will not lead to... 290 00:25:02,687 --> 00:25:05,915 will not cause great distraction, because it is constant. 291 17:27:52,374 --> 00:25:10,499 If you had people screaming around, thing would be more complicated. 292 00:25:10,500 --> 00:25:16,654 You usually want an environment where you do not worry about what is around you. 293 00:25:16,655 --> 00:25:18,871 There may be the occasional distraction, 294 00:25:18,872 --> 00:25:21,632 but mostly you want to have calm periods. 295 00:25:22,362 --> 00:25:25,360 And, at that moment, 296 00:25:25,361 --> 00:25:30,945 what you think depends on the situation. 297 00:25:32,738 --> 00:25:34,246 On the beach, for instance, 298 00:25:34,247 --> 00:25:39,362 I have worked many times trying to refine... 299 00:25:41,584 --> 00:25:46,236 This was work on random matrix multiplications, 300 00:25:46,237 --> 00:25:48,504 which can be seen as dynamic systems. 301 00:25:48,505 --> 00:25:55,773 And the analysis technique I thought could be appropriate... 302 00:25:59,397 --> 00:26:03,987 involved the understanding of certain geometries, or configurations, 303 00:26:07,561 --> 00:26:10,223 on planes of high dimensional complexity, 304 00:26:10,508 --> 00:26:12,909 the different interactions between them.. 305 00:26:13,014 --> 00:26:16,909 how they evolve with the application of certain linear transformations. 306 00:26:16,910 --> 00:26:27,299 So I basically imagined positions for various objects in these spaces, 307 00:26:27,300 --> 00:26:30,148 and how they would be changed by the transformations, 308 00:26:30,149 --> 00:26:35,103 and how these changes could be... 309 00:26:38,127 --> 00:26:42,450 'domesticated' by the technique we were thinking of using. 310 00:26:42,451 --> 00:26:48,850 You had to build certain objects, in the case certain expressions, 311 00:26:48,851 --> 00:26:53,535 which would adapt to possible configurations, and which would evolve... 312 00:26:57,751 --> 00:27:00,254 in a certain direction that would be beneficial to us, 313 00:27:00,255 --> 00:27:04,049 influenced by a certain dynamics - a chaotic one, in this case. 314 00:27:06,577 --> 00:27:07,295 So I did that… 315 00:27:07,296 --> 00:27:13,621 I imagined the types of configuration that were of concern for this technique, 316 00:27:13,622 --> 00:27:14,884 and how I could... 317 00:27:16,340 --> 00:27:21,182 modify the expressions to take this new scenario into account. 318 00:27:22,917 --> 00:27:27,111 And I insisted on doing that without paper, 319 00:27:27,112 --> 00:27:30,934 because it wouldn't be useful for that. 320 00:27:30,935 --> 00:27:35,294 I needed a good understanding of the nature of the obstacle. 321 00:27:35,295 --> 00:27:36,983 In this particular case, 322 00:27:36,984 --> 00:27:41,701 it took a long time to overcome this problem. 323 00:27:41,702 --> 00:27:46,143 Many tries, and conversations with colleagues... 324 00:27:47,718 --> 00:27:48,750 In other situations, 325 00:27:50,349 --> 00:27:54,787 it can be simply a matter of reconsidering something you thought of many times... 326 00:27:54,788 --> 00:27:57,511 and still have no way forward, 327 00:27:58,505 --> 00:28:00,470 and then try again after a while. 328 00:28:00,471 --> 00:28:04,724 Like I said, not always focusing on the problem you don't get, 329 00:28:04,725 --> 00:28:07,159 but it might be worth revisiting later. 330 00:28:07,160 --> 00:28:10,675 Maybe an idea will come out from something else that has matured. 331 00:28:10,676 --> 00:28:12,143 So you try again. 332 00:28:12,144 --> 00:28:18,008 In general you realize that you are still stuck, 333 00:28:19,391 --> 00:28:21,709 but you keep trying, and sometimes a solution comes out. 334 00:28:21,744 --> 00:28:27,514 So you walk, and think again of the problem you were stuck on before, 335 00:28:27,515 --> 00:28:31,911 reflect, understand once more why you couldn't do it… 336 00:28:36,082 --> 00:28:38,545 This last time I went to the beach, for instance, 337 00:28:38,546 --> 00:28:42,545 I was thinking of a problem that often comes to my mind, 338 00:28:45,736 --> 00:28:48,930 and which I always find the same difficulties with. 339 00:28:48,931 --> 00:28:53,078 But I come back to it in the hope I will see something different. 340 00:29:10,131 --> 00:29:13,965 In general, mathematical knowledge... 341 00:29:16,758 --> 00:29:22,089 often ends up being applied to scientific developments, 342 00:29:22,465 --> 00:29:24,440 like in Physics or Chemistry, 343 00:29:24,441 --> 00:29:28,558 and eventually in technological progress. 344 00:29:28,559 --> 00:29:31,311 I would say that, in our current world, 345 00:29:31,312 --> 00:29:35,035 it is quite obvious to see how past Mathematics... 346 00:29:35,319 --> 00:29:41,057 was important for us to have the resources we take for granted today. 347 00:29:41,058 --> 00:29:45,073 So we can imagine the breakthroughs that we achieve in Math today... 348 00:29:45,074 --> 00:29:47,295 will have future repercussions, 349 00:29:47,296 --> 00:29:51,422 even though it is impossible to predict how this will happen, 350 00:29:51,423 --> 00:29:55,019 and what breakthroughs will have the biggest repercussions. 351 00:29:55,020 --> 00:30:00,181 Of course, there is Mathematics that is done... 352 00:30:00,317 --> 00:30:06,469 with clearer goals for immediate application. 353 00:30:06,470 --> 00:30:11,699 You can do research with consequences that are... 354 00:30:13,090 --> 00:30:15,940 immediately calculable for certain technologies… 355 00:30:17,565 --> 00:30:21,838 Certain advances, for instance, in Medicine, 356 00:30:22,750 --> 00:30:25,438 such as techniques in tomography and the like, 357 00:30:25,439 --> 00:30:32,098 are based on research that requires very targeted Mathematical breakthroughs. 358 00:30:35,300 --> 00:30:43,412 That said, I don't think all contributions of a study... 359 00:30:43,541 --> 00:30:48,051 have a direct application in possible technologies. 360 00:30:52,747 --> 00:30:59,284 I would imagine that there is a society of Mathematicians and scientists who interact, 361 00:31:00,606 --> 00:31:06,078 and their ideas interact, and the questions and answers... 362 00:31:06,113 --> 00:31:13,158 of persons working on different things end up reverberating. 363 00:31:13,159 --> 00:31:18,392 So, even if one sees Math... 364 00:31:18,393 --> 00:31:26,741 as a concern merely for its more visible impact on society 365 00:31:26,950 --> 00:31:32,528 - which is not necessarily mine - but even from that standpoint, 366 00:31:32,658 --> 00:31:37,059 the study of a broad spectrum of Mathematics, 367 00:31:37,060 --> 00:31:42,918 going from the purer side to the more applicability-driven side… 368 00:31:43,841 --> 00:31:45,305 It is all part of... 369 00:31:45,977 --> 00:31:48,209 It all interacts in the end. All these ideas interact. 370 00:31:48,244 --> 00:31:51,980 A researcher who studies something more esoteric, 371 00:31:51,981 --> 00:31:55,934 which sometimes seems foolish to observers from the outside… 372 00:31:55,935 --> 00:32:01,216 About objects that look like a game Mathematicians play with themselves… 373 00:32:04,671 --> 00:32:12,383 These studies will often unexpectedly raise questions and techniques... 374 00:32:12,384 --> 00:32:23,303 that slowly propagate to other areas of Mathematics often identified as important, 375 00:32:23,304 --> 00:32:26,964 and sometimes end up being precisely... 376 00:32:26,965 --> 00:32:32,294 what someone with a 'clearer' goal needed. 377 00:32:32,295 --> 00:32:37,781 These ideas go in these directions, and the opposite direction as well. 378 00:32:37,782 --> 00:32:44,956 Many fundamental concepts of interest to Mathematicians who want to "play around", 379 00:32:44,957 --> 00:32:46,934 the 'purer' Mathematicians, 380 00:32:46,935 --> 00:32:52,571 who are more concerned with abstractions, the so-called 'aesthetics of Mathematics', 381 00:32:52,572 --> 00:32:57,568 who want to understand a thing very intrinsically to that thing, 382 00:32:57,569 --> 00:32:59,425 without concern for the outside world, 383 00:32:59,426 --> 00:33:07,271 will also find questions and techniques from research done with clearer goals. 384 00:33:07,272 --> 00:33:12,692 Ideas will go from one side to the other, and vice-versa. 385 00:33:15,444 --> 00:33:19,567 So, an environment where all this research happens concurrently... 386 00:33:19,568 --> 00:33:23,192 is much more productive than if you only had one focus, 387 00:33:23,193 --> 00:33:27,960 and could only look at it in terms of how it can lead you... 388 00:33:27,961 --> 00:33:32,545 to a certain goal within a certain time frame. 389 00:33:32,546 --> 00:33:37,387 I believe many discoveries wouldn't have been made, and won't be made, 390 00:33:37,388 --> 00:33:44,167 without an environment that can drive... 391 00:33:45,346 --> 00:33:49,330 this work in all of these directions at the same time. 392 00:33:49,331 --> 00:33:51,129 So this would be... 393 00:34:20,873 --> 00:34:23,434 I am more concerned with... 394 00:34:24,234 --> 00:34:27,540 I am happy that people are using my work, 395 00:34:30,924 --> 00:34:37,231 but I am more concerned with solving problems that in fact interest me... 396 00:34:37,232 --> 00:34:43,912 and which will interest me due to what I have studied and have been thinking. 397 00:34:43,913 --> 00:34:46,351 This creates a certain 'taste' for Mathematics, 398 00:34:46,352 --> 00:34:54,610 and a trend to try to seek a better understanding of certain parts of Math. 399 00:34:57,574 --> 00:35:02,032 A willingness to understanding something that is often focused on specific issues. 400 00:35:02,033 --> 00:35:10,472 And, in Math, having this understanding often requires solving very specific problems... 401 00:35:10,677 --> 00:35:17,015 that identify the main difficulties associated with that understanding. 402 00:35:20,225 --> 00:35:25,350 Many times, however, you don't work on responding questions asked of you. 403 00:35:25,355 --> 00:35:28,583 Many interesting questions you ask yourself. 404 00:35:30,442 --> 00:35:33,064 It has happened, and sometimes happens, 405 00:35:33,065 --> 00:35:37,900 that as you learn more about a topic, 406 00:35:39,116 --> 00:35:45,588 you see the question as something different from what other people see. 407 00:35:47,524 --> 00:35:50,806 Each one has their own view, 408 00:35:50,807 --> 00:35:55,835 and sees a certain field of Math with their own internal representation, 409 00:35:55,836 --> 00:36:00,900 which is influenced by past experience and knowledge. 410 00:36:00,901 --> 00:36:03,521 Each Mathematician brings their contribution. 411 00:36:03,522 --> 00:36:08,771 I bring mine, which comes from an intercession of many prior things. 412 00:36:08,772 --> 00:36:15,949 In certain circumstances, this position can be particular and special in a certain field, 413 00:36:15,950 --> 00:36:21,289 and that allows me to offer questions and approaches… 414 00:36:21,290 --> 00:36:25,198 that wouldn't have been natural for other people in that area. 415 00:36:25,199 --> 00:36:28,977 This certainly has happened quite clearly... 416 00:36:28,978 --> 00:36:35,684 in my study of Schrodinger operators, or a certain class of them. 417 00:36:35,685 --> 00:36:42,575 I had the ability to pose questions and propose approaches, 418 00:36:42,576 --> 00:36:45,687 and ended up doing a little bit of everything. 419 00:36:48,103 --> 00:36:53,878 I proposed questions that were not natural, but which, when posed, 420 00:36:53,879 --> 00:36:59,285 made it clear that they were essential, and that were needed for progress. 421 00:37:01,266 --> 00:37:03,641 I try to propose questions, 422 00:37:03,642 --> 00:37:07,553 but associated with a possible way to understand it, 423 00:37:07,568 --> 00:37:12,199 with context so that it can be developed. 424 00:37:12,200 --> 00:37:19,788 And then, using my technical characteristics and my individual knowledge, 425 00:37:19,789 --> 00:37:25,717 take that research to its end and find the solution to all this, 426 00:37:25,718 --> 00:37:30,081 leading to a somewhat developed theory... 427 00:37:30,082 --> 00:37:34,374 that will answer the questions that were asked. 428 00:37:34,375 --> 00:37:40,579 Then, when these techniques are developed, they can be used elsewhere, 429 00:37:40,580 --> 00:37:48,545 One would expect, then, that beautiful Mathematical techniques will be reused. 430 00:37:48,546 --> 00:37:54,626 And they tend to - and I find this natural - 431 00:37:56,160 --> 00:37:59,755 become knowledge incorporated by others, 432 00:37:59,955 --> 00:38:03,076 who can use it in varied situations. 433 00:38:04,822 --> 00:38:09,725 It is desirable that this is the natural path of discovery in this field. 434 00:38:24,473 --> 00:38:26,112 I try… 435 00:38:27,946 --> 00:38:32,513 I think people say this more because of my work style. 436 00:38:34,069 --> 00:38:40,063 In a few of the topics I studied, I came sort of from the 'outside'. 437 00:38:43,151 --> 00:38:47,321 I wrote my thesis on a certain field, and when I arrived in Paris, 438 00:38:47,322 --> 00:38:53,640 I found few people who wanted to talk to me about what I was working on. 439 00:38:54,980 --> 00:38:59,593 Then, so I wouldn't keep talking to myself, I wanted to learn what was being done. 440 00:38:59,616 --> 00:39:02,614 But I didn't know a lot about what they were doing. 441 00:39:03,384 --> 00:39:07,441 And when I looked at these fields, 442 00:39:07,442 --> 00:39:09,660 I chose a somewhat… 443 00:39:14,278 --> 00:39:15,794 unconventional approach. 444 00:39:15,795 --> 00:39:23,620 Acceptable, of course, but an unconventional approach to relatively complex things. 445 00:39:23,646 --> 00:39:27,356 Instead of trying to start from the beginning, with simpler questions, 446 00:39:27,357 --> 00:39:29,675 I tried looking at things I would consider important, 447 00:39:29,676 --> 00:39:34,184 but that would tell me something from the point of view I was coming from. 448 00:39:34,962 --> 00:39:40,759 I worked many times with very little understanding of the state of theory, 449 00:39:40,760 --> 00:39:43,508 at least at that time, 450 00:39:43,509 --> 00:39:52,648 and decided to try to look at them with my little knowledge of that particular area, 451 00:39:52,649 --> 00:39:55,726 and see what that tells me. 452 00:39:57,066 --> 00:39:59,019 Sort of thinking from scratch. 453 00:40:00,704 --> 00:40:05,751 This led to certain approaches that were not being used in these areas, 454 00:40:06,496 --> 00:40:12,476 and sometimes, due certainly to some luck and some work, 455 00:40:12,477 --> 00:40:22,001 led to solutions for problems that were somewhat stalled, 456 00:40:22,002 --> 00:40:24,379 that weren't seeing progress in that direction. 457 00:40:24,655 --> 00:40:27,073 That may be why people say that, 458 00:40:27,074 --> 00:40:32,198 because I decided to look at things from a different perspective, 459 00:40:32,199 --> 00:40:35,226 mostly because I didn't know how people were looking at it, 460 00:40:35,227 --> 00:40:36,998 and didn't try too much. 461 00:40:37,877 --> 00:40:44,216 For a while now, I've had this trend of not studying the works of others so much. 462 00:40:44,217 --> 00:40:49,105 I know what has been done, but my main strategy is not to read it in depth. 463 00:40:49,919 --> 00:40:52,699 I try to find out what ideas are in vogue, 464 00:40:52,700 --> 00:40:58,557 but without trying to understand the literature exhaustively. 465 00:41:00,080 --> 00:41:01,657 I get an idea of the fundamental issues, 466 00:41:01,658 --> 00:41:08,536 and try to think of something different that can help to advance these. 467 00:41:08,537 --> 00:41:11,300 I think I would have less to contribute... 468 00:41:11,301 --> 00:41:19,107 if I tried to improve a little on techniques people are already trying to apply. 469 00:41:19,300 --> 00:41:23,390 I think I have a better chance to contribute... 470 00:41:23,391 --> 00:41:28,725 by reflecting on what is not being done, 471 00:41:28,726 --> 00:41:35,311 and on what would be natural for someone not deeply involved in an given area. 472 00:41:35,312 --> 00:41:40,610 I suspect that is what people alluded to when they told you this. 473 00:41:54,626 --> 00:41:56,526 Well, Math... 474 00:41:56,527 --> 00:41:59,660 When you are a child, 475 00:41:59,661 --> 00:42:03,762 you are not really presented to what really is a Mathematical work. 476 00:42:03,763 --> 00:42:09,347 You are introduced to a few numbers first, like I did as a child, 477 00:42:09,348 --> 00:42:14,276 and that they have their characteristics, and some internal logic. 478 00:42:21,718 --> 00:42:26,725 There are a few things that are true and remain true, that don't evolve, 479 00:42:28,705 --> 00:42:30,737 just like other areas of knowledge. 480 00:42:32,051 --> 00:42:35,506 Certain somewhat primitive aspects of Mathematics, 481 00:42:37,792 --> 00:42:39,258 which were all I know, 482 00:42:39,259 --> 00:42:41,525 looked interesting from the start. 483 00:42:42,132 --> 00:42:45,696 It was a very fortuitous thing for my knowledge, 484 00:42:45,697 --> 00:42:52,066 that I really liked to read very basic things about Math, school Math, 485 00:42:52,067 --> 00:42:54,625 that were quite rudimentary, 486 00:42:55,861 --> 00:42:59,966 and I 'didn't dislike' that aspect. 487 00:42:59,967 --> 00:43:06,798 Then I became interested in Math Olympiads, solving Olympiad problems. 488 00:43:06,799 --> 00:43:11,635 And I found those a lot of fun, too. Studying those problems. 489 00:43:11,788 --> 00:43:18,229 And then I started working on more theoretical areas, when I did my Master's, 490 00:43:18,264 --> 00:43:22,101 I had to learn more, which I also found interesting. 491 00:43:22,102 --> 00:43:24,001 This more abstract learning. 492 00:43:24,002 --> 00:43:26,504 And then I started doing research, 493 00:43:26,505 --> 00:43:30,004 with the characteristics I described just now. 494 00:43:30,005 --> 00:43:31,250 And it was interesting. 495 00:43:31,251 --> 00:43:34,664 Throughout those phases, it was possible… 496 00:43:34,665 --> 00:43:40,923 For instance, it's perfectly normal for a great researcher to hate Olympiad problems, 497 00:43:40,924 --> 00:43:42,829 or hate this aspect. 498 00:43:43,698 --> 00:43:45,073 That is perfectly possible. 499 00:43:45,074 --> 00:43:51,798 In my case, in all my life progress until I became a researcher, 500 00:43:52,450 --> 00:43:58,093 maybe because of my story growing up, 501 00:43:58,094 --> 00:43:59,535 it looked very interesting at that time. 502 00:43:59,536 --> 00:44:04,836 Afterwards I lost interest in Math Olympiad problems, for instance, 503 00:44:04,837 --> 00:44:08,635 it doesn't entice me today, but it was important to me at that time. 504 00:44:08,636 --> 00:44:12,319 It is curious that at each moment something interested me, 505 00:44:12,320 --> 00:44:14,459 and allowed me to progress a little bit faster. 506 00:44:14,460 --> 00:44:19,327 There was no moment of "I'm not interested in Mathematics anymore". 507 00:44:19,328 --> 00:44:26,775 "I have lost interest in this area. I don't want this." 508 00:44:26,794 --> 00:44:31,963 This could have make me drift away, 509 00:44:31,964 --> 00:44:33,890 but at each moment I found something interesting, 510 00:44:33,891 --> 00:44:40,732 until I became a researcher, and have had the same interest since then... 511 00:44:43,871 --> 00:44:47,935 to discover new Mathematics and understand certain aspects better, 512 00:44:47,936 --> 00:44:50,288 which is what interests me now. 513 00:45:12,581 --> 00:45:14,306 It's a feature of... 514 00:45:14,307 --> 00:45:19,676 Well, it is an award conceived at a certain time, 515 00:45:19,677 --> 00:45:24,077 and its story is well-known. 516 00:45:25,120 --> 00:45:29,793 The idea was to have an international award, which didn't exist at the time... 517 00:45:29,794 --> 00:45:31,746 It was the first one. 518 00:45:31,747 --> 00:45:37,276 And its message was to convey unity of the Mathematics community. 519 00:45:37,277 --> 00:45:41,956 It would be given by the International Mathematics Congress, in principle open to all, 520 00:45:41,982 --> 00:45:43,107 at a time when there was some… 521 00:45:46,493 --> 00:45:48,438 It was between the two World Wars, 522 00:45:48,439 --> 00:45:51,909 and there were problems with nationalism and things of the sort. 523 00:45:55,780 --> 00:45:55,780 So there are historical reasons for how it was introduced. 524 00:45:59,214 --> 00:46:01,285 The creator of the award, called Fields, 525 00:46:03,726 --> 00:46:07,778 wanted it to be an award given to young people, younger than 40. 526 00:46:07,779 --> 00:46:13,251 It was not meant as an "end-of-career" award to celebrate a body of work, 527 00:46:13,252 --> 00:46:20,306 but rather to highlight promising work, sometimes well fleshed out, 528 00:46:20,307 --> 00:46:25,302 by someone who is active and likely to continue working. 529 00:46:27,286 --> 00:46:34,058 It has characteristics that differ it from a Nobel Prize, 530 00:46:34,059 --> 00:46:38,895 which is usually given as a reward at the end of a career. 531 00:46:38,896 --> 00:46:42,885 I think there are certain positive aspects… 532 00:46:44,051 --> 00:46:45,649 I don't think… 533 00:46:47,902 --> 00:46:51,478 It is not a good inspiration to do good Mathematics, 534 00:46:51,479 --> 00:46:54,023 or good science, or anything, 535 00:46:54,024 --> 00:46:59,891 to seek to obtain some award for it. 536 00:46:59,892 --> 00:47:02,192 I think it is the wrong direction to look, 537 00:47:02,193 --> 00:47:08,178 even because many things are discovered by looking at less promising directions. 538 00:47:09,659 --> 00:47:14,370 By looking at what really interests you, even though it may not interest others, 539 00:47:14,371 --> 00:47:19,074 you end up discovering something that becomes and is seen as fundamental. 540 00:47:19,075 --> 00:47:22,152 It is important that people look at all directions. 541 00:47:22,153 --> 00:47:26,273 If we all look at the direction that looks more likely to be rewarded, 542 00:47:27,067 --> 00:47:30,698 we will have very biased perspectives, and things won't develop as well. 543 00:47:33,460 --> 00:47:39,173 Also, you do better Mathematics if you do something that genuinely interests you. 544 00:47:39,174 --> 00:47:41,331 That is the real motivation. 545 00:47:41,332 --> 00:47:46,218 That, and it is not a very reasonable goal... 546 00:47:46,219 --> 00:47:52,727 to work in depth aiming at something so rare to achieve. 547 00:47:52,728 --> 00:47:54,257 Not many awards like this are given, 548 00:47:54,258 --> 00:47:57,790 they are like a Nobel or the like... 549 00:47:57,791 --> 00:48:00,927 It is interesting that it has an effect... 550 00:48:01,502 --> 00:48:04,274 There is this negative effect, 551 00:48:04,275 --> 00:48:11,838 in that people, when they feel it's possible for them to win this award, 552 00:48:11,839 --> 00:48:16,314 to try to win this award while they can, i.e. under 40, 553 00:48:16,315 --> 00:48:23,538 so, in fact, it is hard to deny that it creates some competition, 554 00:48:23,539 --> 00:48:27,632 but the positive side is that it has an expiration date. 555 00:48:29,559 --> 00:48:33,064 After all, these awards are given, and then fulfill their role. 556 00:48:35,038 --> 00:48:42,666 And for me, their role, be they the Nobel, Fields, and others that are well known, 557 00:48:43,546 --> 00:48:47,000 is to communicate with the public. 558 00:48:47,868 --> 00:48:55,506 It is through these awards that people see progress is being made in science. 559 00:48:55,507 --> 00:48:57,840 So, when people talk… 560 00:48:59,104 --> 00:49:02,402 There is this week of the year when the Nobels are given out, 561 00:49:02,403 --> 00:49:09,366 and people will talk about advances in Medicine, Physics, Chemistry etc. 562 00:49:09,367 --> 00:49:11,124 And this is interesting. 563 00:49:12,582 --> 00:49:14,482 It lets people know that this work is being done, 564 00:49:14,483 --> 00:49:17,832 and how important it is, and that it continues to be done. 565 00:49:17,833 --> 00:49:23,886 So there is this aspect of bringing science closer... 566 17:52:11,179 --> 00:49:26,163 to a community that won't hear this constantly. 567 00:49:26,164 --> 00:49:28,634 That is the main point, I imagine, 568 00:49:28,635 --> 00:49:32,226 the most important contribution of these awards. 569 00:49:32,227 --> 00:49:35,995 Also, like I said, 570 00:49:35,996 --> 00:49:40,891 they have this negative potential to generate harmful competition sometimes. 571 00:49:40,892 --> 00:49:48,700 So the advantage of the Fields medal, in comparison with other career awards, 572 00:49:48,701 --> 00:49:49,918 is that it has a deadline. 573 00:49:49,935 --> 00:49:55,976 When you get to 40 you no longer need to worry about it, regardless of the result. 574 00:49:55,977 --> 00:50:02,187 It fulfills its role, and doesn't have as much of the harmful effect... 575 00:50:02,188 --> 00:50:06,503 in terms of making people continuously obsessed with this goal. 576 00:50:23,788 --> 00:50:27,493 Yes, but I also don't understand a lot of what is done by the community. 577 00:50:28,585 --> 00:50:33,386 This is something you discover relatively quickly as a Mathematician: 578 00:50:35,401 --> 00:50:42,745 the depth and enormous number of different directions people research. 579 00:50:44,179 --> 00:50:47,075 You have characteristics, and preferences… 580 00:50:48,196 --> 00:50:51,000 For instance, just to give you an idea: 581 00:50:51,001 --> 00:50:55,748 I have a line of thought that leads me to approach.... 582 00:50:55,749 --> 00:51:01,503 issues of a general part of Mathematics known as Analysis. 583 00:51:03,110 --> 00:51:04,457 There are estimates, 584 00:51:04,458 --> 00:51:05,944 and you try to obtain notions of physical quantities, 585 00:51:05,945 --> 00:51:10,819 of how large or small things are, relationships, interactions… 586 00:51:10,820 --> 00:51:13,199 But there is some more flexibility. 587 00:51:13,200 --> 00:51:16,045 Other aspects, which I'd call more algebraic, 588 00:51:16,066 --> 00:51:18,962 have rigid equations, manipulated precisely, 589 00:51:19,050 --> 00:51:21,314 that give exact answers all the time... 590 00:51:22,107 --> 00:51:25,996 This is home to another type of thinking, that I find more rigid. 591 00:51:25,997 --> 00:51:29,427 My intuition goes more to the former than the latter, 592 00:51:29,428 --> 00:51:34,415 so I have difficulty trying to understand those, 593 00:51:38,635 --> 00:51:42,008 even things that are not so complicated. 594 00:51:42,009 --> 00:51:45,570 Certainly for the latest advances in these areas, 595 00:51:45,571 --> 00:51:50,097 I lack the framework to be able to say much about them. 596 00:51:50,098 --> 00:51:53,524 So for me it is quite natural that advances that are, 597 00:51:53,525 --> 00:51:57,402 like you said, on the "frontier" of knowledge, 598 00:51:57,403 --> 00:52:02,054 are not accessible to people not working on them. 599 00:52:02,936 --> 00:52:06,324 Yet, aspects... 600 00:52:06,325 --> 00:52:14,773 A few aspects of discoveries end up being gradually conveyed. 601 00:52:14,774 --> 00:52:16,768 This is one of the ideas I put forward. 602 00:52:16,769 --> 00:52:20,316 Some Mathematicians work between two areas, 603 00:52:21,309 --> 00:52:24,244 and they often 'transfer' these ideas. 604 00:52:24,245 --> 00:52:26,074 This communication happens. 605 00:52:26,075 --> 00:52:30,399 Sometimes it needs to go through several Mathematicians to go from here to there, 606 00:52:30,400 --> 00:52:32,791 but it has this potential. 607 00:52:32,792 --> 00:52:34,791 I did that in some directions, 608 00:52:34,792 --> 00:52:39,939 bring some ideas from one subarea of dynamic systems to another, 609 00:52:39,974 --> 00:52:43,805 and this is also done by people working with me in other directions. 610 00:52:48,178 --> 00:52:52,209 I would consider it natural to see this difficulty of communication. 611 00:52:52,210 --> 00:52:55,892 It is part of the complexity of the things we work with, 612 00:52:57,480 --> 00:53:05,939 but at the same time I don't worry much that they will be isolated eternally. 613 00:53:07,349 --> 00:53:12,815 They continue having the potential to influence others... 614 00:53:12,866 --> 00:53:16,973 who may not understand exactly what I do, but can be influenced by a certain path... 615 00:53:17,085 --> 00:53:20,396 by a few ideas developed in this area. 616 00:53:31,629 --> 00:53:37,042 I don't think much in terms of predicting, because if I look back, 617 00:53:37,239 --> 00:53:40,955 I realize I didn't envisage 10 years ago what I am doing now. 618 00:53:41,290 --> 00:53:43,676 I had some idea, but it was basically... 619 00:53:46,077 --> 00:53:50,257 'I'll probably be interested in things that interest me today, 620 00:53:50,258 --> 00:53:54,997 but will certainly be interested in others I cannot predict now'. 621 00:53:54,998 --> 00:53:58,746 For me it is very natural, working with dynamic systems, 622 00:53:58,747 --> 00:54:01,200 to be unable to make such accurate predictions. 623 00:54:01,201 --> 00:54:05,998 It doesn't worry me much not being able to predict as much. 624 00:54:06,800 --> 00:54:12,424 In a more abstract level, or at a different level, 625 00:54:12,425 --> 00:54:17,834 one goal would be to try to maintain the general characteristics… 626 00:54:19,654 --> 00:54:21,229 In Mathematics it is easier. 627 00:54:21,230 --> 00:54:28,148 In other sciences you work close to the technique, and production is direct. 628 00:54:28,149 --> 00:54:33,175 Then you gain managerial responsibilities, 629 00:54:33,176 --> 00:54:38,653 and have to manage other people and their work more than your own. 630 00:54:39,655 --> 00:54:42,485 In Math that doesn't happen often. 631 00:54:42,486 --> 00:54:44,296 You have small teams, 632 00:54:45,808 --> 00:54:47,886 and Mathematicians, as much as they progress, 633 00:54:47,887 --> 00:54:54,789 often tend to work as close to the 'base' as they did when they began. 634 00:54:56,039 --> 00:54:59,882 So my wish, which is not hard to fulfill, 635 00:54:59,883 --> 00:55:05,860 is to continue to work directly 'hands on'. 636 00:55:05,861 --> 00:55:08,145 I am glad that Math is like this. 637 00:55:08,146 --> 00:55:13,180 It would upset me if the consequence of doing good work... 638 00:55:13,181 --> 00:55:16,731 was to be pushed to direct the work of others, 639 00:55:16,732 --> 00:55:21,827 instead of having the pleasure to continuing to make discoveries directly. 640 00:55:34,444 --> 00:55:36,954 It is interesting to think of Mathematics… 641 00:55:40,311 --> 00:55:43,591 Mathematics… often... 642 00:55:44,024 --> 00:55:47,475 Many who did not pursue more knowledge... 643 00:55:47,476 --> 00:55:51,953 often don't know how Mathematicians think of mathematical objects. 644 00:55:51,954 --> 00:55:57,570 Naturally, each will think differently, 645 00:55:57,571 --> 00:56:01,961 but I believe some things are relatively general. 646 00:56:04,507 --> 00:56:07,892 You are not working only with equations and numbers. 647 00:56:07,893 --> 00:56:12,066 There is meaning and there is representation, 648 00:56:12,067 --> 00:56:15,642 basically corresponding to mathematical objects. 649 00:56:20,895 --> 00:56:23,660 These mathematical objects... 650 00:56:23,661 --> 00:56:28,695 They are abstract, they don't have tangible reality, 651 00:56:28,696 --> 00:56:35,862 so each Mathematician has their own interpretation, sees them differently. 652 00:56:35,863 --> 00:56:42,890 Maybe visual, maybe not describable in terms corresponding to senses, 653 00:56:42,891 --> 00:56:46,378 but there is an inner interpretation of that. 654 00:56:49,798 --> 00:56:52,069 And the one creates... 655 00:56:52,070 --> 00:56:58,421 Imagine Mathematics as populated by objects with connections between them, 656 00:57:01,302 --> 00:57:04,605 and with aspects revealed when seen in a certain way. 657 00:57:04,606 --> 00:57:09,371 You have a notion of a rich Mathematical theory, 658 00:57:13,486 --> 00:57:16,330 with connections that appear quite surprisingly, 659 00:57:16,331 --> 00:57:21,150 sometimes between things that did not appear to be related. 660 00:57:21,151 --> 00:57:26,569 And these connections, these intersections... 661 00:57:26,570 --> 00:57:33,872 of many subjects that end up being discovered as you study certain objects, 662 00:57:33,873 --> 00:57:37,799 contribute to this view that they have... 663 00:57:40,858 --> 00:57:42,006 We call it "richness". 664 00:57:42,007 --> 00:57:46,601 And there's an appreciation of this order, this organization… 665 00:57:47,376 --> 00:57:48,348 that takes place. 666 00:57:48,383 --> 00:57:51,181 In general, this is what, for many are people, 667 00:57:51,216 --> 00:57:55,942 tends to lead to a sensation called 'aesthetic appreciation', 668 00:57:55,966 --> 00:58:00,927 and to call these types of objects 'beautiful'. 669 00:58:03,555 --> 00:58:04,738 That is basically it. 670 00:58:04,739 --> 00:58:08,371 This beauty is seen very subjectively by each Mathematician, 671 00:58:08,372 --> 00:58:11,082 and communicated between them. 672 00:58:11,083 --> 00:58:14,532 There is this attempt to communicate, which is interesting. 673 00:58:16,287 --> 00:58:19,511 They have their personal view of objects, 674 00:58:19,512 --> 00:58:21,814 but manage to convey them in a way. 675 00:58:21,815 --> 00:58:25,053 It is imprecise, since you don't convey it totally, 676 00:58:25,054 --> 00:58:31,812 but you do convey a few aspects that enrich the understanding by the other. 677 00:58:31,813 --> 00:58:39,157 But it is sill very individual, how one interprets each universe. 678 01:00:03,148 --> 01:00:08,431 Mathematics always seemed to me to be something... 679 01:00:11,989 --> 01:00:13,578 let's say 'stable', 680 01:00:13,579 --> 01:00:16,624 and which always develops. 681 01:00:16,625 --> 01:00:22,582 The more you learn, the more you find that you ignore as it develops. 682 01:00:23,754 --> 01:00:27,356 Our breakthroughs are much slower... 683 01:00:27,357 --> 01:00:30,886 than the pace of discovery of what we don't know. 684 01:00:31,419 --> 01:00:32,766 On the other hand, 685 01:00:33,673 --> 01:00:37,087 what you can achieve... 686 01:00:38,460 --> 01:00:43,943 becomes part, albeit small, of a growing theory… 687 01:00:46,008 --> 01:00:51,871 And this fact that mathematical knowledge accumulates, 688 01:00:51,872 --> 01:00:56,035 that it is not destructive in its nature… 689 01:00:56,618 --> 01:01:02,133 like can happen in other areas of knowledge, 690 01:01:02,170 --> 01:01:09,322 where theories overlap, or contradict each other at certain times, 691 01:01:09,323 --> 01:01:13,442 or develop, and become obsolete, 692 01:01:15,817 --> 01:01:18,114 and are fully replaced over time. 693 01:01:18,115 --> 01:01:20,079 Mathematical knowledge develops, 694 01:01:21,192 --> 01:01:26,785 but does not invalidate what has been discovered in the past. 695 01:01:26,786 --> 01:01:29,822 There may be more complicated developments, 696 01:01:29,857 --> 01:01:31,933 but it persists quite well. 697 01:01:33,009 --> 01:01:38,660 This is one thing that has always attracted me, and given me some… 698 01:01:38,661 --> 01:01:44,857 I like to think of Math at a moment of tranquility, 699 01:01:44,858 --> 01:01:50,977 and even when I am not in that state to bring some of it, 700 01:01:50,978 --> 01:01:55,038 to think of it as something that 'is there', 701 01:01:55,039 --> 01:01:58,575 that has its characteristics, that has its logic. 702 01:01:58,576 --> 01:02:03,170 It may be difficult sometimes, and it may seem very rigid, 703 01:02:03,171 --> 01:02:08,975 and make you work hard to do things that seem so evident, 704 01:02:11,049 --> 01:02:14,395 but that same work, those same rules, 705 01:02:14,396 --> 01:02:21,724 lead to the fact that when you reach a goal, it is stable. 706 01:02:21,988 --> 01:02:23,237 It has structure. 707 01:02:27,704 --> 01:02:30,782 And that is not something easily found elsewhere. 708 01:02:33,222 --> 01:02:38,868 That is something pleasant to think about. 709 01:02:40,711 --> 01:02:42,528 Maybe at more tense moments, 710 01:02:42,529 --> 01:02:48,786 if you need to separate from anything that may be drawing your attention, 711 01:02:48,787 --> 01:02:53,771 if you manage it, you can be in the mathematical world again, 712 01:02:53,772 --> 01:02:58,633 with its rules, and its stability. 713 01:02:59,064 --> 01:03:01,144 So, it has its moments, 714 01:03:01,169 --> 01:03:03,201 but I wouldn't... 715 01:03:06,037 --> 01:03:12,469 I would say that, emotionally, that has a small role in that sense. 716 01:03:19,034 --> 01:03:20,984 Afraid… I guess I'm not afraid. 717 01:03:20,985 --> 01:03:25,118 You have to be realistic about what you can discover, 718 01:03:25,119 --> 01:03:27,970 and how things happen… 719 01:03:27,971 --> 01:03:28,971 You don't need to... 720 01:03:30,451 --> 01:03:33,483 I don't see it as very problematic when you have moments... 721 01:03:34,365 --> 01:03:37,015 in which it looks like you are not making progress. 722 01:03:37,016 --> 01:03:40,334 That is normal in any long career. 723 01:03:41,735 --> 01:03:44,209 Any mathematician will go through this, 724 01:03:44,210 --> 01:03:50,293 and experiences teaches you that many things depend on... 725 01:03:50,294 --> 01:03:54,689 being at the right moment and having a bit of luck as well. 726 01:03:57,281 --> 01:03:58,682 These oscillations happen. 727 01:03:59,426 --> 01:04:03,919 This type of concern doesn't affect me much. 728 01:04:03,920 --> 01:04:07,807 I think you have to persist in seeking things, 729 01:04:09,223 --> 01:04:14,277 be realistic about the fact that most times you won't get to where you would like to, 730 01:04:16,104 --> 01:04:20,695 remain determined, and appreciate when things work. 731 01:04:22,970 --> 01:04:26,384 In daily work in Math, 732 01:04:27,316 --> 01:04:31,599 most of the time you are working with things you know well, 733 01:04:31,776 --> 01:04:38,120 and developing techniques that you already master, 734 01:04:39,440 --> 01:04:44,231 seeking to take them a bit further. 735 01:04:46,298 --> 01:04:49,891 Most often, this work... 736 01:04:52,592 --> 01:04:54,956 is what is more visible as 'work'. 737 01:04:54,957 --> 01:04:57,922 Like I said, you are often stuck without an idea, 738 01:05:03,267 --> 01:05:05,439 and you wait for this 'unblock'. 739 01:05:05,440 --> 01:05:07,308 So, sometimes it appears you're doing nothing, 740 01:05:07,309 --> 01:05:14,528 then a short moment when you realize something that can make a difference, 741 01:05:14,529 --> 01:05:20,650 and then the more 'conventional' work, 742 01:05:20,651 --> 01:05:28,167 which involves a lot of creativity, but when it's easier to understand what is needed. 743 01:05:28,308 --> 01:05:33,547 It is more 'concrete', perceivable work. 744 01:05:35,312 --> 01:05:38,094 This moment of discovery, 745 01:05:39,896 --> 01:05:44,062 when you feel you understood something different, 746 01:05:44,063 --> 01:05:48,177 is an interesting one emotionally. 747 01:05:48,178 --> 01:05:50,177 It combines several things. 748 01:05:50,178 --> 01:05:56,472 You often get the immediate feeling that it can be important, 749 01:05:56,473 --> 01:06:00,383 but sometimes that's inaccurate. 750 01:06:00,384 --> 01:06:08,215 You become immediately excited about the potential, 751 01:06:08,216 --> 01:06:13,933 of what can unfold from that idea. 752 01:06:13,934 --> 01:06:18,449 You really want it to be fruitful, 753 01:06:18,450 --> 01:06:25,236 and sometimes cling to keeping it alive in the face of difficulties. 754 01:06:25,237 --> 01:06:28,334 At the same time, there are doubts, 755 01:06:28,335 --> 01:06:31,130 and it is part of a Mathematician's work... 756 01:06:31,131 --> 01:06:37,391 to attack your own ideas to find possible faults. 757 01:06:39,487 --> 01:06:44,563 It's an emotionally complicated approach. There are several sides of you. 758 01:06:45,004 --> 01:06:46,849 One side wants the idea to work, 759 01:06:46,850 --> 01:06:53,552 but you have to work on finding the faults in it, 760 01:06:53,553 --> 01:06:56,734 and of course would like not to, 761 01:06:58,039 --> 01:07:01,357 but it is very productive to put a lot of energy into it, 762 01:07:03,071 --> 01:07:06,806 so you're not working with an idea that turns out to be problematic. 763 01:07:08,022 --> 01:07:11,347 Sometimes these moments are complicated… 764 01:07:12,273 --> 01:07:15,612 But this excitement happens at very precise times, 765 01:07:15,613 --> 01:07:19,720 and in general is what precedes this more arduous work. 766 01:07:21,012 --> 01:07:27,287 You need to think about it, but have the feeling something can happen. 767 01:07:31,575 --> 01:07:33,801 I don't believe much... 768 01:07:35,986 --> 01:07:41,072 I don't believe much in looking directly or continuously... 769 01:07:41,073 --> 01:07:43,780 at a very complicated direction. 770 01:07:43,781 --> 01:07:45,721 I think things happens more naturally. 771 01:07:45,722 --> 01:07:50,448 I mentioned that sometimes you could be looking in a different direction... 772 01:07:51,982 --> 01:07:58,331 and you get that 'click', you have an idea and think: 773 01:07:58,332 --> 01:08:03,989 'This could finally be useful for that big problem you considered before'. 774 01:08:03,990 --> 01:08:09,672 So I prefer not to look directly at the problems. 775 01:08:11,832 --> 01:08:13,432 I have done it for some, 776 01:08:13,433 --> 01:08:20,897 but I know of several problems in dynamic systems that are fundamental, 777 01:08:20,898 --> 01:08:26,079 whose understanding and solution would lead to great progress... 778 01:08:26,080 --> 01:08:31,250 in our general understanding of specific dynamic systems that interest me, 779 01:08:32,192 --> 01:08:38,800 but I think it is very hard to predict where you will go, 780 01:08:39,990 --> 01:08:44,948 so I avoid thinking in terms of 'I would really like to solve problem X'. 781 01:08:44,949 --> 01:08:48,410 There are many problems that everybody would like to solve, 782 01:08:48,411 --> 01:08:53,475 but it is not something I define as a major goal. 783 01:09:21,130 --> 01:09:23,590 I find it interesting to talk about things that are non mathematical... 784 01:09:24,286 --> 01:09:27,525 with non-Mathematicians and mathematicians as well. 785 01:09:28,234 --> 01:09:30,321 There are many topics that are interesting… 786 01:09:30,322 --> 01:09:36,677 I don't think I'd have a great need to discuss work with them. 787 01:09:36,678 --> 01:09:42,804 When people can appreciate it, math is one more topic of conversation, 788 01:09:42,805 --> 01:09:49,009 but it's perfectly reasonable for different people to have very different interests, 789 01:09:49,010 --> 01:09:54,270 and I don't see the need to try to talk about these things… 790 01:09:55,642 --> 01:09:58,929 It's natural not to be interested in them. 791 01:10:24,249 --> 01:10:26,818 Well, I have solved many problems... 792 01:10:30,642 --> 01:10:36,019 that resisted approaches for a long time, that is true. 793 01:10:37,348 --> 01:10:39,669 This is a feature of the way I work. 794 01:10:43,786 --> 01:10:47,111 And I like solving problems like that, and to formulate questions. 795 01:10:47,861 --> 01:10:54,280 But I certainly am attracted to some questions not just because they were open, 796 01:10:54,281 --> 01:10:57,799 even because there are so many them, 797 01:10:57,800 --> 01:11:03,947 but some because they cover issues for which I've developed a certain 'affection'. 798 01:11:07,049 --> 01:11:08,850 This brings self-satisfaction, 799 01:11:10,018 --> 01:11:14,327 and it is nice to speak to other Mathematicians about them, 800 01:11:14,328 --> 01:11:17,734 but is not something I need to discuss with everyone. 801 01:11:17,735 --> 01:11:20,390 I don't need everyone to appreciate it. 802 01:11:21,472 --> 01:11:25,784 The most fundamental appreciation for your work must be your own. 803 01:11:25,785 --> 01:11:32,437 If you don't appreciate your own work, you'll have a lot of trouble pursuing it. 804 01:11:32,830 --> 01:11:34,997 This is something you have to seek, and once you get it, 805 01:11:35,069 --> 01:11:39,013 it is nice that your colleagues appreciate it too, 806 01:11:39,014 --> 01:11:41,036 and to talk to them about it, 807 01:11:41,037 --> 01:11:45,090 to have this exchange and hear from them, 808 01:11:45,091 --> 01:11:46,967 but it is a part of it… 809 01:11:48,187 --> 01:11:54,017 I have no need for everyone to appreciate my contributions. 810 01:12:01,717 --> 01:12:02,416 No… 811 01:12:02,649 --> 01:12:08,784 I always work a lot with others, and find people I can talk to. 812 01:12:08,785 --> 01:12:11,933 In some of the work there was a gap... 813 01:12:11,934 --> 01:12:19,465 between the needs to make it progress and what others could contribute, 814 01:12:19,466 --> 01:12:27,042 even because it took many simultaneous traits and I was more suitable to do it alone, 815 01:12:27,043 --> 01:12:31,022 but as time goes by, if you work is interesting, 816 01:12:31,023 --> 01:12:39,268 it's normal for people to follow and investigate how they can use it, 817 01:12:40,663 --> 01:12:43,268 or even improve what we had. 818 01:12:45,434 --> 01:12:48,028 I don't have this feeling... 819 01:12:49,052 --> 01:12:51,354 Mathematicians, they have… 820 01:12:52,844 --> 01:12:57,666 There is that contribution that leads to solutions to problems, 821 01:12:58,297 --> 01:13:03,391 but that often comes from work that didn't bring solutions but introduced ideas, 822 01:13:03,392 --> 01:13:08,469 so I am aware of and greatly appreciate this work that is done, 823 01:13:08,470 --> 01:13:15,194 and is often overlooked by awards etc., 824 01:13:15,195 --> 01:13:18,741 but that is fundamental for the progress of Math. 825 01:13:20,010 --> 01:13:24,700 So I see the contributions of all who worked in this field. 826 01:13:24,701 --> 01:13:29,196 Each of them did something that, if they hadn't, 827 01:13:29,197 --> 01:13:32,644 I wouldn't have done anything or have a field to work on. 828 01:13:35,169 --> 01:13:36,285 I also know that, 829 01:13:36,286 --> 01:13:40,511 even when something I do has technical difficulties... 830 01:13:40,512 --> 01:13:43,729 that means it may take a while to be appreciated, 831 01:13:43,730 --> 01:13:48,728 I do hope it is, and that it will be useful to people. 832 01:13:48,729 --> 01:13:51,538 That it's not something that stays isolated in me, 833 01:13:51,539 --> 01:13:59,313 but that increases the overall mathematical knowledge of the community. 834 01:13:59,314 --> 01:14:00,314 So there's that. 835 01:14:00,315 --> 01:14:01,501 It's not just… 836 01:14:01,502 --> 01:14:06,548 I would be reasonably satisfied to have my own fulfillment... 837 01:14:06,549 --> 01:14:10,485 for having gotten somewhere on a problem whose solution interests me, 838 01:14:10,486 --> 01:14:15,732 but there's this additional component, and I don't particularly feel it is missing. 839 01:14:52,807 --> 01:15:01,527 Between mathematical thought and everyday life? 840 01:15:02,583 --> 01:15:03,588 In that sense? 841 01:15:14,470 --> 01:15:18,590 Depends on the type of problem you're working on. 842 01:15:20,011 --> 01:15:22,274 There are certain situations in which... 843 01:15:22,275 --> 01:15:27,843 you do these internal representations of the object, 844 01:15:27,844 --> 01:15:31,194 and try to look at them differently. 845 01:15:35,768 --> 01:15:42,970 It could be by seeking a side you haven't explored yet but could develop, 846 01:15:42,971 --> 01:15:46,019 or it could be manipulations, 847 01:15:46,020 --> 01:15:54,400 where it is clear, or you have identified something that can lead to a solution, 848 01:15:54,401 --> 01:15:59,020 but which has some internal complexities. 849 01:16:02,010 --> 01:16:05,242 Then you do something close to - albeit quite different - 850 01:16:05,243 --> 01:16:08,339 from what non-Mathematicians call 'calculation', 851 01:16:08,340 --> 01:16:13,401 and then do approximations that we Mathematicians would also 'calculate', 852 01:16:13,402 --> 01:16:16,752 to estimate the consequences of certain processes. 853 01:16:16,753 --> 01:16:20,099 You do certain procedures to manipulate objects, 854 01:16:20,100 --> 01:16:28,520 and see if that reveals any other traits to be explored. 855 01:16:29,672 --> 01:16:33,753 In that process, you are indeed doing a sequence of operations, 856 01:16:33,754 --> 01:16:39,625 and imagining which next operation can be useful, 857 01:16:39,626 --> 01:16:44,974 and all that happens inside your head… 858 01:16:46,629 --> 01:16:48,399 Hard to describe further than that. 859 01:16:48,400 --> 01:16:52,161 And that also varies in each situation. 860 01:17:14,340 --> 01:17:17,831 It is possible to get distracted in a few situations. 861 01:17:17,856 --> 01:17:21,311 Not sure if much more than other people, 862 01:17:21,374 --> 01:17:24,926 but I do have this tendency, when I start thinking of something, 863 01:17:24,927 --> 01:17:27,161 to become somewhat involved. 864 01:17:28,471 --> 01:17:35,635 I think certain activities require some level of exclusive dedication, 865 01:17:35,636 --> 01:17:41,100 especially if it creates certain risks to yourself or to other people, 866 01:17:41,101 --> 01:17:42,636 such as driving. 867 01:17:47,167 --> 01:17:48,893 And I would imagine that there is a trend... 868 01:17:48,894 --> 01:17:54,422 to get comfortable with anything that becomes repetitive, 869 01:17:54,423 --> 01:17:57,519 and in which you develop a feeling of safety. 870 01:17:59,781 --> 01:18:05,194 And that could lead you to think of other things, 871 01:18:05,195 --> 01:18:11,860 initially lightly, but then draw a lot of your attention inadvertently. 872 01:18:14,501 --> 01:18:19,520 That may not cause any consequences, maybe out of luck, 873 01:18:19,886 --> 01:18:26,602 but you are creating the possibility that it will. 874 01:18:28,474 --> 01:18:35,830 There is also the issue that I prefer to live in urban areas... 875 01:18:35,831 --> 01:18:41,619 where the need to have your own car is reduced. 876 01:18:41,620 --> 01:18:48,616 I don't make sacrifices by not driving while living in Rio or in Paris, 877 01:18:48,617 --> 01:18:53,625 while in other cities of the world that could be an issue. 878 01:18:53,626 --> 01:18:57,383 I don't feel this is a problem with my life here, though, 879 01:18:57,857 --> 01:19:03,410 so I think it's not worth risking this, 880 01:19:05,478 --> 01:19:10,025 or not risking it and having to force myself to fully dedicate... 881 01:19:10,026 --> 01:19:16,874 to a relatively repetitive activity feeling like I can't distract from it. 882 01:19:19,225 --> 01:19:21,598 I don't see the need for that. 883 01:19:27,543 --> 01:19:32,505 I think it is easier to answer how I see… 884 01:19:38,081 --> 01:19:41,314 why Math is something interesting to develop, 885 01:19:41,315 --> 01:19:45,395 or why people should study it. 886 01:19:45,396 --> 01:19:48,772 One of the reasons, which I mentioned, 887 01:19:49,809 --> 01:19:55,518 has to do with repercussions and developments... 888 01:19:55,519 --> 01:20:00,690 of technological progress and science in general through Math, 889 01:20:00,691 --> 01:20:06,509 but Math, specially seen from the 'purer' side of it, 890 01:20:06,510 --> 01:20:11,842 is often developed as something close to a form of... 891 01:20:13,381 --> 01:20:15,384 artistic expression. 892 01:20:18,169 --> 01:20:26,874 Maybe it is something that attracts fewer people than music or sculpture, 893 01:20:28,398 --> 01:20:34,233 because understanding Math requires, even for its simpler concepts, 894 01:20:34,234 --> 01:20:42,174 a certain immersion in a prior study that other artistic expressions... 895 01:20:42,175 --> 01:20:52,427 don't require for you to immediately see why it is beautiful or moving. 896 01:20:52,428 --> 01:20:59,644 But, with some preparation and dedication, 897 01:20:59,645 --> 01:21:02,254 it can have the same potential impact. 898 01:21:02,255 --> 01:21:08,554 Like I said, you can appreciate this, this order, 899 01:21:08,555 --> 01:21:14,198 this increasing structuring, and the surprises that appear, 900 01:21:14,199 --> 01:21:21,866 and things that are very similar or very different when you don't expect them to be, 901 01:21:21,867 --> 01:21:27,856 and all these characteristics are... 902 01:21:27,857 --> 01:21:33,120 something that many people can (and do) seek... 903 01:21:35,967 --> 01:21:39,149 to advance in this... 904 01:21:41,801 --> 01:21:46,028 advance in this search for abstract understanding, 905 01:21:46,033 --> 01:21:48,808 and the formation of a mathematical universe. 906 01:21:48,809 --> 01:21:56,150 You are building, or helping to build, this piece… 907 01:21:56,151 --> 01:22:01,913 Maybe not a piece, but this knowledge of this interconnected mathematical universe... 908 01:22:02,297 --> 01:22:06,862 that each one only knows a small part of, but can contribute to. 909 01:22:09,226 --> 01:22:12,629 I see it as quite a natural thing, then, 910 01:22:12,630 --> 01:22:17,860 that it attracts people. 911 01:22:17,861 --> 01:22:22,161 Not all people, since each will be attracted to something different. 912 01:22:23,221 --> 01:22:26,936 But it can be just as attractive, and is, to certain people, 913 01:22:26,937 --> 01:22:31,932 as other completely natural forms of expression. 914 01:22:31,933 --> 01:22:34,094 Math has a history that goes back to… 915 01:22:34,901 --> 01:22:39,806 even before then, but recognizably to the ancient Greeks, 916 01:22:40,282 --> 01:22:46,401 who often sought to learn Math or develop it... 917 01:22:46,544 --> 01:22:52,177 because they considered it something natural to pursue. 918 01:22:55,685 --> 01:22:59,220 As part of the human quest. 919 01:22:59,221 --> 01:23:02,586 It's been like that since the beginning, 920 01:23:02,587 --> 01:23:07,422 and it is important to emphasize that additional importance: 921 01:23:07,423 --> 01:23:15,597 that, in addition to these more perceivable, tangible repercussions, 922 01:23:15,598 --> 01:23:23,173 we should remember that it is more than that. 923 01:23:23,174 --> 01:23:29,440 It incorporates other things that should be appreciated in a more intrinsic manner. 924 01:24:20,076 --> 01:24:24,354 In Math, I see it as you... 925 01:24:25,238 --> 01:24:29,425 finding out more about things that are well-established discoveries. 926 01:24:29,570 --> 01:24:31,174 Of course, people make mistakes. 927 01:24:31,986 --> 01:24:33,273 It is part of the process. 928 01:24:34,481 --> 01:24:37,095 Any person can make mistakes, which may be short-lived, 929 01:24:37,096 --> 01:24:40,726 but other than these errors - and those are often quite limited - 930 01:24:43,280 --> 01:24:49,536 the mathematical knowledge being developed is considered in general to be true. 931 01:24:49,537 --> 01:24:57,346 There is no serious dispute between mathematicians with different interpretations... 932 01:24:57,347 --> 01:25:04,105 regarding the true nature of the status of a certain discovery made. 933 01:25:06,052 --> 01:25:12,281 Mathematics is populated by small truths that correspond to a growing universe, 934 01:25:12,282 --> 01:25:17,886 and so there is no great doubt when one is working. 935 01:25:17,887 --> 01:25:22,218 It is something quite characteristic of our knowledge. 936 01:25:23,814 --> 01:25:29,333 There have been moments of discussions in that sense, 937 01:25:29,334 --> 01:25:33,688 the Mathematicians... 938 01:25:33,689 --> 01:25:39,909 about the foundational issues of Math, including in dialogue with Philosophers... 939 01:25:39,910 --> 01:25:47,949 but we have arrived at notions of the limitations, of where Math can take us. 940 01:25:47,950 --> 01:25:53,210 Ambitions of total formalism have been somewhat abandoned. 941 01:25:53,211 --> 01:25:58,113 The known limitations coming from Logic, coming from Gödel, for instance, 942 01:25:59,761 --> 01:26:07,724 may have destroyed the more exaggerated ambitions of the beginning of the century, 943 01:26:09,143 --> 01:26:10,463 I mean the 20th century, 944 01:26:11,600 --> 01:26:12,640 and are now well understood. 945 01:26:13,844 --> 01:26:19,840 Mathematicians essentially accept the limitations of what they can work on. 946 01:26:20,728 --> 01:26:22,345 They know these limitations. 947 01:26:22,346 --> 01:26:25,642 But they seek to work on certain well-defined objects, 948 01:26:25,643 --> 01:26:31,380 they have the continuous feeling of obtaining something true and permanent. 949 01:26:41,197 --> 01:26:46,482 In simple terms, pure Mathematics is considered to be that which is investigated... 950 01:26:46,483 --> 01:26:51,231 without the goal of achieving immediate application outside Math. 951 01:26:51,232 --> 01:26:55,647 It may find application, and many times it does. 952 01:26:55,648 --> 01:26:57,478 Things that initially belonged... 953 01:26:57,479 --> 01:27:06,241 It is more a way of selecting your research objects and your intention in studying them. 954 01:27:07,412 --> 01:27:13,617 Sometimes the type of Math is the same used in applied and pure Math, 955 01:27:13,701 --> 01:27:15,542 sometimes the objects are the same. 956 01:27:16,075 --> 01:27:21,046 This distinction is based on intent and this leads to some selection, 957 01:27:21,475 --> 01:27:24,323 in that applied Math will work more on certain types of objects, 958 01:27:24,324 --> 01:27:26,185 and pure Math on others. 959 01:27:26,186 --> 01:27:30,596 In the end, you will potentially be using the same things. 960 01:27:31,000 --> 01:27:36,345 If it is Math, the same tools are valid for one side or the other. 961 01:27:36,346 --> 01:27:41,454 But there is this distinction on what you select and what goals you have: 962 01:27:41,455 --> 01:27:48,102 Is your interest intrinsic to Math, or are you seeking direct application of it? 963 01:27:49,960 --> 01:27:51,801 I put myself… 964 01:27:51,802 --> 01:27:57,160 I am more interested in the 'intrinsic interest' side… 965 01:27:57,286 --> 01:28:03,015 I sometimes work on problems directly inspired by physical models etc., 966 01:28:03,016 --> 01:28:08,949 but the reason I select it to work on is not its applicability, 967 01:28:08,950 --> 01:28:12,735 but rather because the associated mathematical theory interests me, 968 01:28:12,736 --> 01:28:16,504 due to the characteristics I mentioned, such as its richness, beauty etc. 969 01:28:55,879 --> 01:28:58,023 At that moment… 970 01:29:00,359 --> 01:29:04,039 When I'm reflecting on issues, 971 01:29:04,494 --> 01:29:09,713 seeking another idea without a very well-defined plan, 972 01:29:10,107 --> 01:29:11,637 and you're not... 973 01:29:13,527 --> 01:29:19,420 You have some hope, but are just seeking a starting point, 974 01:29:19,455 --> 01:29:23,891 that takes relatively little time… 975 01:29:24,358 --> 01:29:26,745 It's like I said, I could be working on something else, 976 01:29:26,746 --> 01:29:31,994 but back then I was working on a question I didn't have a clear path for, 977 01:29:31,995 --> 01:29:35,641 and that occupies relatively little time. 978 01:29:35,642 --> 01:29:41,426 Much more time is spent on mathematical work when... 979 01:29:42,468 --> 01:29:44,853 you start with an idea you already had, 980 01:29:44,854 --> 01:29:50,346 and trying to develop it, and use other techniques... 981 01:29:50,347 --> 01:29:57,084 to make connections in a more predictable way. 982 01:30:01,901 --> 01:30:07,509 This attempt to arrive more consciously at an idea... 983 01:30:07,510 --> 01:30:10,068 without knowing where it will come from... 984 01:30:10,069 --> 01:30:12,147 is something I spent less time on. 985 01:30:12,148 --> 01:30:15,584 I look at it again and check if something has matured, 986 01:30:15,585 --> 01:30:19,754 but in general you get to the conclusion you already had very quickly. 987 01:30:19,755 --> 01:30:22,530 Then, either something appears, 988 01:30:22,531 --> 01:30:26,568 or you realize your manipulations keep leading you back to the same place, 989 01:30:26,569 --> 01:30:30,032 and that you're not getting any further. 990 01:30:30,033 --> 01:30:35,022 I don't consider it very useful to keep persisting in this. 991 01:30:36,495 --> 01:30:41,359 In this problem in particular, recently I've been ending up in the same place. 992 01:30:41,360 --> 01:30:45,776 Even though I try small variations, they all take me to the same place. 993 01:30:45,938 --> 01:30:51,817 When I realize this, I tend to abandon it for a bit longer. 994 01:30:51,818 --> 01:30:54,410 But that is not a large part of my day. 995 01:30:59,648 --> 01:31:02,276 There isn't a set rhythm… 996 01:31:02,277 --> 01:31:04,731 Every day has its characteristics. 997 01:31:04,732 --> 01:31:10,273 I will spare a moment among my daily tasks to think of Math, 998 01:31:10,274 --> 01:31:16,480 but as part of my practical obligations and the like. 999 01:31:16,481 --> 01:31:19,357 Since I don't have a lot of standing obligations… 1000 01:31:22,069 --> 01:31:29,293 Sometimes I will arrange to speak with a student or a colleague, 1001 01:31:29,294 --> 01:31:31,852 and that will occupy a specific part of my day. 1002 01:31:31,853 --> 01:31:36,166 The moment I spare to work alone is when I feel more available, 1003 01:31:36,167 --> 01:31:37,167 but that changes from day to day. 1004 01:31:37,168 --> 01:31:39,559 I don't have fixed times in my schedule. 1005 01:31:39,560 --> 01:31:44,582 I also collaborate a lot with people from other countries on-line, 1006 01:31:44,583 --> 01:31:49,342 so I need to adapt my schedule to time zones etc. 1007 01:31:49,343 --> 01:31:52,879 No need to follow a very precise routine. 1008 01:32:21,684 --> 01:32:23,231 That changes a lot, but yes, 1009 01:32:24,047 --> 01:32:27,237 for a while now I've had the habit of waking up late. 1010 01:32:27,911 --> 01:32:32,902 I usually wake up at around noon or later. 1011 01:32:32,903 --> 01:32:37,496 And… These are my hours. 1012 01:32:37,545 --> 01:32:42,434 Since I don't have the need to wake up in the morning, I don't, 1013 01:32:42,435 --> 01:32:45,162 but on the other hand, I go to sleep later. 1014 01:32:46,183 --> 01:32:47,659 Other than that… 1015 01:32:47,660 --> 01:32:50,856 I wake up and take care of a few practical things, 1016 01:32:50,857 --> 01:32:55,057 like eating, going to the gym… 1017 01:32:55,058 --> 01:32:58,367 I do my chores when it is most convenient. 1018 01:32:59,527 --> 01:33:04,468 And like I said, there are the things that are more 'identifiable' as work, 1019 01:33:04,469 --> 01:33:08,495 and in certain moments you may find yourself thinking… 1020 01:33:08,496 --> 01:33:12,686 It has to be at a clear-headed moment… 1021 01:33:12,687 --> 01:33:20,130 It's hard, at least for me, to say "I will book this time for abstract thinking." 1022 01:33:20,131 --> 01:33:22,941 I may schedule to speak to colleagues for this, 1023 01:33:22,942 --> 01:33:27,654 but when I'm alone, I will choose a moment in which I feel more serene, 1024 01:33:27,655 --> 01:33:35,225 and without other sources of stress that can divide my attention. 1025 01:33:35,226 --> 01:33:39,013 I wait for these moments to happen naturally. 1026 01:33:58,434 --> 01:33:59,834 I don't like... 1027 01:33:59,835 --> 01:34:01,941 I don't think that's necessary, 1028 01:34:01,942 --> 01:34:06,590 but I don't like to plan very carefully how things will happens, 1029 01:34:06,591 --> 01:34:13,197 because I end up wanting to change things at the last minute. 1030 01:34:13,198 --> 01:34:20,123 Every plan I make may not be ideal, and I'll be tempted to change them, 1031 01:34:20,124 --> 01:34:24,574 so I try not to plan too far in advance. 1032 01:34:26,774 --> 01:34:29,799 I have some difficulty organizing in that sense, 1033 01:34:29,800 --> 01:34:36,175 but fortunately my work allows me to keep doing it this way. 1034 01:34:36,176 --> 01:34:39,485 I could do it - and many Mathematicians do it without issue - 1035 01:34:39,486 --> 01:34:42,875 within a very well-established routine. 1036 01:34:42,876 --> 01:34:46,692 But I feel uncomfortable with very rigid things, 1037 01:34:46,693 --> 01:34:52,605 so this would likely have a psychological impact of making me feel restricted, 1038 01:34:52,606 --> 01:34:54,981 and would bother me. 1039 01:34:54,982 --> 01:35:00,235 It is not a prerequisite to do the type of discovery I do. 1040 01:35:00,236 --> 01:35:05,425 I think it is more because of the emotional perception than anything else. 1041 01:35:18,000 --> 01:35:20,269 I find it interesting that in Math... 1042 01:35:20,270 --> 01:35:26,857 we are very accepting of people who can contribute however they can, 1043 01:35:26,858 --> 01:35:37,544 and that we do not put much stock in social conventions or norms. 1044 01:35:37,545 --> 01:35:42,765 They exist, but they do not prevent a person from doing their work. 1045 01:35:42,766 --> 01:35:49,039 In certain professions, people who do not conform to norms, 1046 01:35:49,040 --> 01:35:53,694 such as dress or basic behavior codes, 1047 01:35:53,695 --> 01:35:56,906 won't have the chance to work in them. 1048 01:35:56,907 --> 01:36:01,602 In Math, there might be people who are isolated, 1049 01:36:03,498 --> 01:36:06,892 or who are not very proficient in talking to others, 1050 01:36:06,893 --> 01:36:09,097 but who are contributing in many ways. 1051 01:36:09,098 --> 01:36:18,298 In the end their work will be assessed for its content, for what it brings. 1052 01:36:18,299 --> 01:36:21,030 Since there are no interpretations, 1053 01:36:21,031 --> 01:36:23,269 you don't have the defend your arguments... 1054 01:36:23,270 --> 01:36:26,284 and compare with different arguments and interpretations. 1055 01:36:26,826 --> 01:36:30,009 Normally, in most works, 1056 01:36:30,010 --> 01:36:34,832 you have a theory being developed, 1057 01:36:34,833 --> 01:36:42,645 and it can be assessed despite the rhetorical skills of the person defending it. 1058 01:36:42,646 --> 01:36:44,809 Any other person can then see it, 1059 01:36:44,810 --> 01:36:49,560 if the material is prepared according to the usual standards, 1060 01:36:49,561 --> 01:36:56,620 and say 'Yes, this answers the question' or not regardless of the person's personality. 1061 01:36:56,621 --> 01:37:03,933 This is quite desirable, because it doesn't exclude people that could contribute. 1062 01:37:03,934 --> 01:37:10,499 You don't have to be eccentric, or have difficulties interacting, 1063 01:37:11,810 --> 01:37:15,518 or any other odd trait to do Math, 1064 01:37:15,519 --> 01:37:19,666 but we don't exclude people in that regard. 1065 01:37:20,830 --> 01:37:22,939 There are all sorts of people in Math. 1066 01:37:22,940 --> 01:37:28,146 It is a stereotype to think that the rule is to have odd people, 1067 01:37:28,147 --> 01:37:30,046 it's just that they are accepted. 1068 01:37:30,584 --> 01:37:33,486 They are included, and often contribute, 1069 01:37:33,487 --> 01:37:37,629 and all sorts of people can be represented. 1070 01:38:08,319 --> 01:38:12,179 It depends on what you are investigating about the world. 1071 01:38:14,821 --> 01:38:19,933 There is this difficulty of making predictions in certain time scales, 1072 01:38:19,934 --> 01:38:24,661 and the effect of chaos must be recognized. 1073 01:38:24,662 --> 01:38:28,213 On the other hand, there is great order, 1074 01:38:30,299 --> 01:38:36,121 for instance, simply in the capacity to express physical laws in mathematical terms. 1075 01:38:36,122 --> 01:38:41,053 It is not obvious why that is possible, even in principle. 1076 01:38:41,054 --> 01:38:46,885 The fact that this happens allows for short-term predictability. 1077 01:38:46,886 --> 01:38:50,248 There are simple rules that govern this behavior, 1078 01:38:50,277 --> 01:38:52,613 This is a form of order that has no reason to exist. 1079 01:38:52,693 --> 01:38:54,494 We take it for granted today... 1080 01:38:55,718 --> 01:39:01,182 that certain aspects of Physics will have this type of behavior, 1081 01:39:01,622 --> 01:39:04,356 but it is not at all obvious that it has to be that way. 1082 01:39:05,237 --> 01:39:09,759 This ensures us the possibility of living our every-day lives... 1083 01:39:09,760 --> 01:39:14,177 knowing that nothing completely unusual will happen from a moment to the other. 1084 01:39:14,316 --> 01:39:16,157 There is a certain awareness of that. 1085 01:39:16,158 --> 01:39:17,504 So it depends on the issue. 1086 01:39:17,505 --> 01:39:24,592 You have to recognize in which situations chaos will be predominant, 1087 01:39:24,593 --> 01:39:27,289 and in which order will prevail. 1088 01:39:27,290 --> 01:39:28,661 You don't have to worry much about... 1089 01:39:28,662 --> 01:39:33,786 the fact that tomorrow the Moon will be present in some way or another. 1090 01:39:33,787 --> 01:39:40,803 Maybe full moon, maybe new, but there is not much doubt it will be there. 1091 18:42:28,083 --> 01:39:43,749 But it is an aspect of order we incorporate. 1092 01:39:43,750 --> 01:39:47,489 As for the weather, however, 1093 01:39:47,490 --> 01:39:49,675 you need to be prepared for other things. 1094 01:39:50,685 --> 01:39:54,543 Work in dynamic systems leads you to suspect the possibility that chaos exists, 1095 01:39:54,544 --> 01:39:58,914 but it is important to appreciate that there exists a great order as well. 1096 01:39:58,915 --> 01:40:01,890 It all depends on what you are researching.