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00:00:07 It is a field of mathematics with very ancient roots,
00:00:15 whose objective, put simply,
00:00:22 is to study the evolution in the very long term...
00:00:26 of a system whose evolution in the short term may be given by a simple rule,
00:00:32 but in which even simple short-term rules can cause big issues in the long term.
00:00:42 One very commonly cited example (because people are very familiar with it),
00:00:48 and which was initially defined in the 17th century,
00:00:55 is a planetary system ruled by Newton's gravitational rules.
00:01:01 It doesn't need much more complication to become interesting.
00:01:07 If you have two masses interacting,
00:01:12 you will have what people know as the behavior of elliptical orbits,
00:01:18 which are well known, are regular and persist in time.
00:01:23 But there is a question that has been asked since then,
00:01:27 and which is still not well understood:
00:01:29 If you have 3 systems interacting through the same gravitation laws for the short-term,
00:01:38 which are well defined and understood for a long time now,
00:01:44 what can you say about behavior in the very long term?
00:01:49 This is what interests Mathematicians looking at this system:
00:01:54 If you ignore other limitations,
00:02:02 like a sun exploding or imploding, or something similar,
00:02:10 or other things that may happen in the timescales we can consider,
00:02:14 then, in a potentially limitless timescale ruled only by gravitational rules,
00:02:20 one would like to be able to make interesting conclusions.
00:02:24 Would these planets, which seem to propagate regularly,
00:02:29 start interacting in more complicated ways and escape far beyond where they began?
00:02:36 Thing like this.
00:02:37 That is one example of a question that can be asked,
00:02:44 and there are many other systems.
00:02:46 We can start from modeling,
00:02:48 such as a model with population interactions,
00:02:53 with several populations competing over simple rules,
00:02:58 and try to predict what could happen with the system.
00:03:02 Maybe you want to know what would happen if you made a small intervention in it.
00:03:07 Would it be catastrophic, or potentially catastrophic?
00:03:15 Another thing that is understood…
00:03:19 There are also systems based on purely theoretical models,
00:03:22 used only to explore abstract approaches to dynamic systems,
00:03:27 or sometimes interactions with other fields of Math, like numbers theory...
00:03:35 and a few abstract dynamic systems that don't model physical phenomena,
00:03:40 but only mathematical ones that answer interesting questions...
00:03:44 that don't initially seem to be about dynamic systems.
00:03:48 But what interests people when we talk about...
00:03:55 the potential to say something interesting on long-term behavior,
00:04:00 one major finding from discoveries made along the last century...
00:04:10 is the discovery of the frequency and inevitability of chaotic behaviors.
00:04:19 These are systems in which chaos appears - which may happen even with simple rules -
00:04:27 despite these simple rules, and in certain scenarios,
00:04:29 it is often theoretically impossible to make very...
00:04:39 accurate predictions of what behavior will be at any given time.
00:04:42 In this case, some questions would be hard to answer:
00:04:45 if I have this system, and, say, look at it a million years from now,
00:04:53 at that moment, what would the state of that system be?
00:04:57 Imagine we are measuring something in a physical system,
00:05:03 like the temperature of something, which is evolving, and ask:
00:05:06 What will temperature be at a very precise moment in the very distant future?
00:05:10 That is often a question that one, when faced with chaos,
00:05:14 can't reasonably give intelligent answers to.
00:05:18 On the other hand, you can ask more statistical questions about the system,
00:05:24 which is also something that has been developed.
00:05:29 The idea is, even faced with this unpredictability,
00:05:33 which indeed arises...
00:05:37 in things like the so-called 'butterfly effect', where even small changes...
00:05:42 create changes so large they prevent predictions,
00:05:46 sometimes it is possible to robustly describe not the exact, precise behavior at point X,
00:05:54 but rather 'average behaviors'.
00:05:56 Something like being able to say 'in period X,
00:06:01 average temperature will likely be about Y'.
00:06:05 This is somewhat related to the robustness of statistical laws.
00:06:12 It can be compared to...
00:06:15 a situation in which you are playing a game of heads and tails.
00:06:21 The question of what the next toss will be...
00:06:26 is not one you can say much about.
17:09:18 It's either heads or tails, 50/50.
00:06:34 So you can't predict accurately.
00:06:37 But you can say, quite confidently
00:06:39 that if you do 1,000 tosses, you will have approximately 500 heads and 500 tails,
00:06:46 with error margins that can be well estimated.
00:06:50 You'll have said something useful about the behavior as a whole,
00:06:53 but not about a single toss.
00:06:55 What happens with chaotic systems is interesting...
00:07:00 Heads/tails tosses have inherent randomness.
00:07:06 But sometimes you start with a system deterministic in principle,
00:07:09 but whose intrinsic complications create behavior close to randomness.
00:07:17 And, in fact, the best way to model this behavior, even though it is deterministic,
00:07:22 is through a law that assumes the system is truly random.
00:07:26 It can be approximated quite well by a system very similar to coin tossing.
00:07:32 That allows you to say interesting things in the same way...
00:07:34 about this system's average behavior.
00:07:39 That's an example of a question, and answers, and type of answer...
00:07:43 modern dynamic systems theory can offer when chaos is involved.
00:08:01 I have worked on several aspects of dynamic systems.
00:08:12 I didn't work only on chaotic systems, that was more a general example.
00:08:18 I often work with systems that have a little less chaos in them,
00:08:28 and one of the methods I used the most often...
00:08:33 in different scenarios is the one known as renormalization.
00:08:38 The concept has a certain history,
00:08:45 and represents different things in different areas.
00:08:48 It means one thing in Physics, another in Dynamic Systems,
00:08:54 and maybe others for other fields of Math.
00:08:56 For Dynamic Systems specifically, it is used to try to understand the behavior...
00:09:02 of a system in a certain 'subarea'.
00:09:07 The system acts over a certain space of configurations,
00:09:10 and you would like to focus attention on behavior in a small subset of that space,
00:09:15 but, as usual, over a very long time horizon.
00:09:19 So you obtain a certain system with this new time and space scale,
00:09:26 which you then re-scale - you renormalize -
00:09:31 and imagine your whole time horizon is in fact just one time beat,
00:09:36 and that subarea is in fact your whole space, that it corresponds to the whole.
00:09:42 In certain situations, the system you obtain through this process...
00:09:48 is qualitatively similar to the one you started with,
00:09:56 even though it is not exactly the same.
00:09:58 In these cases, you will find a sort of operation in the space of dynamic systems...
00:10:04 that has certain specific features.
00:10:08 This change of scale on how you observe systems,
00:10:13 going to a smaller space and longer time scale,
00:10:16 leads from one dynamic system to another, but within the same 'class'.
00:10:21 This idea, surprisingly, works quite well...
00:10:31 in certain classes of systems that are not chaotic.
00:10:35 When you consider a chaotic system and change its time scale,
00:10:39 you produce something with much more chaos.
00:10:44 Information loss becomes much faster with a longer time scale.
00:10:51 But for other systems that are more regular than a completely chaotic one,
00:10:56 you have this potential - not always, but in some situations -
00:11:00 of arriving at a system not so different, qualitatively, from what you had before.
00:11:06 Then, as you try to study and achieve longer and deeper time scales for your system,
00:11:13 You begin trying to iterate this process,
00:11:18 and making that operation in the dynamic systems space.
00:11:25 In general, you consider this acting over systems in small dimensional scales.
00:11:30 These are 1- or 2-dimensional spaces - we live in a 3-dimensional one,
00:11:41 but many interesting dynamic systems happen in low-dimensional spaces,
00:11:44 on a plane or on a straight line.
00:11:50 So if you consider these systems,
00:11:52 and look at this action in the space of the systems,
00:11:56 you're looking at them in space that is often of infinite dimensions
00:12:00 What we are studying here, then,
00:12:02 is this operation in a class of systems that live in an infinite-dimension space.
00:12:07 And, interestingly, when a system operates over itself,
00:12:11 you can repeat it - and you are interested in repeating it.
00:12:14 You get a new dynamic system, with a new set of features,
00:12:17 that now acts, that 'lives', in a much bigger space.
00:12:24 What is surprising is that now you're in something much more complicated,
00:12:29 - from, say, one dimension to infinite dimensions -
00:12:32 but it is frequently possible to say very interesting things...
00:12:37 about this infinite-dimension system.
00:12:40 And this repeats often, for several situations...
00:12:45 when we discover this infinite-dimension system now has chaotic features.
00:12:51 And then the methods that are efficient to analyze certain chaotic systems,
00:12:56 including this probabilistic analysis I mentioned,
00:12:59 can be used to later say interesting things...
00:13:02 about the low-dimension system we were interested in at first.
00:13:06 This sort of idea emerged in the 1970s,
00:13:10 and was well developed in several directions in the next decades.
00:13:18 I applied them several times in my work,
00:13:22 on studies of one-dimensional systems, for example,
00:13:25 or certain scenarios on the behavior of billiards,
00:13:29 which is also something I worked on...
00:13:33 and also on certain aspects of certain Schrodinger operators,
00:13:39 which are also modeled on dynamic systems that frequently can also be renormalized.
00:13:57 It's an idea that has been developing for a while.
00:14:02 I applied and developed certain techniques, and incorporated...
00:14:08 For every scenario in which you use this group of ideas,
00:14:12 some features differ,
00:14:14 and you have to integrate them to understand things.
00:14:19 I began working in the low-dimension dynamics of the interval.
00:14:24 These are very well developed applications, associated with some well-known fractals,
00:14:29 such as the Mandelbrot Set and the like.
00:14:37 But then, over time, I became interested in other systems...
00:14:42 that could also be subject to renormalization theories.
00:14:49 You have to understand how these ideas can be applied,
00:14:52 because the questions that it can help answer are different.
00:14:55 For instance, when I began my work on Schrodinger operators,
00:15:00 it was interesting to me how the questions were asked differently.
00:15:05 The ultimate goal wasn't even to understand a given dynamic system.
00:15:10 The system was the means to analyze the associated equation's behavior...
00:15:18 which will correspond to a dynamic, now a non-classical one.
00:15:21 Before, I was talking of classical dynamics, with deterministic laws.
00:15:29 But when I was interested in Schrödinger operators,
00:15:32 the goal was to understand a quantum dynamics phenomenon,
00:15:35 but before arriving at the equation that will say interesting things about it,
00:15:40 you have the option to use this classical dynamic system model...
00:15:47 to understand a piece of the equation.
00:15:48 Actually you have to understand a family of dynamic systems...
00:15:54 associated with the various possible energies for your quantum system.
00:15:59 For each energy there is a different classical system that you can look at...
00:16:05 and will yield a good behavior of that system's functions.
00:16:12 The issue is you need...
00:16:14 I had a long learning process to discover what interested people.
00:16:20 I began with dynamic systems,
00:16:21 but people had specific questions to ask about these operators,
00:16:29 and on what understanding of the dynamics would allow you...
00:16:34 to reach any conclusions on the aspects of that equation at the end.
00:16:40 Renormalization, then, was one part of a theory, but I branched out later.
00:16:46 I started by using it, it was sort of the way I started.
00:16:52 When I began, this was in 2002, 2003, and I realized...
00:16:58 renormalization could be used to better understand some specific problems.
00:17:07 Later, this started yielding interesting results,
00:17:13 such as conclusions about the fractal known as the Hofstadter's butterfly,
00:17:23 merely by using renormalization slightly differently from how it was used before.
00:17:33 Then I went on to other aspects less related to normalization,
00:17:37 which could, after a long time,
00:17:42 be reunified with renormalization for a deeper understanding of the area.
00:17:47 But this took many years.
00:17:50 I still work on it, but there was slow progress for many years…
00:17:55 Then in 2008-09,
00:18:00 I had a better understanding of how to unify these aspects,
00:18:05 including by looking at another area that is also very present in my work:
00:18:10 the Lyapunov exponents.
00:18:13 Specifically, understanding the appearance of 0 or positive Lyapunov exponents.
00:18:18 Lyapunov exponents are certain numbers associated to a dynamic system...
00:18:24 which detect a bit of the instability of the system.
00:18:28 What is the speed at which orbits separate, for instance?
00:18:35 You begin with two configurations, two orbits that are quite close,
00:18:42 and they begin separating at a certain exponential speed,
00:18:45 which is the rapid loss associated with chaotic systems.
00:18:58 I usually work with many potential problems.
00:19:03 I work on one at a given moment,
00:19:05 but I oscillate a lot over a few days or weeks...
00:19:09 thinking in different directions.
00:19:12 What often happens in this work...
00:19:14 is that you arrive at an obstacle where you lack...
00:19:20 ideas on how you can move forward or what the strategy is.
00:19:27 In that scenario...
00:19:30 where will the idea come from that will allow you to 'unblock',
00:19:35 and continue the work on a more…
00:19:41 objective, actually...
00:19:45 in a more comprehensible way?
00:19:47 Once you have an idea to work with,
00:19:48 there are several techniques that can be used,
00:19:52 and many ways to find associations and move ahead towards a solution.
00:19:59 But without a starting point, that gets very hard.
00:20:03 There isn't much to do.
00:20:06 You know that, with current ideas and techniques,
00:20:09 you can't get past a certain point.
00:20:12 When you get to this 'blockage',
00:20:15 there is no formula on how to get out.
00:20:23 There are theories on how discovery happens,
00:20:29 and some say it happens in the subconscious,
00:20:32 and it is not something that can be 'willed' by the person,
00:20:37 but regardless of those, there are interesting possibilities to consider…
00:20:42 You get to a block, you try, and after a while it becomes clear...
00:20:47 you don't have what you need at that time.
00:20:50 Then maybe it is interesting to look at something else,
00:20:53 first psychologically, so you don't feel you're banging your head against a wall,
00:20:57 and also because, if you look into another direction,
00:21:01 you may end up having a new idea, or...
00:21:06 getting in touch with a new idea, without looking directly at the former direction,
00:21:09 and then a new idea can arise,
00:21:12 that you can see as useful to the problem you were working on before.
00:21:18 This happened with me sometimes.
00:21:21 I was working with multiple problems,
00:21:23 and realized what I was working on in a certain context...
00:21:28 helped open the path for progress in another context.
00:21:33 It could give me some…
00:21:38 Sometimes it's just a different perspective on something.
00:21:42 You basically had all elements,
00:21:44 but you needed the right angle to pick the right way to move forward.
00:22:23 That depends a lot on the situation.
00:22:27 Each project has its own traits.
00:22:29 The choice of working entirely without using paper or the like...
00:22:38 is a bit deliberate in some situations,
00:22:42 to try to avoid the temptation of going for a more 'brute force' approach,
00:22:52 more calculation-heavy approach to systems.
00:22:55 I avoid calculating very complicated things,
00:23:00 and if you don't have paper around you,
00:23:04 it limits what you can calculate.
00:23:06 So, instead of calculating,
00:23:07 you need to understand what you are studying better...
00:23:12 until the moment the calculation you need becomes evident.
00:23:18 It becomes so simple that you can do it in your head.
00:23:21 You try to increase your understanding in a more abstract way,
00:23:27 and see how this can be interpreted differently.
00:23:31 More obvious things, like expressions etc.,
00:23:35 can be manipulated so you can look at how they can interact differently.
00:23:40 It is a choice I make,
00:23:44 because I think it tends to be very fruitful.
00:23:51 I think there is a lot that can be done in terms of calculations,
00:23:56 but I usually work in collaboration,
00:23:58 and other people can do the calculations, and sometimes they don't get there.
00:24:02 So I prefer to think a bit of what can be done without calculations,
00:24:05 and sometimes it can merge with the calculation work some of my colleagues did.
00:24:27 I don't really look…
00:24:31 In the situations in which I am working out of the office,
00:24:38 and indeed I do,
00:24:39 either walking, or on the beach…
00:24:43 The goal is not for me to be inspired by my environment.
00:24:48 What is important is that it is a calm environment,
00:24:51 and you can be without a clear distraction.
00:24:57 Interestingly, for instance,
00:24:58 the sound of the ocean will not lead to...
00:25:02 will not cause great distraction, because it is constant.
17:27:52 If you had people screaming around, thing would be more complicated.
00:25:10 You usually want an environment where you do not worry about what is around you.
00:25:16 There may be the occasional distraction,
00:25:18 but mostly you want to have calm periods.
00:25:22 And, at that moment,
00:25:25 what you think depends on the situation.
00:25:32 On the beach, for instance,
00:25:34 I have worked many times trying to refine...
00:25:41 This was work on random matrix multiplications,
00:25:46 which can be seen as dynamic systems.
00:25:48 And the analysis technique I thought could be appropriate...
00:25:59 involved the understanding of certain geometries, or configurations,
00:26:07 on planes of high dimensional complexity,
00:26:10 the different interactions between them..
00:26:13 how they evolve with the application of certain linear transformations.
00:26:16 So I basically imagined positions for various objects in these spaces,
00:26:27 and how they would be changed by the transformations,
00:26:30 and how these changes could be...
00:26:38 'domesticated' by the technique we were thinking of using.
00:26:42 You had to build certain objects, in the case certain expressions,
00:26:48 which would adapt to possible configurations, and which would evolve...
00:26:57 in a certain direction that would be beneficial to us,
00:27:00 influenced by a certain dynamics - a chaotic one, in this case.
00:27:06 So I did that…
00:27:07 I imagined the types of configuration that were of concern for this technique,
00:27:13 and how I could...
00:27:16 modify the expressions to take this new scenario into account.
00:27:22 And I insisted on doing that without paper,
00:27:27 because it wouldn't be useful for that.
00:27:30 I needed a good understanding of the nature of the obstacle.
00:27:35 In this particular case,
00:27:36 it took a long time to overcome this problem.
00:27:41 Many tries, and conversations with colleagues...
00:27:47 In other situations,
00:27:50 it can be simply a matter of reconsidering something you thought of many times...
00:27:54 and still have no way forward,
00:27:58 and then try again after a while.
00:28:00 Like I said, not always focusing on the problem you don't get,
00:28:04 but it might be worth revisiting later.
00:28:07 Maybe an idea will come out from something else that has matured.
00:28:10 So you try again.
00:28:12 In general you realize that you are still stuck,
00:28:19 but you keep trying, and sometimes a solution comes out.
00:28:21 So you walk, and think again of the problem you were stuck on before,
00:28:27 reflect, understand once more why you couldn't do it…
00:28:36 This last time I went to the beach, for instance,
00:28:38 I was thinking of a problem that often comes to my mind,
00:28:45 and which I always find the same difficulties with.
00:28:48 But I come back to it in the hope I will see something different.
00:29:10 In general, mathematical knowledge...
00:29:16 often ends up being applied to scientific developments,
00:29:22 like in Physics or Chemistry,
00:29:24 and eventually in technological progress.
00:29:28 I would say that, in our current world,
00:29:31 it is quite obvious to see how past Mathematics...
00:29:35 was important for us to have the resources we take for granted today.
00:29:41 So we can imagine the breakthroughs that we achieve in Math today...
00:29:45 will have future repercussions,
00:29:47 even though it is impossible to predict how this will happen,
00:29:51 and what breakthroughs will have the biggest repercussions.
00:29:55 Of course, there is Mathematics that is done...
00:30:00 with clearer goals for immediate application.
00:30:06 You can do research with consequences that are...
00:30:13 immediately calculable for certain technologies…
00:30:17 Certain advances, for instance, in Medicine,
00:30:22 such as techniques in tomography and the like,
00:30:25 are based on research that requires very targeted Mathematical breakthroughs.
00:30:35 That said, I don't think all contributions of a study...
00:30:43 have a direct application in possible technologies.
00:30:52 I would imagine that there is a society of Mathematicians and scientists who interact,
00:31:00 and their ideas interact, and the questions and answers...
00:31:06 of persons working on different things end up reverberating.
00:31:13 So, even if one sees Math...
00:31:18 as a concern merely for its more visible impact on society
00:31:26 - which is not necessarily mine - but even from that standpoint,
00:31:32 the study of a broad spectrum of Mathematics,
00:31:37 going from the purer side to the more applicability-driven side…
00:31:43 It is all part of...
00:31:45 It all interacts in the end. All these ideas interact.
00:31:48 A researcher who studies something more esoteric,
00:31:51 which sometimes seems foolish to observers from the outside…
00:31:55 About objects that look like a game Mathematicians play with themselves…
00:32:04 These studies will often unexpectedly raise questions and techniques...
00:32:12 that slowly propagate to other areas of Mathematics often identified as important,
00:32:23 and sometimes end up being precisely...
00:32:26 what someone with a 'clearer' goal needed.
00:32:32 These ideas go in these directions, and the opposite direction as well.
00:32:37 Many fundamental concepts of interest to Mathematicians who want to "play around",
00:32:44 the 'purer' Mathematicians,
00:32:46 who are more concerned with abstractions, the so-called 'aesthetics of Mathematics',
00:32:52 who want to understand a thing very intrinsically to that thing,
00:32:57 without concern for the outside world,
00:32:59 will also find questions and techniques from research done with clearer goals.
00:33:07 Ideas will go from one side to the other, and vice-versa.
00:33:15 So, an environment where all this research happens concurrently...
00:33:19 is much more productive than if you only had one focus,
00:33:23 and could only look at it in terms of how it can lead you...
00:33:27 to a certain goal within a certain time frame.
00:33:32 I believe many discoveries wouldn't have been made, and won't be made,
00:33:37 without an environment that can drive...
00:33:45 this work in all of these directions at the same time.
00:33:49 So this would be...
00:34:20 I am more concerned with...
00:34:24 I am happy that people are using my work,
00:34:30 but I am more concerned with solving problems that in fact interest me...
00:34:37 and which will interest me due to what I have studied and have been thinking.
00:34:43 This creates a certain 'taste' for Mathematics,
00:34:46 and a trend to try to seek a better understanding of certain parts of Math.
00:34:57 A willingness to understanding something that is often focused on specific issues.
00:35:02 And, in Math, having this understanding often requires solving very specific problems...
00:35:10 that identify the main difficulties associated with that understanding.
00:35:20 Many times, however, you don't work on responding questions asked of you.
00:35:25 Many interesting questions you ask yourself.
00:35:30 It has happened, and sometimes happens,
00:35:33 that as you learn more about a topic,
00:35:39 you see the question as something different from what other people see.
00:35:47 Each one has their own view,
00:35:50 and sees a certain field of Math with their own internal representation,
00:35:55 which is influenced by past experience and knowledge.
00:36:00 Each Mathematician brings their contribution.
00:36:03 I bring mine, which comes from an intercession of many prior things.
00:36:08 In certain circumstances, this position can be particular and special in a certain field,
00:36:15 and that allows me to offer questions and approaches…
00:36:21 that wouldn't have been natural for other people in that area.
00:36:25 This certainly has happened quite clearly...
00:36:28 in my study of Schrodinger operators, or a certain class of them.
00:36:35 I had the ability to pose questions and propose approaches,
00:36:42 and ended up doing a little bit of everything.
00:36:48 I proposed questions that were not natural, but which, when posed,
00:36:53 made it clear that they were essential, and that were needed for progress.
00:37:01 I try to propose questions,
00:37:03 but associated with a possible way to understand it,
00:37:07 with context so that it can be developed.
00:37:12 And then, using my technical characteristics and my individual knowledge,
00:37:19 take that research to its end and find the solution to all this,
00:37:25 leading to a somewhat developed theory...
00:37:30 that will answer the questions that were asked.
00:37:34 Then, when these techniques are developed, they can be used elsewhere,
00:37:40 One would expect, then, that beautiful Mathematical techniques will be reused.
00:37:48 And they tend to - and I find this natural -
00:37:56 become knowledge incorporated by others,
00:37:59 who can use it in varied situations.
00:38:04 It is desirable that this is the natural path of discovery in this field.
00:38:24 I try…
00:38:27 I think people say this more because of my work style.
00:38:34 In a few of the topics I studied, I came sort of from the 'outside'.
00:38:43 I wrote my thesis on a certain field, and when I arrived in Paris,
00:38:47 I found few people who wanted to talk to me about what I was working on.
00:38:54 Then, so I wouldn't keep talking to myself, I wanted to learn what was being done.
00:38:59 But I didn't know a lot about what they were doing.
00:39:03 And when I looked at these fields,
00:39:07 I chose a somewhat…
00:39:14 unconventional approach.
00:39:15 Acceptable, of course, but an unconventional approach to relatively complex things.
00:39:23 Instead of trying to start from the beginning, with simpler questions,
00:39:27 I tried looking at things I would consider important,
00:39:29 but that would tell me something from the point of view I was coming from.
00:39:34 I worked many times with very little understanding of the state of theory,
00:39:40 at least at that time,
00:39:43 and decided to try to look at them with my little knowledge of that particular area,
00:39:52 and see what that tells me.
00:39:57 Sort of thinking from scratch.
00:40:00 This led to certain approaches that were not being used in these areas,
00:40:06 and sometimes, due certainly to some luck and some work,
00:40:12 led to solutions for problems that were somewhat stalled,
00:40:22 that weren't seeing progress in that direction.
00:40:24 That may be why people say that,
00:40:27 because I decided to look at things from a different perspective,
00:40:32 mostly because I didn't know how people were looking at it,
00:40:35 and didn't try too much.
00:40:37 For a while now, I've had this trend of not studying the works of others so much.
00:40:44 I know what has been done, but my main strategy is not to read it in depth.
00:40:49 I try to find out what ideas are in vogue,
00:40:52 but without trying to understand the literature exhaustively.
00:41:00 I get an idea of the fundamental issues,
00:41:01 and try to think of something different that can help to advance these.
00:41:08 I think I would have less to contribute...
00:41:11 if I tried to improve a little on techniques people are already trying to apply.
00:41:19 I think I have a better chance to contribute...
00:41:23 by reflecting on what is not being done,
00:41:28 and on what would be natural for someone not deeply involved in an given area.
00:41:35 I suspect that is what people alluded to when they told you this.
00:41:54 Well, Math...
00:41:56 When you are a child,
00:41:59 you are not really presented to what really is a Mathematical work.
00:42:03 You are introduced to a few numbers first, like I did as a child,
00:42:09 and that they have their characteristics, and some internal logic.
00:42:21 There are a few things that are true and remain true, that don't evolve,
00:42:28 just like other areas of knowledge.
00:42:32 Certain somewhat primitive aspects of Mathematics,
00:42:37 which were all I know,
00:42:39 looked interesting from the start.
00:42:42 It was a very fortuitous thing for my knowledge,
00:42:45 that I really liked to read very basic things about Math, school Math,
00:42:52 that were quite rudimentary,
00:42:55 and I 'didn't dislike' that aspect.
00:42:59 Then I became interested in Math Olympiads, solving Olympiad problems.
00:43:06 And I found those a lot of fun, too. Studying those problems.
00:43:11 And then I started working on more theoretical areas, when I did my Master's,
00:43:18 I had to learn more, which I also found interesting.
00:43:22 This more abstract learning.
00:43:24 And then I started doing research,
00:43:26 with the characteristics I described just now.
00:43:30 And it was interesting.
00:43:31 Throughout those phases, it was possible…
00:43:34 For instance, it's perfectly normal for a great researcher to hate Olympiad problems,
00:43:40 or hate this aspect.
00:43:43 That is perfectly possible.
00:43:45 In my case, in all my life progress until I became a researcher,
00:43:52 maybe because of my story growing up,
00:43:58 it looked very interesting at that time.
00:43:59 Afterwards I lost interest in Math Olympiad problems, for instance,
00:44:04 it doesn't entice me today, but it was important to me at that time.
00:44:08 It is curious that at each moment something interested me,
00:44:12 and allowed me to progress a little bit faster.
00:44:14 There was no moment of "I'm not interested in Mathematics anymore".
00:44:19 "I have lost interest in this area. I don't want this."
00:44:26 This could have make me drift away,
00:44:31 but at each moment I found something interesting,
00:44:33 until I became a researcher, and have had the same interest since then...
00:44:43 to discover new Mathematics and understand certain aspects better,
00:44:47 which is what interests me now.
00:45:12 It's a feature of...
00:45:14 Well, it is an award conceived at a certain time,
00:45:19 and its story is well-known.
00:45:25 The idea was to have an international award, which didn't exist at the time...
00:45:29 It was the first one.
00:45:31 And its message was to convey unity of the Mathematics community.
00:45:37 It would be given by the International Mathematics Congress, in principle open to all,
00:45:41 at a time when there was some…
00:45:46 It was between the two World Wars,
00:45:48 and there were problems with nationalism and things of the sort.
00:45:55 So there are historical reasons for how it was introduced.
00:45:59 The creator of the award, called Fields,
00:46:03 wanted it to be an award given to young people, younger than 40.
00:46:07 It was not meant as an "end-of-career" award to celebrate a body of work,
00:46:13 but rather to highlight promising work, sometimes well fleshed out,
00:46:20 by someone who is active and likely to continue working.
00:46:27 It has characteristics that differ it from a Nobel Prize,
00:46:34 which is usually given as a reward at the end of a career.
00:46:38 I think there are certain positive aspects…
00:46:44 I don't think…
00:46:47 It is not a good inspiration to do good Mathematics,
00:46:51 or good science, or anything,
00:46:54 to seek to obtain some award for it.
00:46:59 I think it is the wrong direction to look,
00:47:02 even because many things are discovered by looking at less promising directions.
00:47:09 By looking at what really interests you, even though it may not interest others,
00:47:14 you end up discovering something that becomes and is seen as fundamental.
00:47:19 It is important that people look at all directions.
00:47:22 If we all look at the direction that looks more likely to be rewarded,
00:47:27 we will have very biased perspectives, and things won't develop as well.
00:47:33 Also, you do better Mathematics if you do something that genuinely interests you.
00:47:39 That is the real motivation.
00:47:41 That, and it is not a very reasonable goal...
00:47:46 to work in depth aiming at something so rare to achieve.
00:47:52 Not many awards like this are given,
00:47:54 they are like a Nobel or the like...
00:47:57 It is interesting that it has an effect...
00:48:01 There is this negative effect,
00:48:04 in that people, when they feel it's possible for them to win this award,
00:48:11 to try to win this award while they can, i.e. under 40,
00:48:16 so, in fact, it is hard to deny that it creates some competition,
00:48:23 but the positive side is that it has an expiration date.
00:48:29 After all, these awards are given, and then fulfill their role.
00:48:35 And for me, their role, be they the Nobel, Fields, and others that are well known,
00:48:43 is to communicate with the public.
00:48:47 It is through these awards that people see progress is being made in science.
00:48:55 So, when people talk…
00:48:59 There is this week of the year when the Nobels are given out,
00:49:02 and people will talk about advances in Medicine, Physics, Chemistry etc.
00:49:09 And this is interesting.
00:49:12 It lets people know that this work is being done,
00:49:14 and how important it is, and that it continues to be done.
00:49:17 So there is this aspect of bringing science closer...
17:52:11 to a community that won't hear this constantly.
00:49:26 That is the main point, I imagine,
00:49:28 the most important contribution of these awards.
00:49:32 Also, like I said,
00:49:35 they have this negative potential to generate harmful competition sometimes.
00:49:40 So the advantage of the Fields medal, in comparison with other career awards,
00:49:48 is that it has a deadline.
00:49:49 When you get to 40 you no longer need to worry about it, regardless of the result.
00:49:55 It fulfills its role, and doesn't have as much of the harmful effect...
00:50:02 in terms of making people continuously obsessed with this goal.
00:50:23 Yes, but I also don't understand a lot of what is done by the community.
00:50:28 This is something you discover relatively quickly as a Mathematician:
00:50:35 the depth and enormous number of different directions people research.
00:50:44 You have characteristics, and preferences…
00:50:48 For instance, just to give you an idea:
00:50:51 I have a line of thought that leads me to approach....
00:50:55 issues of a general part of Mathematics known as Analysis.
00:51:03 There are estimates,
00:51:04 and you try to obtain notions of physical quantities,
00:51:05 of how large or small things are, relationships, interactions…
00:51:10 But there is some more flexibility.
00:51:13 Other aspects, which I'd call more algebraic,
00:51:16 have rigid equations, manipulated precisely,
00:51:19 that give exact answers all the time...
00:51:22 This is home to another type of thinking, that I find more rigid.
00:51:25 My intuition goes more to the former than the latter,
00:51:29 so I have difficulty trying to understand those,
00:51:38 even things that are not so complicated.
00:51:42 Certainly for the latest advances in these areas,
00:51:45 I lack the framework to be able to say much about them.
00:51:50 So for me it is quite natural that advances that are,
00:51:53 like you said, on the "frontier" of knowledge,
00:51:57 are not accessible to people not working on them.
00:52:02 Yet, aspects...
00:52:06 A few aspects of discoveries end up being gradually conveyed.
00:52:14 This is one of the ideas I put forward.
00:52:16 Some Mathematicians work between two areas,
00:52:21 and they often 'transfer' these ideas.
00:52:24 This communication happens.
00:52:26 Sometimes it needs to go through several Mathematicians to go from here to there,
00:52:30 but it has this potential.
00:52:32 I did that in some directions,
00:52:34 bring some ideas from one subarea of dynamic systems to another,
00:52:39 and this is also done by people working with me in other directions.
00:52:48 I would consider it natural to see this difficulty of communication.
00:52:52 It is part of the complexity of the things we work with,
00:52:57 but at the same time I don't worry much that they will be isolated eternally.
00:53:07 They continue having the potential to influence others...
00:53:12 who may not understand exactly what I do, but can be influenced by a certain path...
00:53:17 by a few ideas developed in this area.
00:53:31 I don't think much in terms of predicting, because if I look back,
00:53:37 I realize I didn't envisage 10 years ago what I am doing now.
00:53:41 I had some idea, but it was basically...
00:53:46 'I'll probably be interested in things that interest me today,
00:53:50 but will certainly be interested in others I cannot predict now'.
00:53:54 For me it is very natural, working with dynamic systems,
00:53:58 to be unable to make such accurate predictions.
00:54:01 It doesn't worry me much not being able to predict as much.
00:54:06 In a more abstract level, or at a different level,
00:54:12 one goal would be to try to maintain the general characteristics…
00:54:19 In Mathematics it is easier.
00:54:21 In other sciences you work close to the technique, and production is direct.
00:54:28 Then you gain managerial responsibilities,
00:54:33 and have to manage other people and their work more than your own.
00:54:39 In Math that doesn't happen often.
00:54:42 You have small teams,
00:54:45 and Mathematicians, as much as they progress,
00:54:47 often tend to work as close to the 'base' as they did when they began.
00:54:56 So my wish, which is not hard to fulfill,
00:54:59 is to continue to work directly 'hands on'.
00:55:05 I am glad that Math is like this.
00:55:08 It would upset me if the consequence of doing good work...
00:55:13 was to be pushed to direct the work of others,
00:55:16 instead of having the pleasure to continuing to make discoveries directly.
00:55:34 It is interesting to think of Mathematics…
00:55:40 Mathematics… often...
00:55:44 Many who did not pursue more knowledge...
00:55:47 often don't know how Mathematicians think of mathematical objects.
00:55:51 Naturally, each will think differently,
00:55:57 but I believe some things are relatively general.
00:56:04 You are not working only with equations and numbers.
00:56:07 There is meaning and there is representation,
00:56:12 basically corresponding to mathematical objects.
00:56:20 These mathematical objects...
00:56:23 They are abstract, they don't have tangible reality,
00:56:28 so each Mathematician has their own interpretation, sees them differently.
00:56:35 Maybe visual, maybe not describable in terms corresponding to senses,
00:56:42 but there is an inner interpretation of that.
00:56:49 And the one creates...
00:56:52 Imagine Mathematics as populated by objects with connections between them,
00:57:01 and with aspects revealed when seen in a certain way.
00:57:04 You have a notion of a rich Mathematical theory,
00:57:13 with connections that appear quite surprisingly,
00:57:16 sometimes between things that did not appear to be related.
00:57:21 And these connections, these intersections...
00:57:26 of many subjects that end up being discovered as you study certain objects,
00:57:33 contribute to this view that they have...
00:57:40 We call it "richness".
00:57:42 And there's an appreciation of this order, this organization…
00:57:47 that takes place.
00:57:48 In general, this is what, for many are people,
00:57:51 tends to lead to a sensation called 'aesthetic appreciation',
00:57:55 and to call these types of objects 'beautiful'.
00:58:03 That is basically it.
00:58:04 This beauty is seen very subjectively by each Mathematician,
00:58:08 and communicated between them.
00:58:11 There is this attempt to communicate, which is interesting.
00:58:16 They have their personal view of objects,
00:58:19 but manage to convey them in a way.
00:58:21 It is imprecise, since you don't convey it totally,
00:58:25 but you do convey a few aspects that enrich the understanding by the other.
00:58:31 But it is sill very individual, how one interprets each universe.
01:00:03 Mathematics always seemed to me to be something...
01:00:11 let's say 'stable',
01:00:13 and which always develops.
01:00:16 The more you learn, the more you find that you ignore as it develops.
01:00:23 Our breakthroughs are much slower...
01:00:27 than the pace of discovery of what we don't know.
01:00:31 On the other hand,
01:00:33 what you can achieve...
01:00:38 becomes part, albeit small, of a growing theory…
01:00:46 And this fact that mathematical knowledge accumulates,
01:00:51 that it is not destructive in its nature…
01:00:56 like can happen in other areas of knowledge,
01:01:02 where theories overlap, or contradict each other at certain times,
01:01:09 or develop, and become obsolete,
01:01:15 and are fully replaced over time.
01:01:18 Mathematical knowledge develops,
01:01:21 but does not invalidate what has been discovered in the past.
01:01:26 There may be more complicated developments,
01:01:29 but it persists quite well.
01:01:33 This is one thing that has always attracted me, and given me some…
01:01:38 I like to think of Math at a moment of tranquility,
01:01:44 and even when I am not in that state to bring some of it,
01:01:50 to think of it as something that 'is there',
01:01:55 that has its characteristics, that has its logic.
01:01:58 It may be difficult sometimes, and it may seem very rigid,
01:02:03 and make you work hard to do things that seem so evident,
01:02:11 but that same work, those same rules,
01:02:14 lead to the fact that when you reach a goal, it is stable.
01:02:21 It has structure.
01:02:27 And that is not something easily found elsewhere.
01:02:33 That is something pleasant to think about.
01:02:40 Maybe at more tense moments,
01:02:42 if you need to separate from anything that may be drawing your attention,
01:02:48 if you manage it, you can be in the mathematical world again,
01:02:53 with its rules, and its stability.
01:02:59 So, it has its moments,
01:03:01 but I wouldn't...
01:03:06 I would say that, emotionally, that has a small role in that sense.
01:03:19 Afraid… I guess I'm not afraid.
01:03:20 You have to be realistic about what you can discover,
01:03:25 and how things happen…
01:03:27 You don't need to...
01:03:30 I don't see it as very problematic when you have moments...
01:03:34 in which it looks like you are not making progress.
01:03:37 That is normal in any long career.
01:03:41 Any mathematician will go through this,
01:03:44 and experiences teaches you that many things depend on...
01:03:50 being at the right moment and having a bit of luck as well.
01:03:57 These oscillations happen.
01:03:59 This type of concern doesn't affect me much.
01:04:03 I think you have to persist in seeking things,
01:04:09 be realistic about the fact that most times you won't get to where you would like to,
01:04:16 remain determined, and appreciate when things work.
01:04:22 In daily work in Math,
01:04:27 most of the time you are working with things you know well,
01:04:31 and developing techniques that you already master,
01:04:39 seeking to take them a bit further.
01:04:46 Most often, this work...
01:04:52 is what is more visible as 'work'.
01:04:54 Like I said, you are often stuck without an idea,
01:05:03 and you wait for this 'unblock'.
01:05:05 So, sometimes it appears you're doing nothing,
01:05:07 then a short moment when you realize something that can make a difference,
01:05:14 and then the more 'conventional' work,
01:05:20 which involves a lot of creativity, but when it's easier to understand what is needed.
01:05:28 It is more 'concrete', perceivable work.
01:05:35 This moment of discovery,
01:05:39 when you feel you understood something different,
01:05:44 is an interesting one emotionally.
01:05:48 It combines several things.
01:05:50 You often get the immediate feeling that it can be important,
01:05:56 but sometimes that's inaccurate.
01:06:00 You become immediately excited about the potential,
01:06:08 of what can unfold from that idea.
01:06:13 You really want it to be fruitful,
01:06:18 and sometimes cling to keeping it alive in the face of difficulties.
01:06:25 At the same time, there are doubts,
01:06:28 and it is part of a Mathematician's work...
01:06:31 to attack your own ideas to find possible faults.
01:06:39 It's an emotionally complicated approach. There are several sides of you.
01:06:45 One side wants the idea to work,
01:06:46 but you have to work on finding the faults in it,
01:06:53 and of course would like not to,
01:06:58 but it is very productive to put a lot of energy into it,
01:07:03 so you're not working with an idea that turns out to be problematic.
01:07:08 Sometimes these moments are complicated…
01:07:12 But this excitement happens at very precise times,
01:07:15 and in general is what precedes this more arduous work.
01:07:21 You need to think about it, but have the feeling something can happen.
01:07:31 I don't believe much...
01:07:35 I don't believe much in looking directly or continuously...
01:07:41 at a very complicated direction.
01:07:43 I think things happens more naturally.
01:07:45 I mentioned that sometimes you could be looking in a different direction...
01:07:51 and you get that 'click', you have an idea and think:
01:07:58 'This could finally be useful for that big problem you considered before'.
01:08:03 So I prefer not to look directly at the problems.
01:08:11 I have done it for some,
01:08:13 but I know of several problems in dynamic systems that are fundamental,
01:08:20 whose understanding and solution would lead to great progress...
01:08:26 in our general understanding of specific dynamic systems that interest me,
01:08:32 but I think it is very hard to predict where you will go,
01:08:39 so I avoid thinking in terms of 'I would really like to solve problem X'.
01:08:44 There are many problems that everybody would like to solve,
01:08:48 but it is not something I define as a major goal.
01:09:21 I find it interesting to talk about things that are non mathematical...
01:09:24 with non-Mathematicians and mathematicians as well.
01:09:28 There are many topics that are interesting…
01:09:30 I don't think I'd have a great need to discuss work with them.
01:09:36 When people can appreciate it, math is one more topic of conversation,
01:09:42 but it's perfectly reasonable for different people to have very different interests,
01:09:49 and I don't see the need to try to talk about these things…
01:09:55 It's natural not to be interested in them.
01:10:24 Well, I have solved many problems...
01:10:30 that resisted approaches for a long time, that is true.
01:10:37 This is a feature of the way I work.
01:10:43 And I like solving problems like that, and to formulate questions.
01:10:47 But I certainly am attracted to some questions not just because they were open,
01:10:54 even because there are so many them,
01:10:57 but some because they cover issues for which I've developed a certain 'affection'.
01:11:07 This brings self-satisfaction,
01:11:10 and it is nice to speak to other Mathematicians about them,
01:11:14 but is not something I need to discuss with everyone.
01:11:17 I don't need everyone to appreciate it.
01:11:21 The most fundamental appreciation for your work must be your own.
01:11:25 If you don't appreciate your own work, you'll have a lot of trouble pursuing it.
01:11:32 This is something you have to seek, and once you get it,
01:11:35 it is nice that your colleagues appreciate it too,
01:11:39 and to talk to them about it,
01:11:41 to have this exchange and hear from them,
01:11:45 but it is a part of it…
01:11:48 I have no need for everyone to appreciate my contributions.
01:12:02 I always work a lot with others, and find people I can talk to.
01:12:08 In some of the work there was a gap...
01:12:11 between the needs to make it progress and what others could contribute,
01:12:19 even because it took many simultaneous traits and I was more suitable to do it alone,
01:12:27 but as time goes by, if you work is interesting,
01:12:31 it's normal for people to follow and investigate how they can use it,
01:12:40 or even improve what we had.
01:12:45 I don't have this feeling...
01:12:49 Mathematicians, they have…
01:12:52 There is that contribution that leads to solutions to problems,
01:12:58 but that often comes from work that didn't bring solutions but introduced ideas,
01:13:03 so I am aware of and greatly appreciate this work that is done,
01:13:08 and is often overlooked by awards etc.,
01:13:15 but that is fundamental for the progress of Math.
01:13:20 So I see the contributions of all who worked in this field.
01:13:24 Each of them did something that, if they hadn't,
01:13:29 I wouldn't have done anything or have a field to work on.
01:13:35 I also know that,
01:13:36 even when something I do has technical difficulties...
01:13:40 that means it may take a while to be appreciated,
01:13:43 I do hope it is, and that it will be useful to people.
01:13:48 That it's not something that stays isolated in me,
01:13:51 but that increases the overall mathematical knowledge of the community.
01:13:59 So there's that.
01:14:00 It's not just…
01:14:01 I would be reasonably satisfied to have my own fulfillment...
01:14:06 for having gotten somewhere on a problem whose solution interests me,
01:14:10 but there's this additional component, and I don't particularly feel it is missing.
01:14:52 Between mathematical thought and everyday life?
01:15:02 In that sense?
01:15:14 Depends on the type of problem you're working on.
01:15:20 There are certain situations in which...
01:15:22 you do these internal representations of the object,
01:15:27 and try to look at them differently.
01:15:35 It could be by seeking a side you haven't explored yet but could develop,
01:15:42 or it could be manipulations,
01:15:46 where it is clear, or you have identified something that can lead to a solution,
01:15:54 but which has some internal complexities.
01:16:02 Then you do something close to - albeit quite different -
01:16:05 from what non-Mathematicians call 'calculation',
01:16:08 and then do approximations that we Mathematicians would also 'calculate',
01:16:13 to estimate the consequences of certain processes.
01:16:16 You do certain procedures to manipulate objects,
01:16:20 and see if that reveals any other traits to be explored.
01:16:29 In that process, you are indeed doing a sequence of operations,
01:16:33 and imagining which next operation can be useful,
01:16:39 and all that happens inside your head…
01:16:46 Hard to describe further than that.
01:16:48 And that also varies in each situation.
01:17:14 It is possible to get distracted in a few situations.
01:17:17 Not sure if much more than other people,
01:17:21 but I do have this tendency, when I start thinking of something,
01:17:24 to become somewhat involved.
01:17:28 I think certain activities require some level of exclusive dedication,
01:17:35 especially if it creates certain risks to yourself or to other people,
01:17:41 such as driving.
01:17:47 And I would imagine that there is a trend...
01:17:48 to get comfortable with anything that becomes repetitive,
01:17:54 and in which you develop a feeling of safety.
01:17:59 And that could lead you to think of other things,
01:18:05 initially lightly, but then draw a lot of your attention inadvertently.
01:18:14 That may not cause any consequences, maybe out of luck,
01:18:19 but you are creating the possibility that it will.
01:18:28 There is also the issue that I prefer to live in urban areas...
01:18:35 where the need to have your own car is reduced.
01:18:41 I don't make sacrifices by not driving while living in Rio or in Paris,
01:18:48 while in other cities of the world that could be an issue.
01:18:53 I don't feel this is a problem with my life here, though,
01:18:57 so I think it's not worth risking this,
01:19:05 or not risking it and having to force myself to fully dedicate...
01:19:10 to a relatively repetitive activity feeling like I can't distract from it.
01:19:19 I don't see the need for that.
01:19:27 I think it is easier to answer how I see…
01:19:38 why Math is something interesting to develop,
01:19:41 or why people should study it.
01:19:45 One of the reasons, which I mentioned,
01:19:49 has to do with repercussions and developments...
01:19:55 of technological progress and science in general through Math,
01:20:00 but Math, specially seen from the 'purer' side of it,
01:20:06 is often developed as something close to a form of...
01:20:13 artistic expression.
01:20:18 Maybe it is something that attracts fewer people than music or sculpture,
01:20:28 because understanding Math requires, even for its simpler concepts,
01:20:34 a certain immersion in a prior study that other artistic expressions...
01:20:42 don't require for you to immediately see why it is beautiful or moving.
01:20:52 But, with some preparation and dedication,
01:20:59 it can have the same potential impact.
01:21:02 Like I said, you can appreciate this, this order,
01:21:08 this increasing structuring, and the surprises that appear,
01:21:14 and things that are very similar or very different when you don't expect them to be,
01:21:21 and all these characteristics are...
01:21:27 something that many people can (and do) seek...
01:21:35 to advance in this...
01:21:41 advance in this search for abstract understanding,
01:21:46 and the formation of a mathematical universe.
01:21:48 You are building, or helping to build, this piece…
01:21:56 Maybe not a piece, but this knowledge of this interconnected mathematical universe...
01:22:02 that each one only knows a small part of, but can contribute to.
01:22:09 I see it as quite a natural thing, then,
01:22:12 that it attracts people.
01:22:17 Not all people, since each will be attracted to something different.
01:22:23 But it can be just as attractive, and is, to certain people,
01:22:26 as other completely natural forms of expression.
01:22:31 Math has a history that goes back to…
01:22:34 even before then, but recognizably to the ancient Greeks,
01:22:40 who often sought to learn Math or develop it...
01:22:46 because they considered it something natural to pursue.
01:22:55 As part of the human quest.
01:22:59 It's been like that since the beginning,
01:23:02 and it is important to emphasize that additional importance:
01:23:07 that, in addition to these more perceivable, tangible repercussions,
01:23:15 we should remember that it is more than that.
01:23:23 It incorporates other things that should be appreciated in a more intrinsic manner.
01:24:20 In Math, I see it as you...
01:24:25 finding out more about things that are well-established discoveries.
01:24:29 Of course, people make mistakes.
01:24:31 It is part of the process.
01:24:34 Any person can make mistakes, which may be short-lived,
01:24:37 but other than these errors - and those are often quite limited -
01:24:43 the mathematical knowledge being developed is considered in general to be true.
01:24:49 There is no serious dispute between mathematicians with different interpretations...
01:24:57 regarding the true nature of the status of a certain discovery made.
01:25:06 Mathematics is populated by small truths that correspond to a growing universe,
01:25:12 and so there is no great doubt when one is working.
01:25:17 It is something quite characteristic of our knowledge.
01:25:23 There have been moments of discussions in that sense,
01:25:29 the Mathematicians...
01:25:33 about the foundational issues of Math, including in dialogue with Philosophers...
01:25:39 but we have arrived at notions of the limitations, of where Math can take us.
01:25:47 Ambitions of total formalism have been somewhat abandoned.
01:25:53 The known limitations coming from Logic, coming from Gödel, for instance,
01:25:59 may have destroyed the more exaggerated ambitions of the beginning of the century,
01:26:09 I mean the 20th century,
01:26:11 and are now well understood.
01:26:13 Mathematicians essentially accept the limitations of what they can work on.
01:26:20 They know these limitations.
01:26:22 But they seek to work on certain well-defined objects,
01:26:25 they have the continuous feeling of obtaining something true and permanent.
01:26:41 In simple terms, pure Mathematics is considered to be that which is investigated...
01:26:46 without the goal of achieving immediate application outside Math.
01:26:51 It may find application, and many times it does.
01:26:55 Things that initially belonged...
01:26:57 It is more a way of selecting your research objects and your intention in studying them.
01:27:07 Sometimes the type of Math is the same used in applied and pure Math,
01:27:13 sometimes the objects are the same.
01:27:16 This distinction is based on intent and this leads to some selection,
01:27:21 in that applied Math will work more on certain types of objects,
01:27:24 and pure Math on others.
01:27:26 In the end, you will potentially be using the same things.
01:27:31 If it is Math, the same tools are valid for one side or the other.
01:27:36 But there is this distinction on what you select and what goals you have:
01:27:41 Is your interest intrinsic to Math, or are you seeking direct application of it?
01:27:49 I put myself…
01:27:51 I am more interested in the 'intrinsic interest' side…
01:27:57 I sometimes work on problems directly inspired by physical models etc.,
01:28:03 but the reason I select it to work on is not its applicability,
01:28:08 but rather because the associated mathematical theory interests me,
01:28:12 due to the characteristics I mentioned, such as its richness, beauty etc.
01:28:55 At that moment…
01:29:00 When I'm reflecting on issues,
01:29:04 seeking another idea without a very well-defined plan,
01:29:10 and you're not...
01:29:13 You have some hope, but are just seeking a starting point,
01:29:19 that takes relatively little time…
01:29:24 It's like I said, I could be working on something else,
01:29:26 but back then I was working on a question I didn't have a clear path for,
01:29:31 and that occupies relatively little time.
01:29:35 Much more time is spent on mathematical work when...
01:29:42 you start with an idea you already had,
01:29:44 and trying to develop it, and use other techniques...
01:29:50 to make connections in a more predictable way.
01:30:01 This attempt to arrive more consciously at an idea...
01:30:07 without knowing where it will come from...
01:30:10 is something I spent less time on.
01:30:12 I look at it again and check if something has matured,
01:30:15 but in general you get to the conclusion you already had very quickly.
01:30:19 Then, either something appears,
01:30:22 or you realize your manipulations keep leading you back to the same place,
01:30:26 and that you're not getting any further.
01:30:30 I don't consider it very useful to keep persisting in this.
01:30:36 In this problem in particular, recently I've been ending up in the same place.
01:30:41 Even though I try small variations, they all take me to the same place.
01:30:45 When I realize this, I tend to abandon it for a bit longer.
01:30:51 But that is not a large part of my day.
01:30:59 There isn't a set rhythm…
01:31:02 Every day has its characteristics.
01:31:04 I will spare a moment among my daily tasks to think of Math,
01:31:10 but as part of my practical obligations and the like.
01:31:16 Since I don't have a lot of standing obligations…
01:31:22 Sometimes I will arrange to speak with a student or a colleague,
01:31:29 and that will occupy a specific part of my day.
01:31:31 The moment I spare to work alone is when I feel more available,
01:31:36 but that changes from day to day.
01:31:37 I don't have fixed times in my schedule.
01:31:39 I also collaborate a lot with people from other countries on-line,
01:31:44 so I need to adapt my schedule to time zones etc.
01:31:49 No need to follow a very precise routine.
01:32:21 That changes a lot, but yes,
01:32:24 for a while now I've had the habit of waking up late.
01:32:27 I usually wake up at around noon or later.
01:32:32 And… These are my hours.
01:32:37 Since I don't have the need to wake up in the morning, I don't,
01:32:42 but on the other hand, I go to sleep later.
01:32:46 Other than that…
01:32:47 I wake up and take care of a few practical things,
01:32:50 like eating, going to the gym…
01:32:55 I do my chores when it is most convenient.
01:32:59 And like I said, there are the things that are more 'identifiable' as work,
01:33:04 and in certain moments you may find yourself thinking…
01:33:08 It has to be at a clear-headed moment…
01:33:12 It's hard, at least for me, to say "I will book this time for abstract thinking."
01:33:20 I may schedule to speak to colleagues for this,
01:33:22 but when I'm alone, I will choose a moment in which I feel more serene,
01:33:27 and without other sources of stress that can divide my attention.
01:33:35 I wait for these moments to happen naturally.
01:33:58 I don't like...
01:33:59 I don't think that's necessary,
01:34:01 but I don't like to plan very carefully how things will happens,
01:34:06 because I end up wanting to change things at the last minute.
01:34:13 Every plan I make may not be ideal, and I'll be tempted to change them,
01:34:20 so I try not to plan too far in advance.
01:34:26 I have some difficulty organizing in that sense,
01:34:29 but fortunately my work allows me to keep doing it this way.
01:34:36 I could do it - and many Mathematicians do it without issue -
01:34:39 within a very well-established routine.
01:34:42 But I feel uncomfortable with very rigid things,
01:34:46 so this would likely have a psychological impact of making me feel restricted,
01:34:52 and would bother me.
01:34:54 It is not a prerequisite to do the type of discovery I do.
01:35:00 I think it is more because of the emotional perception than anything else.
01:35:18 I find it interesting that in Math...
01:35:20 we are very accepting of people who can contribute however they can,
01:35:26 and that we do not put much stock in social conventions or norms.
01:35:37 They exist, but they do not prevent a person from doing their work.
01:35:42 In certain professions, people who do not conform to norms,
01:35:49 such as dress or basic behavior codes,
01:35:53 won't have the chance to work in them.
01:35:56 In Math, there might be people who are isolated,
01:36:03 or who are not very proficient in talking to others,
01:36:06 but who are contributing in many ways.
01:36:09 In the end their work will be assessed for its content, for what it brings.
01:36:18 Since there are no interpretations,
01:36:21 you don't have the defend your arguments...
01:36:23 and compare with different arguments and interpretations.
01:36:26 Normally, in most works,
01:36:30 you have a theory being developed,
01:36:34 and it can be assessed despite the rhetorical skills of the person defending it.
01:36:42 Any other person can then see it,
01:36:44 if the material is prepared according to the usual standards,
01:36:49 and say 'Yes, this answers the question' or not regardless of the person's personality.
01:36:56 This is quite desirable, because it doesn't exclude people that could contribute.
01:37:03 You don't have to be eccentric, or have difficulties interacting,
01:37:11 or any other odd trait to do Math,
01:37:15 but we don't exclude people in that regard.
01:37:20 There are all sorts of people in Math.
01:37:22 It is a stereotype to think that the rule is to have odd people,
01:37:28 it's just that they are accepted.
01:37:30 They are included, and often contribute,
01:37:33 and all sorts of people can be represented.
01:38:08 It depends on what you are investigating about the world.
01:38:14 There is this difficulty of making predictions in certain time scales,
01:38:19 and the effect of chaos must be recognized.
01:38:24 On the other hand, there is great order,
01:38:30 for instance, simply in the capacity to express physical laws in mathematical terms.
01:38:36 It is not obvious why that is possible, even in principle.
01:38:41 The fact that this happens allows for short-term predictability.
01:38:46 There are simple rules that govern this behavior,
01:38:50 This is a form of order that has no reason to exist.
01:38:52 We take it for granted today...
01:38:55 that certain aspects of Physics will have this type of behavior,
01:39:01 but it is not at all obvious that it has to be that way.
01:39:05 This ensures us the possibility of living our every-day lives...
01:39:09 knowing that nothing completely unusual will happen from a moment to the other.
01:39:14 There is a certain awareness of that.
01:39:16 So it depends on the issue.
01:39:17 You have to recognize in which situations chaos will be predominant,
01:39:24 and in which order will prevail.
01:39:27 You don't have to worry much about...
01:39:28 the fact that tomorrow the Moon will be present in some way or another.
01:39:33 Maybe full moon, maybe new, but there is not much doubt it will be there.
18:42:28 But it is an aspect of order we incorporate.
01:39:43 As for the weather, however,
01:39:47 you need to be prepared for other things.
01:39:50 Work in dynamic systems leads you to suspect the possibility that chaos exists,
01:39:54 but it is important to appreciate that there exists a great order as well.
01:39:58 It all depends on what you are researching.