English subtitles for clip: File:Math genius Artur Avila about calming chaos-VPRO-The Mind of the Universe.ogv

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.