What's so sexy about math? | Cédric Villani

671,922 views ・ 2016-06-28

TED


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翻译人员: Hancheng Li 校对人员: Nanyuan Chen
00:12
What is it that French people do better than all the others?
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法国人在什么方面做得比别人好呢?
00:18
If you would take polls,
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如果你去做个投票调查,
00:20
the top three answers might be:
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排名前三的或许是:
00:22
love, wine and whining.
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爱情,红酒,发牢骚。
00:26
(Laughter)
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(笑声)
00:27
Maybe.
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可能吧。
00:29
But let me suggest a fourth one:
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但我还想提出第四个答案:
00:31
mathematics.
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数学。
00:33
Did you know that Paris has more mathematicians
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你们知道巴黎的数学家比世界上 其它任何一个城市
00:36
than any other city in the world?
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都要多吗?
00:38
And more streets with mathematicians' names, too.
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而且以数学家的名字命名的街道数量也更多。
00:42
And if you look at the statistics of the Fields Medal,
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如果你查查菲尔兹奖的统计,
00:45
often called the Nobel Prize for mathematics,
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它也经常被称作“诺贝尔数学奖”,
00:48
and always awarded to mathematicians below the age of 40,
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只授予40岁以下的数学家,
00:52
you will find that France has more Fields medalists per inhabitant
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你会发现,就人均获奖数量来说,
法国是世界第一。
00:56
than any other country.
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00:58
What is it that we find so sexy in math?
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我们在数学中到底 发现了什么让人着迷的东西?
01:02
After all, it seems to be dull and abstract,
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毕竟它看上去那么无聊、抽象,
01:05
just numbers and computations and rules to apply.
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只是数字、计算、定理而已。
01:10
Mathematics may be abstract,
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数学可能很抽象,
01:12
but it's not dull
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但是它并不无聊,
01:13
and it's not about computing.
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而且它并不都是计算。
01:16
It is about reasoning
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它是有关逻辑的推理,
01:17
and proving our core activity.
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让我们的所作所为都有理有据。
01:20
It is about imagination,
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它有关丰富的想象,
01:22
the talent which we most praise.
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我们最常歌颂的人类天赋。
01:24
It is about finding the truth.
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它还有关真理的追寻。
01:27
There's nothing like the feeling which invades you
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当你苦思冥想数月之后,
01:30
when after months of hard thinking,
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终于找到问题的正确解法那一刻,
01:32
you finally understand the right reasoning to solve your problem.
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那种感受真的无与伦比。
01:37
The great mathematician André Weil likened this --
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伟大的数学家安德雷·韦依 把这种感受比作——
01:40
no kidding --
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不开玩笑——
01:41
to sexual pleasure.
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比作性快感。
01:44
But noted that this feeling can last for hours, or even days.
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但是他还说这种感受 可以持续数小时甚至数天。
01:50
The reward may be big.
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这种回报可能难以估量。
01:53
Hidden mathematical truths permeate our whole physical world.
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隐藏的数学规律 渗透在我们整个物质世界中。
01:57
They are inaccessible to our senses
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我们的感官不能捕捉它们,
02:00
but can be seen through mathematical lenses.
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但可以通过数学的放大镜观察到。
02:04
Close your eyes for moment
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闭上眼睛一小会儿,
02:05
and think of what is occurring right now around you.
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想一想你周围此时此刻正在发生的事。
02:10
Invisible particles from the air around are bumping on you
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看不见的空气分子 在不断地撞击你,
02:13
by the billions and billions at each second,
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每秒钟数十亿次,
02:16
all in complete chaos.
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完全是混乱无序的状态。
02:19
And still,
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然而,
它们的行为可以被数学物理学精准地预测。
02:20
their statistics can be accurately predicted by mathematical physics.
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02:25
And open your eyes now
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睁开你的双眼,
02:28
to the statistics of the velocities of these particles.
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看看这些分子的速率分布统计。
02:32
The famous bell-shaped Gauss Curve,
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这是著名的钟形高斯曲线,
02:35
or the Law of Errors --
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也可以叫做误差律——
02:37
of deviations with respect to the mean behavior.
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描述分子平均行为的一些偏差。
02:41
This curve tells about the statistics of velocities of particles
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这条曲线告诉我们 粒子的速率分布情况,
02:45
in the same way as a demographic curve
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正如一条人口统计曲线
02:48
would tell about the statistics of ages of individuals.
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能够告诉我们人口的年龄分布情况。
02:52
It's one of the most important curves ever.
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它是最重要的一条曲线之一。
02:56
It keeps on occurring again and again,
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它的规律不断地重复,
02:59
from many theories and many experiments,
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在诸多理论与实验中呈现,
03:01
as a great example of the universality
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它是数学的普适性的体现,
03:05
which is so dear to us mathematicians.
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而这种性质对我们数学家至关重要。
03:09
Of this curve,
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关于这个曲线,
03:10
the famous scientist Francis Galton said,
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著名的科学家弗朗西斯·高尔顿说:
“如果古希腊人知道这个规律, 他们一定会把它神化的。
03:14
"It would have been deified by the Greeks if they had known it.
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03:19
It is the supreme law of unreason."
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它是无理性的至高准则。”
03:23
And there's no better way to materialize that supreme goddess than Galton's Board.
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高尔顿板就是 把这个“神灵”实体化的最佳体现。
03:31
Inside this board are narrow tunnels
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在这个板子里有一些狭道,
03:34
through which tiny balls will fall down randomly,
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一些掉落的小球会随机通过这里,
03:40
going right or left, or left, etc.
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有些往右,有些往左。
03:46
All in complete randomness and chaos.
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完全是随机的、混乱的。
03:50
Let's see what happens when we look at all these random trajectories together.
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让我们看看这些随机路线 会呈现怎样的规律。
03:56
(Board shaking)
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(板子摇动)
04:01
This is a bit of a sport,
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这其实算是锻炼身体,
04:04
because we need to resolve some traffic jams in there.
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因为我们得疏通一些拥堵的状况。
04:11
Aha.
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啊哈。
04:13
We think that randomness is going to play me a trick on stage.
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看来随机性要在这个舞台上 跟我开个小玩笑了。
04:19
There it is.
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好了!
04:22
Our supreme goddess of unreason.
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这个无理性的至高无上的神,
04:24
the Gauss Curve,
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高斯曲线,
04:26
trapped here inside this transparent box as Dream in "The Sandman" comics.
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被囚禁在了这个透明的盒子里, 就像《睡魔》漫画里的梦魇一样。
04:34
For you I have shown it,
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我向各位展示了这个规律,
04:37
but to my students I explain why it could not be any other curve.
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但向我的学生,我要解释 为什么它不可能是任何其它的曲线。
04:43
And this is touching the mystery of that goddess,
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这就近乎揭开了这个神灵的面纱,
把一个美丽的巧合 变成一个赏心悦目的数学解释。
04:46
replacing a beautiful coincidence by a beautiful explanation.
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04:51
All of science is like this.
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一切的科学都是这样的。
04:54
And beautiful mathematical explanations are not only for our pleasure.
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漂亮的数学解法并不只是 为了我们自己开心。
04:59
They also change our vision of the world.
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它们同样改变了我们看世界的视角。
05:03
For instance,
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举个例子,
05:04
Einstein,
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爱因斯坦、
05:05
Perrin,
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佩兰、
05:06
Smoluchowski,
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斯莫鲁霍夫斯基,
05:07
they used the mathematical analysis of random trajectories
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他们对粒子的随机轨迹 进行了数学分析,
05:11
and the Gauss Curve
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再加上高斯曲线,
05:13
to explain and prove that our world is made of atoms.
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他们解释并证明了 我们的世界由原子组成。
05:19
It was not the first time
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这并不是第一次,
05:21
that mathematics was revolutionizing our view of the world.
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数学已经多次颠覆了我们的世界观。
05:25
More than 2,000 years ago,
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两千多年前,
05:27
at the time of the ancient Greeks,
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在古希腊的时代,
05:31
it already occurred.
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颠覆已经发生了。
05:33
In those days,
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在那个时代,
05:35
only a small fraction of the world had been explored,
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人们只探索了世界的很小一部分,
05:38
and the Earth might have seemed infinite.
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而地球看上去无边无际。
05:42
But clever Eratosthenes,
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但是聪明的埃拉托色尼
05:43
using mathematics,
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利用数学工具,
05:45
was able to measure the Earth with an amazing accuracy of two percent.
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成功的测量了地球的大小, 误差只有惊人的2%。
05:51
Here's another example.
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还有另一个例子。
05:54
In 1673, Jean Richer noticed
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在1673年,让·里谢注意到
05:58
that a pendulum swings slightly slower in Cayenne than in Paris.
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一个单摆在卡宴(译注:南美法属圭亚那首都, 近赤道)比在巴黎摆动得稍稍慢一些。
06:06
From this observation alone, and clever mathematics,
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只用这一个现象, 以及一些巧妙的数学推导,
06:10
Newton rightly deduced
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牛顿正确地推断出
06:13
that the Earth is a wee bit flattened at the poles,
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地球在两极地区稍稍扁一些,
06:18
like 0.3 percent --
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大概只有0.3%,
06:20
so tiny that you wouldn't even notice it on the real view of the Earth.
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这种细微的差别在 观察地球全貌时根本无法发现。
06:26
These stories show that mathematics
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这些故事说明了,
数学能够让我们超越自己的直觉,
06:30
is able to make us go out of our intuition
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06:35
measure the Earth which seems infinite,
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测量看似不可测的地球尺寸,
观察看不见的原子,
06:39
see atoms which are invisible
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06:41
or detect an imperceptible variation of shape.
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或是检测肉眼不可识别的微小形变。
06:44
And if there is just one thing that you should take home from this talk,
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如果你们只能从我的演讲中 了解到一样东西,
06:48
it is this:
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那应该是这个:
06:49
mathematics allows us to go beyond the intuition
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数学让我们超越人类直觉,
06:54
and explore territories which do not fit within our grasp.
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并且探索我们所无法触及的领域。
06:59
Here's a modern example you will all relate to:
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这有个例子各位都非常熟悉:
07:03
searching the Internet.
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网上搜索。
07:06
The World Wide Web,
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万维网,
07:07
more than one billion web pages --
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有着超过十亿个网页,
07:09
do you want to go through them all?
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难道你想全部搜索一遍吗?
07:11
Computing power helps,
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计算机可能有帮助,
07:13
but it would be useless without the mathematical modeling
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但是如果没有了数学模型, 它就是一堆废铁,
07:16
to find the information hidden in the data.
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无法搜寻数据中隐藏的信息。
07:20
Let's work out a baby problem.
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让我们做一道很简单的题。
07:23
Imagine that you're a detective working on a crime case,
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想象你是一个侦探, 正在调查一个犯罪案件,
07:27
and there are many people who have their version of the facts.
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很多人参与其中,并且各执一词。
07:32
Who do you want to interview first?
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你想先询问谁呢?
07:34
Sensible answer:
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合理的答案是:
07:36
prime witnesses.
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主要的目击者。
07:38
You see,
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想想看,
07:40
suppose that there is person number seven,
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假设有一位7号证人,
07:44
tells you a story,
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告诉了你一件事情,
07:45
but when you ask where he got if from,
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但当你问他从哪里听说的,
07:47
he points to person number three as a source.
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他说3号证人是消息来源。
07:50
And maybe person number three, in turn,
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有可能3号证人
07:52
points at person number one as the primary source.
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也相应地指向1号证人作为主要消息来源。
07:56
Now number one is a prime witness,
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现在1号证人是主要目击者了,
07:58
so I definitely want to interview him -- priority.
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所以我一定想要先去采访他。
08:02
And from the graph
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从这幅图中,
08:03
we also see that person number four is a prime witness.
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我们同样看到4号证人 是一位主要目击者。
08:06
And maybe I even want to interview him first,
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我可能更想先去采访他,
08:09
because there are more people who refer to him.
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因为他被提及的次数比1号还要多。
08:12
OK, that was easy,
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好吧,这还算简单的,
08:15
but now what about if you have a big bunch of people who will testify?
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但是如果有一大堆要作证的人, 那该怎么办呢?
08:20
And this graph,
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在这幅图中,
08:22
I may think of it as all people who testify in a complicated crime case,
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我可以把它当作 一件复杂案件的所有证人,
08:27
but it may just as well be web pages pointing to each other,
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但也可以把它看作是 互相链接的网页,
08:31
referring to each other for contents.
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互相引用其中的内容。
08:34
Which ones are the most authoritative?
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哪些网页最有权威性呢?
08:37
Not so clear.
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还不太清楚。
08:40
Enter PageRank,
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于是PageRank问世了,
它是谷歌最早的基石之一。
08:42
one of the early cornerstones of Google.
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08:45
This algorithm uses the laws of mathematical randomness
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这种算法运用了 数学随机性的定律,
08:49
to determine automatically the most relevant web pages,
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来自动判断哪些网页关联最多,
08:53
in the same way as we used randomness in the Galton Board experiment.
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与我们在高尔顿板实验中 运用随机性的方法一样。
08:59
So let's send into this graph
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那就把一堆小小的数码玻璃珠
09:01
a bunch of tiny, digital marbles
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放到这个图表中,
09:04
and let them go randomly through the graph.
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让它们随机的在图中穿行。
09:08
Each time they arrive at some site,
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每当它们到达某个网页,
09:10
they will go out through some link chosen at random to the next one.
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它们就会随机选择一个链接, 然后跳转到另一页。
09:14
And again, and again, and again.
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一遍又一遍重复。
09:16
And with small, growing piles,
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用这些小小的光点,
我们记录下每个网页 被访问的次数,
09:18
we'll keep the record of how many times each site has been visited
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09:21
by these digital marbles.
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就用这些数码珠子。
09:24
Here we go.
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开始喽。
09:25
Randomness, randomness.
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一切随机。
09:27
And from time to time,
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随着时间推移,
09:29
also let's make jumps completely randomly to increase the fun.
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为了增强趣味性, 这些跳跃是完全随机的。
09:34
And look at this:
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看看这个:
09:36
from the chaos will emerge the solution.
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在一片混乱中产生了一个答案。
09:39
The highest piles correspond to those sites
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这里最高的几堆对应着
09:41
which somehow are better connected than the others,
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那些相对来说链接更多的网页,
09:45
more pointed at than the others.
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被引用更多次的网页。
09:47
And here we see clearly
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在这里我们清晰地看到,
09:49
which are the web pages we want to first try.
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哪一些是我们最想先看的网页。
09:53
Once again,
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再一次,
09:54
the solution emerges from the randomness.
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问题的解答来源于随机性。
09:57
Of course, since that time,
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当然,从那以来,
10:00
Google has come up with much more sophisticated algorithms,
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谷歌已经发明出 数不胜数的复杂算法,
10:03
but already this was beautiful.
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但是这个算法已经很漂亮了。
10:06
And still,
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然而,
10:08
just one problem in a million.
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这只是沧海一粟。
10:10
With the advent of digital area,
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随着数字领域的飞速发展,
越来越多的问题 需要用数学分析来解决,
10:13
more and more problems lend themselves to mathematical analysis,
187
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10:18
making the job of mathematician a more and more useful one,
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让数学家这个工作 变得越来越实用,
10:23
to the extent that a few years ago,
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以至于大约几年前,
10:25
it was ranked number one among hundreds of jobs
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它在数百个职业中排名第一,
10:29
in a study about the best and worst jobs
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这份排名是有关最好和最差的职业,
10:33
published by the Wall Street Journal in 2009.
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由华尔街日报在2009年发表。
10:37
Mathematician --
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数学家——
10:39
best job in the world.
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世界上最好的工作。
10:41
That's because of the applications:
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这是因为它应用广泛:
10:44
communication theory,
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通讯理论、
10:46
information theory,
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信息理论、
10:48
game theory,
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博弈论、
压缩传感、
10:50
compressed sensing,
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10:51
machine learning,
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机器学习、
10:53
graph analysis,
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图表分析、
10:54
harmonic analysis.
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谐波分析。
10:56
And why not stochastic processes,
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2640
还有随机过程、
10:59
linear programming,
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线性编程、
11:00
or fluid simulation?
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2028
或者是流体模拟。
11:03
Each of these fields have monster industrial applications.
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以上每一个领域都有 规模巨大的工业应用。
11:07
And through them,
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透过它们可以看出,
11:08
there is big money in mathematics.
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668386
1999
数学的商机是无限的。
11:11
And let me concede
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2040
我必须承认,
11:13
that when it comes to making money from the math,
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谈到从数学中赚钱,
11:15
the Americans are by a long shot the world champions,
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美国人可是遥遥领先全世界,
11:19
with clever, emblematic billionaires and amazing, giant companies,
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有一群标志性绝顶聪明的领导者, 还有让人大开眼界的商业巨头,
11:24
all resting, ultimately, on good algorithm.
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归根结底都不约而同地依赖好的算法。
11:29
Now with all this beauty, usefulness and wealth,
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数学兼具着美、实用性, 以及无限商机,
11:33
mathematics does look more sexy.
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2284
它似乎的确更有魅力了。
11:36
But don't you think
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但是你千万别以为
11:38
that the life a mathematical researcher is an easy one.
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数学家的生活很轻松。
11:42
It is filled with perplexity,
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这种生活满是混乱,
11:46
frustration,
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满是沮丧,
11:48
a desperate fight for understanding.
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是追求真知的绝望之战。
11:51
Let me evoke for you
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让我向各位讲述
11:54
one of the most striking days in my mathematician's life.
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我的数学生涯中最特别的一天。
11:58
Or should I say,
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1151
或者我应该说,
11:59
one of the most striking nights.
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最特别的一晚。
12:02
At that time,
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那个时候,
12:03
I was staying at the Institute for Advanced Studies in Princeton --
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3151
我待在普林斯顿大学的 高等研究所里,
这里曾是爱因斯坦多年的家,
12:07
for many years, the home of Albert Einstein
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12:09
and arguably the most holy place for mathematical research in the world.
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4428
也很可能是世界上 数学研究的神圣之颠。
12:14
And that night I was working and working on an elusive proof,
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在那个晚上我苦思冥想, 寻找一个非常隐晦的证明,
12:18
which was incomplete.
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它十分的不完整。
12:21
It was all about understanding
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它都是有关于了解
12:23
the paradoxical stability property of plasmas,
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等离子体的矛盾稳定特性,
12:27
which are a crowd of electrons.
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1958
这里指的是一团电子云。
12:30
In the perfect world of plasma,
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2736
在等离子体的理想世界,
12:33
there are no collisions
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是没有任何碰撞的,
12:34
and no friction to provide the stability like we are used to.
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而且没有任何摩擦力, 使其像我们习惯的那么稳定。
12:39
But still,
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然而,
12:40
if you slightly perturb a plasma equilibrium,
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如果你轻微打破等离子体平衡,
12:43
you will find that the resulting electric field
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你会发现相应产生的电场
12:46
spontaneously vanishes,
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会自发的消失,
12:48
or damps out,
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1975
或者是减弱,
12:50
as if by some mysterious friction force.
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3294
好像受到了某种神秘摩擦力的影响。
12:54
This paradoxical effect,
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这种矛盾的特性,
12:56
called the Landau damping,
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叫做朗道阻尼,
12:58
is one of the most important in plasma physics,
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是等离子物理中 最重要的现象之一,
13:01
and it was discovered through mathematical ideas.
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而且它是由数学思想推导出来的。
13:04
But still,
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1151
然而,
13:06
a full mathematical understanding of this phenomenon was missing.
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对此现象的完整数学理解还不完善。
13:10
And together with my former student and main collaborator Clément Mouhot,
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我和我的前学生、主要合作者 克莱蒙·穆奥合作,
13:15
in Paris at the time,
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我们那时在巴黎,
13:16
we had been working for months and months on such a proof.
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我们为了寻找这个证法 已经工作了好几个月。
13:21
Actually,
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1335
实际上,
13:23
I had already announced by mistake that we could solve it.
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我还错误地宣布 我们可以解决这个问题。
13:27
But the truth is,
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然而事实上,
13:29
the proof was just not working.
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2147
那种证法完全无效。
13:32
In spite of more than 100 pages of complicated, mathematical arguments,
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即使是一百多页的复杂数学推导,
13:36
and a bunch discoveries,
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1690
还有一大堆的新发现,
13:38
and huge calculation,
258
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1267
巨大的计算量,
13:39
it was not working.
259
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依然得不出个所以然。
13:41
And that night in Princeton,
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在普林斯顿的那个晚上,
13:42
a certain gap in the chain of arguments was driving me crazy.
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证明中的一个小缺口让我近乎疯狂。
13:47
I was putting in there all my energy and experience and tricks,
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4593
我对它使出浑身解数,
13:52
and still nothing was working.
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1742
但是依旧没有进展。
13:54
1 a.m., 2 a.m., 3 a.m.,
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3882
凌晨一点、两点、三点,
13:58
not working.
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1308
毫无进展。
14:00
Around 4 a.m., I go to bed in low spirits.
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4321
大概凌晨四点的时候, 我无精打采的上床。
14:05
Then a few hours later,
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然而几个小时后,
14:08
waking up and go,
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1151
我从床上爬起来,
14:09
"Ah, it's time to get the kids to school --"
269
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3357
“啊,该送孩子们上学了。”……
14:12
What is this?
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1151
这是什么?
14:14
There was this voice in my head, I swear.
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2142
我发誓,我的脑袋里有个声音。
14:16
"Take the second term to the other side,
272
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1913
“把第二项移到等式另一边,
14:18
Fourier transform and invert in L2."
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1919
“傅里叶展开然后在L2域反变换。”
14:21
(Laughter)
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1151
(笑声)
14:22
Damn it,
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1702
可恶!
14:24
that was the start of the solution!
276
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2113
这就是解法的开始啊!
14:27
You see,
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1151
你们看到,
14:28
I thought I had taken some rest,
278
868694
2283
我觉得我自己在休息,
但实际上我的大脑还在持续思考。
14:31
but really my brain had continued to work on it.
279
871001
3388
14:35
In those moments,
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1597
在那些时刻,
14:36
you don't think of your career or your colleagues,
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2601
你不会想到你的职业生涯 或是你的同事,
14:39
it's just a complete battle between the problem and you.
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这只是你自己与问题之间的斗争。
14:44
That being said,
283
884098
1328
但说到这里,
14:45
it does not harm when you do get a promotion in reward for your hard work.
284
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3949
你因为工作努力而受到奖赏 当然是很好的事情。
14:49
And after we completed our huge analysis of the Landau damping,
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889808
5160
在我们完成了朗道阻尼方面的 重大研究后,
14:54
I was lucky enough
286
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1615
我很幸运地
14:56
to get the most coveted Fields Medal
287
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3030
获得了我梦寐以求的菲尔兹奖,
14:59
from the hands of the President of India,
288
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2867
我从印度总统手中接过此奖,
15:02
in Hyderabad on 19 August, 2010 --
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902576
3920
那是在2010年8月19日, 在海德拉巴城——
15:07
an honor that mathematicians never dare to dream,
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3251
这是一个数学家不敢奢望的奖项,
15:10
a day that I will remember until I live.
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2399
我也会将这天永远铭记在心。
15:14
What do you think,
292
914366
1447
对于这样的情况,
15:15
on such an occasion?
293
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2141
你们觉得怎样呢?
很自豪,对吧?
15:18
Pride, yes?
294
918002
1150
15:19
And gratitude to the many collaborators who made this possible.
295
919791
3640
还有对主要合作者的感激之情。
15:24
And because it was a collective adventure,
296
924304
2212
而且因为这是一个集体研究,
15:26
you need to share it, not just with your collaborators.
297
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4142
你需要把成果公开, 而非只是与合作者共享。
15:31
I believe that everybody can appreciate the thrill of mathematical research,
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5692
我相信每个人都可以欣赏 数学研究的刺激感,
15:37
and share the passionate stories of humans and ideas behind it.
299
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4318
并且分享精彩研究过程中的人和事。
15:42
And I've been working with my staff at Institut Henri Poincaré,
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4774
我在昂利·庞加莱研究所 与我的团队工作,
15:47
together with partners and artists of mathematical communication worldwide,
301
947292
5181
还有一些其他的合伙人、 世界各地的数学交流艺术家,
15:52
so that we can found our own, very special museum of mathematics there.
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4587
于是我们就可以创立我们自己的, 特殊的数学博物馆。
15:58
So in a few years,
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958537
1777
再过几年,
16:00
when you come to Paris,
304
960885
1577
当你到巴黎来,
16:02
after tasting the great, crispy baguette and macaroon,
305
962486
5658
在你们品尝过美味酥脆的 法国长面包和马卡龙(蛋白杏仁饼甜点)之后,
16:08
please come and visit us at Institut Henri Poincaré,
306
968168
3663
请各位也来我们的 昂利·庞加莱研究所转一转,
16:11
and share the mathematical dream with us.
307
971856
2515
与我们共享一个数学的梦。
16:14
Thank you.
308
974395
1151
谢谢。
16:15
(Applause)
309
975570
7000
(掌声)
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