陶哲轩谈数学合作

作者:匿名用户
链接:https://www.zhihu.com/question/52820579/answer/132420009
来源:知乎

我不可能讲别人,只说一说我自己的研究,至少有一半的文章,我是与一个或多个作者合作的,我最好的工作也出自于这些合作合作文章,他们是真正合作的文章。

当然,每一个数学家有他(她)自己独有的研究风格,而且这种多样性对数学整体来讲是非常健康的。

 

I can’t speak for others, but as for my own research, at least half of my papers are joint with one or more authors, and amongst those papers that I consider among my best work, they are virtually all joint.

Of course, each mathematician has his or her own unique research style, and this diversity is a very healthy thing for mathematics as a whole. But I think 21st century mathematics differs from 19th and early 20th century mathematics in at least two important respects. Firstly, the advent of modern communication technologies, most notably the internet, has made it significantly easier to collaborate with other mathematicians who are not at the same physical location. (Most of my collaborations, for instance, would be non-existent, or at least significantly less productive, without the internet.) One can imagine the next generation of technologies having an even stronger impact in this direction (with this project possibly being an example; other extant examples include Wikipedia and the Online Encyclopedia of Integer Sequences).

Secondly, the main focus of mathematical activity has shifted significantly towards interdisciplinary work spanning several fields of mathematics, as opposed to specialist work which requires deep knowledge of just one field of mathematics, and for such problems it is more advantageous to have more than one mathematician working on the problem. (Admittedly, much of 19th century mathematics was similarly interdisciplinary, but mathematics had a much smaller diameter back then, and it was possible for a good mathematician to master the state of the art in several subfields simultaneously. This is significantly more difficult to do nowadays.)

The largest collaboration I have been in to date has involved five people – but already the dynamics of research change dramatically at that scale (especially when all five people are in the same room at once). One can toss an idea out there and have it debated by two other collaborators, while a fourth makes comments and corrections from the sidelines, and a fifth takes notes. Connections are made much faster, errors are detected quicker, and thoughts are clarified much more efficiently (often, I find one of my collaborators acting as a “translator” to distill an excited inspiration of another). It may not be “magic”, but it is certainly productive, and actually quite a lot of fun.

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最大熵原理

信息熵定义为

H(x)=-\int_{R^n} f(x)\log f(x) dx

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William Thurston

Mathematics is a process of staring hard enough with enough perseverence at at the fog of muddle and confusion to eventually break through to improved clarity. I’m happy when I can admit, at least to myself, that my thinking is muddled, and I try to overcome the embarrassment that I might reveal ignorance or confusion.

Over the years, this has helped me develop clarity in some things, but I remain muddled in many others.

I enjoy questions that seem honest, even when they admit or reveal confusion, in preference to questions that appear designed to project sophistication.

 

It’s not mathematics that you need to contribute to. It’s deeper than that: how might you contribute to humanity, and even deeper, to the well-being of the world, by pursuing mathematics? Such a question is not possible to answer in a purely intellectual way, because the effects of our actions go far beyond our understanding. We are deeply social and deeply instinctual animals, so much that our well-being depends on many things we do that are hard to explain in an intellectual way. That is why you do well to follow your heart and your passion. Bare reason is likely to lead you astray. None of us are smart and wise enough to figure it out intellectually.
The product of mathematics is clarity and understanding. Not theorems, by themselves. Is there, for example any real reason that even such famous results as Fermat’s Last Theorem, or the Poincare conjecture, really matter? Their real importance is not in their specific statements, but their role in challenging our understanding, presenting challenges that led to mathematical developments that increased our understanding.

The world does not suffer from an oversupply of clarity and understanding (to put it mildly). How and whether specific mathematics might lead to improving the world (whatever that means) is usually impossible to tease out, but mathematics collectively is extremely important.

I think of mathematics as having a large component of psychology, because of its strong dependence on human minds. Dehumanized mathematics would be more like computer code, which is very different. Mathematical ideas, even simple ideas, are often hard to transplant from mind to mind. There are many ideas in mathematics that may be hard to get, but are easy once you get them. Because of this, mathematical understanding does not expand in a monotone direction. Our understanding frequently deteriorates as well. There are several obvious mechanisms of decay. The experts in a subject retire and die, or simply move on to other subjects and forget. Mathematics is commonly explained and recorded in symbolic and concrete forms that are easy to communicate, rather than in conceptual forms that are easy to understand once communicated. Translation in the direction conceptual -> concrete and symbolic is much easier than translation in the reverse direction, and symbolic forms often replaces the conceptual forms of understanding. And mathematical conventions and taken-for-granted knowledge change, so older texts may become hard to understand.

In short, mathematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary. Revolutionary change does matter, but revolutions are few, and they are not self-sustaining — they depend very heavily on the community of mathematicians.

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教学之美

I was drawn to the teaching profession because I enjoy working with children.I view teaching as a constant challenge that is seldom,if ever,boring.The realization that a teacher can introduce children to new worlds of information and open up new possibilities daily was motivation in itself for me to select teaching as a profession many years ago.

What I try to do every day in my class is be a good listener,to guide instruction rather than lecture,and be willing to use varied methodologies to foster learning.I also strongly believe in establishing and maintaining good communication with parents.This can foster a positive learning atmosphere in the classroom and can only enhance instruction.
Teaching to me is not a nine to five occupation.It is a privilege and a tremendous responsibility.I believe in placing my students as a priority during the school year and have a very support family who accepts the fact that my teaching is important.
I would be a fool to say that I know everything there is about teaching.I am a life-long learner and want my students to become the same as well.

I do set high standards in my classroom and will not accept minimal effort from anyone including myself.If I am to expect a one hundred percent effort from my students,then it is only fair to say that my students can expect the same from their teacher.

I really try to make teaching an interesting and personally rewarding experience for my students while at the same time trying to have fun with what we are doing.

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陶的数学之路

数学是一个多层面的学科,我们的经验和品味会随着时间和阅历的改变而改变。

当我上小学时,我被数学中形式计算的抽象美所吸引,并且醉心于利用简单规则得到非平凡答案的超常能力。

当我是一个中学生时,我把数学竞赛当做一种运动,通过采取巧妙的设计数学谜题并且寻找正确的“技巧”来解决这些问题,乐此不疲。

当我是一个本科生时,第一次瞥见丰富深刻而又令人着迷的理论以及存在于现代核心数学的结构,这让我心存敬畏。

当我是一个研究生时,我体会到了拥有自己的研究计划的自豪感,以及通过原创的讨论来重新解决之前公开的问题的满足感。

直到现在作为职业数学研究开始学术生涯,我开始观察蕴含于现代数学理论背后的直觉与动机,当意识到非常复杂艰深的结果常常由非常简单寻常的例子来指引,常常让我欣喜不已。掌握这些原理并且突然发现一大批数学是怎样启示与阐释,这真是一个非凡的“原来如此”经历。

数学中仍然有许多值得去发现的东西,直到最近,我已经掌握了足够多的数学领域,并且开始朝着数学作为一个整体的学科而努力,同时联系其他科学等学科。

Mathematics is a multifaceted subject, and our experience and appreciation of it changes with time and experience.

As a primary school student, I was drawn to mathematics by the abstract beauty of formal manipulation, and the remarkable ability to repeatedly use simple rules to achieve non-trivial answers.

As a high school student, competing in mathematics competitions, I enjoyed mathematics as a sport, taking cleverly designed mathematical puzzle problems and searching for the right “trick” that would unlock each one.

As an undergraduate, I was awed by my first glimpses of the rich, deep, and fascinating theories and structures which lie at the core of modern mathematics today.

As a graduate student, I learnt the pride of having one’s own research project, and the unique satisfaction that comes from creating an original argument that resolved previously open question.

Upon starting my career as a professional research mathematician, I began to see the intuition and motivation that lay behind the theories and problems of modern mathematics, and was delighted when realizing how even very complex and deep results are often at heart be guided by very simple, even common-sensical, principles. The “Aha!” experience of grasping one of these principles, and suddenly seeing how it illuminates and informs a large body of mathematics, is a truly remarkable one.

And there are yet more aspects of mathematics to discover; it is only recently for me that I have grasped enough fields of mathematics to begin to get a sense of the endeavour of modern mathematics as a unified subject, and how it connects to the sciences and other disciplines.

我并没有任何魔力 。我看一个问题,看到它和我之前看到的的很类似,我认为之前的想法也许可以用来解决现在的问题。如果不能,那么你会认为一个小的技巧会使得问题改观但仍然不完全正确。

我与数学玩耍,直到我搞清楚发生了什么。大多数人面对一个数学问题,并且试图直接解决这个问题。即使他们已经完全解决了问题,但是也许并非真正明白其中的道理。

在我写出全部细节之前,我会先提出一个策略。一旦你有一个策略,一个非常复杂的问题就可以分解成许多小的问题。我从未满足刚刚解决的问题。我想看看如果改变一些会发生什么,仍然凑效吗?

如果你足够有经验,你就会理解的越深刻。不久,当相似的事情伴随而来的时候,你就会知道行不行。这与你是否聪明敏捷无关。

这有一点像爬山:如果你非常强壮而且很快,并且有很多绳索,这会帮助你,但是你需要设计出好的路线是你到达目的地。

快速计算并且知道许多事实,这类似于一个攀岩者拥有力气、速度以及好的工具。你仍然需要一个计划-这会是很困难的部分-使得你看到更加广阔的图景。

 

 

I don’t have any magical ability. I look at a problem, and if it looks something like one I’ve done before; I think maybe the idea that worked before will work here. If nothing’s working out; then you think of a small trick that makes it a little better but still is not quite right.

I play with the problem, and after a while, I figure out what’s going on. Most people, faced with a math problem, will try to solve the problem directly. Even if they get it, they might not understand exactly what they did.

Before I work out any details, I work on the strategy. Once you have a strategy, a very complicated problem can split up into a lot of mini-problems. I’ve never really been satisfied with just solving the problem. I want to see what happens if I make some changes; will it still work?

If you experiment enough, you get a deeper understanding. After a while, when something similar comes along, you get an idea of what works and what doesn’t work. It’s not about being smart or even fast.

It’s like climbing a cliff: If you’re very strong and quick and have a lot of rope, it helps, but you need to devise a good route to get up there.

Doing calculations quickly and knowing a lot of facts are like a rock climber with strength, quickness and good tools. You still need a plan — that’s the hard part — and you have to see the bigger picture.

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我眼中的数学之美

一千个读者心中就有一千个哈姆莱特,“数学之美”也是这样,正如每个人的审美不一样,不同的人由于教育经历,人生阅历,对于数学之美也不经相同。最近读了吴军博士的《数学之美》,作者以一个信息技术科学家的视角,向读者展现了数学在中文信息处理、语音识别、搜索引擎等领域的应用之美,娓娓道来,深入浅出。作者引言培根的名言:美德就如同华贵的宝石,在朴素的衬托下最先华贵。数学的好处也恰恰在于一个好的方法,常常是最简单明了的方法。作者不知一次地强调数学是“道”,而其他技巧则是“术”。

我看的是第一版,大约两年前就买下了,翻一翻和之前理解的传统意义上讲数学之美的书不同,中间有很多计算机的内容,于是就束之高阁。最近由于Google的Alphago与国际围棋冠军李世乭的人机大战,人工智能和机器学习炒得很热,再加上这学期带的软件学院的高等数学,老想着让他们能对数学赶点兴趣,于是就想起吴军的这本书,再次打开变手不释卷,一周空隙便把它读完了。感触很深,之前学纯数学的通病就是看不起应用数学,而且这种鄙视感一直延续到博士快毕业,直到03年5月份听鄂维南院士一个关于应用数学的报告,才开始思考这个问题,纯粹数学到底比应用数学“高等”在哪里,应用数学也有很有趣的问题,甚至用的好比起套几篇文章意义大的多。后来到了中科院应用数学所,接触的概率论或者应用数学的同事多了,更加慢慢接受这个转变,现在想来,那会的清高真是自欺欺人啊。

大多数学生都会问这样一个问题,学这么多数学有什么用,吴军书里也提到他当时学线性代数的时候也不知道要算那么多矩阵行列式干什么,直到他学了自然语言处理和搜索才直到大有用处。回想我当学生那会,比较单纯,根本就没有问有没有用这个问题,觉得美就足够了,甚至大一第一学期抄了一本关于“数学之美”的书,那会就是觉得美,内容当然不太明白,也许正是这种朦朦胧胧对数学的感情促使着我与数学的缘分走过了13年而且会一直走下去,假如那会就问有没有用,答案肯定是没用,这样自然会产生厌烦情绪,说不定就中途放弃了。现在给学生讲,你们之所以觉得没用,是因为你们还不知道怎么用,把这本吴军的《数学之美》推荐给他们也许会起到一点作用,正如自己当年抄“数学之美”那样。

罗素说,数学如同雕像,亦或是夜晚远处的星星,有一种遥远而冷峻的美(第一次听这句话应该归与大学近世代数荆老师)。这是数学家和哲学家兼诺贝尔文学奖获得者感受到的数学之美,随着年龄的增加,阅历的丰富,数学之美也在逐渐变化,如同一个男孩对美女的定义一样。少年时,觉得外表清纯可爱就是美,结构对称简单就是数学美;长大后,觉得“美有两种,一种是动人的方程,一种是你泛着倦意淡淡的笑容”;再后来,认为除了外表美,还要有心灵美,能提出一个大的框架或者解决一个公开问题(比如朗兰兹纲领和庞加莱猜想);而现在,陪伴是最长情的告白,平凡最美,自己正真参与进去,哪怕是一点点小小的突破,就是我眼中的数学之美。

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Scalar curvature equation on warped products manifolds

In this aritcle by Dobarro and Dozo, the authors give another method calculating the scalar curvatrue on warped products manifolds by making using of conformal change.

Let M = (M^m,g) and N = (N^n,h) be two Riemann manifolds. For \phi\in C^{\infty}(M), \phi> 0 on M, we consider the warped product

\displaystyle M\times_{\phi} N = ((M\times N)^{m+n}, g + \phi^2 h)

and show the relationship between the scalar curvatures R_g on M, R_h on N and R_{\phi} on M \times_{\phi} N. This relationship is a nonlinear partial differential equation satisfied by a power of the weight \phi.

Theorem:  Let R_g, R_h and R_{\phi} denote the scalar curvature on M, N and M \times_{\phi} N respectively. Then the following equality holds:

\displaystyle  -\frac{4n}{n+1}\Delta_gu+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u\hfill (1)

where u=\phi^{\frac{n+1}2}. Namely,

\displaystyle  R_{\phi}=R_g+\frac{R_h}{\phi^2}-2n\frac{\Delta\phi}{\phi}-n(n-1)\frac{|\nabla_g\phi|^2}{\phi^2}.

Proof: Write g_{\phi}\doteqdot g+\phi^2h=\phi^2(\phi^{-2}g+h), so g_{\phi} is conformal to \tilde{g}_{\phi}\doteqdot\phi^{-2}g+h on M\times N and \phi^{-2}g is conformal to g on M.

Suppose m \ge 3, we apply Yamabe equation in M to obtain that \phi satisfies

\displaystyle  -\frac{4(m-1)}{m-2}\Delta_gv+R_gv=v^{\frac{n+2}{n-2}}\widetilde{R}_g\hfill (2)

with \phi^2=v^{\frac4{m-2}}, where \widetilde{R}_g denotes the scalar curvature on (M^m,\phi^{-2}g).
As m+n\ge3, we use Yamabe equation in M\times N. Hence \phi also satisfies

\displaystyle  -\frac{4(m+n-1)}{m+n-2}\Delta_{\tilde{g}_{\phi}}w+\widetilde{R}_{g_{\phi}}w={R}_{g_{\phi}}w^{\frac{m+n+2}{m+n-2}}\hfill (3)

with w^{\frac4{m+n-2}}=\phi^2 where \widetilde{R}_{g_{\phi}} denotes the scalar curvature on (M^m+N^n,\tilde{g}_{\phi}) and \Delta_{\tilde{g}_{\phi}} the corresponding laplacian.

From w\in C^{\infty}(M), we deduce that

\displaystyle  \Delta_{\tilde{g}_{\phi}}w=\Delta_{\phi^{-2}g+h}w=\Delta_{\phi^{-2}g}w.

Working in local coordinates

\displaystyle  \Delta_{\phi^{-2}g}w=\frac1{\sqrt{|\phi^{-2}g|}}\partial_i\left((\phi^{-2}g)^{ij}\sqrt{|\phi^{-2}g|}\partial_jw\right)

where |\phi^{-2}g|=\det(\phi^{-2}g_{ij})=\phi^{-2m}|g| and (\phi^{-2}g)^{ij}=\phi^2g^{ij}, hence

\displaystyle\begin{gathered}  \Delta_{\phi^{-2}g}w=\phi^m\frac1{\sqrt{|g|}}\partial_i\left(\phi^{2-m}g^{ij}\sqrt{|g|}\partial_jw\right)\hfill\\  \qquad=v^{\frac{-2m}{m-2}}\frac1{\sqrt{|g|}}\partial_i\left(v^2g^{ij}\sqrt{|g|}\partial_jw\right)\hfill\\  \qquad=\left(v\Delta_gw+2g^{ij}\partial_iv\partial_jw\right)v^{-\frac{n+2}{n-2}}.  \end{gathered}

On the other hand,

\displaystyle  \Delta_g(vw)=v\Delta_gw+w\Delta_gv+2g^{ij}\partial_iv\partial_jw.

Hence

\displaystyle  v^{\frac{n+2}{n-2}}\Delta_{\phi^{-2}g}w=\Delta_g(vw)-w\Delta_gv.\hfill(4)

We have

\displaystyle  \widetilde{R}_{g_{\phi}}=\widetilde{R}_g+R_h\hfill(5)

By using of this in (3)  and multiplying by v^{\frac{n+2}{n-2}}, we obtain from (4)

\displaystyle  -\frac{4(m+n-1)}{m+n-2}\big(\Delta_g(vw)-w\Delta_gv\big)+v^{\frac{n+2}{n-2}}(\widetilde{R}_g+R_h)={R}_{g_{\phi}}w^{\frac{m+n+2}{m+n-2}}v^{\frac{n+2}{n-2}}\hfill (6)

From (2) we arrive at

\displaystyle\begin{gathered}  -\frac{4(m+n-1)}{m+n-2}\Delta_g(vw)-\frac{4n}{(m+n-2)(m-2)}w\Delta_gv\hfill\\  \qquad+R_g(vw)+R_hwv^{\frac{n+2}{n-2}}=R_{\phi}(vw)w^{\frac4{m+n-2}}v^{\frac4{m-4}}=R_{\phi}(vw).  \end{gathered}

Recalling that w^{\frac4{m+n-2}}v^{\frac4{m-4}}=\phi^2\phi^{-2}=1 and denoting u =\phi^{\frac{n+1}2}, replacing in terms of u in this last equality, then multiplying by u^{1/n+1}, we obtain

\displaystyle\begin{gathered}  -\frac{4(m+n-1)}{m+n-2}\Delta_g(u^{\frac n{n+1}})u^{\frac1{n+1}}-\frac{4n}{(m+n-2)(m-2)}u^{\frac{m+n-1}{n+1}}\Delta_gu^{\frac{2-m}{n+1}}\hfill(7)\\  \quad+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u.  \end{gathered}

For any \alpha\neq0, u\in C^{\infty}(M), u>0 such that

\displaystyle  \Delta_gu^{\alpha}=\alpha(\alpha-1)u^{\alpha-2}|\nabla_gu|^2+\alpha u^{\alpha-1}\Delta_gu.

Choosing \alpha=\frac n{n+1} and \frac{2-m}{n+1} in (7), we obtain

\displaystyle  -\frac{4n}{n+1}\Delta_gu+R_gu+R_hu^{\frac{n-3}{n+1}}=R_{\phi}u.

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