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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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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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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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。








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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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.


\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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