Things add up.

Review Notes for Real and Complex Analysis by Walter Rudin

This is the Notes I made when preparing for an exam on Real Analysis (MATH5011).



Speed of Mathematicians

Today I read the famous article  on the proof of Poincaré’s conjecture, Manifold Destiny on The New Yorker. A brilliant and colorful story and you can find an informative Wiki-page on it, the Controversy section is very interesting.

Personally, I found it quite curious why so many people got interested in this story of a battle for credit. Just as Pereman himself put it,

Everybody understood that if the proof is correct then no other recognition is needed.

However, one quote of a Russian mathematician, Burago, caught my eyes. When describing Pereman, he said something like,

He was not fast. Speed means nothing. Math doesn’t depend on speed. It is about deep.

This reminded me of one of my professors (he was once an associate professor at Berkley), who told me that he was surprised by the harsh time constraint of exams at my university. Back in Berkley, he mentioned, they always give students much-more-than-needed time. Actually, he conducted some two-hour exams on Algebra this semester and many students finished within 15 minutes.

This suggests a contrast between mathematics education in the U.S., where deep and independent thoughts are encouraged, and in China, where more emphasize is put on mastering the techniques (Well, to do things fast, you have to be quite familiar with all the well-established results and too often, you are left with no time to develop your independent thoughts or work on what you find intriguing). Personally, I have a preference for the former. After all, mathematics is not just a bunch of techniques but about imagination, which is what really distinguishes great ones and the ordinary. However, your imagination and thoughts sometimes become a burden if you have to struggle with lots of exercises.

Moreover, it is not those who solve problems that push mathematics forward but those who raise good problems (take Fermat and Poincaré as examples, whose problems actually engines the development of mathematics for centuries). So it is hard to justify the emphasize on techniques but not imagination and creative thoughts.

Maybe this also explains why many Chinese work so hard in the science field but few contributes originative ideas.

Trajectory of Some Points on Harmonic Functions

In this {\mathit{Open Problem}} catalogue I plan to keep a record of some of the problems I think about. By {\mathit{Open}} I do not indicate that they are open in the real sense, nor do I pretend they have any deep implications. They are just some problems I find interesting and have had no answer up to the point I publish the posts.

Today I was thinking about PDEs and came across this simple property of harmonic functions:

Theorem 1 {Mean Value Property}Suppose {\Omega} is an open set in {\Re^{2}} and let {\mathit{u}} be a function of class {\mathcal{C}^2} with {\Delta\mathit{u}=0} in {\Omega}. If the closure of the disc centered at {(x,y)} of radius {R} is contained in {\Omega}, then

\displaystyle \mathit{u}(x,y)=\frac{1}{2\pi}\int_{0}^{2\pi}\mathit{u}(x+r\cos\theta,y+r\sin\theta)d\theta \ \ \ \ \ (1)

for all {0\le r\le R}.

Now suppose {\mathit{u}} is such a function in {\Re^2} and {r} and {R} be as stated in the theorem.

Then given any {r_0}, there exists some point, say, {(x_0,y_0)} on the circle centered at {(x,y)} with radius {r_0} such that {\mathit{u}(x_0,y_0)=\mathit{u}(x,y)}, followed from {\mathit{Mean Value Theorem}} of integral.

So my question is, what is the trajectory of such points when {r} ranges over all admissible values.

I do not have an answer yet, and maybe it is related to the {\mathit{Maximal Principle}} of PDEs.

Measure Theory and Some Trends in Modern Analysis (II)

The second trend is even more obvious in measure theory, namely, the interplay between stages and actors.

By stages, I mean number systems, or, in general, sets; and actors refer to functions and transformations.

In classical analysis, these two are separated and handled quite differently in nature. Sets are like sites on which one constructs things and functions are those buildings constructed, though closely related to each other, these two are distinct and they do not interact.

However, in measure theory, things are different. These two act intimately with each other and this interaction is quite substantial in the theory in the way that it ties functions closely to the topology or other properties of the underlying space.

On the one hand, many functions are defined according to properties of sets/ spaces while others are constructed just to demonstrate something about the space, among which characteristic functions serve as good examples.

On the other hand, sets are manipulated with respect to certain functions. As an example, consider the trick when the whole space is partitioned into preimages of functions. Another good example comes from Riesz Representation Theorem for locally compact Hausdorff spaces and positive linear functionals, where the interplay of stages and actors is so apparent and plays such an central role.

This phenomenon somehow reminds me of Einstein’s Special Relativity, which states that time-space is not something constant, but keeps changing, due to the energy and matter inside.

Also, the idea that given any measure and any function, a new measure can be given by just taking their product also resembles principles in Special Relativity that length, area, or volume are subject to change within different frame of reference. But this is beyond the scale of this post.

Measure Theory and Some Trends in Analysis (I)

It occurred to me that measure theory, especially Lebesgue‘s theory suggests some trends of the development of modern mathematical analysis.

  • Lose of Control of Objects
  • Interplay between the Stage and the Actors

For the first one, I mean that, comparing to their predecessors, modern mathematicians are more willing to sacrifice some control over objects (sets, functions, etc.) in return for more useful or elegant theories. Maybe this is their surrender after being tortured for so many years by wild creatures released by themselves, but this reluctant give-up does lead to many deep results. After all, the whole point of Lebesgue‘s theory is supported by the philosophy that some small things are negligible.

Take Littlewood‘s famous three principles as an example:

There are three principles, roughly expressible in the following terms: Every set is nearly a finite sum of intervals; every function is nearly continuous; every convergent sequence of functions is nearly uniformly convergent.

For layman, these nearly and roughly things do not sound very mathematical. Anyway, mathematics is all about rigor and among all subjects of mathematics, analysis requires the most of it. It is difficult to imagine generating rigor results while you lose such an extent of control with these nearly and roughly.

However, these principles are highly appreciated and demonstrates a very deep understanding of mathematics that can only be possessed by a top-class mathematician, those with a sense of how to perceive the abstract through intuition and how to balance the requirement of rigor and the desire for beauty. Only after many years of training can one observe that without some control lost, elegance just cannot be achieved.

Similar examples are more than abundant: from convergence to convergence a.e. or in measure, from strong topology to weak topology, etc. Without these, insights will be lost. Many beautiful results would be drowned in ugly details and others would never be discovered at all.

And this is one of the trends of modern analysis: lose some control to achieve elegance and beauty.