Announcement

I would like to announce a study group to learn things from Professor Tao’s blog. Since this is only a thought, I still don’t have any idea how this works exactly. The hardworking to learn maths systematically and the decisiveness for long term work are obviously needed. Anyone who is interested can contact me. Any reader could quote this article to help me to other places in the Internet. Thank you so much.

My name is Liu Xiaochuan, and my email address is lxc1984 at gmail.com

My own experiences on maths learing

Years ago, I don’t have a good habit in maths learning. Among al the problems, I believe the most serious one is that what we learned are split apart from each other. This problem turns out to be a big problem no matter in undergraduate or graduate education in China. (and maybe the same situation happens in other countries.)

From high school, this habit was with me for a long time. When I was learning mathematics at that time, I have this confusion that the maths only contains such a little content. All the textbooks are written to be too self-contained to ask proper questions for learners. Besides, the Entrance Examinations in China makes the lives of teenagers miserable. Even the teachers stop students from asking questions. One memory of mime is that I got interested in on maths problem in the physics class and write something to my teacher. The teacher is a nice lady. But she take my writings and said she would give them back to me after The Examination.

When I was studying maths in a university as a undergraduate student, the same problem happens. Once when I was learning a course, I thought about a problem for quite a long time, which turned out to be a basic one in “another” maths course. All the classmates and the teachers around me told me to forget it until we have chance to learn that course. But mathematics, as a whole, should not be split into quite different subjects. Even different subjects have connections from each other. So, some wield things happens. Some students in mathematics department can get really high score in each course, and afterwards they can remember surprisingly little maths.

When I graduated in 2006, I thought things could get better. I was wrong. For example, if some student majoring in discrete mathematics want to take a geometry class, he or she would become quite curious to others. For the same reason, the students majoring in geometry generally don’t learn things about algebra and those majoring in algebra generally don’t learn things about topology, ether. I myself took a course in PDE in my university and was the only one who are not majoring in that field. There is not qualifying exams for graduate students in China, so they can only learn things seemingly related and still can survive.

The advices given by Tao

Please visit this page of Tao’s blog by a click Career advice

The importance of doing exercises

The one single important thing to do in learning maths is to do exercises. The posts in Terry’s blog contains lots of exercises. No matter how our future group works, one thing would be certain, that is, to do exercise as much as possible. In fact, during the past year, I myself learned a lot when I was trying to do all the exercises in Tao’s course: ergodic theory. Some exercises are important part of the the whole proof, others are an analogy of the proof s of the theorems. Only few are the direct use of the theorems. Terry wants the readers to grasp the idea the complicated proofs as a whole. Sometimes one can get feeling of doing reaserch.

Another thing is to write down as much as possible. It is seemingly a waste of time, but in fact people always find their mistakes while trying writing things dowm.

The following is an example:

Exercise 14 of the Notes 8 of the course: Real Analysis:Let X be a topological space. Show that X is compact if and only if every net has a convergent subnet. (Hint: equate both properties of X with the finite intersection property, and review the proof of Theorem 1). Similarly, show that a subset E of X is relatively compact if and only if every net in E has a subnet that converges in X. (Note that as not every compact space is sequentially compact, this exercise shows that we cannot enforce injectivity of \phi in the definition of a subnet.

ProofWe only prove the first conclusion.

\Rightarrow“: Suppose for contradiction that the net \{x_\alpha\}_{\alpha\in A} is without convergent subnet. Then an easy implication shows for any x_0\in X, there exists an open neighborhood V_{x_0} of x_0 such that some \alpha\in A satisfies that for all \beta \geq \alpha, x_\beta\notin V_{x_0}. All the V_x form a open cover of the space X, and thus give rise to a finite number of V_x which is a finite cover. Accordingly there exists a \alpha_x for each x. Then since we can choose a \alpha^* larger than each \alpha_x, the contradiction follows since there is no place to live for all the elements larger than \alpha^*.

\Leftarrow“: Suppose for contradiction that (F_\alpha)_{\alpha \in A} is a collection of closed subsets of X such that any finite subcollection of sets has non-empty intersection, and the intersection of the entire collection is empty. Then choose from all the finite subcollection of sets (F_\alpha)_{\alpha \in A} an element x_{\alpha_1,\cdots \alpha_k}. Define (\alpha_1,\cdots,\alpha_k)\leq (\alpha_1',\cdots,\alpha_l') iff \cap_{i=1}^kF_{\alpha_i}\subseteq \cap_{j=1}^lF_{\alpha_j'} and this forms a net. Observe that this net has no convergent sebnet, otherwise there would be some x in all the F_\alpha, a contradiction.

The way to communicate

Thanks to the Internet, we can communicate with each other easily through email and this would be the major way for us to discuss things. Anyway, the way to all these things is of the least importance. The key to accomplish this project is the enthusiasm from the heart.

By the way, we insist that we should write letters to each other inEnglish.

Ideal

Nicolas Bourbaki is a great group which is started by several young mathematicians. The readers will understand my reason of mentioning them.

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