倡议

我以个人的名义,倡议建立一个小组,以学习Terence Tao博客中的帖子为主要活动。由于这还是一个初始的想法,我还没有一个很具体的计划。但是系统的学习总是要耗费精力,而长期的坚持需要毅力,这都是不言自明的。有兴趣的朋友请与我联系。关注的朋友请尽量转载。刘小川 lxc1984 at gmail.com

个人经历和学习经验

(参考我的帖子:经验一二三给学数学的你中国的学生差在哪哦大学数学书等等

我自认在学习数学的过程中是走过很多弯路的。

而在各种不良低效的学习上的问题之中,我觉得最需要改进的问题是,我们所学的各个分支的严重的割裂。无论是各个数学分支,还是英文的教与学;无论是大学本科或者是研究生教育,这个问题一直是个大问题。

我觉得这是从中学以来的一个习惯。我在中学学数学的时候,就有过一个错觉,以为整个数学学科内容就那么一点点。教材实在是编写的太过自洽了,很封闭,让人问不出问题。而且有高考的指挥棒,老师们也都阻止大家问问题。我个人记忆中一个事情是,在物理课上我为一个有关的数学问题着迷。自己写了点东西给物理老师看,这位老师是个女士,她把我写的东西收下了,告诉我要高考之后再说。

后来上大学,课程的安排仍然有这个问题。整个气氛上把一切都割裂开来。比如我在学习一门数学课的时候,对一个习题想了很久之后,发现是另一门课程的基本的练习。于是所有人都告诉我等学那门课的时候再说。但是数学是一个整体。甚至其他学科,学科与学科之间都是相互密切关联的。我觉得不应该这样割裂开学习。本科的时候,学生的最重要的目的是把每门考试的分数提高,所以各个高校中都有一些奇怪的高分者,他们每门考试的成绩高的惊人,但是考试过后就会忘得一干二净。大家或多或少都受这个影响。

研究生阶段,该现象依然存在。假设我是学离散数学的,如果去上门几何课,就变得很奇怪。学分析的就完全不懂代数,学代数的未必懂拓扑。中国没有博士资格考试,我听说美国是有的,但是也不很清楚具体的情况。我去年在学校中选了一门偏微分方程的课程,结果所有的听课人中好像只有我一个不是本专业的。

这些事情在我看来都是一个问题,学一个知识的时候不想其他的,学到其他的有忘了这个,学英文的时候不想着用,用的时候又抓耳挠腮的找不到库存。这些我都经历过,而且目前还在跟自己旧的不好的习惯做斗争。

Tao在数学研究中的经验以及给年轻人的建议

我翻译过两篇文章:做数学一定要是天才吗? 我如何安排时间

更多的请登录Terry本人的博客中的一页:Career advice

做题目的重要性:

我一直认为,学数学,做题是要多重要有多重要。Terry的帖子中做大的特点就是习题够多。无论我们这个未来的学习小组具体做法是怎样的,我都坚持每个人必须要长期养成做题的好习惯。其实我个人在学习遍历论的过程中就有了较深的体会,这些习题很多都是证明的重要过程。Terry一直试图在让读者能够真正跟上一些复杂证明的思路。真正对这些习题下过功夫之后,是可以明显的感觉到收获的。甚至有些习题做起来很有科研的感觉。

另外一个方面就是尽量的写出来。虽然写出来表面上浪费时间,但并不是这样。往往以个人在动笔的时候会发现自己一些思维上的漏洞。而且自己写的东西也会印象深刻。下面是一个例子:(latex码)

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.

交流的方式:

感谢互联网给这件事情带来了无比的方便,当然email将会是最主要的讨论方式。无论怎样,手段本身是十分的不重要,关键是热情,是年轻人的冲劲:比如学起了一门课程,就不能以任何理由中途退却。

一个另外的想法就是在电子邮件中坚持用英文通信。

理想:

Bourbaki学派的最初组织人就是一群年轻人,提起他们这群人我是什么意思,我想读者比我更明白,呵。

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6 Comments on “倡议”

  1. percy li Says:

    i am a Chinese exchange student in Washington.d.c. i go to terry’s blog a lot but these math was too abstract for me. I once mentioned one of the typo in his book “sloving mathematical problems” and he really replied me. he was a great teacher. i would love to join the group if i were 5 years older…

  2. liuxiaochuan Says:

    Dear percy li:
    The way that Professor Tao helps the Maths-learners around all over the world, in my opinion, is something just like Euler centuries ago. But the chinese students are seemingly less concerned.

    As to your questions, I suggest you just try learning the posts which you are most interested. From my own experience, though it is quite difficult form the begining, you can learn much more during the process. Also, your will be more confident after some time. Anyway, you can just e-mail me at any time.

  3. percy li Says:

    thanks for reply.
    i am just in high school. i take calculus in our high school and i am reading his book on analysis(undergraduate course). his book is far more easy to read than the books i brought in China but actually i really learned a lot.

  4. purplelightning Says:

    I will try to go furture in math study though I am still a freshman in college.And I hope I will have the opportunity to enter the team years later.

  5. nouoo Says:

    我是一位自学数学的学生,本科物理出身,虽然基础不好,但希望能加入你们。

  6. le Xi Says:

    I also want to join you to improve mathematics.


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