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Thursday, April 24, 2014

散散心

同一班好耐冇見嘅師兄行山。其中一個喺 BU 做緊 research assistant professor,話如果有困難嘅話可以去佢果到萬五蚊咁做兩年 RA。非常感激呢番好意,但呢類 RA 工對我前途應該冇太大幫助 ...。暫時盡力去搵中大嘅工,留喺果到學習喺科大比較難學到嘅野。

Wednesday, April 16, 2014

漫長艱辛嘅等 offer 期結束,要搵工 lu

最近都係冇 offer,但又難怪,自己寫 personal statement 求求其其 (幾乎間間學校用同一份 ...),報學校又報得少,toefl 同 gre math 又考得唔好,冇得怨,因為須要再改進嘅地方實在太多,只好下年再黎同認真少少。

期望可以搵到一份可以留喺大學到繼續讀書嘅工。經 email 同一個較熟嘅中大 professor 傾過後,有機會去 CU。期望去到喇...,因為好想喺果到跟真係做緊 analysis 嘅 professor 學野,聽一 d 真正自己想聽嘅 seminar。

當初寧願等多年都唔應該入科大,喺科大幾乎冇野學到 (當然我仍然好感激 kin li 幫我 R 個 mphil 位返黎)。

開始要備戰下學期考 toefl 了。Peter 送左我一大堆 download 返黎嘅電映,我要開始俾心機煲哂佢同學多 d 生字。

Wednesday, April 2, 2014

Connected Components

In the past I learnt connected components in a quite high generality. Special case is of particular importance, the following are the few used implicitly in complex analysis on \C.
Fact 1. If U is open in a locally connected topological space X, then each of its connected components is open. In particular, if K is closed, \C\setminus K is a countable union of open connected components.
Proof. Suppose C\subseteq X is a connected component, let x\in C, then there is a connected neighborhood V, x\in V. Since V\cap C\neq \emptyset, and since V,C\subseteq X, it must happen that V\subseteq C, as desired.\qed
Fact 2. If D is open connected, then D is a component of \C\setminus \partial D.
Proof. Let D\subseteq U\subseteq \C\setminus \partial D with U connected, we show that D=U. Since U=D\cup (U\setminus D). As U\cap \partial D=\emptyset, thus we have U=D\cup (U\setminus \ol{D}). Since U is connected, and both D and U\setminus \ol D are open, one of them must be empty, and since D\neq \emptyset, we have U=D, as desired.\qed
Fact 3. If X is locally path connected, then connected subsets are path-connected.
Proof. Let U\subseteq X be connected, fix x\in U, then the set U_x of points that can be connected with x by a path in U is both open and closed. Since x and x can be connected by a path in U, U_x\neq \emptyset, thus U_x=U, as desired.\qed
Fact 4. If C is a connected component of a locally connected topological subspace X of some Y, then \partial C\subseteq \partial X. A typical example of X is open subset \C\setminus K of \C. This is a particularly important example in potential theory as it pops up quite frequently and naturally when using maximum principle for subharmonic functions (since \partial C\subseteq \partial (\C\setminus K) =\partial K).
Proof 1. To show \ol C\setminus C^\circ \subseteq \ol X\setminus X^\circ, we pick x\in \ol C. We now show that if x\in X^\circ, then x\in C^\circ. Suppose x\in X^\circ, then there is a connected neighborhood V\subseteq X of x. Now V\cap C\neq \emptyset, thus V\subseteq C, so x\in C^\circ.

Now we have shown that \ol C\setminus C^\circ \subseteq \ol C\setminus X^\circ, thus \partial C\subseteq \partial X.\qed

Proof 2 (When C is just a component of \mathbb P\setminus K with K compact). This is a very common scenario. Take x\in \partial C. For the sake of contradiction, suppose that x\not\in \partial(\mathbb P \setminus K)=\partial K, then there is a path connected open V, with x\in V, such that V\cap \partial(\mathbb P\setminus K)=\emptyset. Now V\cap C\neq \emptyset (since x\in \ol C) and as C must be open, V\cup C is a connected open set.

We expect V\cap K=\emptyset, if not, then there is v\in V\cap K, joining this point with a c\in C\subseteq \mathbb P\setminus K using a path in V\cup C, we get a point in (V\cup C)\cap (\partial K) = C\cap \partial K\subseteq C\cap K=\emptyset, a contradiction. We conclude that V\subseteq \mathbb P\setminus K.

Therefore V\cup C is an open connected subset of \mathbb P\setminus K, and as C is a component of \mathbb P\setminus K, we have C\cup V=C, so V\subseteq C, a contradiction since V is a neighborhood of x\in \partial C.\qed

Tuesday, April 1, 2014

反思

臨就黎畢業,PhD application 唔敢抱樂觀態度,再諗返,究竟係我喺科大學到學得到 d 乜?我可以好肯定咁同自己講,幾乎冇野學到!我真正感覺到自己知識有正增長嘅時期係 year 1,喺中大讀書嘅時期。

作為一個做分析嘅人,喺科大讀書只可以話係一埸悲劇。仲 active 喺 research in analysis 嘅 professor 得返一個,十個 seminar 九個同 analysis 無關。反觀人地中大個 IMS 不斷有 mini course,seminar 同 analysis 有關,我真係羨慕到不得了,但又唔能夠下下都走過去中大聽。

如果我年半前係跟一個 active 喺 research 嘅 professor,話唔定我已有一年真真正正嘅 research experience,已經有 paper 喺手,已經學到好多有對我有意義,有趣同深奧嘅知識,可惜喺科大嘅第二年學到 (及被迫去學) 嘅野對 analysis 及對我做緊嘅 complex analysis research 一 d 幫助都冇。

有志做 analysis 嘅人,唔應該入科大。入得科大,唔好諗住做 analysis。真係喺科大揀左做 analysis 嘅話,只好願上天保祐。