不知为何,突然想起测度论里的不可测度的维塔利集合

复制以我写的知乎文章

我在知乎上写的目前竟是一些有关于美国华人和ABC和犹太人的政治话题,自己快成了民族活动家了,其实对于学理工科的人而言,民族活动家比较贬义。民族活动家似的言论与活动,尤其在美国,其实是自然被有能力的人所藐视的,这原则很简单,它根本就是不“专业”的表现,甚至可以说是一种流氓耍诬赖的作为。在美国,中国人政治上都是特别老实的,从来不闹事儿,不抗议,就服从性的低调的埋头苦干。相反,我看到过一位根本不黑但有黑人血统的数学研究生,他的数学水平其实很差的,与其他人相比,可是他却公开的支持Black Lives Matter,然后学校媒体却非常支持他,以他宣传自己的diversity,公布的视频里还有他说I didn’t have to think about race。有意思的是他根本不黑,要他不说,其实都看不出来他是黑人。所预料,这些在学校没人敢说的,说了都怕给自己惹麻烦,其实好多人都为此感到不满,但不得不不了了之,最终政治赢者是谁就毫无疑问了。

Continue reading “不知为何,突然想起测度论里的不可测度的维塔利集合”

Advertisements

Big Picard theorem

I’ve been asked to prove the Big Picard theorem, assuming the fundamental normality test. Assuming the latter, it is a very short proof, and I could half-ass with that. I don’t like writing up stuff that I don’t actually understand for the sake of doing so. There’s little point, and if I’m going to actually write up a proof of it, I’ll do so for real, which means that I go over the fundamental normality test in its entirety.

Continue reading “Big Picard theorem”

On grad school, science, academia, and also a problem on Riemann surfaces

I like mathematics a ton and I am not bad at it. In fact, I am probably better than many math graduate students at math, though surely, they will have more knowledge than I do in some respects, or maybe even not that, because frankly, the American undergrad math major curriculum is often rather pathetic, well maybe largely because the students kind of suck. In some sense, you have to be pretty clueless to be majoring in just pure math if you’re not a real outlier at it, enough to have a chance at a serious academic career. Of course, math professors won’t say this. So we have now an excess of people who really shouldn’t be in science (because they much lack the technical power or an at least reasonable scientific taste/discernment, or more often both) adding noise to the job market. On this, Katz in his infamous Don’t Become a Scientist piece writes:

Continue reading “On grad school, science, academia, and also a problem on Riemann surfaces”

Construction of Riemann surfaces as quotients

There is a theorem in Chapter 4 Section 5 of Schlag’s complex analysis text. I went through it a month ago, but only half understood it, and it is my hope that passing through it again, this time with writeup, will finally shed light, after having studied in detail some typical examples of such Riemann surfaces, especially tori, the conformal equivalence classes of which can be represented by the fundamental region of the modular group, which arise from quotienting out by lattices on the complex plane, as well as Fuchsian groups.

In the text, the theorem is stated as follows.

Theorem 4.12.  Let \Omega \subset \mathbb{C}_{\infty} and G < \mathrm{Aut}(\mathbb{C}_{\infty}) with the property that

  • g(\Omega) \subset \Omega for all g \in G,
  • for all g \in G, g \neq \mathrm{id}, all fixed points of g in \mathbb{C}_{\infty} lie outside of \Omega,
  • for all K \subset \Omega compact, the cardinality of \{g \in G | g(K) \cap K \neq \phi\} is finite.

Under these assumptions, the natural projection \pi : \Omega \to \Omega / G is a covering map which turns \Omega/G canonically onto a Riemann surface.

The properties essentially say that the we have a Fuchsian group G acting on \Omega \subset \mathbb{C}_{\infty} without fixed points, excepting the identity. To show that quotient space is a Riemann surface, we need to construct charts. For this, notice that without fixed points, there is for all z \in \Omega, a small pre-compact open neighborhood of z denoted by K_z \subset \Omega, so that

g(\overline{K_z} \cap \overline{K_z}) = \emptyset \qquad \forall g \in G, g \neq \mathrm{id}.

So, in K_z no two elements are twice represented, which mean the projection \pi : K_z \to K_z is the identity, and therefore we can use the K_zs as charts. The gs as Mobius transformations are open maps which take the K_zs to open sets. In other words, \pi^{-1}(K_z) = \bigcup_{g \in G} g^{-1}(K_z) with pairwise disjoint open sets g^{-1}(K_z). From this, the K_zs are open sets in the quotient topology. In this scheme, the gs are the transition maps.

Finally, we verify that this topology is Hausdorff. Suppose \pi(z_1) \neq \pi(z_2) and define for all n \geq 1,

A_n = \left\{z \in \Omega | |z-z_1| < \frac{r}{n}\right\} \subset \Omega

B_n = \left\{z \in \Omega | |z-z_2| < \frac{r}{n}\right\} \subset \Omega

where r > 0 is sufficiently small. Define K = \overline{A_1} \cup \overline{B_1} and suppose that \pi(A_n) \cap \pi(B_n) \neq \emptyset for all n \geq 1. Then for some a_n \in A_n and g_n \in G we have

g_n(a_n) \in B_n \qquad \forall n \geq 1.

Since g_n(K) \cap K has finite cardinality, there are only finitely many possibilities for g_n and one of them therefore occurs infinitely often. Pass to the limit n \to \infty and we have g(z_1) = z_2 or \pi(z_1) = \pi(z_2), a contradiction.

 

Variants of the Schwarz lemma

Take some self map on the unit disk \mathbb{D}, f. If f(0) = 0, g(z) = f(z) / z has a removable singularity at 0. On |z| = r, |g(z)| \leq 1 / r, and with the maximum principle on r \to 1, we derive |f(z)| \leq |z| everywhere. In particular, if |f(z)| = |z| anywhere, constancy by the maximum principle tells us that f(z) = \lambda z, where |\lambda| = 1. g with the removable singularity removed has g(0) = f'(0), so again, by the maximum principle, |f'(0)| = 1 means g is a constant of modulus 1. Moreover, if f is not an automorphism, we cannot have |f(z)| = |z| anywhere, so in that case, |f'(0)| < 1.

Cauchy’s integral formula in complex analysis

I took a graduate course in complex analysis a while ago as an undergraduate. However, I did not actually understand it well at all, to which is a testament that much of the knowledge vanished very quickly. It pleases me though now following some intellectual maturation, after relearning certain theorems, they seem to stick more permanently, with the main ideas behind the proof more easily understandably clear than mind-disorienting, the latter of which was experienced by me too much in my early days. Shall I say it that before I must have been on drugs of something, because the way about which I approached certain things was frankly quite weird, and in retrospect, I was in many ways an animal-like creature trapped within the confines of an addled consciousness oblivious and uninhibited. Almost certainly never again will I experience anything like that. Now, I can only mentally rationalize the conscious experience of a mentally inferior creature but such cannot be experienced for real. It is almost like how an evangelical cannot imagine what it is like not to believe in God, and even goes as far as to contempt the pagan. Exaltation, exhilaration was concomitant with the leap of consciousness till it not long after established its normalcy.

Continue reading “Cauchy’s integral formula in complex analysis”

Implicit function theorem and its multivariate generalization

The implicit function theorem for a single output variable can be stated as follows:

Single equation implicit function theorem. Let F(\mathbf{x}, y) be a function of class C^1 on some neighborhood of a point (\mathbf{a}, b) \in \mathbb{R}^{n+1}. Suppose that F(\mathbf{a}, b) = 0 and \partial_y F(\mathbf{a}, b) \neq 0. Then there exist positive numbers r_0, r_1 such that the following conclusions are valid.

a. For each \mathbf{x} in the ball |\mathbf{x} - \mathbf{a}| < r_0 there is a unique y such that |y - b| < r_1 and F(\mathbf{x}, y) = 0. We denote this y by f(\mathbf{x}); in particular, f(\mathbf{a}) = b.

b. The function f thus defined for |\mathbf{x} - \mathbf{a}| < r_0 is of class C^1, and its partial derivatives are given by

\partial_j f(\mathbf{x}) = -\frac{\partial_j F(\mathbf{x}, f(\mathbf{x}))}{\partial_y F(\mathbf{x}, f(\mathbf{x}))}.

Proof. For part (a), assume without loss of generality positive \partial_y F(\mathbf{a}, b). By continuity of that partial derivative, we have that in some neighborhood of (\mathbf{a}, b) it is positive and thus for some r_1 > 0, r_0 > 0 there exists f such that |\mathbf{x} - \mathbf{a}| < r_0 implies that there exists a unique y (by intermediate value theorem along with positivity of \partial_y F) such that |y - b| < r_1 with F(\mathbf{x}, y) = 0, which defines some function y = f(\mathbf{x}). Continue reading “Implicit function theorem and its multivariate generalization”

A nice consequence of Baire category theorem

In a complete metric space X, we call a point x for which \{x\} is open an isolated point. If X is countable and there are no isolated points, we can take \displaystyle\cap_{x \in X} X \setminus x = \emptyset, with each of the X \setminus x open and dense, to violate the Baire category theorem. From that, we can arrive at the proposition that in a complete metric space, no isolated points implies that the space uncountable, and similarly, that countable implies there is an isolated point.