## 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.

## 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:

## Elliptic functions

I am writing this as a way to go through in detail the section on elliptic functions in Schlag’s book.

Proposition 4.14.  Let $\Lambda = \{m\omega_1 + n\omega_2 | m,n \in \mathbb{Z}\}$ and set $\Lambda^* = \Lambda \setminus \{0\}$. For any integer $n \geq 3$, the series Continue reading “Elliptic functions”

## 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 $K_z$s as charts. The $g$s as Mobius transformations are open maps which take the $K_z$s 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 $K_z$s are open sets in the quotient topology. In this scheme, the $g$s 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.

## Weierstrass products

Long time ago when I was a clueless kid about the finish 10th grade of high school, I first learned about Euler’s determination of $\zeta(2) = \frac{\pi^2}{6}$. The technique he used was of course factorization of $\sin z / z$ via its infinitely many roots to

$\displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z}{n\pi}\right)\left(1 + \frac{z}{n\pi}\right) = \displaystyle\prod_{n=1}^{\infty} \left(1 - \frac{z^2}{n^2\pi^2}\right)$.

## 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.