Problem 1 (yau-contest 2011)
Section: Analysis and Differential Equations

Let $H^2(\Delta)$ be the space of holomorphic functions in the unit disk $\Delta = \{\lvert z\rvert < 1\}$ such that $\int_{\Delta} \lvert f\rvert^2\,\lvert dz\rvert^2 < \infty$. Prove that $H^2(\Delta)$ is a Hilbert space and that for any $r < 1$, the map $T : H^2(\Delta) \to H^2(\Delta)$ given by $Tf(z) := f(rz)$ is a compact operator.

Problem 2 (yau-contest 2011)
Section: Analysis and Differential Equations

For any continuous function $f(z)$ of period $1$, show that the equation
$$\frac{d\varphi}{dt} = 2\pi\varphi + f(t)$$
has a unique solution of period $1$.

Problem 3 (yau-contest 2011)
Section: Analysis and Differential Equations

Let $h(x)$ be a $C^{\infty}$ function on the real line $\mathbb{R}$. Find a $C^{\infty}$ function $u(x,y)$ on an open subset of $\mathbb{R}^2$ containing the $x$-axis such that $u_x + 2u_y = u^2$ and $u(x,0) = h(x)$.

Problem 4 (yau-contest 2011)
Section: Analysis and Differential Equations

Let $S = \{ x \in \mathbb{R} \mid \lvert x - p/q \rvert \le c/q^{\beta},\ \text{for infinitely many relatively prime } p,q \in \mathbb{Z},\ q>0,\ \text{for a fixed } c>1 \}$. Show that $S$ is uncountable and its measure is zero.

Problem 5 (yau-contest 2011)
Section: Analysis and Differential Equations

Let $sl(n)$ denote the set of all $n\times n$ real matrices with trace equal to zero and let $SL(n)$ be the set of all $n\times n$ real matrices with determinant equal to one. Let $\varphi(z)$ be a real analytic function defined in a neighborhood of $z=0$ of the complex plane $\mathbb{C}$ satisfying the conditions $\varphi(0)=1$ and $\varphi'(0)=1$.
(a) If $\varphi$ maps any near zero matrix in $sl(n)$ into $SL(n)$ for some $n\ge 3$, show that $\varphi(z)=\exp(z)$.
(b) Is the conclusion of (a) still true in the case $n=2$? If it is true, prove it. If not, give a counterexample.

Problem 6 (yau-contest 2011)
Section: Analysis and Differential Equations

Use mathematical analysis to show that:
(a) $e$ and $\pi$ are irrational numbers;
(b) $e$ and $\pi$ are also transcendental numbers.