Problem 1 (yau-contest 2010)
Section: Geometry and Topology

Let $S^n \subset \mathbb{R}^{n+1}$ be the unit sphere, and $\mathbb{R}^n \subset \mathbb{R}^{n+1}$ the equator $n$-plane through the center of $S^n$. Let $N$ be the north pole of $S^n$. Define a mapping $\pi : S^n\backslash\{N\} \to \mathbb{R}^n$ called the stereographic projection that takes $A \in S^n\backslash\{N\}$ into the intersection $A' \in \mathbb{R}^n$ of the equator $n$-plane $\mathbb{R}^n$ with the line which passes through $A$ and $N$. Prove that the stereographic projection is a conformal change, and derive the standard metric of $S^n$ by the stereographic projection.

Problem 2 (yau-contest 2010)
Section: Geometry and Topology

Let $M$ be a (connected) Riemannian manifold of dimension 2. Let $f$ be a smooth non-constant function on $M$ such that $f$ is bounded from above and $\Delta f \ge 0$ everywhere on $M$. Show that there does not exist any point $p \in M$ such that $f(p) = \sup\{ f(x) : x \in M \}.$

Problem 3 (yau-contest 2010)
Section: Geometry and Topology

Let $M$ be a compact smooth manifold of dimension $d$. Prove that there exists some $n \in \mathbb{Z}^+$ such that $M$ can be regularly embedded in the Euclidean space $\mathbb{R}^n$.

Problem 4 (yau-contest 2010)
Section: Geometry and Topology

Show that any $C^\infty$ function $f$ on a compact smooth manifold $M$ (without boundary) must have at least two critical points. When $M$ is the 2-torus, show that $f$ must have more than two critical points.

Problem 5 (yau-contest 2010)
Section: Geometry and Topology

Construct a space $X$ with $H_0(X)=\mathbb{Z}$, $H_1(X)=\mathbb{Z}_2 \times \mathbb{Z}_3$, $H_2(X)=\mathbb{Z}$, and all other homology groups of $X$ vanishing.

Problem 6 (yau-contest 2010)
Section: Geometry and Topology

(a). Define the degree $\deg f$ of a $C^\infty$ map $f : S^2 \longrightarrow S^2$ and prove that $\deg f$ as you present it is well-defined and independent of any choices you need to make in your definition.

(b). Prove in detail that for each integer $k$ (possibly negative), there is a $C^\infty$ map $f : S^2 \longrightarrow S^2$ of degree $k$.