# Accute result

Any set of vectors $v_1, \dots, v_n$ in some Euclidean space such that any pair of them lie at an obtuse angle are either linearly independent or have a linear combination with non-negative coefficients equal to zero.

# Lines in flags

Let $F_1 \subseteq \dots \subseteq F_n$ and $F'_1 \subseteq \dots \subseteq F'_n$ be complete flags over an $n$-dimensional vector space $V$. Show that there is a set of $n$ lines $L_1,\dots, L_n$ compatible with the two flags, in the sense that there exists a permutation $\sigma \in S_n$ such that $F_i = L_1 \oplus \dots \oplus L_i$ and $F'_i = L_{\sigma(1)}\oplus \dots \oplus L_{\sigma(i)}$.

# Kill the Killing form

Let $\mathfrak g$ be a Lie algebra and $K$ its Killing form. Is the kernel of $K$ always equal to the radical of $\mathfrak g$?

# Centers

Let $G$ be a connected Lie subgroup of $\mathrm{GL}_n$. Then $A\in G$ commutes with all of $G$ iff it commutes with all of $\mathfrak{g}$.

# Bounding the projective plane

Is there a compact 3-manifold having $\Bbb{R}\mathrm{P}^2$ as boundary?

# (Not) on a plane

A plane flies 100 km north, 100 km east, 100 km south and then 100 km west. Give a necessary and sufficient condition for it to land on the same spot where it took off.

# No Eckmann-Hilton

Prove the following two statements and conclude that the fundamental group of a (connected) topological group $G$ is abelian:

1) A discrete normal subgroup of a connected topological group is central.
2) If $p:\hat{G}\to G$ is the universal covering, the total space $\hat{G}$ admits a topological group structure such that $p$ is a group morphism. The kernel of $p$ is then isomorphic to the fundamental group of $G$.