# Infinite dimensional Schur

Let $A$ be an algebra over $\mathbb{C}$ and let $V$ be an irreducible representation of $A$ with at most countable basis. Then any homomorphism of representations $\phi:V\to V$ is a scalar operator.

# Euler & Lie

Any compact connected Lie group has Euler characteristic zero. What are the possible values for the Euler characteristic of a connected, but not necessarily compact Lie group?

# Your degree is a chance for expansion

Let $X$ and $Y$ be compact oriented smooth manifolds of the same dimension, with $Y$ connected. Let $f : X \to Y$ be a smooth function. Take $y \in Y$ a regular value of $f$ and, for $x \in f^{-1}(y)$, define $o(x)$ as $+1$ if $f$ preserves orientation at $x$ and as $-1$ if it reverses the orientation. We define the degree of $f$ as the number $d(f) = \sum_{x\in f^{-1}(y)}o(x)$.

Prove the following extension theorem: Assume $Y=S^k$ is a $k$-dimensional sphere and $X=\partial W$ is the boundary of a compact oriented $(k+1)$-manifold $W$. There exists a smooth extension $F:W\to Y$ of $f$ if and only if $d(f)=0$.

# Measure my balls

Suppose $X$ is a compact metric space and $\mu$ is a finite Borel measure on $X$. Is the measure $\mu$ determined by the value of the measure of the balls?

# Measure this

Let $\mu,\nu$ be non-negative, finite Borel measures on $\mathbb{R}^n$ that are singular with respect to each other. Find $\displaystyle\lim_{r\to 0}\dfrac{\nu(B(x,r))}{\mu(B(x,r))}$ for $\mu$-almost every $x$ and $\nu$-almost every $x$.

# This’ a glue problem

Every $CW$-complex is homotopy equivalent to (the realization of) a simplicial complex.

# Small rings

Any field that is finitely generated as a ring is a finite field.