Let be an algebra over and let be an irreducible representation of with at most countable basis. Then any homomorphism of representations 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 and be compact oriented smooth manifolds of the same dimension, with connected. Let be a smooth function. Take a regular value of and, for , define as if preserves orientation at and as if it reverses the orientation. We define the degree of as the number .

Prove the following extension theorem: Assume is a -dimensional sphere and is the boundary of a compact oriented -manifold . There exists a smooth extension of if and only if .

# Measure my balls

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

# Measure this

Let be non-negative, finite Borel measures on that are singular with respect to each other. Find for -almost every and -almost every .

# This’ a glue problem

Every -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.