Let be metric spaces such that is connected and is complete and locally compact. Let be a uniformly equicontinuous family of functions from to such that there exists with the property that is precompact. Prove that all points have property .

# Be rational

Most of representation theory results work out for algebraically closed fields. For instance, we can classify irreducible representations for the dihedral group with elements over . Classify all irreducible representations for over .

# Fundamental theorem of algebra…?

Let be the cyclic group of order and consider the group algebra. Prove that the number of solutions to the equation for is for .

# Separating the étalé space

We briefly recall the construction of the étalé space of a sheaf. Given a sheaf over a space , the associated étalé space is a topological space which, as a set, is the disjoint union of the stalks of , with the topology induced by the basis , where is an open set in , and is the germ at of the element .

Given a topological space , consider the sheaf of real-valued continuous functions on , which we denote . It is more or less easy to see that the associated étalé space is not Hausdorff iff the space has the following property :

*There exist continuous functions , where is an open set, and a point such that the germs of and at are different, but nevertheless for any open neighborhood of there exists an open set such that .*

For instance, has property , and it’s easy to see that is inherited by subspaces and products which contain any factor. This, for instance, shows that subspaces of are spaces satisfying this property.

Is there any characterization of all topological spaces for which the étalé space associated with the sheaf of real-valued continuous functions is not Hausdorff?

# Infinite dimensional Schur

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 .