In large dimensions, the only known compact, simply connected Riemannian manifolds with positive sectional curvature are spheres and projective spaces. The natural metrics on these spaces have large isometry groups, so it is natural to consider highly symmetric metrics when searching for new examples. On the other hand, there are many topological obstructions to a manifold admitting a positively curved metric with large symmetry. I will discuss a new obstruction in this setting. This is joint work with Manuel Amann (KIT).

A subset of the real line is called a Cantor set if it is compact,
perfect, and nowhere dense. Cantor sets arise in many areas; in this
talk we will discuss their relevance in the spectral theory of
Schr\"odinger operators. We discuss several results showing that the
spectrum of such an operator is a Cantor set, from the discovery of the
first example by Moser to a genericity result by Avila, Bochi, and
Damanik. A Cantor measure is a probability measure on the real line
whose topological support is a Cantor set. A primary example in the
spectral theory context is the density of states measure in situations
where the spectrum is a Cantor set. A conjecture of Simon claims a
strict inequality between the dimensions of the set and the measure for
the Fibonacci potential. If time permits, we will discuss a recent
result of Damanik, Gorodetski, and Yessen, which establishes this
conjecture in full generality.

Advisor: Martin Zeman

Given a polynomial map between two vector spaces over a field,
how many values can it miss? The lecture will present a number of
new results on this question. They were inspired by the work of
Daqing Wan, and obtained jointly with Michiel Kosters (Leiden).