We shall describe $\cap_m \operatorname{ran} M_m^*$, where $m$ ranges over the cyclic multipliers of the DruryArveson space $H_d^2$, and $M_m$ denotes multiplication by $m$ on $H_d^2$. I will try to convince the audience that there is some interesting functional analysis behind the description.

This is joint work with Alexander Aleman, Michael Hartz and Stefan Richter.

Let $V$ be an analytic subvariety of a domain $\Omega$ in $\mathbb{C}^{n}$. When does $V$ have the property that every bounded holomorphic function $f$ on $V$ has an extension to a bounded holomorphic function on $\Omega$ with the same norm?
An obvious sufficient condition is if $V$ is a holomorphic retract of $\Omega$. We shall discuss for what domains $\Omega$ this is also necessary.
This is joint work with Łukasz Kosiński.