*Abstract: We consider two kinds of group invariance properties for rational functions mapping the unit sphere in complex Euclidean space to the unit sphere in a space of typically larger dimension. Let $\Gamma$ be a subgroup of the automorphismgroup of the source ball. A map $f$ is $\Gamma$ invariant if $f \circ \gamma = f$ for all $\gamma$ in $\Gamma$, and is Hermitian $\Gamma$ invariant if, for each such $\gamma$ there is a target automorphism $\psi$ with $\psi \circ f = f \circ \gamma$. To each map $f$ we associate its (maximal) Hermitian invariant group.The first kind of invariance is very restricted; for example, a nonconstant map can be $\Gamma$ invariant only if $\Gamma$ is cyclic and represented in a short list of ways. But, in joint work with Ming Xiao, we show that every finite group can be represented as a subgroup of the unitary group that is in fact the invariant group of a polynomial sphere map.*

*Host: John McCarthy*