Abstract: Given a graph $G$ cellularly embedded in a closed surface $S$, an automorphism of $G$ is called a cellular automorphism of $G$ in $S$ when, loosely speaking, it takes facial boundary walks to facial boundary walks. I will describe how we constructed complete catalogs of all irreducible cellular automorphisms of the sphere, projective plane, torus, Klein bottle, and three-crosscaps surface for a particular notion of reducibility related to taking minors.\par We have also determined concrete procedures sufficient for constructing all possible self-dual embeddings in any closed surface $S$ given a catalog of all irreducible cellular automorphisms in $S$. I will illustrate by way of examples some of these procedures and some resulting self-dual graphs.\par

This talk will be accessible to graduate students.