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Abstract

Various incarnations of Stefan Bergman's notion of representative coordinates will be given that are useful in a variety of contexts. Bergman wanted his coordinates to map to canonical regions, but they fail to do this for multiply connected regions. We show, however, that it is possible to define generalized Bergman coordinates that map multiply connected domains to quadrature domains which satisfy a long list of desirable properties, making them excellent candidates to be called Bergman representative domains. We also construct a kind of Bergman coordinate that maps a domain to an algebraic variety in $\mathbb C^2$ in a natural way, and thereby show that Bergman-style coordinates can be used to convert problems in conformal mapping to problems in algebraic geometry. Many of these results generalize routinely to finite Riemann surfaces.