This thesis concerns real algebraic curves. Equivalently one may consider compact Riemann surfaces together with an anti-holomorphic involution. The basic problem is to find the minimal degree of a map from a real algebraic curve to a genus zero curve. Motivation for studying this problem comes on the one hand from the theory of non-orientable minimal surfaces (soap films) and on the other hand from classical algebraic geometry. This work focuses on the case in which the curve has a real point-- equivalently, the anti-holomorphic involution of the Riemann surface has a fixed point. In this case the genus zero curve will always be one dimensional projective space.
The analogous problem for algebraic curves over the complex numbers dates from about 1870. Brill and Noether proposed a solution which was verified rigorously by a variety of different techniques between 1960 and 1985. The minimal degree of a map to a genus zero curve depends on the choice of the curve. However, for "most" curves the answer is always the same, namely [(g+3)/2], where g=genus and  indicates the integer part of a rational number. This number is called the Brill-Noether number for the given genus. Every curve of a given genus admits a map to a genus zero curve whose degree is less than or equal to the Brill-Noether number.
The behaviour of real curves is more complicated. There exist real curves of most topological types in genus 4 and 8 which do not admit any map to a genus zero curve of degree less than or equal to the Brill-Noether number. In fact this phenomenon occurs in an open subset of moduli in the classical topology. On the other hand there exist other open subsets in the same moduli spaces where the curves do admit maps to genus zero curves whose degree is the Brill-Noether number. The behaviour in genus 3, 5, 6, and 7 is quite different--here the degree of the minimal map is always at most the Brill-Noether number. In fact it is shown that this phenomenon holds for an infinite, albeit rather sparse set of genuses. Precisely what happens in most high genuses is still an open question.