Symmetries of compact Riemann surfaces or, equivalently, real forms of complex algebraic curves

Resumen:

It is well known that a complex algebraic curve (smooth, projective and irreducible) C can be viewed as a compact Riemann surface X = X_C and the fact that such a curve has a real form (real equations) is equivalent to the fact that the corresponding surface is symmetric. Furthermore, the symmetries non-conjugate in the group of all automorphisms of X give rise to the non-isomorphic, over the reals, real forms of the initial curve. In addition, the topology of the set of points fixed by a symmetry gives the precise information about the topology of the corresponding real form.

In this talk we outline, I hope in a form accessible for a general audience, how the quantitative and qualitative aspects of the study of real forms of complex algebraic curves can be reduced to certain problems concerning combinatorial group theory, due to the equivalences mentioned before, the Riemann uniformization theorem and the knowledge of discrete subgroups of the group of isometries of the hyperbolic plane. We shall give a survey concerning the theory.

The talk is planing to be in Spanish and transparencies in English.