The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmueller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra---especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory. The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given.
Riemann surface for the function ƒ(z) = √z. The two horizontal axes represent the real and imaginary parts of z, while the vertical axis represents the real part of √z. For the imaginary part of √z, rotate the plot 180° around the vertical axis.