Goals of the Lecture: - To understand the notion of homotopy of paths in a topological space - To understand concatenation of paths in a topological space - To sketch how the set of fixed-end-point (FEP) homotopy classes of loops at a point becomes a group under concatenation, called the First Fundamental Group - To look at examples of fundamental groups of some common topological spaces - To realise that the fundamental group is an algebraic invariant of topological spaces which helps in distinguishing non-isomorphic topological spaces - To realise that a first classification of Riemann surfaces can be done based on their fundamental groups by appealing to the theory of covering spaces. Topics: Path or arc in a topological space, initial or starting point and terminal or ending point of a path, path as a map, geometric path, parametrisation of a geometric path, homotopy, continuous deformation of maps, product topology, equivalence of paths under homotopy, fixed-end-point (FEP) homotopy, concatenation of paths, constant path, binary operation, associative binary operation, identity element for a binary operation, inverse of an element under a binary operation, first fundamental group, topological invariant.
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.