Goals of Lecture 11: - To see that the formation of the fundamental group is a covariant functorial operation, from the category whose objects are pointed topological spaces and whose morphisms are base-point-preserving continuous maps, to the category whose objects are groups and whose morphisms are group homomorphisms - To deduce the path-lifting property for a covering map as a consequence of the Covering Homotopy Theorem - To deduce from the Covering Homotopy Theorem that the fundamental group of a covering space can be identified naturally with a subgroup of the fundamental group of the space being covered - To note that the inverse image of a point (fibre over a point) under a covering map may be identified with the space of cosets of the fundamental group (based at a point fixed above) inside the fundamental group at the point below - To note that the universal covering of a space may be pictured as a fibration consisting of fundamental groups over that space Topics: Covering Homotopy Theorem, stationary homotopy, lifting of a homotopy, path-lifting property, category, objects of a category, morphisms of a category, covariant functor, functorial operation, fundamental group as a covariant functor, pointed topological space, group action, transitive action, fundamental group, universal covering, subgroup, cosets of a subgroup in a group

1. The Idea of a Riemann Surface
2. Simple Examples of Riemann Surfaces
3. Maximal Atlases and Holomorphic Maps of Riemann Surfaces
4. Riemann Surface Structure on a Cylinder
5. Riemann Surface Structure on a Torus
6. Riemann Surface Structures on Cylinders and Tori via Covering Spaces
7. Möbius Transformations Make up Fundamental Groups of Riemann Surfaces
8. Homotopy and the First Fundamental Group
9. A First Classification of Riemann Surfaces
10. The Importance of the Path-lifting Property
11. Fundamental groups as Fibres of the Universal covering Space
12. The Monodromy Action
13. The Universal covering as a Hausdorff Topological Space
14. The Construction of the Universal Covering Map
15. Universality of the Universal Covering
16. The Fundamental Group of the base as the Deck Transformation Group
17. The Riemann Surface Structure on the Topological Covering of a Riemann Surface
18. Riemann Surfaces with Universal Covering the Plane or the Sphere
19. Classifying Complex Cylinders Riemann Surfaces
20. Möbius Transformations with a Single Fixed Point
21. Möbius Transformations with Two Fixed Points
22. Torsion-freeness of the Fundamental Group of a Riemann Surface
23. Characterizing Riemann Surface Structures on Quotients of the Upper Half
24. Classifying Annuli up to Holomorphic Isomorphism
25. Orbits of the Integral Unimodular Group in the Upper Half-Plane
26. Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
27. Local Actions at the Region of Discontinuity of a Kleinian Subgroup
28. Quotients by Kleinian Subgroups give rise to Riemann Surfaces
29. The Unimodular Group is Kleinian
30. The Necessity of Elliptic Functions for the Classification of Complex Tori
31. The Uniqueness Property of the Weierstrass Phe-function
32. The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function
33. The Values of the Weierstrass Phe function at the Zeros of its Derivative
34. The Construction of a Modular Form of Weight Two on the Upper Half-Plane
35. The Fundamental Functional Equations satisfied by the Modular Form of Weight
36. The Weight Two Modular Form assumes Real Values on the Imaginary Axis
37. The Weight Two Modular Form Vanishes at Infinity
38. The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
39. Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
40. The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
41. Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
42. The Fundamental Region in the Upper Half-Plane for the Unimodular Group
43. A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
44. Moduli of Elliptic Curves
45. Punctured Complex Tori are Elliptic Algebraic Affine Plane
46. The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
47. Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
48. Complex Tori are the same as Elliptic Algebraic Projective Curves

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.