An Introduction to Riemann Surfaces and Algebraic Curves Video Lectures

An Intro to Riemann Surfaces and Algebraic Curves
'An Introduction to Riemann Surfaces and Algebraic Curves' Video Lectures by Dr. T.E. Venkata Balaji from IIT Madras
"An Intro to Riemann Surfaces and Algebraic Curves" - Video Lectures
1. The Idea of a Riemann Surface
2. Simple Examples of Riemann Surfaces
3. Maximal Atlases and Holomorphic Maps of Riemann Surfaces
4. A Riemann Surface Structure on a Cylinder
5. A Riemann Surface Structure on a Torus
6. Riemann Surface Structures on Cylinders and Tori via Covering Spaces
7. Moebius 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. Completion of the Construction of the Universal Covering: Universality of the Universal Covering
16. Completion of the Construction of the Universal Covering: 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 with Universal Covering the Complex Plane
20. Characterizing Moebius Transformations with a Single Fixed Point
21. Characterizing Moebius 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-Plane with Abelian Fundamental Groups
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 of Moebius Transformations
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 associated to a Lattice in the Plane
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 are nonvanishing Analytic Functions on the Upper Half-Plane
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 Two on the Upper Half-Plane
36. The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane
37. The Weight Two Modular Form Vanishes at Infinity
38. The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
39. A 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. A 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 Cubic Curves in Complex 2-Space
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
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