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source: nptelhrd 2013年6月13日
Mathematics - An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves by Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. For more details on NPTEL visit http://www.nptel.iitm.ac.in/syllabus/111106044/
01 The Idea of a Riemann Surface 57:13
02 Simple Examples of Riemann Surfaces 57:47
03 Maximal Atlases and Holomorphic Maps of Riemann Surfaces 50:58
04 A Riemann Surface Structure on a Cylinder 54:56
05 A Riemann Surface Structure on a Torus 48:26
06 Riemann Surface Structures on Cylinders and Tori via Covering Spaces 56:44
07 Moebius Transformations Make up Fundamental Groups of Riemann Surfaces 48:34
08 Homotopy and the First Fundamental Group 53:35
09 A First Classification of Riemann Surfaces 49:03
10 The Importance of the Path-lifting Property 57:49
11 Fundamental groups as Fibres of the Universal covering Space 56:52
12 The Monodromy Action 53:33
13 The Universal covering as a Hausdorff Topological Space 1:01:02
14 The Construction of the Universal Covering Map 55:26
15A Completion of the Construction of the Universal Covering 37:29
15B Completion of the Construction of the Universal Covering: The Fundamental Group 43:47
16 The Riemann Surface Structure on the Topological Covering of a Riemann Surface 59:12
17 Riemann Surfaces with Universal Covering the Plane or the Sphere 1:18:54
18 Classifying Complex Cylinders Riemann Surfaces 1:01:21
19 Characterizing Moebius Transformations with a Single Fixed Point 56:08
20 Characterizing Moebius Transformations with Two Fixed Points 1:01:38
21 Torsion-freeness of the Fundamental Group of a Riemann Surface 46:25
22 Characterizing Riemann Surface Structures on Quotients of the Upper Half 1:12:59
23 Classifying Annuli up to Holomorphic Isomorphism 45:18
24 Orbits of the Integral Unimodular Group in the Upper Half-Plane 1:15:20
25 Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions 1:05:23
26 Local Actions at the Region of Discontinuity of a Kleinian Subgroup 1:11:25
27 Quotients by Kleinian Subgroups give rise to Riemann Surfaces 50:51
28 The Unimodular Group is Kleinian 1:06:11
29 The Necessity of Elliptic Functions for the Classification of Complex Tori 48:15
30 The Uniqueness Property of the Weierstrass Phe-function 1:08:15
31 The First Order Degree Two Cubic Ordinary Differential Equation satisfied 1:06:12
32 The Values of the Weierstrass Phe function at the Zeros of its Derivative 49:24
33 The Construction of a Modular Form of Weight Two on the Upper Half-Plane 55:50
34 The Fundamental Functional Equations satisfied by the Modular Form of Weight 54:34
35 The Weight Two Modular Form assumes Real Values on the Imaginary Axis 56:55
36 The Weight Two Modular Form Vanishes at Infinity 50:47
37A The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity 43:36
37B A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal 50:24
38 The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve 59:17
39 A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant 51:28
40 The Fundamental Region in the Upper Half-Plane for the Unimodular Group 1:16:24
41 A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once 49:46
42 Moduli of Elliptic Curves 1:08:12
43 Punctured Complex Tori are Elliptic Algebraic Affine Plane 1:00:32
44 The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve 1:09:14
45A Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
45B Complex Tori are the same as Elliptic Algebraic Projective Curves 36:19
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