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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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