An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves

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Course Curriculum

The Idea of a Riemann Surface Details 57:13
Simple Examples of Riemann Surfaces Details 57:47
Maximal Atlases and Holomorphic Maps of Riemann Surfaces Details 50:58
A Riemann Surface Structure on a Cylinder Details 54:56
A Riemann Surface Structure on a Torus I Details 48:26
Riemann Surface Structures on Cylinders and Tori via Covering Spaces II Details 56:44
Moebius Transformations Make up Fundamental Groups of Riemann Surfaces Details 48:34
Homotopy and the First Fundamental Group Details 53:35
A First Classification of Riemann Surfaces Details 49:3
The Importance of the Path-lifting Property Details 57:49
Fundamental groups as Fibres of the Universal covering Space Details 56:52
The Monodromy Action Details 53:33
The Universal covering as a Hausdorff Topological Space Details 1:1:2
The Construction of the Universal Covering Map Details 55:26
Completion of the Construction of the Universal Coveringl Details 37:29
Completion of the Construction of the Universal Covering: The Fundamental Group I Details 43:47
The Riemann Surface Structure on the Topological Covering of a Riemann Surface Details 59:12
Riemann Surfaces with Universal Covering the Plane or the Sphere I Details 1:18:54
Classifying Complex Cylinders Riemann Surfaces Details 1:1:21
Characterizing Moebius Transformations with a Single Fixed Point Details 56:8
Characterizing Moebius Transformations with Two Fixed Points I Details 1:1:38
Torsion-freeness of the Fundamental Group of a Riemann Surface Details 46:25
Characterizing Riemann Surface Structures on Quotients of the Upper Half Details 1:12:59
Classifying Annuli up to Holomorphic Isomorphism Details 45:18
Orbits of the Integral Unimodular Group in the Upper Half-Plane Details 1:15:20
Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions Details 1:5:23
Local Actions at the Region of Discontinuity of a Kleinian Subgroup Details 1:11:25
Quotients by Kleinian Subgroups give rise to Riemann Surfaces Details 50:51
The Unimodular Group is Kleinian Details 1:6:11
The Necessity of Elliptic Functions for the Classification of Complex Tori Details 48:15
The Uniqueness Property of the Weierstrass Phe-function Details 1:8:15
The First Order Degree Two Cubic Ordinary Differential Equation satisfied Details 1:6:12
The Values of the Weierstrass Phe function at the Zeros of its Derivative Details 49:24
The Construction of a Modular Form of Weight Two on the Upper Half-Plane Details 55:50
The Fundamental Functional Equations satisfied by the Modular Form of Weight Details 54:34
The Weight Two Modular Form assumes Real Values on the Imaginary Axis Details 56:55
The Weight Two Modular Form Vanishes at Infinity I Details 50:47
The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity III Details 43:36
A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Details 50:24
The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve Details 59:17
A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant Details 51:28
The Fundamental Region in the Upper Half-Plane for the Unimodular Group I Details 1:16:24
A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once Details 49:46
Moduli of Elliptic Curves Details 1:8:12
Punctured Complex Tori are Elliptic Algebraic Affine Plane Details 1:32
The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve Details 1:9:14
Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two Details 43:29
Complex Tori are the same as Elliptic Algebraic Projective Curves I Details 36:19

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