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NPTEL :: Mathematics - An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves
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Course Co-ordinated by :
IIT Madras
Course Available from :
14-August-2013
NPTEL
Mathematics
An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves (Video)
The Idea of a Riemann Surface
Modules / Lectures
Definitions and Examples of Riemann Surfaces
The Idea of a Riemann Surface
Simple Examples of Riemann Surfaces
Maximal Atlases and Holomorphic Maps of Riemann Surfaces
A Riemann Surface Structure on a Cylinder
A Riemann Surface Structure on a Torus
Classification of Riemann Surfaces
Riemann Surface Structures on Cylinders and Tori via Covering Spaces
Moebius Transformations Make up Fundamental Groups of Riemann Surfaces
Homotopy and the First Fundamental Group
A First Classification of Riemann Surfaces
Universal Covering Space Theory
The Importance of the Path-lifting Property
Fundamental groups as Fibres of the Universal covering Space
The Monodromy Action
The Universal covering as a Hausdorff Topological Space
The Construction of the Universal Covering Map
Completion of the Construction of the Universal Covering: Universality of the Universal Covering
Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group
Classifying Moebius Transformations and Deck Transformations
The Riemann Surface Structure on the Topological Covering of a Riemann Surface
Riemann Surfaces with Universal Covering the Plane or the Sphere
Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane
Characterizing Moebius Transformations with a Single Fixed Point
Characterizing Moebius Transformations with Two Fixed Points
Torsion-freeness of the Fundamental Group of a Riemann Surface
Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups
Classifying Annuli up to Holomorphic Isomorphism
The Riemann Surface Structure on the Quotient of the Upper Half-Plane by the Unimodular Group
Orbits of the Integral Unimodular Group in the Upper Half-Plane
Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations
Quotients by Kleinian Subgroups give rise to Riemann Surfaces
The Unimodular Group is Kleinian
Doubly-Periodic Meromorphic (or) Elliptic Functions
The Necessity of Elliptic Functions for the Classification of Complex Tori
The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane
The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function
The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing Analytic Functions on the Upper Half-Plane
A Form Modular for the Congruence-mod-2 Subgroup of the Unimodular Group on the Upper Half-Plane
The Construction of a Modular Form of Weight Two on the Upper Half-Plane
The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half-Plane
The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane
The Weight Two Modular Form Vanishes at Infinity
The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
The Elliptic Modular J-invariant and the Moduli of Complex 1-dimensional Tori (or) Elliptic Curves
The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
The Fundamental Region in the Upper Half-Plane for the Unimodular Group
A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
Moduli of Elliptic Curves
Complex 1-dimensional Tori are Projective Algebraic Elliptic Curves
Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space
The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
Complex Tori are the same as Elliptic Algebraic Projective Curves
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