An Introduction to Riemann Surfaces and Algebraic Curves: Complex 1-Tori and Elliptic Curves EF79Q5HV1A

URI	`m2://res/EF79Q5HV1A`
Kind	course
Creators	T.E. Venkata Balaji (instructor)
Date	2013-06-13
MSC	14-XX
Scope	in
Ids	course 03v0r5n49:111106044

Expressions

e1: edition

Language	en
Date	2013-06-13

m1: page, IIT Madras (record). nptel 111106044
m2: page, 2013-06-13 (record). nptel 111106044

Contents

part 1: Definitions and Examples of Riemann Surfaces
session 1: The Idea of a Riemann Surface
session 2: Simple Examples of Riemann Surfaces
session 3: Maximal Atlases and Holomorphic Maps of Riemann Surfaces
session 4: A Riemann Surface Structure on a Cylinder
session 5: A Riemann Surface Structure on a Torus
part 2: Classification of Riemann Surfaces
session 6: Riemann Surface Structures on Cylinders and Tori via Covering Spaces
session 7: Moebius Transformations Make up Fundamental Groups of Riemann Surfaces
session 8: Homotopy and the First Fundamental Group
session 9: A First Classification of Riemann Surfaces
part 3: Universal Covering Space Theory
session 10: The Importance of the Path-lifting Property
session 11: Fundamental groups as Fibres of the Universal covering Space
session 12: The Monodromy Action
session 13: The Universal covering as a Hausdorff Topological Space
session 14: The Construction of the Universal Covering Map
session 15: Completion of the Construction of the Universal Covering: Universality of the Universal Covering
session 16: Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group
part 4: Classifying Moebius Transformations and Deck Transformations
session 17: The Riemann Surface Structure on the Topological Covering of a Riemann Surface
session 18: Riemann Surfaces with Universal Covering the Plane or the Sphere
session 19: Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane
session 20: Characterizing Moebius Transformations with a Single Fixed Point
session 21: Characterizing Moebius Transformations with Two Fixed Points
session 22: Torsion-freeness of the Fundamental Group of a Riemann Surface
session 23: Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups
session 24: Classifying Annuli up to Holomorphic Isomorphism
part 5: The Riemann Surface Structure on the Quotient of the Upper Half-Plane by the Unimodular Group
session 25: Orbits of the Integral Unimodular Group in the Upper Half-Plane
session 26: Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
session 27: Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations
session 28: Quotients by Kleinian Subgroups give rise to Riemann Surfaces
session 29: The Unimodular Group is Kleinian
part 6: Doubly-Periodic Meromorphic (or) Elliptic Functions
session 30: The Necessity of Elliptic Functions for the Classification of Complex Tori
session 31: The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane
session 32: The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function
session 33: The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing Analytic Functions on the Upper Half-Plane
part 7: A Form Modular for the Congruence-mod-2 Subgroup of the Unimodular Group on the Upper Half-Plane
session 34: The Construction of a Modular Form of Weight Two on the Upper Half-Plane
session 35: The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half-Plane
session 36: The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane
session 37: The Weight Two Modular Form Vanishes at Infinity
session 38: The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity
session 39: A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane
part 8: The Elliptic Modular J-invariant and the Moduli of Complex 1-dimensional Tori (or) Elliptic Curves
session 40: The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
session 41: A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
session 42: The Fundamental Region in the Upper Half-Plane for the Unimodular Group
session 43: A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
session 44: Moduli of Elliptic Curves
part 9: Complex 1-dimensional Tori are Projective Algebraic Elliptic Curves
session 45: Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space
session 46: The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
session 47: Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
session 48: Complex Tori are the same as Elliptic Algebraic Projective Curves

Catalogue records

nptel: 111106044 (manifestation)
nptel_video: 111106044 (manifestation)