IIT Madras Course , Prof. T.E. Venkata Balaji

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IIT Madras Course , Prof. T.E. Venkata Balaji

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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 httpwww.nptel.iitm.ac.insyllabus111106044Goals of the Lecture- To look at the set of all possible Riemann surface structures on a cylinder and theneed for a method to distinguish between them- To explain the motivation for the use of the Theory of Covering Spaces to distinguishRiemann surface structures- To motivate the notion of a covering map by examples- To get introduced to the fact (called General Uniformisation) that any Riemann surfaceis the quotient (via a covering map) of a suitable simply connected Riemann surface- To understand the idea of the Fundamental group and where it fits into our discussionKeywords for Lecture 6Cylinder, punctured plane, punctured unit disc, annulus, Riemanns Theorem on removable singularities, covering map, covering space, pathwise connected, locally pathwise connected, admissible neighbourhood or admissible open set, universal covering space or simply connected covering space, fundamental group, uniformisation of a general Riemann surface

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IITMadras delivers the above video lessons under NPTEL program, there are more than 6000+ nptel video lectures by other IITs as well.
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- 1.The Idea of a Riemann Surface
- 2.Simple Examples of Riemann Surfaces
- 3.Maximal Atlases and Holomorphic Maps of Riemann Surfaces
- 4.A Riemann Surface Structure on a Cylinder
- 5.A Riemann Surface Structure on a Torus I
- 6.Riemann Surface Structures on Cylinders and Tori via Covering Spaces II
- 7.Moebius Transformations Make up Fundamental Groups of Riemann Surfaces
- 8.Homotopy and the First Fundamental Group
- 9.A First Classification of Riemann Surfaces
- 10.The Importance of the Path-lifting Property
- 11.Fundamental groups as Fibres of the Universal covering Space
- 12.The Monodromy Action
- 13.The Universal covering as a Hausdorff Topological Space
- 14.The Construction of the Universal Covering Map
- 15.Completion of the Construction of the Universal Coveringl
- 16.Completion of the Construction of the Universal Covering The Fundamental Group I
- 17.The Riemann Surface Structure on the Topological Covering of a Riemann Surface
- 18.Riemann Surfaces with Universal Covering the Plane or the Sphere I
- 19.Classifying Complex Cylinders Riemann Surfaces
- 20.Characterizing Moebius Transformations with a Single Fixed Point
- 21.Characterizing Moebius Transformations with Two Fixed Points I
- 22.Torsion-freeness of the Fundamental Group of a Riemann Surface
- 23.Characterizing Riemann Surface Structures on Quotients of the Upper Half
- 24.Classifying Annuli up to Holomorphic Isomorphism
- 25.Orbits of the Integral Unimodular Group in the Upper Half-Plane
- 26.Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions
- 27.Local Actions at the Region of Discontinuity of a Kleinian Subgroup
- 28.Quotients by Kleinian Subgroups give rise to Riemann Surfaces
- 29.The Unimodular Group is Kleinian
- 30.The Necessity of Elliptic Functions for the Classification of Complex Tori
- 31.The Uniqueness Property of the Weierstrass Phe-function
- 32.The First Order Degree Two Cubic Ordinary Differential Equation satisfied
- 33.The Values of the Weierstrass Phe function at the Zeros of its Derivative
- 34.The Construction of a Modular Form of Weight Two on the Upper Half-Plane
- 35.The Fundamental Functional Equations satisfied by the Modular Form of Weight
- 36.The Weight Two Modular Form assumes Real Values on the Imaginary Axis
- 37.The Weight Two Modular Form Vanishes at Infinity I
- 38.The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity III
- 39.A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal
- 40.The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve
- 41.A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant
- 42.The Fundamental Region in the Upper Half-Plane for the Unimodular Group I
- 43.A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once
- 44.Moduli of Elliptic Curves
- 45.Punctured Complex Tori are Elliptic Algebraic Affine Plane
- 46.The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve
- 47.Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two
- 48.Complex Tori are the same as Elliptic Algebraic Projective Curves I

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