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Description
An Introduction to Riemann Surfaces and Algebraic Curves. Instructor: Dr. T. E. Venkata Balaji, Department of Mathematics, IIT Madras. The subject of algebraic curves (equivalently compact Riemann surfaces) has its origins going back to the work of Riemann, Abel, Jacobi, Noether, Weierstrass, Clifford and Teichmuller. It continues to be a source for several hot areas of current research. Its development requires ideas from diverse areas such as analysis, PDE, complex and real differential geometry, algebra  especially commutative algebra and Galois theory, homological algebra, number theory, topology and manifold theory.
The course begins by introducing the notion of a Riemann surface followed by examples. Then the classification of Riemann surfaces is achieved on the basis of the fundamental group by the use of covering space theory and uniformisation. This reduces the study of Riemann surfaces to that of subgroups of Moebius transformations. The case of compact Riemann surfaces of genus 1, namely elliptic curves, is treated in detail. The algebraic nature of elliptic curves and a complex analytic construction of the moduli space of elliptic curves is given. (from nptel.ac.in)
Course Curriculum

 Lecture 01 – The Idea of a Riemann Surface Unlimited
 Lecture 02 – Simple Examples of Riemann Surfaces Unlimited
 Lecture 03 – Maximal Atlases and Holomorphic Maps of Riemann Surfaces Unlimited
 Lecture 04 – A Riemann Surface Structure on a Cylinder Unlimited
 Lecture 05 – A Riemann Surface Structure on a Torus Unlimited

 Lecture 06 – Riemann Surface Structures on Cylinders and Tori via Covering Spaces Unlimited
 Lecture 07 – Moebius Transformations Make Up Fundamental Groups of Riemann Surfaces Unlimited
 Lecture 08 – Homotopy and the First Fundamental Group Unlimited
 Lecture 09 – A First Classifications of Riemann Surfaces Unlimited

 Lecture 10 – The Importance of the Pathlifting Property Unlimited
 Lecture 11 – Fundamental Groups as Fibres of the Universal Covering Space Unlimited
 Lecture 12 – The Monodromy Action Unlimited
 Lecture 13 – The Universal Covering as a Hausdorff Topological Space Unlimited
 Lecture 14 – The Construction of the Universal Covering Map Unlimited
 Lecture 15 – Completion of the Construction of the Universal Covering Unlimited
 Lecture 15B – Completion of the Construction of the Universal Covering Unlimited

 Lecture 16 – Riemann Surface Structure on the Topological Covering of a Riemann Surface Unlimited
 Lecture 17 – Riemann Surfaces with Universal Covering the Plane or the Sphere Unlimited
 Lecture 18 – Classifying Complex Cylinders: Riemann Surfaces with Universal Covering … Unlimited
 Lecture 19 – Characterizing Moebius Transformations with a Single Fixed Point Unlimited
 Lecture 19 – Characterizing Moebius Transformations with a Single Fixed Point Unlimited
 Lecture 20 – Characterizing Moebius Transformations with Two Fixed Point Unlimited
 Lecture 21 – Torsionfreeness of the Fundamental Group of a Riemann Surface Unlimited
 Lecture 22 – Characterizing Riemann Surface Structures on Quotients of the Upper HalfPlane … Unlimited
 Lecture 23 – Classifying Annuli up to Holomorphic Isomorphism Unlimited

 Lecture 24 – Orbits of the Integral Unimodular Group in the Upper HalfPlane Unlimited
 Lecture 25 – Galois Coverings are Precisely Quotients by Properly Discontinuous Free Actions Unlimited
 Lecture 26 – Local Actions at the Region of Discontinuity of a Kleinian Subgroup of … Unlimited
 Lecture 27 – Quotients by Kleinian Subgroups Give Rise to Riemann Surfaces Unlimited
 Lecture 28 – The Unimodular Group is Kleinian Unlimited

 Lecture 29 – The Necessity of Elliptic Functions for the Classification of Complex Tori Unlimited
 Lecture 30 – The Uniqueness Property of the Weierstrass Phefunction Associated to … Unlimited
 Lecture 31 – The First Order Degree Two Cubic Ordinary Differential Equation Satisfied by … Unlimited
 Lecture 32 – The Values of the Weierstrass Phefunction at the Zeros of its Derivative … Unlimited

 Lecture 33 – The Construction of a Modular Form of Weight Two on the HalfPlane Unlimited
 Lecture 34 – The Fundamental Functional Equations Satisfied by the Modular Form of … Unlimited
 Lecture 35 – The Weight Two Modular Form Assumes Real Values on the Imaginary Axis in … Unlimited
 Lecture 36 – The Weight Two Modular Form Vanishes at Infinity Unlimited
 Lecture 37 – The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity Unlimited
 Lecture 37B – A Suitable Restriction of the Weight Two Modular Form is a Holomorphic … Unlimited

 Lecture 38 – The Jinvariant of a Complex Torus (or) of an Algebraic Elliptic Curve Unlimited
 Lecture 39 – A Fundamental Region in the Upper HalfPlane for the Elliptic Modular Jinvariant Unlimited
 Lecture 40 – The Fundamental Region in the Upper HalfPlane for the Unimodular Group Unlimited
 Lecture 41 – A Region in the Upper HalfPlane Meeting Each Unimodular Orbit Exactly Once Unlimited
 Lecture 42 – Moduli of Elliptic Curves Unlimited

 Lecture 43 – Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in … Unlimited
 Lecture 44 – The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve Unlimited
 Lecture 45 – Complex Projective 2Space as a Compact Complex Manifold of Dimension Two Unlimited
 Lecture 45B – Complex Tori are the same as Elliptic Algebraic Projective Curves Unlimited
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