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September 23, 2023

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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 Path-lifting 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 – Torsion-freeness of the Fundamental Group of a Riemann Surface Unlimited
    • Lecture 22 – Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane … Unlimited
    • Lecture 23 – Classifying Annuli up to Holomorphic Isomorphism Unlimited
    • Lecture 24 – Orbits of the Integral Unimodular Group in the Upper Half-Plane 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 Phe-function Associated to … Unlimited
    • Lecture 31 – The First Order Degree Two Cubic Ordinary Differential Equation Satisfied by … Unlimited
    • Lecture 32 – The Values of the Weierstrass Phe-function at the Zeros of its Derivative … Unlimited
    • Lecture 33 – The Construction of a Modular Form of Weight Two on the Half-Plane 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 J-invariant of a Complex Torus (or) of an Algebraic Elliptic Curve Unlimited
    • Lecture 39 – A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-invariant Unlimited
    • Lecture 40 – The Fundamental Region in the Upper Half-Plane for the Unimodular Group Unlimited
    • Lecture 41 – A Region in the Upper Half-Plane 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 2-Space 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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