# An Introduction to Riemann Surfaces and Algebraic Curves (USB)

**SKU**

Media Storage Type : 32 GB USB Stick

NPTEL Subject Matter Expert : Dr. T.E. Venkata Balaji

NPTEL Co-ordinating Institute : IIT Madras

NPTEL Lecture Count : 48

NPTEL Course Size : 23 GB

NPTEL PDF Text Transcription : Available and Included

NPTEL Subtitle Transcription : Available and Included (SRT)

**Lecture Titles:**

Lecture 1 - The Idea of a Riemann Surface

Lecture 2 - Simple Examples of Riemann Surfaces

Lecture 3 - Maximal Atlases and Holomorphic Maps of Riemann Surfaces

Lecture 4 - A Riemann Surface Structure on a Cylinder

Lecture 5 - A Riemann Surface Structure on a Torus

Lecture 6 - Riemann Surface Structures on Cylinders and Tori via Covering Spaces

Lecture 7 - Moebius Transformations Make up Fundamental Groups of Riemann Surfaces

Lecture 8 - Homotopy and the First Fundamental Group

Lecture 9 - A First Classification of Riemann Surfaces

Lecture 10 - The Importance of the Path-lifting Property

Lecture 11 - Fundamental groups as Fibres of the Universal covering Space

Lecture 12 - The Monodromy Action

Lecture 13 - The Universal covering as a Hausdorff Topological Space

Lecture 14 - The Construction of the Universal Covering Map

Lecture 15 - Completion of the Construction of the Universal Covering: Universality of the Universal Covering

Lecture 16 - Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group

Lecture 17 - The Riemann Surface Structure on the Topological Covering of a Riemann Surface

Lecture 18 - Riemann Surfaces with Universal Covering the Plane or the Sphere

Lecture 19 - Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane

Lecture 20 - Characterizing Moebius Transformations with a Single Fixed Point

Lecture 21 - Characterizing Moebius Transformations with Two Fixed Points

Lecture 22 - Torsion-freeness of the Fundamental Group of a Riemann Surface

Lecture 23 - Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups

Lecture 24 - Classifying Annuli up to Holomorphic Isomorphism

Lecture 25 - Orbits of the Integral Unimodular Group in the Upper Half-Plane

Lecture 26 - Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions

Lecture 27 - Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations

Lecture 28 - Quotients by Kleinian Subgroups give rise to Riemann Surfaces

Lecture 29 - The Unimodular Group is Kleinian

Lecture 30 - The Necessity of Elliptic Functions for the Classification of Complex Tori

Lecture 31 - The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane

Lecture 32 - The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function

Lecture 33 - The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing Analytic Functions on the Upper Half-Plane

Lecture 34 - The Construction of a Modular Form of Weight Two on the Upper Half-Plane

Lecture 35 - The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half-Plane

Lecture 36 - The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane

Lecture 37 - The Weight Two Modular Form Vanishes at Infinity

Lecture 38 - The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity

Lecture 39 - A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane

Lecture 40 - The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve

Lecture 41 - A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant

Lecture 42 - The Fundamental Region in the Upper Half-Plane for the Unimodular Group

Lecture 43 - A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once

Lecture 44 - Moduli of Elliptic Curves

Lecture 45 - Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space

Lecture 46 - The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve

Lecture 47 - Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two

Lecture 48 - Complex Tori are the same as Elliptic Algebraic Projective Curves

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