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

- Institution: IIT Madras
- Department: Mathematics
- Teacher: Dr. T.E. Venkata Balaji
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106044)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106044)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

## Lectures

1. [The Idea of a Riemann Surface](https://invisiblenetwork.ai/learn/nptel/111106044/2ba55dc369cc) (Video)
2. [Simple Examples of Riemann Surfaces](https://invisiblenetwork.ai/learn/nptel/111106044/304fba566056) (Video)
3. [Maximal Atlases and Holomorphic Maps of Riemann Surfaces](https://invisiblenetwork.ai/learn/nptel/111106044/1519f5db9a73) (Video)
4. [A Riemann Surface Structure on a Cylinder](https://invisiblenetwork.ai/learn/nptel/111106044/d7a5be45cc98) (Video)
5. [A Riemann Surface Structure on a Torus](https://invisiblenetwork.ai/learn/nptel/111106044/39c0e959fda5) (Video)
6. [Riemann Surface Structures on Cylinders and Tori via Covering Spaces](https://invisiblenetwork.ai/learn/nptel/111106044/9d9fda992277) (Video)
7. [Moebius Transformations Make up Fundamental Groups of Riemann Surfaces](https://invisiblenetwork.ai/learn/nptel/111106044/21095efe24cd) (Video)
8. [Homotopy and the First Fundamental Group](https://invisiblenetwork.ai/learn/nptel/111106044/c03dc5e3d1d4) (Video)
9. [A First Classification of Riemann Surfaces](https://invisiblenetwork.ai/learn/nptel/111106044/d12e13a743a6) (Video)
10. [The Importance of the Path-lifting Property](https://invisiblenetwork.ai/learn/nptel/111106044/82d5a47b01f7) (Video)
11. [Fundamental groups as Fibres of the Universal covering Space](https://invisiblenetwork.ai/learn/nptel/111106044/22a6fb5121a3) (Video)
12. [The Monodromy Action](https://invisiblenetwork.ai/learn/nptel/111106044/e56ba762fc86) (Video)
13. [The Universal covering as a Hausdorff Topological Space](https://invisiblenetwork.ai/learn/nptel/111106044/9d2240d41ae5) (Video)
14. [The Construction of the Universal Covering Map](https://invisiblenetwork.ai/learn/nptel/111106044/4ad3492fafb4) (Video)
15. [Completion of the Construction of the Universal Covering: Universality of the Universal Covering](https://invisiblenetwork.ai/learn/nptel/111106044/1bd3f56e0155) (Video)
16. [Completion of the Construction of the Universal Covering: The Fundamental Group of the base as the Deck Transformation Group](https://invisiblenetwork.ai/learn/nptel/111106044/7dfb968c4c7d) (Video)
17. [The Riemann Surface Structure on the Topological Covering of a Riemann Surface](https://invisiblenetwork.ai/learn/nptel/111106044/7342caead306) (Video)
18. [Riemann Surfaces with Universal Covering the Plane or the Sphere](https://invisiblenetwork.ai/learn/nptel/111106044/77e6716252c8) (Video)
19. [Classifying Complex Cylinders: Riemann Surfaces with Universal Covering the Complex Plane](https://invisiblenetwork.ai/learn/nptel/111106044/17c2e966f2e4) (Video)
20. [Characterizing Moebius Transformations with a Single Fixed Point](https://invisiblenetwork.ai/learn/nptel/111106044/478aeec2db83) (Video)
21. [Characterizing Moebius Transformations with Two Fixed Points](https://invisiblenetwork.ai/learn/nptel/111106044/6cbfbd8fdf55) (Video)
22. [Torsion-freeness of the Fundamental Group of a Riemann Surface](https://invisiblenetwork.ai/learn/nptel/111106044/155cf40808f7) (Video)
23. [Characterizing Riemann Surface Structures on Quotients of the Upper Half-Plane with Abelian Fundamental Groups](https://invisiblenetwork.ai/learn/nptel/111106044/ad14fad57b16) (Video)
24. [Classifying Annuli up to Holomorphic Isomorphism](https://invisiblenetwork.ai/learn/nptel/111106044/f5f513f39394) (Video)
25. [Orbits of the Integral Unimodular Group in the Upper Half-Plane](https://invisiblenetwork.ai/learn/nptel/111106044/328fd00add35) (Video)
26. [Galois Coverings are precisely Quotients by Properly Discontinuous Free Actions](https://invisiblenetwork.ai/learn/nptel/111106044/cb17d1a25137) (Video)
27. [Local Actions at the Region of Discontinuity of a Kleinian Subgroup of Moebius Transformations](https://invisiblenetwork.ai/learn/nptel/111106044/6fff6d6dbc57) (Video)
28. [Quotients by Kleinian Subgroups give rise to Riemann Surfaces](https://invisiblenetwork.ai/learn/nptel/111106044/00d93f9907ac) (Video)
29. [The Unimodular Group is Kleinian](https://invisiblenetwork.ai/learn/nptel/111106044/77fba628e8b5) (Video)
30. [The Necessity of Elliptic Functions for the Classification of Complex Tori](https://invisiblenetwork.ai/learn/nptel/111106044/873ebe39e9d4) (Video)
31. [The Uniqueness Property of the Weierstrass Phe-function associated to a Lattice in the Plane](https://invisiblenetwork.ai/learn/nptel/111106044/528c1455b907) (Video)
32. [The First Order Degree Two Cubic Ordinary Differential Equation satisfied by the Weierstrass Phe-function](https://invisiblenetwork.ai/learn/nptel/111106044/ed9740f3f331) (Video)
33. [The Values of the Weierstrass Phe-function at the Zeros of its Derivative are nonvanishing Analytic Functions on the Upper Half-Plane](https://invisiblenetwork.ai/learn/nptel/111106044/65f2c7162240) (Video)
34. [The Construction of a Modular Form of Weight Two on the Upper Half-Plane](https://invisiblenetwork.ai/learn/nptel/111106044/ed1fa0209e55) (Video)
35. [The Fundamental Functional Equations satisfied by the Modular Form of Weight Two on the Upper Half-Plane](https://invisiblenetwork.ai/learn/nptel/111106044/3fb7dab4d5a8) (Video)
36. [The Weight Two Modular Form assumes Real Values on the Imaginary Axis in the Upper Half-plane](https://invisiblenetwork.ai/learn/nptel/111106044/1714b57191af) (Video)
37. [The Weight Two Modular Form Vanishes at Infinity](https://invisiblenetwork.ai/learn/nptel/111106044/24b40dab12e6) (Video)
38. [The Weight Two Modular Form Decays Exponentially in a Neighbourhood of Infinity](https://invisiblenetwork.ai/learn/nptel/111106044/1ddccf1a8e27) (Video)
39. [A Suitable Restriction of the Weight Two Modular Form is a Holomorphic Conformal Isomorphism onto the Upper Half-Plane](https://invisiblenetwork.ai/learn/nptel/111106044/73ac26f72fac) (Video)
40. [The J-Invariant of a Complex Torus (or) of an Algebraic Elliptic Curve](https://invisiblenetwork.ai/learn/nptel/111106044/8963ec9cc428) (Video)
41. [A Fundamental Region in the Upper Half-Plane for the Elliptic Modular J-Invariant](https://invisiblenetwork.ai/learn/nptel/111106044/fbd89ac21b11) (Video)
42. [The Fundamental Region in the Upper Half-Plane for the Unimodular Group](https://invisiblenetwork.ai/learn/nptel/111106044/ce554ca1d977) (Video)
43. [A Region in the Upper Half-Plane Meeting Each Unimodular Orbit Exactly Once](https://invisiblenetwork.ai/learn/nptel/111106044/e2e34c6001ce) (Video)
44. [Moduli of Elliptic Curves](https://invisiblenetwork.ai/learn/nptel/111106044/784ec54a3727) (Video)
45. [Punctured Complex Tori are Elliptic Algebraic Affine Plane Cubic Curves in Complex 2-Space](https://invisiblenetwork.ai/learn/nptel/111106044/e60460469055) (Video)
46. [The Natural Riemann Surface Structure on an Algebraic Affine Nonsingular Plane Curve](https://invisiblenetwork.ai/learn/nptel/111106044/ab11c39cf3c3) (Video)
47. [Complex Projective 2-Space as a Compact Complex Manifold of Dimension Two](https://invisiblenetwork.ai/learn/nptel/111106044/001039ff36cd) (Video)
48. [Complex Tori are the same as Elliptic Algebraic Projective Curves](https://invisiblenetwork.ai/learn/nptel/111106044/00482a67f785) (Video)