# The Geometry of Möbius Transformations

- Institution: IISER Mohali
- Department: Mathematics
- Teacher: Prof. Krishnendu Gongopadhyay
- Level: Undergraduate
- Language: English
- Subjects: Mathematics
- Source: [NPTEL](https://nptel.ac.in/courses/111106001)
- Licence: [CC BY-NC-SA](https://nptel.ac.in/faq) (Non-commercial use only.)
- Syllabus: [at the source](https://nptel.ac.in/courses/111106001)
- Who can watch: Tweens profiles and older (sources give no ratings; placed by Invisible Network from the source's own words)

Explore the elegant world where complex analysis meets geometry! This course offers a deep dive into the geometry of Möbius transformations which are fundamental symmetries the Riemann sphere. Whether one is interested in complex analysis, geometry, or the foundations of modern mathematical physics, this course provides a journey through one of mathematics' most beautiful structures. Basic prerequisite that is needed to follow this course would be familiarity with complex numbers, basic group theory, general topology and introductory real analysis.

Intended audience: BS-MS students, Ph.D students, Anyone with a Bachelor level Math background who wants to learn this.

Prerequisites: Basic linear algebra, general topology and group theory.

## Lectures

1. [Introduction](https://invisiblenetwork.ai/learn/nptel/111106001/b3377f6603d0) (YouTube)
2. [Complex numbers](https://invisiblenetwork.ai/learn/nptel/111106001/ad75a458e2d3) (YouTube)
3. [Euler's forumula](https://invisiblenetwork.ai/learn/nptel/111106001/03c80c4e3010) (YouTube)
4. [Orthogonal matrices](https://invisiblenetwork.ai/learn/nptel/111106001/7caec4f32dbe) (YouTube)
5. [Euclidean isometries](https://invisiblenetwork.ai/learn/nptel/111106001/49a7dd1aba65) (YouTube)
6. [Classification of isometries](https://invisiblenetwork.ai/learn/nptel/111106001/1ad0212066f9) (YouTube)
7. [An example](https://invisiblenetwork.ai/learn/nptel/111106001/2f646e0c7540) (YouTube)
8. [Complex differentiability-I](https://invisiblenetwork.ai/learn/nptel/111106001/72e3ba8bc0ef) (YouTube)
9. [Complex differentiability-II](https://invisiblenetwork.ai/learn/nptel/111106001/cb283c8d0b95) (YouTube)
10. [Cauchy-Riemann Eqn](https://invisiblenetwork.ai/learn/nptel/111106001/cd4fddb115b5) (YouTube)
11. [Cauchy-Riemann revisited](https://invisiblenetwork.ai/learn/nptel/111106001/8d5f6e6bd19d) (YouTube)
12. [Complex logarithm](https://invisiblenetwork.ai/learn/nptel/111106001/6ef6b8ff08bd) (YouTube)
13. [Complex Integration](https://invisiblenetwork.ai/learn/nptel/111106001/612fe796e6cb) (YouTube)
14. [Cauchy Integral Formula](https://invisiblenetwork.ai/learn/nptel/111106001/a372c6c98cb4) (YouTube)
15. [Power Series Expansion](https://invisiblenetwork.ai/learn/nptel/111106001/db472e74d066) (YouTube)
16. [Zeros of holomorphic fn](https://invisiblenetwork.ai/learn/nptel/111106001/a39b5ba1233e) (YouTube)
17. [Open Maping thm](https://invisiblenetwork.ai/learn/nptel/111106001/7d5aeffe08e9) (YouTube)
18. [Singularity](https://invisiblenetwork.ai/learn/nptel/111106001/4299eea2ac16) (YouTube)
19. [Poles](https://invisiblenetwork.ai/learn/nptel/111106001/209677fdc010) (YouTube)
20. [Essential singularities](https://invisiblenetwork.ai/learn/nptel/111106001/33cdd9740c45) (YouTube)
21. [Residues](https://invisiblenetwork.ai/learn/nptel/111106001/fb1c8d0b5b73) (YouTube)
22. [The residue theorem](https://invisiblenetwork.ai/learn/nptel/111106001/8c6577c0ba47) (YouTube)
23. [The Riemann Sphere](https://invisiblenetwork.ai/learn/nptel/111106001/4636be49780a) (YouTube)
24. [Meromorphic functions](https://invisiblenetwork.ai/learn/nptel/111106001/89d01054afb1) (YouTube)
25. [Rational functions](https://invisiblenetwork.ai/learn/nptel/111106001/538889057eb7) (YouTube)
26. [degree of rational functions](https://invisiblenetwork.ai/learn/nptel/111106001/85bddb4e63bf) (YouTube)
27. [degree theorem and orders](https://invisiblenetwork.ai/learn/nptel/111106001/1fe8528694d0) (YouTube)
28. [Mobius transformation](https://invisiblenetwork.ai/learn/nptel/111106001/e029059fad8d) (YouTube)
29. [Mobius transformation-II](https://invisiblenetwork.ai/learn/nptel/111106001/295f5d032c98) (YouTube)
30. [anti-automorphism](https://invisiblenetwork.ai/learn/nptel/111106001/37fbd1fd1b40) (YouTube)
31. [Generators of Mobius group](https://invisiblenetwork.ai/learn/nptel/111106001/38d33d009831) (YouTube)
32. [Circles on Riemann Sphere](https://invisiblenetwork.ai/learn/nptel/111106001/3bcbfdc078e8) (YouTube)
33. [Mobius transformation & circles](https://invisiblenetwork.ai/learn/nptel/111106001/53abcc5139b0) (YouTube)
34. [Cross Ratios](https://invisiblenetwork.ai/learn/nptel/111106001/64a65571fb38) (YouTube)
35. [Permutation and cross ratios](https://invisiblenetwork.ai/learn/nptel/111106001/1ded478ac55a) (YouTube)
36. [Circles and Cross Ratios](https://invisiblenetwork.ai/learn/nptel/111106001/2c4aa59c5be9) (YouTube)
37. [Group action](https://invisiblenetwork.ai/learn/nptel/111106001/3487cda0e5cf) (YouTube)
38. [Stabiliser of a disc & circle](https://invisiblenetwork.ai/learn/nptel/111106001/b9dc0114f6ec) (YouTube)
39. [Standard forms](https://invisiblenetwork.ai/learn/nptel/111106001/d6ec35f936a2) (YouTube)
40. [classification of Mobius transformation](https://invisiblenetwork.ai/learn/nptel/111106001/01fbb8d2167a) (YouTube)
41. [Commutator](https://invisiblenetwork.ai/learn/nptel/111106001/8201c7c83858) (YouTube)
42. [Commutator and fixed points](https://invisiblenetwork.ai/learn/nptel/111106001/7f3b17b1646a) (YouTube)
43. [Commuting Motions](https://invisiblenetwork.ai/learn/nptel/111106001/12badf0ab4d9) (YouTube)
44. [Centralizers](https://invisiblenetwork.ai/learn/nptel/111106001/918d230678de) (YouTube)
45. [Conformality of stereographic projection](https://invisiblenetwork.ai/learn/nptel/111106001/ad75bc5ced1e) (YouTube)
46. [Conformal automorphisms](https://invisiblenetwork.ai/learn/nptel/111106001/792c7f1e5ec3) (YouTube)
47. [Conformal maps of spheres](https://invisiblenetwork.ai/learn/nptel/111106001/950eea4ff448) (YouTube)
48. [Rotations of Riemann sphere](https://invisiblenetwork.ai/learn/nptel/111106001/968a679b4649) (YouTube)
49. [Subgroup of elliptic elements](https://invisiblenetwork.ai/learn/nptel/111106001/aa273323297d) (YouTube)
50. [Subgroup of elliptic elements-II](https://invisiblenetwork.ai/learn/nptel/111106001/5eb980600bfa) (YouTube)
51. [Finite group of rotations](https://invisiblenetwork.ai/learn/nptel/111106001/ba870d468487) (YouTube)
52. [Elements of arc length](https://invisiblenetwork.ai/learn/nptel/111106001/44a9f4f0c5d0) (YouTube)
53. [hyperbolic upper half model](https://invisiblenetwork.ai/learn/nptel/111106001/50b2768f49c5) (YouTube)
54. [Geodesics](https://invisiblenetwork.ai/learn/nptel/111106001/7718df4ca45b) (YouTube)
55. [Hyperbolic isometries](https://invisiblenetwork.ai/learn/nptel/111106001/ae46ef7dc54e) (YouTube)
56. [Concluding words](https://invisiblenetwork.ai/learn/nptel/111106001/777fbfff19f4) (YouTube)