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The Geometry of Möbius Transformations

Prof. Krishnendu Gongopadhyay · IISER Mohali

Start with lecture 1 At NPTEL

About

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

Week 1

  1. Introduction YouTube
  2. Complex numbers YouTube
  3. Euler's forumula YouTube
  4. Orthogonal matrices YouTube
  5. Euclidean isometries YouTube
  6. Classification of isometries YouTube

Week 2

  1. An example YouTube
  2. Complex differentiability-I YouTube
  3. Complex differentiability-II YouTube
  4. Cauchy-Riemann Eqn YouTube

Week 3

  1. Cauchy-Riemann revisited YouTube
  2. Complex logarithm YouTube
  3. Complex Integration YouTube
  4. Cauchy Integral Formula YouTube

Week 4

  1. Power Series Expansion YouTube
  2. Zeros of holomorphic fn YouTube
  3. Open Maping thm YouTube
  4. Singularity YouTube

Week 5

  1. Poles YouTube
  2. Essential singularities YouTube
  3. Residues YouTube
  4. The residue theorem YouTube

Week 6

  1. The Riemann Sphere YouTube
  2. Meromorphic functions YouTube
  3. Rational functions YouTube
  4. degree of rational functions YouTube
  5. degree theorem and orders YouTube

Week 7

  1. Mobius transformation YouTube
  2. Mobius transformation-II YouTube
  3. anti-automorphism YouTube
  4. Generators of Mobius group YouTube

Week 8

  1. Circles on Riemann Sphere YouTube
  2. Mobius transformation & circles YouTube
  3. Cross Ratios YouTube
  4. Permutation and cross ratios YouTube
  5. Circles and Cross Ratios YouTube

Week 9

  1. Group action YouTube
  2. Stabiliser of a disc & circle YouTube
  3. Standard forms YouTube
  4. classification of Mobius transformation YouTube

Week 10

  1. Commutator YouTube
  2. Commutator and fixed points YouTube
  3. Commuting Motions YouTube
  4. Centralizers YouTube

Week 11

  1. Conformality of stereographic projection YouTube
  2. Conformal automorphisms YouTube
  3. Conformal maps of spheres YouTube
  4. Rotations of Riemann sphere YouTube
  5. Subgroup of elliptic elements YouTube
  6. Subgroup of elliptic elements-II YouTube

Week 12

  1. Finite group of rotations YouTube
  2. Elements of arc length YouTube
  3. hyperbolic upper half model YouTube
  4. Geodesics YouTube
  5. Hyperbolic isometries YouTube
  6. Concluding words YouTube

Syllabus and materials

At NPTEL, as the course publishes them.

Credits and licence

Institution
IISER Mohali
Department
Mathematics
Teacher
Prof. Krishnendu Gongopadhyay
Level
Undergraduate
Language
English
Subjects
Mathematics
Source
NPTEL
Licence
CC BY-NC-SA, as NPTEL states it. Non-commercial use only. NPTEL's terms
Attribution
NPTEL states: "Distributed under Creative Commons Attribution-ShareAlike / CC BY - NC - SA", beside a BY-NC-SA badge. Coordinated by IIT Madras.
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