Instructor: | David Gu |

Email: | gu@cs.stonybrook.edu |

Date: | Every Lecture 8:30-10:00am EST |

Duration: | Start From July 4th |

Voov Meeting ID: | 933 8415 7259 |

Live Streaming: | Use a web browser to open the live streaming link online.conformalgeometry.org |

Participants: | Public to general audience |

**Abstract:**
This course will cover fundamental concepts and theorems in algebraic topology, surface differential geometry, Riemann surface theory and geometric partial differential equations, as well as optimal transportation theory; it also covers the computational methods for surface fundamental group, homology group, harmonic maps, meromorphic differentials, foliation, conformal mapping, quasi-conformal mapping and Ricci flow. Their applications in Computer Graphics, Computer Vision, Visualization, Geometric Modeling, Networking, Medical Imaging and Deep Learning will be briefly introduced as well.

- Computational Conformal Geometry by Gu and Yau, High Education Press and International Press, 2007.
- Computational Conformal Geometry - Theory by Gu and Yau, High Education Press and International Press, 2020.
- Optimal Transportation - Theory and Computation by Lei and Gu, High Education Press, 2021.

- Classical and Discrete Differential Geometry, Theory, Applications and Algorithmsby Gu and Saucan, CRC Press, 2022
- Ricci Flow for Shape Analysis and Surface Registration - Theories, Algorithms and Applications by Zeng and Gu, Springer 2013.
- Conformal Geometry - Computational Algorithms and Engineering Applicationsby Jin, Gu, He and Wang, Springer 2018.
- Variational Principles for Discrete Surfaces by Luo, Gu and Dai, High Education Press and International Press, 2007.

No. | Date | Title | Download |

Week One | Algebraic Topology | ||

1. | 07/04/2022 | Introduction to Computational Conformal Geometry | [pdf] |

2. | 07/05/2022 | Optimal Transportatiton Theory in AI | [pdf] |

3. | 07/06/2022 | Algebraic Topology: Fundamental group, Covering space | [pdf] |

4. | 07/07/2022 | Algebraic Topology: Simplicial Homology and Cohomology | [pdf] |

Assignment 1. | 07/15/2022 | Algorithm Implementation for Fundamental Group | [Data Structure] [Algorithm] [Skeleton Code] [Demo] |

Theoretic Proof for Homology Group | [Problem Set] | ||

Week Two | Differential Topology | ||

5. | 07/11/2022 | Geometric Programming : Halfedge Data Structure and Algorithmic Design Principle | [Data Structure][Algorithm][demo] |

6. | 07/12/2022 | Fixed point, Poincare-Hopf Index theorem, Characteristic Class of Fiber Bundle | [pdf] |

7. | 07/13/2022 | deRham Cohomology, Hodge Decomposition | [pdf] |

8. | 07/15/2022 | Surface Immersion Regular Homotopy | [pdf] |

Assignment 2. | 07/22/2022 | Algorithm Implementation for Hodge Decomposition | [Algorithm] [Skeleton Code] [Demo] |

Theoretic Proofs for Differential Topology | [Problem Set] | ||

Week Three | Riemannian Metric Structure | ||

9. | 07/18/2022 | Geometric Programming : Hodge Decomposition | [Algorithm] [demo] |

10. | 07/19/2022 | Surface Differential Geometry, Movable Frame Method | [pdf] |

11. | 07/20/2022 | Yamabe Equation and Geodesics | [pdf] |

12. | 07/21/2022 | Surface Harmonic Maps | [pdf] |

Assignment 3. | 07/22/2022 | Algorithm Implementation for Hodge Decomposition | [Algorithm] [Skeleton Code] [Demo] |

Theoretic Proofs for Surface Differential Geometry | [Problem Set] | ||

Week Four | Conformal Structure | ||

13. | 07/25/2022 | Conformal Modules via Geometric Complex Analysis | [pdf] |

14. | 07/26/2022 | Circle Domain Mapping: Koebe's Theorem' | [pdf] |

15. | 07/27/2022 | Convergence Analysis of Koebei's Iteration | [pdf] |

16. | 07/28/2022 | Surface Uniformization | [pdf] |

Assignment 4. | 07/29/2022 | Algorithm Implementation for Koebe Iteration | [Algorithm] [Skeleton Code] [Demo] |

Theoretic Proofs for Conformal Structure | [Problem Set] | ||

Week Five | 3D Vision | ||

17. | 08/01/2022 | 3D Acquisition Algorithmic Pipeline | [pdf] |

18. | 08/02/2022 | Phase Shifting Structured Light Method | [pdf] |

19. | 08/03/2022 | Camera Calibration, Point Cloud Fusion and Surface Reconstruction | [pdf] |

Assignment 5. | 08/05/2022 | Algorithm Implementation for Stereo-matching using Phase Shifting Structured Light | [Algorithm] [Skeleton Code] [Demo] |

Week Six | Persistent Homology | ||

20. | 08/08/2022 | Persistent Homology | [pdf] |

21. | 08/09/2022 | Combinatorial Map | [pdf] |

22. | 08/10/2022 | Abelian Differential | [pdf] |

Assignment 6. | 08/12/2022 | Handle and Tunnel Loops based on persistent homology | [Algorithm] [Skeleton Code] [Demo] |

Week Seven | Riemann Surface Theory | ||

23. | 08/15/2022 | Abel Differential and Mesh Generation | [pdf] |

24. | 08/16/2022 | Abel-Jacobi Theory | [pdf] |

25. | 08/17/2022 | Riemann-Roch Theory | [pdf] |

26. | 08/18/2022 | Algebraic Function Field on Riemann Surfaces | [pdf] |

Assignment 7. | 08/22/2022 | Verification of Abel-Jacobi Theorem | [Algorithm] [Data] |

Week Eight | Surface Ricci Flow | ||

27. | 08/22/2022 | Discrete Surface Ricci Flow | [pdf] |

28. | 08/24/2022 | Generalized Discrete Surface Ricci Flow | [pdf] |

29. | 08/26/2022 | Hyperbolic Geometry | [pdf] |

30. | 08/27/2022 | Discrete Dynamic Yamabe flow | [pdf] |

Assignment 8. | 08/28/2022 | Surface Hyperbolic Structure | [Algorithm] [Data] |

Week Nine | Optiaml Transportation | ||

31. | 08/29/2022 | Duality Theory | [pdf] |

32. | 09/05/2022 | Convex Geometric View | [pdf] |

33. | 09/07/2022 | Spherical Optimal Transportation | [pdf] |

34. | 09/09/2022 | Fluid Dynamics View | [pdf] |

35. | 09/10/2022 | Computational Methods | [pdf] |

Assignment 9. | 09/10/2022 | Euclidean Optimal Transportation | [Algorithm] [Skeleton Code] [Demo Video] |

This lecture briefly introduces the concept of conformal mapping, uniformization theorem, main types of computational algorithms and direct applications in graphics, vision, geometric modeling, networking, medical imaging and deep learning.