2020 Lecture Series: Computational Conformal Geometry

Instructor: David Gu
Email: gu@cs.stonybrook.edu
Date:Every Friday and Saturday 9:00-10:30pm EST
Duration:From July 3rd to September 4th
Zoom ID:871 6057 8498
Live Streaming: Use a web browser to open the live streaming link online.conformalgeometry.org
Host:Yau Mathematics Science Center of Tsinghua University and Beijing Yanxi Lake Applied Mathematics Institute
Participants:Public to general audience for free

Abstract: This course will cover fundamental concepts and theorems in algebraic topology, surface differential geometry, Riemann surface theory and geometric partial differential equations; 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.


Reference books:

Assignments: This lecture series will offer elementary library, the students are encouraged to implement some of the fundamental algorithms, such as computational topology, harmonic map and optimal transport map. Teaching assistants will answer the equestions and offer some helps for coding.

Online Demo: The online demo is written using WebGL, you can observe the conformal mappings of 3D surfaces interactively. More videos, images can be accessed by scanning the bar codes in the text book.

Lecture Slides

The lecture slides of the previous year can be found here.

1.07/03/2020Introduction to Computational Conformal GeometryCh 1[pdf] [video]
2.07/04/2020Algebraic Topology: Fundamental group, Covering spaceCh 2 [pdf][video]
Assignment 1.07/05/2020Mesh Halfedge Data Structure and Algorithm for Cut GraphCh 2 [pdf][pdf][Skeleton Code]
3.07/10/2020Algebraic Topology: Homology and Cohomology, CW-cell DecompositionCh 4, Ch 5[pdf][video]
4.07/11/2020Fixed point, Poincare-Hopf Index theorem, Characteristic Class of Fiber BundleCh 4, Ch 5[pdf][video]
Assignment 2.07/16/2020Discrete Surface Harmonic MapCh 23, Ch 24 [pdf][Skeleton Code]
5.07/17/2020Differential Topology: deRham Cohomology, Hodge DecompositionCh 6[pdf][video]
6.07/18/2020Surface Differential Geometry: fundamental forms, Movable frame method, Gauss-Bonnet theoremCh 18[pdf][video]
7.07/24/2020Isothermal Coordinates and GeodesicsCh 21, Ch 22, Ch 23[pdf][video]
8. Assignment 3.07/25/2020Hodge Decomposition and Riemann MappingCh 23, Ch 24 [pdf][Skeleton Code][video]
9.07/31/2020Harmonic maps and Conformal MapsCh 21, Ch 22, Ch 23[pdf][video]
10. 08/01/2020Conformal Module via Geometric Complex AnalysisCh 8, Ch 9, Ch 10, Ch 11, Ch 13[pdf][video]
Assignment 4.08/03/2020Spherical Harmonic MappingCh 23, Ch 24 [pdf][C++ Skeleton Code][Matlab Skeleton Code]
11.08/07/2020Optimal Transportation : Duality TheoremHandout[pdf] [video]
12.08/08/2020Opitmal Transportation : Convex Geometric ViewHandout[pdf][video]
Assignment 5.08/13/2020Optimal Transportation MapHandout [pdf][C++ Skeleton Code]
13.08/14/2020Optimal Transportation: Fluid Dynamics ViewHandout[pdf][video]
14.08/15/2020Circle Domain MappingCh 14, Ch 15[pdf][video]
Assignment 6.08/21/2020Circular Slit Map and Koebe's IterationHandout [pdf][C++ Skeleton Code]
15.08/21/2020Koebe's IterationCh 14, Ch 15[pdf] [video]
16.08/22/2020Surface UniformizationCh 16[pdf] [video]
17.08/28/2020Persistent HomologyCh 4[pdf] [video]
18.08/29/2020Combinatorial Maps Ch 25[pdf] [video]
Assignment 7.08/30/2020Handle and Tunnel Loops based on persistent homologyHandout [pdf][C++ Skeleton Code]
19.09/04/2020Discrete Surface Ricci FlowCh 33, Ch 34[pdf] [video]
20.09/05/2020Generalized Discrete Surface Ricci FlowCh 33, Ch 34[pdf] [video]
21.09/11/2020Hyperbolic GeometryCh 30, Ch 31[pdf] [video]
22.09/12/2020Uniqueness, existence of the solution to discrete Yamabe flowCh 35[pdf] [video]
Assignment 8.09/12/2020Surface Hyperbolic StructureHandout [pdf][Data Set]
2309/18/2020Summary of Computational Conformal GeometryHandout [pdf][video]
24.09/26/2020Guest Lecture: Dr. Hang Si, Delaunay Triangulations in the Plane Handout[pdf] [video][Demo]
25.09/27/2020Guest Lecture: Dr. Hang Si, Triangular Mesh Generation in the PlaneHandout[pdf] [video]
Assignment One.10/20/2020Incremental Algorithm for Convex HullHandout [pdf][C++ Skeleton Code]
26.10/24/2020Guest Lecture: Prof. Xiaoming Du, Teichmuller Space Handout[pdf] [video]
27.10/25/2020Guest Lecture: Prof. Xiaoming Du, Mapping Class GroupHandout[pdf] [video]
Assignment Two.10/27/2020Delaunay Triangulation and Voronoi DiagramHandout [pdf][C++ Skeleton Code]
Assignment Three.11/06/2020Geometric Variational Algorithm for Optimal Transportation MapHandout [pdf][C++ Skeleton Code][video][data set][bindary demo]
Assignment Four.11/20/2020Geometric Variational Algorithm for Spherical Optimal Transportation MapHandout [pdf][C++ Skeleton Code][video][data set][bindary demo]
28.11/28/2020Guest Lecture:Dr. Fei Sun, Algebraic curves, Applications of Riemann-Roch TheoryHandout [pdf] [pdf][video]
29.11/29/2020Guest Lecture:Dr. Fei Sun, Sheaf CohomologyHandout [pdf] [pdf][video]
30.12/20/2020Guest Lecture:Dr. Xiaoming Du, The hyperbolic structure of the figure-eight knot complement and history of the geometrization theorem for 3-manifoldsHandout [pdf][video]
Assignment Five.12/25/2020Incremental Algorithm for 4D Convex HullHandout [pdf][C++ Skeleton Code]
Assignment Six.01/03/20213D Voronoi DiagramHandout [pdf][C++ Skeleton Code]
Riemann Surface, meromrophic functions, holomorphic quadratic differentials, foliationsCh 25, Ch 26
Meromorphic differentials, Abel-Jacobi Theorem, Ch 25
Riemann-Roch TheoremCh 25
Quasi-conformal map, Beltrami equationCh 29

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.

Discussion Group: By scanning the following bar codes to join the discussion group: