Computational Conformal Geometry
Lecturer
Xianfeng David Gu (顾险峰, Professor)
Date
2024-09-23 ~ 2024-11-30
Schedule
Weekday Time Venue Online ID Password
Mon,Wed,Sat 20:10-21:45 A7-201 Zoom 637 734 0280 BIMSA
Prerequisite
具有多元微积分、线性代数、初等抽象代数的基础；最好具备C++编程能力，熟悉OpenGL。The students are required to be familiar with multivariate calculus, linear algebra, elementary abstract algebra, and complex analysis. It is preferred to know C++ geometric programming and OpenGL for visualization.
Introduction
计算共形几何是现代数学与计算机科学交叉的学科，在课程设置上采用基础数学理论与计算机算法并重的路线。课程涵盖基础数学本科的代数拓扑、黎曼几何、黎曼面理论和微分几何课程。同时，为了解释计算方法，我们也介绍一些研究生水平的数学课程，包括调和映照理论、Teichmuller空间理论、曲面Ricci 流理论和三维流形的拓扑理论。近年来，依随AI的蓬勃发展，我们系统地介绍了凸微分几何中的Minkowski-Alexandrov理论，以及等价的Monge-Kantorovich-Brenier最优传输理论。
“计算共形几何”课程介绍新发展起来的理论和算法。在代数拓扑中，我们介绍了同伦群、单纯同调群、持续同调群的计算方法；在黎曼几何中，我们介绍了霍奇分解算法，调和微分de Rham上同调群的算法，曲面间调和映射的算法，测地线的算法；在凸微分几何中，我们介绍了从高斯曲率重建凸曲面的Minkowski问题的解法，根据高斯曲率重建开放凸曲面的Alexandrov问题的解法，反射镜面、透射镜面设计问题的解法，这些算法等价于最优传输问题的解法；在黎曼面理论中，核心是两类算法：一类是基于单值化定理的离散曲面Ricci流算法。这类算法是迄今为止唯一的能够通过目标高斯曲率得到黎曼度量的算法，而绝大多数工程几何问题，最终都归结为求取某个具有特殊性质的黎曼度量问题；另外一类算法是基于Abel-Jacobi理论的计算Abel微分的方法：各种共形映射、共形不变量的计算最终归为全纯一形式的计算；曲面上的各种叶状结构都归结为全纯二次微分的计算；曲面上的各种实射影结构归结为全纯三次微分的计算；曲面上的四边形网格归结为黎曼面上的亚纯四次微分的计算。我们将这些主要的算法布置成家庭作业，让同学们通过C++编程深化对理论的理解，同时增强处理实际几何问题的工程能力。
Computational Conformal Geometry is an interdisciplinary field that combines modern topology and geometry with computer science. The course focuses on the main concepts and theorems in mathematical fields and the corresponding computational algorithms, and their applications in engineering and medical fields.
Syllabus
The course outline is as follows, depending on the schedule, some conents may not be covered.
1. Algebraic Topology
Fundamental group, covering space, simplicial homology, simplicial cohomology, persistent homology, Von-Kampen Seifert theorem, Brower and Lefschetz fixed point theorem, Poincar?e-Hopf Index theorem, Fiber bundle, characteristic class via obstruction.
2. Differential Topology
Exterior calculus, de Rham cohomology, Stokes theorem, Hodge decomposition theorem,
3. Surface Differential Geometry
Movable frame method, First fundamental form, second fundamental form, principle curvature, Gaussian curvature, mean curvature, Weigarten transform, Gauss’ theorem Egregium, Gauss-Bonnet theorem, connection, covariant differential, geodesics, minimal surface, Yamabe equation,
4. Harmonic Mapping
Harmonic function, mean value theorem, maximal value principle, Rado theorem, Hopf differential, surface harmonic mapping, spherical harmonic mapping, Yau surface harmonic mapping theorem, Graphvalued harmonic map, surface foliation
5. Geometric Complex Analysis
conformal module, Riemann mapping theorem, extremal length, slit mapping, Koebe theorem, Hilbert theorem, Kobe iteration, Klein-Poincar?e-Koebe uniformization theorem,
6. Discrete Surface Ricci Flow
Hamilton’s Ricci flow, surface Ricci flow theorem, derivative cosine law, discrete conformal equivalence, discrete surface Ricci flow theorem,
7. Riemann Surface Theory
Abel differential, Abel-Jacobi theorem, Riemann-Roch theorem, meromorphic function field, holomorphic line bundle, sheaf Cech cohomology, plane algebraic curves
8. Convex Geometry
Minkowski theorem, Alexandrov theorem, Alexandrov convex cap theorem, Alexandrov convex polytope theorem,
9. Optimal Transportation
Monge-Kantorovich theory, Brenier theorem, spherical optimal transportation theorem, McCann transportation, Benamou-Brenier theorem, geometric variational method, linearization of Monge-Ampere equation
10. 3-Manifold
Mayer-Vietoris theorem, homology of 3-manifolds, surface mapping class group, Heegaard splitting and Kirby diagram, prime decomposition and Haken hierarchy, JSJ decomposition, Thurston’s geometrization, mostow rigidity theorem, hyperbolic knot theory, hyperbolic 3-manifold, hyperbolic discrete surface Ricci flow, discrete ricci flow for polyhedral surface
Video Public
Yes
Notes Public
Yes
Audience
Undergraduate, Advanced Undergraduate, Graduate, Postdoc, Researcher
Language
Chinese
Lecturer Intro
David Gu is a New York Empire Innovation Professor at the Department of Computer Science, Stony Brook University. He received his Ph.D degree from the Department of Computer Science, Harvard University in 2003， supervised by the Fields medalist, Prof. Shing-Tung Yau and B.S. degree from the Tsinghua University, Beijing, China in 1995.
His research focuses on applying modern geometry in engineering and medical fields. With Prof. Yau and his collaborators, David systematically develops discrete theories and computational algorithms in the interdisciplinary field: Computational Conformal Geometry, Computational Optimal Transportation, and applied them in engineering and medical imaging fields.
TA
Yi Liu (刘熠, Postdoc)
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