Discrete Differential Geometry and Conformal Geometry
Student No.:50
Time:Mon./Wed. 15:10-17:00
Instructor:Xianfeng David Gu,Jian Sun  
Place:Lecture Hall, Floor 3, Jin Chun Yuan West Building (近春园西楼3层报告厅)
Starting Date:2012-7-4
Ending Date:2012-8-22

Course description:


Computational conformal geometry is an emerging interdisciplinary field, which combines modern geometry and computer science, and is applied in a broad range of fields in engineering, medicine and fundamental sciences. Professor Shing-Tung Yau and his students play important roles in the development of computational conformal geometry.


Computational conformal geometry is in the intersection of many fields in pure mathematics, such as Riemann surface theory, complex analysis, algebraic topology, algebraic curve, differential geometry, partial differential equation and so on. Especially, it has close relation with geometric analysis. This course will cover the following fundamental concepts and theories: homology, homotopy and covering space in algebraic topology; Riemann mapping, uniformization and conformal welding; Riemann surface, Hodge theory, Abel differential, Riemann-Roch theorem, conformal module, Tiechmuller space, quasi-conformal mapping theory; differential geometry of surfaces, exterior calculus, Gauss-Bonnet theorem, characteristic class; hyperbolic geometry, surface Ricci flow etc.


The course will cover the following computational algorithms: representation of discrete surface, discrete exterior calculus, homology groups, homotopy groups, covering space, harmonic maps, harmonic differential forms, holomorphic differential forms, conformal structure, discrete conformal mapping, conformal module, discrete Ricci flow, uniformization metric, Teichmuller coordinates, quasi-conformal mapping, conformal welding, affine and projective structures of surfaces, persistence homology and its stability, and so on.


The course will introduce the following engineering applications: in computer graphics, mesh parameterization, texture mapping, vector field design; in computer vision, surface tracking and matching with large deformations, geometric database indexing, shape analysis; in geometric modeling, manifold spline theory, mesh-spline conversion; in medical imaging, conformal brain mapping, virtual colonoscopy; in computational geometry, homotopy detection, graph embedding; in wireless sensor network, routing based on metric design, load balancing; in computational topology, computing homology basis and persistence diagram and so on. Classical applications of conformal geometry in elasticity mechanics, fluid, electric field computation will be briefly introduced as well.


The course is designed for most senior undergraduates and junior graduates with engineering backgrounds. The students are required to have the back knowledge of linear algebra and multi-variable calculus.


Programming skills in C++ is preferred, but not required. The course will teach relatively profound and advanced concepts in pure mathematics and modern geometry using the language of engineers. Most deep theorems, such as Gauss-Bonnet theorem, Hodge theory, uniformization theorem and surface Ricci flow will be introduced and proven using combinatorial or discrete methods, which do not require professional mathematics training. Most abastract concepts and theorems, such as homotopy group, Abel differentials, conformal mappings, constant curvature metrics can be computed directly using the algorithms covered in the class, and visualized using computer graphics techniques. This course emphasizes implementation skills. The students can directly "see" the concepts and theorems by accomplishing the programming assignments, and grasp the intuition, improve the learning efficiency.


The algorithms covered by the course have great practical values. Simens R&D has liscensed the virtual colonosocpy patent based on conformal geometry with a million dollar. Because conformal geometry is one of the fundamental structures of nature, it must have much broader applications in engineering fields. We hope by propagating the knowledge of computational conformal geometry, more applications will be found, more conjectures will be proven, and this emerging field will be advanced accordingly.




Calculus, Linear Algebra and Computer Programming


Reference for the course:


[1] “Computational Conformal Geometry”by S.-T. Yau and X. Gu, published by International Press and High Education Press.


[2] “Computational topology: an introduction” by Herbert Edelsbrunner, John Harer, published by AMS