Introduction to Computational Manifolds

Course overview

The course is divided into three main parts: foundations, constructions, and applications. Foundations will cover introductory notions of differential geometry, topology, and manifolds, and will introduce a constructive, and yet equivalent, notion of manifold called parametric pseudo-manifold. The focus is on the theoretical aspects of the constructive definition, which will serve as a common ground for the surface constructions described next. Constructions will present the main manifold-based surface constructions available in the literature. The focus of the lectures is on the computational aspects of those constructions. The goal is to “practice” the underlying notions studied before. Finally, the Applications part will cover some applications in graphics and engineering, such as surface modeling and fluid flow simulation, using manifold-based constructions.

Pre-requisites

Instructors

Lectures

Textbooks

• Loring W. Tu. An introduction to Manifolds. Springer-Verlag, 2nd edition, 2010.
• Manfredo P. do Carmo. Differential Geometry of Curves and Surfaces. Prentice-Hall, 1976.
• James R. Munkres. Topology. Prentice-Hall, 2nd edition, 2000.
• Jean H. Gallier. Geometric methods and Applications. Springer-Verlag, 2nd edition, 2011.

• Elon L. Lima. Variedades Diferenciáveis. Publicações Matemáticas, IMPA, 2008.
• Elon L. Lima. Elementos de Topologia Geral. SBM, 2010.
• Manfredo P. do Carmo. Geometria Diferencial de Curvas e Superfícies. SBM, 3rd edition, 2005.

• Cindy M. Grimm and Denis Zorin. Surface modeling and parameterization with manifolds. In ACM SIGGRAPH 2006 Courses (SIGGRAPH ’06), pages 1–81, New York, NY, USA, 2006. ACM Press.

Assignments

Grading

• homeworks - 30%
• project - 40%
• presentation - 30%

Syllabus

Supplementary material