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
pseudomanifold. 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 manifoldbased 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 manifoldbased constructions.

Prerequisites
Linear algebra, multivariate calculus, and programming.
Knowledge of differential geometry, topology, and analysis are desirable but not required.

Instructors
Name

Office

Email

Phone

Office Hours

Jean Gallier

242E

jean@cis.upenn.edu

(21) 2529 5050

We. 1011:30AM, Th. 4:305:30PM

Gustavo Nonato

242E

gnonato@icmc.usp.br

(21) 2529 5050

TBA

Luiz Velho

354

lvelho@impa.br

(21) 2529 5154

TBA

Marcelo Siqueira

242E

mfsiqueira@dimap.ufrn.br

(21) 2529 5050

We. 1011:30AM, Th. 4:305:30PM 

Lectures
Tuesdays and Thursdays, 1.30PM to 3PM, Conference Room 3, IMPA, Rio de Janeiro, RJ, Brazil

Textbooks
We will provide class notes and slides for the course, but we strongly
encourage you to refer to the supplementary material described below.
In the “foundations” of the course, we will cover introductory chapters
of classic textbooks on manifolds, topology and differential geometry:
 Loring W. Tu. An introduction to Manifolds. SpringerVerlag, 2nd edition, 2010.
 Manfredo P. do Carmo. Differential Geometry of Curves and Surfaces. PrenticeHall, 1976.
 James R. Munkres. Topology. PrenticeHall, 2nd edition, 2000.
 Jean H. Gallier. Geometric methods and Applications. SpringerVerlag, 2nd edition, 2011.
Portuguese readers can also refer to the following books:
 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.
For the “constructions” part of the course, read the following SIGGRAPH tutorial (download it from here):
 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.
We also encourage you to download and read the original papers where
the manifoldbased surface constructions we will cover are described.
We will refer you to the URL of PDF files of the papers during the
lectures.
For the “applications” part of the course, you will be
referred to papers related to the applications described in class. 
Assignments
Homework 
Due Date

Remarks

Homework 1 (PDF)

Sep. 20, 2011

The statement of problem 4 has been fixed.

Homework 2 (PDF)

Sep. 29, 2011

The statement of problem 4(c) has been fixed.

Homework 3 (PDF)

Nov. 01, 2011

Typos in problems 3 and 4 have been fixed.


Grading
Your grade will consist of 3 components:
 homeworks  30%
 project  40%
 presentation  30%

Syllabus
Day 
Topic

Slides

Instructor

Sep. 06, 2011

Curves

PDF (09/05/2011)

Jean Gallier

Sep. 08, 2011

A review of multivariate calculus

PDF (09/13/2011)

Jean Gallier

Sep. 13, 2011

Surfaces

PDF (09/16/2011)

Jean Gallier

Sep. 15, 2011

Manifolds

PDF (09/20/2011)

Jean Gallier

Sep. 20, 2011

Manifolds

PDF (09/22/2011)

Jean Gallier

Sep. 22, 2011

Manifolds

PDF (09/22/2011)

Jean Gallier

Sep. 27, 2011

Parametric pseudomanifolds

PDF (10/20/2011)

Jean Gallier

Sep. 29, 2011

Constructions

PDF (10/20/2011)

Marcelo Siqueira

Oct. 04, 2011

Constructions

PDF (10/20/2011)

Marcelo Siqueira

Oct. 06, 2011

Constructions

PDF (10/20/2011)

Marcelo Siqueira

Oct. 11, 2011

Constructions

PDF (10/31/2011)

Marcelo Siqueira

Oct. 13, 2011

Constructions

PDF (10/20/2011)

Marcelo Siqueira

Oct. 18, 2011

NO CLASS  attend the expert's seminar



Oct. 20, 2011

NO CLASS  attend the expert's seminar



Oct. 25, 2011

Constructions

PDF (10/25/2011)

Marcelo Siqueira

Oct. 27, 2011

Constructions

PDF (11/25/2011)

Marcelo Siqueira

Nov. 01, 2011

Splines, subdivision, and manifolds

PDF (11/09/2011)

Luiz Velho

Nov. 03, 2011

Applications

PDF (11/09/2011)

Luiz Velho

Nov. 08, 2011

Manifold harmonics and applications

PDF (11/09/2011)

Gustavo Nonato

Nov. 10, 2011

Manifold harmonics and applications

PDF (11/09/2011)

Gustavo Nonato

Nov. 15, 2011

NO CLASS  holiday



Nov. 17, 2011

Student's presentations

ZIP (11/25/2011)


Nov. 22, 2011

Student's presentations

ZIP (11/25/2011)


Nov. 24, 2011

Invited talks

ZIP (11/25/2011)



Supplementary material
Day

Handout

Remarks

Sep. 29, 2011

NB (Mathematica 8)

Code for the examples shown on Sep. 29, 2011

Oct. 11, 2011

NB (Mathematica 8)

Code for the examples shown on Oct. 11, 2011

Oct. 13, 2011

NB (Mathematica 8)

Code for the examples shown on Oct. 13, 2011

Oct. 25, 2011

NB (Mathematica 8)

Code for the 2D bump function shown in class

Oct. 27, 2011

ZIP

Code for creating and sampling PPS's

Nov. 25, 2011

NB (Mathematica 8)

Code for the examples shown on Oct. 27, 2011

