CIS 610, Spring 2005

Advanced Geometric Methods in Computer Science

Additional Class(es): Monday April 25, noon to 2:00pm
Tuesday April 27, 2:00-3:30pm
Same class room (Towne 309)

This version (Spring 2005) will be devoted mostly to

Group Actions, Manifolds, Lie Groups, Lie Algebras, Riemannian Manifolds, with Applications to Computer Vision and Robotics

One of our main goals will be to build enough foundations to understand some recent work in 2D-Shape Analysis, Diffusion tensors and Shape statistics (in medical imaging).

In particular, among our ultimate goals, we aim to discuss some work of David Mumford:

** 2D-Shape Analysis using Conformal Mappings   (pdf) ,

Thomas Fletcher and Sarang Joshi:

** Principal geodesic analysis on symmetric spaces: statistics of diffusion tensors   (pdf)   and

Thomas Fletcher, Conglin Lu and Sarang Joshi:

** Statistics of shape via principal geodesic analysis on Lie groups   (pdf)


** Jianbo Shi and Kostas Daniilidis will make guest appearances!

Course Information
January 3, 2005


Towne 309, M,W, noon-1:30pm


Jean H. Gallier, MRE 476, 8-4405, jean@saul

Office Hours: , or TBA



Basic Knowledge of linear algebra and geometry (talk to me).


There will be no official textbook(s) but I will use material from several sources including my book (abbreviated as GMA)


Problem Sets (4 or 5), project(s), or presentation.

A Word of Advice :

Expect to be held to high standards, and conversely! In addition to transparencies, I will distribute lecture notes. Please, read the course notes regularly, and start working early on the problems sets. They will be hard! Take pride in your work. Be clear, rigorous, neat, and concise. Preferably, use a good text processor, such as LATEX, to write up your solutions. You are allowed to work in small teams of at most three. We will have special problems sessions, roughly every two weeks, during which we will solve the problems together. Be prepared to present your solutions at the blackboard. I am hard to convince, especially if your use blatantly ``handwaving'' arguments.

Brief description:

This course covers some basic material on (Riemannian) manifolds, group actions, Lie groups and Lie algebras, keeping in mind applications of these theories to computer vision, medical imaging, robotics, machine learning and control theory. The treatment will be rigorous but I will try very hard to convey intuitions and to give many examples illustrating all these concepts.

Tentative Syllabus

Next semester (Spring 2005), I intend to cover (1)-(7) below. Depending on time, I would like to cover (8) (which deals with the purely ``topological side'' of (5)) but I don't know if this will be possible!
  1. Review of spectral theorems in Euclidean geometry and Hermitian spaces (Chapter 11 of GMA)
  2. Review of singular value decomposition (SVD), polar form, least squares and PCA (Chapter 12 of GMA)
  3. Basics of classical groups (Chapter 14 of GMA)
    • The exponential map. The groups GL(n,\reals), SL(n,\reals), O(n,\reals), SO(n,\reals), the Lie algebras gl(n, \reals), sl(n, \reals), o(n), so(n) and the exponential map.
    • Symmetric matrices, symmetric positive definite matrices and the exponential map.
    • The groups GL(n,\complex), SL(n,\complex), U(n), SU(n), the Lie algebras gl(n, \complex), sl(n, \complex), u(n), su(n) and the exponential map.
    • Hermitian matrices, Hermitian positive definite matrices and the exponential map.
    • The group SE(n), the Lie algebra se(n) and the exponential.
    • ``Baby theory'' of Lie groups and Lie algebras
    Items (4)-(6) and (8) below will be taken from GAMLGLA.
  4. Review of Groups and Group Actions
    • Groups
    • Group Actions and Homogeneous Spaces, I
    • The Lorentz Groups O(n, 1), SO(n, 1) and SO_0(n, 1)
    • More on O(p, q)
    • Topological Groups
  5. Manifolds, Tangent Spaces, Cotangent Space
    • Manifolds
    • Tangent Vectors, Tangent Spaces, Cotangent Spaces
    • Tangent and Cotangent Bundles, Vector Fields
    • Submanifolds, Immersions, Embeddings
    • Integral Curves, Flow, One-Parameter Groups
  6. Lie Groups, Lie Algebra, Exponential Map
    • Lie Groups and Lie Algebras
    • Left and Right Invariant Vector Fields, Exponential Map
    • Homomorphisms, Lie Subgroups
    • The Correspondence Lie Groups--Lie Algebras
    • More on the Lorentz Group SO_0(n, 1)
    • More on the Topology of O(p, q) and SO(p, q)
    Item (7) will be taken from Kuhnel's book and do Carmo's first book (listed below).
  7. Introduction to Riemannian Manifolds
    • Riemannian Metrics
    • Affine connections
    • Riemannian connections
    • Geodesics
    • Curvature (tensor, sectional, Ricci), if time permits!
  8. Introduction to Combinatorial Topology, if time permits!
    • Review of basic affine concepts (affine combinations, affine independence, affine frames).
    • Simplices and simplicial complexes
    • Topology of simplicial complexes, stars, links
    • Pure complexes, triangulations
    • Combinatorial surfaces and triangulations
    • Delaunay Triangulations

Additional References:

Lie Groups:

Lie groups, Lie algebras, and representations, Hall, Brian, Springer (GTM No. 222)

Lie Groups. An introduction through linear Linear groups, Wulf Rossmann, Oxford Science Publications, 2002

An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, Arvanitoyeogos, Andreas, AMS, SML, Vol. 22, 2003

Lectures on Lie Groups and Lie Algebras, Carter, Roger and Segal, Graeme and Macdonald, Ian, Cambridge University Press, 1995

Lie Groups, Duistermaat, J.J. and Kolk, J.A.C., Springer Verlag, Universitext, 2000

Lie Groups Beyond an Introduction, Knapp, Anthony W., Birkhauser, Progress in Mathematics, Vol. 140, Second Edition, 2002

Theory of Lie Groups I, Chevalley, Claude, Princeton University Press, first edition, Eighth printing, Princeton Mathematical Series, No. 8, 1946

Foundations of Differentiable Manifolds and Lie Groups, Warner, Frank, Springer Verlag, GTM No. 94, 1983

Introduction to Lie Groups and Lie Algebras, Sagle, Arthur A. and Walde, Ralph E., Academic Press, 1973

Representation of Compact Lie Groups, Br\"ocker, T. and tom Dieck, T., Springer Verlag, GTM, Vol. 98, 1985

Elements of Mathematics. Lie Groups and Lie Algebras, Chapters 1--3, Bourbaki, Nicolas, Springer, 1989

Introduction a la Theorie des Groupes de Lie Classiques, Mneimne', R. and Testard, F., Hermann, 1997

Manifolds and Differential Geometry:

Differential Geometry. Curves, Surfaces, Manifolds, Wolfgang Kuhnel, AMS, SML, Vol. 16, 2002

Riemannian Geometry, Do Carmo, Manfredo, Birkhauser, 1992.

Differential Geometry of Curves and Surfaces, Do Carmo, Manfredo P., Prentice Hall, 1976.

Riemannian Geometry. A beginner's guide, Frank Morgan, A.K. Peters, 1998, Second Edition

A Panoramic View of Riemannian Geometry, Marcel Berger, Springer, 2003, First Edition.

Geometry of Differential Forms, Shigeyuki Morita, AMS, Translations of Mathematical Monographs, Vol. 201, First, Edition.

Modern Differential Geometry of Curves and Surfaces, Gray, A., CRC Press, 1997, Second Edition

Geometry (General):

Ge'ome'trie 1, English edition: Geometry 1, Berger, Marcel, Universitext, Springer Verlag, 1990

Ge'ome'trie 2, English edition: Geometry 2, Berger, Marcel, Universitext, Springer Verlag, 1990

Metric Affine Geometry, Snapper, Ernst and Troyer Robert J., Dover, 1989, First Edition

A vector space approach to geometry, Hausner, Melvin, Dover, 1998

Geometry, Audin, Michele, Universitext, Springer, 2002

Geometry, A comprehensive course, Pedoe, Dan, Dover, 1988, First Edition

Introduction to Geometry, Coxeter, H.S.M. , Wiley, 1989, Second edition

Geometry And The Immagination, Hilbert, D. and Cohn-Vossen, S., AMS Chelsea, 1932

Methodes Modernes en Geometrie, Fresnel, Jean , Hermann, 1996

Computational Line Geometry, Pottman, H. and Wallner, J., Springer, 2001

Topological Geometry, Porteous, I.R., Cambridge University Press, 1981


A course in convexity, Barvinok, Alexander, AMS, (GSM Vol. 54), 2002

Lectures on Polytopes, Gunter Ziegler, Springer (GTM No. 152), 1997

Convex Polytopes, Branko Grunbaum, Springer (GTM No. 221), 2003, Second Edition

Polyhedra, Peter Cromwell, Cambridge University Press, 1999

Convex Sets, Valentine, Frederick, McGraw-Hill, 1964

Convex Analysis, Rockafellar, Tyrrell, Princeton University Press, 1970

Computational Geometry (Voronoi diagrams, Delaunay triangulations):

Geometry and Topology for Mesh Generation, Edelsbrunner, Herbert, Cambdridge U. Press, 2001

Algorithmic Geometry, Boissonnat, Jean-Daniel and Yvinnec, Mariette (Bronniman, H., translator), Cambridge U. Press, 2001

Computational Geometry in C, O'Rourke, Joseph, Cambridge University Press, 1998, Second Edition

Applied Math, Numerical Linear Algebra:

Introduction to the Mathematics of Medical Imaging, Charles L. Epstein, Prentice Hall, 2004

Introduction to Applied Mathematics, Strang, Gilbert, Wellesley Cambridge Press, 1986, First Edition

Linear Algebra and its Applications, Strang, Gilbert, Saunders HBJ, 1988, Third Edition

Applied Numerical Linear Algebra, Demmel, James, SIAM, 1997

Numerical Linear Algebra, L. Trefethen and D. Bau, SIAM, 1997

Matrices, Theory and Applications, Denis Serre, Springer, 2002

Matrix Analysis, R. Horn and C. Johnson, Cambridge University Press, 1985

Introduction to Matrix Analysis , Richard Bellman, SIAM Classics in Applied Mathematics, 1995

Matrix Computations, G. Golub and C. Van Loan, Johns Hopkins U. Press, 1996, Third Edition

Geometry And Music

In mathematics, and especially in geometry, beautiful proofs have a certain ``music.'' I will play short (less than 2mn) pieces of classical music, or Jazz, whenever deemed appropriate by you and me!

Some Slides and Notes

Papers and Talks Suitable for a Project

The table of contents of my book can be found by clicking there:

** Table of contents

For more information, visit
Geometric Methods and Applications For Computer Science and Engineering

Back to Gallier Homepage

published by:

Jean Gallier