Additional Class(es): Monday April 25, noon to 2:00pm
Tuesday April 27, 2:003:30pm
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
2DShape Analysis,
Diffusion tensors and
Shape statistics (in medical imaging).
In particular, among our
ultimate goals, we aim to discuss some work of David Mumford:
2DShape 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
Coordinates:
Towne 309, M,W, noon1:30pm
Instructor:
Jean H.
Gallier, MRE 476, 84405, jean@saul
Office Hours: , or TBA
TBA
Prerequesites:
Basic Knowledge of linear algebra and geometry
(talk to me).
Textbook:
There will be no official textbook(s) but I will use material from
several sources including my book (abbreviated as GMA)
Grades:
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!

Review of spectral theorems in Euclidean geometry and Hermitian spaces
(Chapter 11 of GMA)

Review of singular value decomposition (SVD), polar form, least
squares and PCA
(Chapter 12 of GMA)

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.

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

Manifolds, Tangent Spaces, Cotangent Space

Manifolds

Tangent Vectors, Tangent Spaces, Cotangent Spaces

Tangent and Cotangent Bundles, Vector Fields

Submanifolds, Immersions, Embeddings

Integral Curves, Flow, OneParameter Groups

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 GroupsLie 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).

Introduction to Riemannian Manifolds

Riemannian Metrics

Affine connections

Riemannian connections

Geodesics

Curvature (tensor, sectional, Ricci), if time permits!

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 13,
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 CohnVossen, 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
Convexity:
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, McGrawHill, 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, JeanDaniel 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
 Motivations, Problems and Goals
(slides, pdf)

(slides, ppt)

(slides, keynote)
 Spectral Theorems (Symmetric, SkewSymmetric, Normal matrices)
(slides, ps)

(slides, pdf)
 Polar Form and SVD
(slides, ps)

(slides, pdf)
 Least Squares, SVD, Pseudo Inverse, PCA
(slides, ps)

(slides, pdf)
 Lie Groups and Lie Algebras, the exponential map, part I
(slides, ps)

(slides, pdf)
 Lie Groups and Lie Algebras, the exponential map, part II
(slides, ps)

(slides, pdf)
 Review of Groups and Group Actions, I
(slides, ps)

(slides, pdf)
 The Lorentz Groups O(n, 1), SO(n, 1), SO_0(n, 1),
Topological Groups
(slides, ps)

(slides, pdf)
 Manifolds, Part II
(slides, ps)

(slides, pdf)

Lie Groups and Lie Algebras, the exponential map, part III
(ps)

(pdf)

Notes on Group Actions, Manifolds, Lie Groups and Lie Algebras
(html)
 On the Early History of the Singular Value Decomposition,
by G.W. Stewart
(pdf)
 Lecture Notes on Differentiable Manifolds, Geometry of Surfaces, etc.,
by Nigel Hitchin
(html)
 An Introduction to Riemannian Geometry, by S. Gudmundsson
(html)
 Appendices I and II of Lectures on Matrices, by
J.H.M Wedderburn (1937)
(pdf)

Remarks on the Cayley representation of orthogonal matrices and
on making matrices invertible by perturbing the diagonal
(ps)

(pdf)
 Bibliography (from book)
(ps)

Clifford algebras, Clifford groups, and the groups
Pin and Spin (notes)
(ps)

(pdf)
 ``Semisecret'' Notes on algebraic geometry and algebra
(Algebra, html)

(Algebraic geometry, html)

(Complex algebraic geometry, html)
 Basic Linear Algebra, Determinant
(notes)
Papers and Talks Suitable for a Project
 Computing Exponentials of Skew Symmetric Matrices and Logarithms of
Orthogonal Matrices,
by J. Gallier and Dianna Xu
(pdf)
 An SVDBased Projection Method for Interpolation on SE(3),
by Calin Belta and Vijay Kumar
(pdf)
 Canonical Tensor Decomposition, by P. Comon
(pdf)
 Orthogonal Tensor Decomposition, by Tamara Kolda
(pdf)
 The Generalized Higher order Singular Value Decomposition ...,
by Vandewalle, De Lathauwer and P. Comon
(pdf)
 Multiple Analysis of Image Ensembles: TensorFaces, by
Alex Vasilescu and D. Terzopoulos
(pdf)
 TensorTextures: Multilinear ImageBased Rendering, by
Alex Vasilescu and D. Terzopoulos
(pdf)
 Tensors and Component Analysis, talk by Musawir Ali
(ppt)
 Principal geodesic analysis on symmetric spaces: statistics of diffusion
tensors, by Thomas Fletcher and Sarang Joshi
(pdf)
 Principal geodesic analysis for the study of nonlinear statistics of
shape,
by Thomas Fletcher, Conglin Lu, Steve Pizer and Sarang Joshi
(pdf)
 Statistics of shape via principal geodesic analysis on Lie groups,
by Thomas Fletcher, Conglin Lu and Sarang Joshi
(pdf)
 A differential geometric approach to the geometric mean of
symmetric positivedefinite matrices,
by Maher Moakher
(pdf)
 A Riemannian Framework for Tensor Computing,
by Xavier Pennec, Pierre Fillard and Nicholas Ayache
(pdf)
 2DShape Analysis using Conformal Mappings
by David Mumford and E. Sharon
(pdf)
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
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