This version (Spring 2008) will be devoted mostly to
Manifolds, Lie Groups, Lie Algebras and
Riemannian Geometry,
with Applications to
Medical Imaging and Surface Reconstruction
One of our main goals will be to build enough foundations
to understand some recent work in
Diffusion tensors and
Shape statistics (in medical imaging).
In particular, among our goals, we aim to discuss various papers of the
Asclepios Research Group headed by Nicholas Ayache.
Asclepios, INRIA Sophia Antipolis
(html)
In particular:
Vincent Arsigny's dissertation:
Processing Data in Lie Groups:
An Algebraic Approach. Application to NonLinear
Registration and Diffusion Tensor MRI
(html)
(pdf)
Arsigny, Fillard, Pennec, Ayache:
Geometric Means in a Novel Vector Space Structure on
Symmetric PositiveDefinite Matrices.
:
(pdf)
Arsigny, Fillard, Pennec, Ayache:
LogEuclidean Metrics for Fast and Simple Calculus on Diffusion Tensors
:
(pdf)
Arsigny, Pennec, Ayache:
Polyrigid and Polyaffine Transformations: a Novel Geometrical Tool to
Deal with NonRigid Deformations  Application to the
registration of histological slices
:
(pdf)
Fillard, Pennec, Arsigny, Ayache:
Clinical DTMRI Estimation, Smoothing and Fiber Tracking with LogEuclidean Metrics
:
(pdf)
Xavier Pennec:
Statistical Computing on Manifolds for Computational Anatomy
:
(html)
(pdf)
Xavier Pennec:
Intrinsic Statistics on Riemannian Manifolds:
Basic Tools for Geometric Measurements.
:
(pdf)
Xavier Pennec, P. Fillard and N. Ayache:
A Riemannian Framework for Tensor Computing
:
(pdf)
Ayache, Clatz, Delingette, Malandain, Pennec, Sermesant:
Asclepios: a Research Project at INRIA for the Analysis and Simulation of Biomedical Images
:
(pdf)
Durrleman, Pennec, Trouve, Ayache:
Measuring Brain Variability via Sulcal Lines Registration: a Diffeomorphic Approach
:
(pdf)
Course Information
February 25, 2008
Coordinates:
Towne 315, M,W, noon1:30pm
Instructor:
Jean H.
Gallier, GRW 476, 84405, jean@cis.upenn.edu
Office Hours: , or 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 (3 or 4), 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 manifolds, Lie groups
Lie algebras and Riemannian Geometry,
keeping in mind applications of
these theories to medical imaging, computer vision and
machine learning.
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 2008),
I intend to cover (1)(8) below.

Introduction to Lie groups, Lie algebras and manifolds

Review of Groups and Group Actions

Groups

Group Actions and Homogeneous Spaces, I

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

Partitions of unity

Orientation of manifolds

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

Riemannian Manifolds and Connections

Riemannian metrics

Connections on manifolds

Parallel transport

Connections compatible with a metric; LeviCivita connections

Geodesics on Riemannian Manifolds

Geodesics, local existence and uniqueness

The exponential map

Complete Riemannian manifolds, HopfRinow Theorem, CutLocus

The calculus of variation applied to geodesics

Curvature in Riemanian Manifolds

The curvature tensor

Sectional curvature

Ricci curvature

Isometries and local isometries

Riemannian covering maps

The second variation formula and the index form

Jacobi fields

Applications of Jacobi fields and conjugate points

Cut locus and injectivity radius: some properties

Metrics and curvature on Lie groups

Left and right invariant metrics

Biinvariant metrics

Connections and curvature of leftinvariant metrics

The Killing form
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,
Brocker, 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.
Geometry of Differential Forms,
Shigeyuki Morita, AMS, Translations of Mathematical Monographs, Vol. 201,
First Edition.
Riemannian Geometry,
Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine,
Springer Verlag, Universitext, 2004, Third Edition.
SemiRiemannian Geometry With Applications to Relativity,
Barrett O'Neill, Academic Press, 1983, First Edition.
A Panoramic View of Riemannian Geometry,
Marcel Berger, Springer, 2003, First Edition.
Geometry VI. Riemannian Geomety,
M.M. Postnikov, Springer, Encyclopaedia of Mathematical Sciences, Vol. 91,
2001, First Edition.
Riemannian Geometry,
Takashi Sakai, AMS, Translations of Mathematical Monographs,
Vol. 149, 1997, First Edition.
Morse Theory,
John Milnor, Princeton University Press, Annals of Mathematics Studies,
No. 51, 1969, Third Printing.
Riemannian Geometry. A beginner's guide,
Frank Morgan, A.K. Peters, 1998, Second 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
 GMA, Preface and Chapter 1
(pdf)
 GMA, Chapter 6
(pdf)
 GMA, Chapter 11
(pdf)
 GMA, Chapter 12
(pdf)
 GMA, Chapter 14
(pdf)
 GMA, Bibiography
(pdf)
 Problems, Questions and Motivations
(slides, pdf)
 Lie Groups and Lie Algebras, the exponential map, part I
(slides, pdf)
 Lie Groups and Lie Algebras, the exponential map, part II
(slides, pdf)
 Sir Walter Synnot Manifold
(jpg)
 Review of Groups and Group Actions
(slides, pdf)
 The Lorentz Groups O(n, 1), SO(n, 1), SO_0(n, 1),
Topological Groups
(slides, pdf)
 Manifolds, general case
(slides, pdf)

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

Riemannian manifolds, connections, parallel transport,
LeviCivita connections
(pdf)

Geodesics on Riemannian manifolds
(pdf)

The LogEuclidean Framework
(pdf)

Notes on Differential Geometry and Lie Groups
(html)

Logarithms and Square Roots of Real Matrices (Some Notes)
(pdf)

Construction of C^{\infty} Surfaces From Triangular Meshes Using
Parametric PseudoManifolds
(with Marcelo Siqueira and Dianna Xu)
(pdf)

Notes on Spherical Harmonics and Linear Representations of Lie Groups
(pdf)

Talk in Undergrad. Colloquium, Penn Math Department, 03/26/2008
(pdf)

The derivation of the exponential map of matrices, by
G. M. Tuynman
(pdf)
 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)
 ``Semisecret'' Notes on algebraic geometry and algebra
(Algebra, html)

(Algebraic geometry, html)

(Complex algebraic geometry, html)
 INRIA Macros
(tar.gz)
Papers and Talks Suitable for a Project
 Modeling surfaces of arbitrary topology using manifolds,
by C. Grimm and J. Hughes
(pdf)
 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)
 Approximating the logarithm of a matrix ..., by
S.H. Cheng, N. Higham, C. Kenny and L. Laub
(pdf)
 The scaling and squaring method for the matrix exponential revisited, by
N. Higham
(pdf)
 Algorithms for the matrix pth root, by
D. Bini, N. Higham and B. Meini
(pdf)
 Condition estimates for matrix functions, by
C. Kenny and A. Laub
(pdf)
 Computing real square roots of a real matrix, by
N. Higham
(pdf)
 On the existence and uniqueness of the real logarithm of a matrix, by
W. Culver
(pdf)
 The range of A^{1}A^* in GL(n, C), by
DePrima and Johnson
(pdf)
 Nineteen Dubious ways to compute the exponential of a matrix ..., by
Moler and Van Loan
(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)
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