CIS 610, Spring 2011
Brief description:
This course covers
some basic material on manifolds,
Riemannian Geometry, and Harmonic Analysis,
keeping in mind applications of
these theories to medical imaging, computer vision, computer graphics 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.
Syllabus:

Manifolds, Tangent Spaces, Cotangent Space

Introduction to manifolds

Manifolds (formal definition using charts, etc.)

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

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

Spherical Harmonics

Introduction: Spherical Harmonics on the Circle

Spherical Harmonic on the 2Sphere

The LaplaceBeltrami Operator

Harmonic Polynomials, Spherical Harmonics and L^2(S^n)

Spherical Functions and Representations of Lie Groups

Reproducing Kernels and Zonal Spherical Harmonics

More on the Gegenbauer Polynomials

The FunkHecke Formula
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