# 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:

1. 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, One-Parameter Groups
• Partitions of unity
• Orientation of manifolds
2. Riemannian Manifolds and Connections
• Riemannian metrics
• Connections on manifolds
• Parallel transport
• Connections compatible with a metric; Levi-Civita connections
3. Geodesics on Riemannian Manifolds
• Geodesics, local existence and uniqueness
• The exponential map
• Complete Riemannian manifolds, Hopf-Rinow Theorem, Cut-Locus
• The calculus of variation applied to geodesics
4. 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
5. Spherical Harmonics
• Introduction: Spherical Harmonics on the Circle
• Spherical Harmonic on the 2-Sphere
• The Laplace-Beltrami 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 Funk-Hecke Formula

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