CIS 610, Summer 1, 2013
Brief description:
This course covers
some basic material on linear algebra, manifolds,
Riemannian metrics, Lie groups and Lie algebras,
keeping in mind applications of
these theories to robotics, computer vision, and
machine learning.
The treatment will be rigorous but
we will try very hard to convey intuitions and to give many
examples illustrating all these concepts.
Syllabus:
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Linear Algebra
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Bilinear forms;
Nondegenerate pairings; duality
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Inner products and spectral theory of normal operators
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Special families of operators: symmetric, orthogonal, unitary, etc.
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Polar Decomposition and SVD
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Skew-symmetric bilinear forms; Symplectic operators and mechanics
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The matrix exponential
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The surjectivity of exp: so(n) -> SO(n)
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Rodrigues formula
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The surjectivity of exp: su(n) -> SU(n)
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Symmetric positive definite matrices (SPD).
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The bijection exp: S(n) -> SPD
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Manifolds, Tangent Spaces, Cotangent Space
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Introduction to manifolds
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Manifolds in R^n (charts, etc.)
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Tangent Vectors, Tangent Space, Cotangent Spaces
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Vector Fields
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Submanifolds, Immersions, Embeddings
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Integral Curves, Flow, One-Parameter Groups
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A glimpse at
abstract manifolds (charts, atlases, etc);
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Partitions of unity
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Orientation of manifolds
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Lie Groups, Lie Algebra, Exponential Map
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Lie Groups and Lie Algebras
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Left and Right Invariant Vector Fields, Exponential Map
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Homomorphisms, Lie Subgroups
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The Correspondence Lie Groups--Lie Algebras
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Metrics and curvature on Lie groups
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Left and right invariant metrics
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Bi-invariant metrics
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Connections and curvature of left-invariant metrics
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The Killing form
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Semisimple Lie algebras; a glimpse at the classification theorem
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Specific Groups, Manifolds and Applications
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SO(3), SE(3), RP^3 (Vision)
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Grassmann manifolds G(m,n) (learning)
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Symplectic Geometry in the service of mechanics (an unassuming
introduction)
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Lagrangians and Hamiltonians in some classical examples
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