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Lectures
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Sections in Text
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Brief
Description
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Day 1: 9/25
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Chapters 1.1, 1.2, 1.3, 1.4, 1.5
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Definition of a Manifold. Ways to construct manifolds: connected sum, gluing of manifolds along the boundary, surgery. A sphere is a manifold. |
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Day 2: 9/27
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properties of Manifolds and Chapter 3.1
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General poperties of manifolds. Smooth manifolds, compatible charts, altlases and differentiable structures.
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Day 3: 9/30
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Chapter 3.2
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Open equivalence relations, projective spaces and Grassman manifolds, homeomorphisms between G(k,n) and G(n-k, n)
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Day 4: 10/2
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Chapters 3.3, 3.4
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Smooth mappings, smooth partition of a unit, diffeomorphisms. Constant rank mappings: immersions and submersions.
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Day 5: 10/4
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Chapter 3.5
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Submanifolds, regular submanifolds, preimages of a point in the constant rank mappings.
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Day 6: 10/7
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Chapter 3.6
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Lie groups, subroups of Lie groups, Sl(n), SO(n) as Lie groups, the kernel of a homomorphism of Lie groups is a Lie group.
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Day 7: 10/9
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Chapters 3.7-3.8
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Group Actions, the actions of SO(n), Gl(n), O(n) on R^n. Quotient space of a Lie group by the acting subgroup.
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Day 8: 10/11
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Chapters 3.8
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Actions of discrete groups on Manifolds, quotient manifolds, discrete subgroups of Lie groups.
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Day 9: 10/14
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Chapter 3.9
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Coverings and deck transformations, normal coverings
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Day 10: 10/16
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Chapter 4.1
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Tangent vectors, dimension of the tangent space to a manifold, isomorphism between R^n and the space of derivations of the space of the germs of smooth functions on R^n at a point
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Day 11: 10/18
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Chapter 4.2
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Vector fields, F-invariant vector fields, parallelizable manifolds, parallelizability of Lie groups, Euler class
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Day 12: 10/21
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Finish Chapter 4.2 and start Chapter 4.3
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Homomorphism of Lie groups induces a natural mapping of left-invariant vector fields. One parameter groups acting on a manifold, infinitesemal generators and their invariance under the action, integral curves. Local one parameter actions.
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Day 13: 10/23
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Finish Chapter 4.3
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Behavior of the flow around the non-singular point of the generating vector field.
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Day 14: 10/25
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Chapter 4.4
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Existence theorem for the ordinary differenetial equations. Every smooth vector field on a manifold $M$ is a generator of some local one-parameter action on $M$.
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Day 15: 10/28
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Chapter 4.5
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Examples of one-parameter groups acting on manifolds. Bounded trajectories are closed. Completeness of vector fields on compact manifolds. Left invariant vector fields on Lie groups are complete.
Bijection between one parameter subgroups of a Lie group $G$ and the tangent space $T_e(G)$. Index of a singular point of the vector field.
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Day 16: 10/30
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Finish Chapter 4.5, Chapter 4.6
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Examples of one parameter subgroups.
One parameter subrgoups of $Gl_n(R)$ and the exponent map on M_n(R). One parameter subgroups of a subgroup of a Lie group. One parameter subgroups in $O(n)$ and the dimension of $O(n)$. The exponent map on general Lie groups.
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Day 17: 11/1*
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Start Chapter 4.7
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(Classes meet in
x-hour instead of Friday)
Definition of a Lie algebra. Examples of Lie algebras: Lie algebras of matricies and vector fields on a manifold. Lie derivative of a vector field.
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Day 18: 11/4
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Day 19: 11/6
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Day 20: 11/8
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Day 21: 11/11
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Day 22: 11/13
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Day 23: 11/15
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Day 24: 11/18
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Day 25: 11/20
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Day 26: 11/22
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Day 27: 11/25
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Thanksgiving break 11/27 and
12/1
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Day 28: 12/2
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Day 29: 12/4
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