Math 124
Fall 2010
Current Problems in Topology
Lecture Plan
This lecture plan is tentative and will be updated irregularly. The homework page
will be updated on the regular basis
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Lectures |
Sections in Text |
Brief Description |
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Wednesday September 22 |
Chapter 1 |
Manifolds and Submanifolds, examples and properties,
Manifolds with boundary |
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Friday September 24 No class. Instead we have an x-hour on Tuesday September 28 |
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Monday September 27 |
Chapter 2 |
Differentiable structures,
atlases, smooth manifolds |
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Tuesday September 28 x-hour instead of the class on Thursday September 24 |
Chapter 2 |
Critical point, regular
value, expression of functions in different coordinate systems |
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Wednesday September 29 |
Chapter 2 |
Measure zero sets, Sard Theorem |
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Friday October 1 |
Chapter 2 |
Presentations of functions of
fixed rank, immersion, immersed and imbedded submanifolds, |
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Monday October 4 Note that Tuesday October 5 is the final day to establish
the course load |
Chapter 2 |
Boundary of manifold as an
imbedded submanifold. Preimage of a point under a constant rank mapping is a submanifold |
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Wednesday October 6 Final day for electing to use the Non-Recording Option. |
Chapter 2 |
GL(n), SL(n) as submanifolds. Covers, Refinements of
covers. |
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Friday October 8 |
Chapter 2 |
Partitions of unity and their
applications,imbeddings of manifolds into Euclidian
spaces |
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Monday October 11 |
Chapter 3 |
Tangent vectors, fibers and
projection maps, n-dimensional vector bundles |
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Wednesday October
13 |
Chapter 3 |
Equivalence of vector bundles,
trivial vector bundles, bundle maps, examples of
trivial and nontrivial bundles |
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Friday October 15 |
Chapter 3 |
Sections of vector bundles, vector fields, Euler class |
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Monday October 18 |
Chapter 3 |
Euler characteristic of the
manifold is the Euler class of the
tangent bundle TM, tangent vectors as derivations |
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Wednesday October 20 |
Chapter 3 |
Orientation of a bundle, orientability, summation of bundles and
induced bundles (briefly if time permits) |
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Friday October 22 |
Chapter 4 |
Dual bundle, Cotangent
bundle, coordinate description for the differential, |
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Monday October 25 Middle of the term presentation and
discussion should be done in the period Monday October 25- Friday October 29 |
Chapter 4 |
Covariant tensor fields,
tensors in local coordinates |
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Tuesday October 26 x-hour instead of the class on Friday October 28 |
Chapter 4 |
Contraction of tensors,
covariant and contravariant functors |
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Wednesday October 26 |
Chapter 5 |
Integral curves, existence of
solution theorems for differential equations without proofs, flows |
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Friday October 28 No class. Instead we have an x-hour on Tuesday October 26.
Have a nice homecoming weekend J |
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Monday November 1 Note that Tuesday November 2 is the final day for
dropping the fourth course |
Chapter 5 |
Flows with compact support,
straightening of a vector field |
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Wednesday November 3 |
Chapter 5 |
Lie derivative and its
properties, Lie bracket of vector fields and Lie algebra |
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Friday November 5 |
Chapter 6 |
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Monday November 8 |
Chapter 6 |
Distributions, integrable distributions |
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Wednesday November 10 |
Chapter 6 |
Frobenius Theorem and maximal integral submanifolds |
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Friday November 12 Final day to withdraw from a course |
Chapter 7 |
Differential forms, |
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Monday November 15 |
Chapter 7 |
Differential of a form,
closed and exact forms |
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Tuesday November 16 x-hour instead of the class on Friday November 19 |
Chapter 7 |
Cohomology groups |
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Wednesday November
17 |
Chapter 8 |
Bordism groups |
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Friday November 19 No class, instead we have an x-hour on Tuesday November 16 |
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Monday November 22 |
Chapter 8 |
Integration Stokes Theorem |
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Thanksgiving recess:
starts at 5:50 PM on Tuesday November 23
and ends at 7:45 AM on Monday November 29 |
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Monday November 29 |
Chapter 8 |
Itegration Stokes Theorem |
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Wednesday December 1 |
Chapter 8 |
Piring between bordism groups and cohomology |
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Thursday December 2-Friday December 3 Pre-Examination Break |
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End of the term presentation and
discussion should be done in the period Thursday December 2 – Tuesday
December 6 Final Examinations begin on Saturday December 4 and end
on Wednesday December 8 |
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