| Week 1 (Sep 30-Oct 04) |
|---|
| Monday |
- Example: projective space.
- Definition of pseudo-group of transformations.
- Definition of a manifold.
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| Wednesday |
- Symmetric algebra.
- Tensor algebra.
- Introduction to tensor products.
- Introduction to exterior algebra.
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| Week 2 (Oct 07-11) |
|---|
| Monday |
- Examples.
- Product manifolds, submanifolds, manifolds defined by equations in another manifold.
- Quotient manifolds.
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| Wednesday |
- More on the tensor and exterior algebra.
|
TBA (support class) |
- Definition of a manifold, smooth maps and diffeomorphisms.
- Problems 1 and 4.
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| Week 3 (Oct 14-18) |
|---|
| Monday |
- Tangent vectors, tangent spaces, vector fields.
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| Wednesday |
- Interpretation of exterior algebra via multilinear, alternating maps.
|
TBA (support class) |
- Sub-manifolds, Inverse function theorem, PS1 P2-3.
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| Week 4 (Oct 21-25) |
|---|
| Monday |
- Vector fields, bracket.
- Differential, beginning of definition of immersion, submersion, embedding.
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| Wednesday |
|
TBA (support class) |
- Review of tangent vectors and vector fields.
- Example.
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| Week 5 (Oct 28-Nov 01) |
|---|
| Monday |
- Immersions, submersions, embeddings.
- (Local) 1-parameter subgroups of (local) diffeomorphisms.
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| Wednesday |
- Flows.
- Complete vector fields.
|
TBA (support class) |
- Problem Sheet 2, Problems 1 and 3.
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| Week 6 (Nov 04-08) |
|---|
| Monday |
- Ehresmann's Theorem.
- Complements and Q&A during breather week.
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| Wednesday |
- Go over definition of tensor, exterior and symmetric algebra.
- Informal motivation for $k$-forms.
- Informal motivation for vector bundles.
- Go over some parts of an exam question.
- Complements and Q&A during breather week.
|
TBA (support class) |
- Partitions of unity, PS2 P4.
- Tensor products and multilinear maps.
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| Week 7 (Nov 11-15) |
|---|
| Monday |
- Cotangent space (definition).
- Smooth 1-forms, total differential, cotangent bundle.
- Digression on tangent bundle to $S^2$.
- Vector bundles.
|
| Wednesday |
- Vector bundles.
- Cocycles.
- Differential $r$-forms.
|
TBA (support class) |
- "Baby Pre-image theorem".
- Tensor products and exterior squares.
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| Week 8 (Nov 18-22) |
|---|
| Monday |
- Pull-backs of differential forms.
- Integrals and Orientability.
|
| Wednesday |
- Exterior differentiation.
|
TBA (support class) |
- Differential 2-forms and higher order forms.
- Exterior derivative and $d \circ d = 0$.
- PS3 P3.
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| Week 9 (Nov 25-29) |
|---|
| Monday |
- Stokes' Theorem for manifolds without boundary.
- Manifolds with boundary.
- An orientation on $M$ induces an orientation on $\partial M$.
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| Wednesday |
- Orientation on $\partial [0,1]$.
- Stokes' Theorem.
|
TBA (support class) |
- Pull-back and differentials.
- PS3 P3.
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| Week 10 (Dec 02-06) |
|---|
| Monday |
- Brouwer’s fixed point theorem.
- Q&A: germs vs functions, quotient manifolds, remarks about invariance of domain.
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| Wednesday |
- Q&A: partitions of unity, immersions, differentials,...
|
TBA (support class) |
- Orientability and examples.
- Classical notation for integrals.
- Interior product.
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