| Week 1 (Sep 30-Oct 04) |
|---|
| Monday |
- Example: projective space.
- Definition of pseudo-group of transformations.
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| Tuesday |
- Definition of a manifold.
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| Wednesday |
- Introduction to tensor products.
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| Week 2 (Oct 07-11) |
|---|
| Monday |
- Examples.
- Open submanifolds, product manifolds, submanifolds.
|
| Tuesday |
- More on submanifolds, manifolds defined by equations in another manifold.
|
| Wednesday |
- Universal properties, tensor algebra.
|
Friday (support class) |
- Definition of a manifold, smooth maps and diffeomorphisms.
- Problems 1 and 4.
|
| Week 3 (Oct 14-18) |
|---|
| Monday |
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| Tuesday |
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| Wednesday |
- Symmetric algebra.
- Introduction to exterior algebra.
- Interpretation of exterior algebra via multilinear, alternating maps.
|
Friday (support class) |
- Tensor products, local charts of manifolds.
|
| Week 4 (Oct 21-25) |
|---|
| Monday |
- Tangent spaces, vector fields, bracket (beginning).
|
| Tuesday |
- Differential, beginning of definition of immersion, submersion, embedding.
|
| Wednesday |
- Immersions, submersions, embeddings.
|
Friday (support class) |
- Review of tangent vectors and vector fields, commuting vector fields and P2-2.
|
| Week 5 (Oct 28-Nov 01) |
|---|
| Monday |
- (Local) 1-parameter subgroups of (local) diffeomorphisms.
- Flows.
|
| Tuesday |
- Local flows and vector fields.
- Complete vector fields.
|
| Wednesday |
- Cotangent space (definition).
- Smooth 1-forms, total differential, tangent bundle.
|
Friday (support class) |
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| Week 6 (Nov 04-08) |
|---|
| Monday |
- Partitions of unity, part 1.
|
| Tuesday |
- Partitions of unity, part 2.
- Digression on analytic functions vs smooth functions.
|
| Wednesday |
|
Friday (support class) |
- Tangent vectors to surfaces in $\mathbb{R}^3$.
- Riemannian metrics and pull-backs.
|
| Week 7 (Nov 11-15) |
|---|
| Monday |
- Vector bundles.
- Rank 1 bundles on $S^1$.
|
| Tuesday |
- Differential $r$-forms.
- Beginning of pull-backs of differential forms.
|
| Wednesday |
- Pull-backs of differential forms.
- Orientability.
|
Friday (support class) |
- "Baby Pre-image theorem".
- Tensor products and exterior squares.
|
| Week 8 (Nov 18-22) |
|---|
| Monday |
- Integrals and Orientability.
|
| Tuesday |
- Orientability of $S^n$, non-orientability of $\mathbb{P}^{2n}_{\mathbb{R}}$.
- Exterior differentiation.
|
| Wednesday |
- Exterior differentiation commutes with pull-back.
|
|
|
Friday (support class) |
- Differential 2-forms and higher order forms.
- Exterior derivative and $d \circ d = 0$.
- PS3 P3.
|
| Week 9 (Nov 25-29) |
|---|
| Monday |
- Stokes' Theorem for manifolds without boundary.
- Manifolds with boundary.
- An orientation on $M$ induces an orientation on $\partial M$.
|
| Wednesday |
- Orientation on $\partial [0,1]$.
- Stokes' Theorem.
|
Friday (support class) |
- Pull-back and differentials.
- PS3 P3.
|
| Week 10 (Dec 02-06) |
|---|
| Monday |
- Brouwer’s fixed point theorem.
- Q&A: germs vs functions, quotient manifolds, remarks about invariance of domain.
|
| Wednesday |
- Q&A: partitions of unity, immersions, differentials,...
|
Friday (support class) |
- Orientability and examples.
- Classical notation for integrals.
- Interior product.
|