Lecture summaries

Here are brief summaries of the lectures.

1/26:  Introduction.  The syllabus.   The idea of differential forms and manifolds.  Some highlights we will prove; generalized Stokes' Theorem.  Smooth maps and manifolds.  Reference: Textbook, 1.1.

1/31: Products of manifolds are manifolds.  Examples of manifolds.  Submanifolds.  Derivatives, part 1: the tangent space.  Reference: Textbook, 1.2

2/2: Derivatives.  More examples.  The inverse function theorem.  Functions between manifolds in co-ordinates.  Choosing co-ordinates for which a local diffeomorphism is the identity.  Immersions.  Reference: Textbook,  1.2-1.3.

2/7: The canonical immersion.  Choosing good local co-ordinates for an immersion.  Examples.  When is the image of an immersion a submanifold?  Proper maps.  Reference: Textbook, 1.3.

2/9: The image of a proper one-to-one immersion is a submanifold (proof).  Submersions.  The canonical submersion.  Choosing good local co-ordinates for a submersion.  The regular value theorem.  Examples: spheres, the orthogonal group.  Reference: Textbook, 1.3-1.4.

2/14:  Proof that the orthogonal group is a manifold.  Proof of the regular value theorem.  What is the tangent space to the pre-image of a regular value?  Transversal maps.  Reference: Textbook, 1.4-1.5.

2/16: Examples and non-examples of transverse intersections.  The dimension of a transverse intersection.  Proof that the pre-image of a submanifold under a transversal map is a submanifold.  Stability.  Reference: Textbook,  1.5-1.6.

2/21: Submanifolds are locally cut out by independent functions.  Homotopies and stable properties.  Examples of stable properties: "The Stability Theorem".  Immersions from a compact manfifold are stable (sketch of proof).  Reference: Textbook, 1.4, 1.6.

2/28: Measure zero subsets of manifolds.  Sard's Theorem (statement and some remarks on the proof, but not the proof).  Examples of regular values and critical values.  Reference: Textbook, 1.7

End of material for Midterm 1

3/2: The tangent bundle.  Whitney's embedding theorem.  Proof that compact manifolds embed into R^{2k+1}.  Reference: Textbook, 1.8

3/7: Midterm 1

3/9: Manifolds with boundary.  Reference: Textbook, 2.1

3/14: Transvesality and Sard's Theorem for manifolds with boundary.  Classification of one-manifolds.  The fixed point theorem.   Some remarks on classifications in higher dimensions (not tested).  Reference: Textbook, 2.1-2.2

3/16: Integrals and signed volumes.  Tensors and alternating tensors.  The tensor product and the wedge product.  Examples.  Reference: Textbook, 4.2

Spring Break