# 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: Transversality 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

3/28: Differential forms.  What does it mean for a differential form to be smooth?  Pull-back and wedge product.  Examples.  Reference: Textbook, 4.3

3/30: Proofs of statements from the previous two lectures; writing alternating tensors in terms of 1-tensors and wedge-product.  Reference: Textbook, 4.2-4.3.

4/4: How to compute pull-backs.  The pull-back of a smooth differential form is smooth.  The support of a differential form.  The integral of a differential form supported in a parametrizable open set.  The change of variables formula.  Reference: Textbook, 4.3-4.4.

4/6: Orientations.  More about the integral of a differential form supported in a parametrizable open set.  Integrals over curves in R^3 and line integrals.  Reference: Textbook, 3.2, 4.4

4/11: Integrals of two-forms and surface integrals.

End of material for midterm 2.

More 4/11 lecture: Partitions of unity; the circle example.  Definition of the integral of a compactly supported differential form, via partitions of unity.  Proof that the integral if well-defined.  Reference: Textbook, 1.8, 4.4

Midterm 2

4/18: Discussion of the midterm.  The exterior derivative and its properties.  The statement of Stokes' Theorem.  Reference: Textbook, 4.5, 4.7.

4/20: An orientation of a manifold determines an orientation of its boundary.  The proof of Stokes' Theorem.  Stokes' Theorem and the Fundamental Theorems of Vector Calculus.  Reference: Textbook, 3.2, 4.7

4/25:  The signs in Stokes' Theorem.  The degree theorem.  Homotopy invariance of the degree.  Reference: Textbook, 4.7, 4.8

4/27:  Proof of the fundamental theorem of algebra.  The degree and the number of pre-images of a generic point.  Proof of the degree theorem, part 1.  Reference: Textbook, 2.4 (just the part about the fundamental theorem of algebra), 3.3 (just the part about the definition of the degree, right at the top of p. 109), 4.8

5/2:  Proof of the degree formula.  Reference:  Textbook, 4.8

5/4:  Vector fields, flows, and diffeomorphisms taking one point to another.  The beachball theorem.  Reference: Section 2 of https://ncatlab.org/nlab/show/flow+of+a+vector+field, the first page of http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handouts/hairyball.pdf  and then use the degree theorem, the "homogeneity lermma" in https://math.uchicago.edu/~may/REU2017/MilnorDiff.pdf

5/9: Ways to show that two manifolds are not diffeomorphic.  Closed and exact forms and their significance.  The Poincare Lemma.  The two-sphere is not diffeomorphic to the two-torus; the punctured plane is not diffeomorphic to the plane.   Reference:  Textbook, 4.6 (though we did not use the language of cohomology, we just looked at considerations of closed and exact forms)

End of material for the final exam