# 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