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