1/28 The syllabus. Introduction to the course. Intuition behind manifolds: some manifolds, some manifolds-with-boundary, and some objects that are not manifolds. Open sets and smooth functions. Reference: Textbook 1.1, supplementary notes on topology of Euclidean space on the course webpage
1/30 Smooth functions on arbitrary subsets. Diffeomorphisms. The definition of a manifold. Parametrizations. Products of manifolds are manifolds. Reference: Textbook 1.1
2/4 The tangent space. Derivatives of maps. The chain rule. Reference: Textbook 1.2
2/6 More about the chain rule. Invariance of dimension under diffeomorphism. The inverse function theorem. Writing a function in co-ordinates. Immersions and submersions. Reference: Textbook, 1.3
2/11 Proof of the local immersion theorem. Immersions at a point are immersions in a neighborhood of that point. When is the image of an immersion a submanifold? Some non-examples. Reference: Textbook, 1.3
2/13 Closed and compact sets. Proper maps. Proof that the image of a proper injective immersion is a manifold. An injective immersion whose image is not a manifold; more examples and non-examples. Reference: Textbook, 1.3
2/18 Regular values. The preimage of a regular value is a manifold. Textbook, 1.4
2/20 Transversality. Examples. Reference: Textbook, 1.5
2/25 Ideas behind the proofs of transversality. Homotopy and stability. Statement of the stability theorem. Reference: Textbook, 1.6
2/27 Proof of the stability theorem. Dense sets and measure zero sets. Statement of Sard's theorem. Reference: Textbook, 1.7
3/4 Measure zero subsets of manifolds. Examples and non-examples. Applications of Sard's Theorem: non-existence of surjective maps from a lower dimensional manifold to a higher dimensional one; embedding k-dimensional manifolds into R^{2k+1} (statement). Reference: Textbook, 1.7, 1.8
END OF MATERIAL FOR MIDTERM 1
3/6 Proof that any compact k-dim, manifold embeds into R^{2k+1}. A few words about the proof of Sard's theorem. Reference: Textbook, 1.8
3/11 Midterm 1
3/13 Stokes' Theorem as motivation for some of the next topics. Manifolds with boundary; tangent spaces. Reference: Textbook, 4.1, 2.1
Spring Break
3/25 Orienting a vector space; orientable and non-orientable manifolds. Reference: Textbook, 3.2
3/27 Tensors on a vector space; alternating tensors; the tensor product and the wedge product; the exterior algebra; bases for the space of alternating p-tensors. Reference: Textbook, 4.2
4/1 More about the exterior algebra; proofs of theorems. Reference: Textbook, 4.2
4/3 The definition of a differential form. Smooth differential forms. The dx_i. What are all the differential forms on open sets of R^k? Reference: Textbook, 4.3
4/8 The definition of the integral of a differential form; partitions of unity. Reference: Textbook, 4.4, see 1.8 for partition of unity stuff
4/10 Wrapping up loose ends and examples: integrals of one-forms over curves, integrals of two-forms over surfaces, change of parametrization and change of variables. The exterior derivative on 0-forms. Reference: Textbook 4.4. Here is a note I wrote expanding on 4.4. Here is a nice review of change of variables.