Instructor: Asher Auel

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Syllabus

Week Date Topics Reading Work
1 Tue 17 Sep Finite dimensional vector space review. Bases. Change of basis. Subspaces. Sums and direct sums. FIS 1.1-1.6, DF 11.1, R Ch. 1, L III.5, XIII.1-XIII.2
Thu 19 Sep Linear maps and matrices. Quotient spaces. Rank-nullity. FIS 2.1-2.5, DF 11.2, R Ch. 2, L XIII.3
2 Tue 24 Sep Exact sequences. Factoring through. Resolutions. Complexes. (Co)homology. DF 17.1 (pp. 777-778) Home Work 1
Thu 26 Sep Dual space. Transpose. Annihilators. FIS 2.6, 6.8, DF 11.3
3 Tue 01 Oct Bilinear forms. Orthogonal and symplectic groups. FIS 6.1-6.6, 6.8 Home Work 2
Thu 03 Oct Orthogonality. Diagonalization. Inner product spaces. FIS 6.1-6.6, 6.8
4 Tue 08 Oct Spectral theorem. Tensor products. DF 11.5, 10.4, R, Ch. 14 pp. 355–378 Home Work 3
Thu 10 Oct More tensor products.
5 Tue 15 Oct Symmetric and exterior powers. Multilinear forms. Determinants. R, Ch. 14 pp. 379-406 Home Work 4
Thu 17 Oct Algebras. Graded algebra and homogeneous ideals. Tensor, symmetric, exterior algebras. Categories. R, Ch. 14 pp. 385-406, Ch. 18 pp. 451-469, DF Appendix II, L I.11
6 Tue 22 Oct Functors. Natural transformations. DF Appendix II, L I.11 Home Work 5
Thu 24 Oct Universal properties. Adjoints. Modules. DF Appendix II, L I.11, DF 10.1-10.4
7 Tue 29 Oct Free modules. Quotient modules. Isomorphism theorems. FIS 7.1-7.4, DF 12.1-12.3 Home Work 6
Thu 31 Oct Projective modules. Modules over PID. DF 10.5, 17.1
8 Tue 05 Nov Modules over PID. Smith normal form. Canonical form. DF 10.5, 17.1
Thu 07 Nov Tensor products. Flat modules. Projective resolutions. DF 10.4-5, 17.1
9 Tue 12 Nov Chain homotopy category. DF 10.4-10.5, 17.1, L III.1-4
Thu 14 Nov Derived functors. Tor and Ext. DF 17.1, L III.1-4
10 Tue 19 Nov Representation theory. DF 18.1-18.3, 19.1