## MAT 531: Topology & Geometry, II

### Announcements

Final Exam: Info, Course Overview, S06 exam/solutions, S10 exam/solutions, S11 exam/solutions
Midterm: Info, S06 exam/solutions, S10 exam/solutions, S11 exam/solutions.

General information about the course (TA's OHs updated 02/04)

Notes for Lectures 1-7 (02/18 update: Sections 10,12 slightly expanded). This is likely the end of the notes.
Bonus: 1 HW pt for each typo you find, 3pts for minor error, 5pts for significant error, 10pts for very significant error (only 1 bonus per typo/error, to the first person who let's me know; the points will be added to your problem set scores)

### Course Instructor

Name: Aleksey Zinger     E-mail: azinger@math     Phone: 432-8288
Office: Math Tower 3-111     Office Hours: W 9-10:30am in P-143, 10:30am-12noon in 3-111

### Tentative Schedule and Homework Assignments

Please read the assigned sections from the book before the corresponding lecture.

 Date Topic Read Problem Set 2/1 Smooth manifolds 1.0-1.6; Notes 1,2 ps1; solutions 2/3 Tangent vectors and differentials 1.12-1.24; Notes 3,4 2/8 Submanifolds and Inverse Function Theorem 1.27-1.36; Notes 5 ps2; solutions 2/10 Implicit Function Theorems 1.37-1.40; Notes 6 2/15 Vector bundles 1.25,1.44,1.45; Notes 7-10 ps3; solutions 2/17 1.54-1.59,2.1-2.13,4.1; Notes 10-12 2/22 Vector fields 1.41-1.43,1.46-1.50,1.53 ps4; solutions 2/24 Frobenius Theorem 1.51,1.52,1.54-1.64 3/1 The differential 2.14-2.23 ps5 solutions 3/3 Frobenius Theorem (2nd version); Lie derivative 2.26-2.32,2.24,2.25 3/8 de Rham cohomology of Rn 4.13-4.15,4.18,4.194.4-4.6 ps6 solutions 3/10 Integration on singular chains 4.6,4.7,4.16,4.17 3/15 Integration on oriented manifolds 4.1-4.3,4.8-4.10; Notes 1-12 3/17 Review 3/22 no class b/c of midterm on Wednesday 3/23 Midterm, 5-6:30pm, in Harriman 115 3/24 (Co-)Chain complexes 5.16,5.17 ps7 solutions 3/29 Sheafs and presheafs 5.1-5.3,5.5-5.8 3/31 Sheafs and presheafs, cont'd 5.4,5.11 4/5 Cech Cohomology 5.33 ps8 solutions 4/7 de Rham Theorem (weak version) 5.10-5.12, 5.28-5.30 GH p43-45top 4/12 Free resolutions and cohomology 5.18-5.25,5.275.31,5.32,5.34-5.38 ps9 solutions 4/14 de Rham Isomorphism Theorem 4/19,21 No Class 4/26 Hodge Decomposition Theorem 6.1-6.3,6.7-6.14 ps10 solutions 4/28 Elliptic operators 6.4-6.6,6.28,6.34-6.36 5/3 Applications of elliptic regularity 6.8,6.31 5/5 Proof of elliptic regularity 6.29,6.32,6.33 ps11 solutions 5/10 Cohomology of non-compact manifolds Spivak, pp363-371 5/12 Review 5/22 Final Exam, 1-3:30pm, Math P-131