## MAT 531: Topology & Geometry, II

### Announcements

Final Exam: Info, Course Overview, Spring 06 exam and solutions, Spring 10 exam and solutions
Midterm: Info, Spring 06 exam and solutions, Spring 10 exam and solutions

General information about the course (TA's OHs updated 01/26)

Notes on Vector Bundles

Notes on Lectures 1-5
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: Tu 2:20-3:50pm, W 9-10:30am in P-143

### Tentative Schedule and Homework Assignments

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

 Date Topic Read Problem Set 1/26 Smooth manifolds 1.0-1.6 ps1 1/28 Tangent vectors and differentials 1.12-1.24 2/2 Vector bundles 1.25; Notes 1-3 ps2 2/4 Submanifolds and Inverse Function Theorem 1.27-1.36; Notes 4.1-4.4 2/9 Implicit Function Theorems 1.37-1.40 ps3 2/11 Vector fields 1.41-1.43,1.46-1.50,1.53 2/16 Frobenius Theorem 1.44,1.45,1.51,1.52,1.54-1.61 ps4 2/18 Integral submanifolds; some algebra 1.62-1.64,2.1-2.13 2/23 The differential 2.14-2.23 ps5 2/25 Frobenius Theorem (2nd version); Lie derivative 2.26-2.32,2.24,2.25 3/2 de Rham cohomology of Rn 4.13-4.15,4.18,4.194.4-4.6 ps6 3/4 Integration on singular chains 4.6,4.7,4.16,4.17 3/9 Integration on oriented manifolds 4.1-4.3,4.8-4.10; Notes 4-6 3/11 Review 3/16 no class b/c of midterm W 3/17 Midterm: 5-6:20pm 3/18 (Co-)Chain complexes 5.16,5.17 ps7 3/23 Sheafs and presheafs 5.1-5.3,5.5-5.8 3/25 Sheafs and presheafs, cont'd 5.4,5.11 3/30,4/1 No Class 4/6 Cech Cohomology 5.33 ps8 4/8 De Rham Theorem (weak version) 5.10-5.12, 5.28-5.30 GH p43-45top 4/13 Free resolutions and cohomology 5.18-5.25,5.275.31,5.32,5.34-5.38 ps9 4/15 De Rham Isomorphism Theorem 4/20 Hodge Decomposition Theorem 6.1-6.3,6.7-6.14 ps10 4/22 Elliptic operators 6.4-6.6,6.28,6.34-6.36 4/27 Applications of elliptic regularity 6.8,6.31 4/29 Proof of elliptic regularity 6.29,6.32,6.33 ps11 5/4 Cohomology of non-compact manifolds Spivak, pp363-371 5/6 Review 5/16 Final Exam, 12:30-3pm

This page is maintained by Aleksey Zinger.