Math 545-546 2022, Assignments

When you are officially enrolled in the course, go to the course Canvas site to see and submit homework assignments. For convenience before Canvas opens the course site, or if your enrollment is delayed, the first few assignments will be posted here.

First Reading Report

This first Reading Report has special prompts (at the end below) and special submission rules. First, if you have questions or requests for class discussion about Chapter 1, email them (to arms at math or jmarms at uw) as soon as possible, because we will be done with this chapter by the time the report is due. Second, you may submit your report, like the later reports, as an email to (arms at math or jmarms at uw); or you may make an appointment to give me (JMA) your report on Zoom, so I have a chance to get to know you. Either way, please do so by Sunday, January 9, at 6 PM.

Read the following in ISM.

• In the Preface, read at least the paragraph on notation on p. vii: "I should say ... efficiency later."
• Look over Appendices B and C, and if any section doesn't seem immediately and completely obvious, read more carefully and do the exercises. Exceptions: You may skip Proposition B.57 and its proof, and may skip the proofs of the Inverse and Implicit Function Theorems. You may postpone the section on multiple integrals until we get to Chapter 16.
• In Chapter 1, there is little or nothing new (after Math 544) in pp. 3-10, so you may skim this quickly. Starting at the bottom of p. 10, read as usual: Your first read may skip details, but be sure to work through all proofs and exercises sometime soon after that. Exception: The proof that Grassman Manifolds (Example 1.36) are in fact smooth manifolds is complicated (though elementary) and not important later, so you may skip the details if you wish. We will be able to prove this much more easily later (much later!) in the book. It is included here because Jack likes to have a significant application in every chapter, and to show a rather nontrivial example of a smooth atlas can be proved with elementary techniques. If you choose to study this proof and have questions, feel free to ask about it in office hours.

You don't need to read Appendix A because you just took Math 544. You may use either this appendix or ITM when you need to cite topological results. There is one new topic, Lipschitz continuity (starting after Exercise A.46, through Exercise A.49) which you may consult when and if needed.

For this Reading Report only, respond to the following prompts.

• Why are you taking Manifolds? How do you hope to use the material you learn in this sequence?
• Was there anything in Appendices B and C that wasn't review for you, or about which you have questions? If yes, give details.
• (Optional) Tell me a little bit about your mathematical background.

Week 2 Problem Set

W Problems problems you should think through (but not write up formally and not hand in).

• All exercises in the reading assigned for first report.
• 1-7(ab). Assigned W so you don't have to write them up, but can use them in the HI problem 1-7(cd).

HI Problems (to be written up carefully to submit online in Gradescope by Friday, January 14, by 6 PM.

• B.22defg (in Appendix B). These are assigned partly so you are more likely to remember where to look them up when you need them later in the course.
• 1-7(cd).
• 1-8.
• Problem 0.1.

Homework Information and Guidelines.

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