HOMEWORK
MTH 675 — Winter 2023

Ground Rules

Suggested Reading

It is to your advantage to skim suggested readings as soon as possible. However, do not expect to master this material the first time around. Don't worry; we'll cover it in class, after which the readings should make more sense. But be warned: we will not always cover the material the same way it is presented in the books.

Written Work

It is your job to explain your work to me clearly and completely. Here are some guidelines:

• Write legibly, or use a word processor;
• Do not use scratch paper;
• Explain at least briefly what you're trying to do, don't just do it;
• Use complete sentences;
• Equations should (mostly) be contained in sentences;
• Feel free to use technology when appropriate, but this must be clearly documented. If possible, attach a printout of your session, together with a brief explanation of what you did.

You may discuss homework problems with anyone you like, and you may use any reference materials you like. However, you must write up the solutions in your own words, and you must indicate what help you used. Late homework will be corrected as a courtesy to you, but can earn at most half credit.

Assignments

Assigned problems and practice exercises will be linked directly to the schedule page.

Assignments (labeled "HW") are due at the beginning of class on the date on which they appear in the schedule.

Hard copy is preferred, but assignments may also be submitted via email.

Please either use software such as $\LaTeX$, or a scanner or scanning app, not a camera app, and please submit PDF.
(OneNote users should be aware that it exports large PDF files.
Please shrink before mailing, which probably requires converting vector graphics to bitmaps.)

Practice exercises should not be turned in, but may be discussed in class.

Suggested reading for each topic is listed on the schedule page.

Additional comments regarding suggested readings (and, in some cases, additional practice problems) appear below.

Comments and Suggestions
(HP is Pollatsek's Lie group book; GO is our octonions book; GELG is my online Lie group book)

Week 10

Suggested reading:

Skim as much of §11 of GO as you can.

Comments:

No homework this week...

Week 9

Suggested reading:

Read the new section §6.8 of GELG on Clifford algebras over division algebras, and skim §6.9.

Then read the new §7 of GELG on magic squares. (This chapter is still being written!)

Skim the rest of §9 in GO.

Comments:

No homework this week...

Week 8

Suggested reading:

Read the new sections §6.6–6.7 of GELG, on classifying Clifford algebras.

Skim §9.1–9.2 and §15.1–15.2 of GO on using division algebras to represent orthogonal matrices.

Week 7

Suggested reading:

Read Chapter 6 §6.1–6.5 of GELG, on Clifford algebras.

Comments:

This chapter is new, and may need improvement. Comments welcome!

Week 6

Suggested reading:

Read the last two sections of Chapter 5 of GELG.

Comments:

No homework this week...

Week 5

Suggested reading:

Read the first two sections of Chapter 5 of GELG, and skim the rest of the chapter.

Comments:

The suggested exercises for Friday is needed for the homework. We will go over it during class on Friday in some detail.
If you are not present in class on Friday, please include your solutions to both exercises along with the assigned homework.

Week 4

Suggested reading:

Read the last 3 sections of Chapter 3 of GELG, and skim Chapter 5 of GELG.

Comments:

The second assigned homework problem contains three Challenges.

• These Challenges are optional for students in MTH 405.
• Students in MTH 675 should attempt at least one of the challenges
• In either case, explain your reasoning carefully regardless of whether you are able to answer the question.

Week 3

Suggested reading:

Read Section 6.2 of HP.

Read Chapter 4 of GELG.
The first section need only be skimmed.

Comments:

Clarification: The "spin-2" representation of $\su(2)$ is the one whose maximum eigenvalue is $2$.

Week 2

Suggested reading:

Read Chapter 3 of GO, and skim Chapters 6 and 7.

Read Chapters 1 and 3 of GELG.
(Stop at the section on the Killing form, which you should skim.)

Students in MTH 67x may wish to pay special attention to §3.2 and the embedded examples A.1.2 and A.1.4.

Week 1

Suggested reading:

Skim Chapters 1 and 3 of HP.
Stop and read any bits that seem interesting, unfamiliar, or both, but don't try to master all the details.

Additional practice problems:

HP: §3.1: 1b, 2b, 3ac (pp. 46–47)
(You are strongly discouraged from using t as a parameter...)

Comments:

The first assigned homework problem is a combination of problems HP: §1.4: 16–17.