Advanced Math Program, Algebra

This is the webpage for the algebra portion of the Advanced Mathematics Program (AMP), being organized by Jonathan Alcaraz and by Mike Pierce. We’ll meet each weekday July 15–25, from 10am to 1pm in Skye 268. If you have any questions or inquiries, email Jonathan or me at

             

A couple things of that are important to note: (1) You are not being graded for your performance in this program, and you are not being judged on your ability. We’re just here to bolster your mathematical skills and prepare you for your upper division courses. (2) The list of exercises we ask you to think about each day will be long; we have no expectation that you respond to all of them. Just think about whichever ones you get to during the day, and then you have an excess of exercises in case you’d like to keep practicing algebra on your own later. (3) One of the focuses of the AMP is to get you more experience writing mathematical proofs. While you might have discovered some good proof-writing practices if you’ve taken linear algebra or set theory or discrete math, it can also be helpful to just read about how to write proof explicitly. Here are a few documents that explain just that.

• Proofs and Mathematical Reasoning by Agata Stefanowicz – Very thorough. The math should be familiar, so you can skim this one for the bits about writing proofs.
• Mathematical Proofs: Where to Begin and How to Write Them by Phillip Milner – A nice and concise document, albeit poorly typeset.
• Practical Suggestions for Mathematical Writing by Bjorn Poonen – This is about writing mathematics in general and not just writing proofs, but this still has some good pointers.
• Advice on Mathematical Writing by Keith Conrad – Also good, but more about general mathematical writing and typesetting.

Program Outline, Exercises, & Lecture Notes

Here are the topics we’re talking about each day in the program, our own notes for those lectures that might be helpful, and the exercises and proofs that you can think about each day after the lecture. Ideally we’ll get the notes posted before the day of each lecture, which will afford you the opportunity to think about the material rather than furiously take notes if you’d like. All the exercises were written collaboratively and the source can be found here on Overleaf.

• Day One – Set Theory & Functions (Exercises)
• Day Two – Binary Operations & Groups (Exercises)
• Day Three – Abelian Groups & Subgroups (Exercises)
• Day Four – Playing with Groups (Exercises)
• Day Five – Permutation (Symmetric) Groups (Exercises)
• Day Six – Isomorphisms & Homomorphisms (Exercises)
• Day Seven – Cosets & Normal Subgroups (Exercises)
• Day Eight – Quotient Groups and the Isomorphism Theorem (Exercises)