# What is a good way to broach the subject of abstract algebra, to a student in Calc 1, or pre-calc?

Background

I want to introduce my students to some big names in mathematics, and one of the names I want to bring up is Richard Borcherds, known for his contributions to the fields of number theory, group theory, and abstract algebra.

EDIT: I should have mentioned. This is a group of students who have expressed an interest in pursuing math as their major or minor in higher education.

As a part of this introduction, I want to introduce the students to these three fields as well. Number theory and group theory I have covered. But I'm looking for some nice ways to introduce abstract algebra to someone who hasn't gotten that far in their education yet.

Question

Are there any ways I can relate abstract algebra to the pre-calc algebra they already know?

What are some good examples of real-life problems that can be solved with abstract algebra?

Even with the phasing out of visualizing things, in favor of axioms and abstractions, are there any examples of things that can be visualized?

• Why Borcherds, and not Galois, Cayley, Sylvester, etc.? If it's because Borcherds's work impacts current exotic physics theories and such, I suspect this will have as much relevance and meaning to the students as telling them that rpyc-377 is relevant to ei9-t-49. If you feel you have to do this, I recommend taking your cues from Sawyer's book and Stein's book and Stewart's book and Kramer's book and the many other similar such books. Feb 21, 2022 at 13:18
• What do you mean by abstract algebra? You seem to exclude group theory from it, which seems weird to me, so possibly you have something quite specific in mind? Feb 21, 2022 at 13:25
• not an either/or situation --- I realize that (not just now, but when I wrote my earlier comment), but ordinarily I would expect several dozen (maybe over 100) names to consider before getting to someone like Borchers. Now that you've added to the context that most of these students will actually be math majors/minors later in college, maybe Borcherds as an example of a currently active mathematician (and fields medal winner) makes a little more sense. Feb 21, 2022 at 16:06
• Just a comment, since my remarks relate only to group theory, which you've covered already. Borcherds' most famous work relates to the moonshine conjectures, which have to do with group theory and modular forms. This work also makes use of (infinite dimensional) Lie algebras, as well as more esoteric objects. One thing you might be able to talk about with your students is the modular group, which is an infinite group consisting of certain integer 2x2 matrices, and how it relates to rational tangles. See this post for some student activities. Feb 22, 2022 at 4:00
• Two more quick remarks: Lie algebras intersect with the general area of matrix algebras. One potentially accessible topic is the different ways of representing 3-D rotations by matrices, including the representation as 2x2 complex matrices using the Cayley-Klein parameters. This representation is widely used in computer graphics. Borcherds' work also involves the study of higher dimensional lattices, which are closely related to error correcting codes. For an accessible example, one might discuss the Hamming codes. This also provides a connection with finite fields, another algebra topic. Feb 22, 2022 at 15:24

I think it's nice to be able to give some knowledge of further courses in math, of a very "who's who in the zoo" sort. By which I mean descriptive, not axiomatic, and not even a detailed or working description. Just some extremely light and vague concept to attach to a word. And then let it go.

Not sure why you are pushing this guy, Richard Borcherds. I've never heard of him. I trust you that he's a big wheel in some way and interesting to people further advanced in their studies. But I think it's "A Bridge Too Far" for high school. Be a little wary of sharing your passion, what interests YOU, with what is good for your students. I think you're better off showing them the Horizon video on Andrew Wiles.

https://www.dailymotion.com/video/x223gx8 (probably copyright violating, but you can either buy a physical copy of the video or license or...just decide to steal it.)

In terms of awareness of math topics fields, I actually think what's closer to interest to them is something that talks about the terms and how they connect to school courses, further down the line. Not so much, where's active research going on. (Yeah, I get that you probably are interested in coal face of research...but remember what I said about pushing your interests versus thinking about audience.) We get questions here all the time about "what comes after Calculus BC" (US). Showing a lack of understanding even of 3rd semester calculus and ODEs, that most engineers go on to take a year later.

I think for AA, just knowing that it's like high school algebra but more...uh...abstract is kind of enough. Tell them that it's a core course that math majors take in undergrad, but engineers and scientists almost always don't...don't need it. And that it's related to how they proved that 5th level and higher polynomials don't have a "quadratic formula"-style solution. [This connects to high school algebra and teases that the topic actually did something.] I include GT within AA. You really don't need to get complicated enough to differentiate.

And then, let it go. Teach them the pre-calc they need to know. Honest, even just knowing the (names of, not really even the content) basic future math courses in undergrad for science/engineering (who are much more numerous than math majors) and math majors, is an above and beyond enrichment. Then get back to teaching the actual topic you're supposed to.

• It is a bridge too far for high school. The students I'm tutoring are students that have expressed an interest in majoring or minoring in math further down the line, and during a conversation with them, I learned that one of the big questions they have is "what does higher level math look like?", so this is part of a small project I run with them to create something of a map - or as you put it, a "who's who in the zoo" of math topics. Borcherds is one of the names I want to throw in there because for his contributions, he got a Fields medal, and recently started teaching on YouTube.
– Alec
Feb 21, 2022 at 14:15
• Thanks for the clarification. I still "Bayesian bet" you're overdoing it. For example, how much of your time should be dedicated to this sort of enrichment versus coaching on their actual subjects (what do the paying parents expect)? Or for that matter, what percent of the students will really go on to math Ph.D.'s? (Not just told you, "maybe I will major in math".) But you're gonna do what you're gonna do. See this all the time. Wiles and Tao are enough...and keep it super fluffy and pop science. And teach them some damned pre-calc! Feb 21, 2022 at 14:42
• Thanks for your overwhelming concern, but their path through the curriculum is well taken care of. I feel like you're going off on this "you're wasting their time" tangent now, and it's not really productive to the question I asked.
– Alec
Feb 22, 2022 at 9:58