Math 4130 (Modern Algebra II), Spring 2023

Math 4130 (Modern Algebra II), Spring 2023


"Mathematics, rightly viewed, possesses not only truth, but supreme beauty." --Bertrand Russell

Symmetry, as wide or narrow as you may define its meaning, is one idea by which man through the ages has tried to comprehend and create order, beauty, and perfection. --Hermann Weyl

"If you are someone who prefers large vistas and powerful theories, then it is essential to be able to test general results by applying them to simple examples....where one can do concrete calculations, sometimes with elaborate formulas, that help to make the general theory understandable. They keep your feet on the ground...A good example is a thing of beauty. It shines and convinces. It gives insight and understaning. It provides the bedrock of belief." --Sir Michael Atiyah, in Advice to a young mathematician.

About the class

Group theory is the study of symmetry, and it is one of the most beautiful areas in all of mathematics. It arises in puzzles, visual arts, music, nature, the physical and life sciences, computer science, cryptography, and of course, all throughout mathematics. Yet, it is usually taught with little to no visuals, which is a travesty. The primary reason for this is that every establish books does it this way, and instructors tend to teach it the way the learned it. Without available resources, this will never change. And I'm working to change this. My vision is that in 20 years, students will find it incredulous that this subject used to be taught non-visually. Just like how no one would teach calculus without graphs. Visuals are as essential in group theory as pictures are in fields such as crystallography or art history.

We will not use a textbook for this class because one does not yet exist. (I started writing it in October 2019, and should finish it this year) A few fantastic sources that helped inspire this class include a 2009 general-audience book called Visual Group Theory (VGT), by Nathan Carter. The renowned mathematician Steven Strogatz at Cornell, calls it One of the best introductions to group theory -- or to any branch of higher math -- I've ever read. VGT has 300 color illustrations, and focuses on the intuition behind the difficult concepts in group theory. Another source is a free e-book called An inquiry-based approach to abstract algebra, by Dana Ernst. This follows the "Visual Group Theory" approach, but is more rigorous and proof-based. However, most of the proofs are not provided; you are supposed to fill them in. This is what the "inquiry-based" part means.

In class, we will play with the Rubik's cube. We will analyze art freises, wallpapers, chemical molecules, and Archimedian solids. At the end of the semester, you will truly understand groups and rings, subgroups, cosets, products and quotients, homomorphisms, group actions, conjugacy classes, centralizers, normalizers, semidirect products, theorems by Lagrange, Cayley, Cauchy, and Sylow, and what Évariste Galois stayed up until dawn writing the night before his untimely death in a duel at age 20, that remains one of the most celebrated achievements in all of mathematics, and which provided the framework necessary to elegantly solve several classic mathematical mysteries of the ancient Greeks. In the end, you will leave with a new appreciation of the beauty, and difficulty, of an area of mathematics you never dreamt existed.

Class essentials

Resources

Homework

Homework should be written up carefully and concisely. Please write in complete sentences. Part of your grade will be based on the presentation and clarity of your answers. Enough of the problem statements should be copied down so that your homework solutions are self-contained and the textbook is not needed to read, understand, and grade them. Along with assignment, I will post "supplemental material" consisting of blank images that you are free to use, rather than redraw by hand.

Lecture notes

See an explanation below for the story behind these, and why they are a new and improved version of what I thought had converged to something that I would never change! I will eventually record YouTube lectures to go along with these, but not this semester.

Exams


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