TMWYF: A visual tour of the beauty of group theory (Matthew Macauley)

Group theory is all about symmetry, and is one of the most beautiful subjects in all of mathematics. However, it's almost never presented that way. We teach it out of books with hardly any pictures, written by old white men from Ivy League-ish schools, for younger versions of themselves...because that's how it was taught to us. In this talk, I will summarize an entire semester of group theory in pictures, with the goal of convincing you that these are as necessary to the subject as drawing graphs are to calculus. I will show you new ways to think about classic topics like homomorphisms and group actions, why the isomorphism theorems are always done wrong (they're out of order, and there should be 5 of them). I will explain how subgroups and quotients are like stalactites and stalagmites in a cave, how purple unicorns arise in groups, how actions can be thought of as operating a switchboard, how to build semidirect products with rewirings, and how solvable and nilpotent groups can be described by climbing and chopping lattices. I will show you some tantalizingly simple ways to modify everyday dihedral and quaternion groups, to get beautiful new friends like the dicyclic, quasidihedral, semiabelian, and Pauli groups. This talk should be widely accessible to anyone who loves mathematics, and I especially hope to make new connections with people in the math education field.