Koch Takes on Abstract Algebra in Newest Book

Math is hard. Alan Koch knows this. He also thinks it’s cool. 

For many people, the application of daily math looks like splitting a bill with your friends or preparing an annual tax return. Other applications can be a bit more abstract. 

As a mathematician, Koch aims to simplify the subject, especially in a concentration such as abstract algebra, making it more digestible and engaging for his audience. That characteristic makes Koch a consistently beloved member of the faculty at Agnes Scott College. 

“He knows how to keep classes engaged and makes everything understandable,” says Caitlin Weaver ‘21. “Professor Koch is the reason I was a math major.”

In a new book, “Hopf Algebras and Galois Module Theory,” Koch helps explain the increasing use for abstract algebra specific to the Galois module theory with six scholars in the field. 

Abstract algebra, Koch says, can be summarized as algebra concerning collections of “possibly abstract” objects upon which arithmetic can be done. “For example, if we take a square, we can ask: how can we manipulate the square in such a way as to have it occupy the same position in space? This would lead us to think about things such as rotations and reflections of the square.” 

Hopf algebra is just a tiny component of abstract algebra, a topic Koch became interested in during his senior year at the University of Vermont. Before 2016, Hopf algebra did not have much common use, or as Koch says, it was “math for math’s sake.” 

But that all changed six years ago when a published paper connected the Hopf algebras covered in “Hopf Algebras and Galois Module Theory” and the Yang-Baxter equation. The latter, also known as the star-triangle relation, is of importance in statistical mechanics. The application of this connection has since found many uses across both the areas of physics and mathematics.

The fact that the application came well after the theory was developed isn’t a new phenomenon but is reasonably common in the world of mathematics. “Number theory was considered worthless until it became fundamental in computing…non-Euclidean geometry was known long before it was needed to describe curved space,” Koch adds. 

To learn more about this research, check out the new book here. 

Click here to read a 2021 faculty feature on Koch. 

Click here for information on Dr. Koch’s upcoming Mathvenger Hunt, a part of the Atlanta Science Festival.