Despite overwhelmingly popular knowledge, there are worlds in which ab is not equal to ba and, sometimes, (ab)c is not equal to a(bc). It's fascinating what can or cannot happen when we relax the rules for multiplication, and I hope to share a little about this aspect of algebra and why I find it interesting.

This peculiar behavior is described by mathematicians in terms of axioms; we begin by assuming particular conditions to be true and use only these to build an elaborate structure of theorems and properties. Despite the esoteric idea of stating some properties and studying what happens, these sets, called algebras, turn up in (arguably) real life. Even though these are certainly interesting and entertaining ideas in their own right, mathematicians aren't just making them up for fun; these types of structures are used model to specific physical applications, particularly in particle and quantum physics.

This talk will begin in the integers and end up in sets where our standard notions of multiplication do not work. We'll discuss the axioms required for rings and algebras, see some examples of rings with elements that behave badly, and discuss how these rings differ from our familiar ideas of numbers. Because I'll focus on examples, very little prerequisite knowledge will be required.