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Episode Info: Kevin Knudson: Welcome to My Favorite Theorem. I’m your cohost Kevin Knudson, professor of mathematics at the University of Florida. I am joined by cohost number 2. Evelyn Lamb: I am Evelyn Lamb. I’m a freelance math and science writer in Salt Lake City. So how are you? KK: I’m okay. And by the way, I did not mean to indicate that you are number 2 in this. EL: Only alphabetically. KK: That’s right. Yeah. Things are great. How are things in Salt Lake? EL: Pretty good. I had a fantastic weekend. Basically spent the whole thing reading and singing, so yeah, it was great. KK: Good for you. EL: Yeah. KK: I didn’t do much. I mopped the floors. EL: That’s good too. My floors are dirty. KK: That’s okay. Dirty floors, clean…something. So today we are pleased to have Chawne Kimber on the show. Chawne, do you want to introduce yourself? Chawne Kimber: Sure. Hi, I’m a professor at Lafayette College. I got my Ph.D. a long time ago at University of Florida. KK: Go Gators! CK: Yay, woo-hoo. I work in lattice-ordered groups. KK: Lattice-ordered groups, very cool. I should probably know what those are, but maybe we’ll find out what they are today. So yeah, let’s get into it. What’s your favorite theorem, Chawne? CK: Okay, so maybe you don’t like this, but it’s a suite of theorems. KK: Even better. EL: Go for it. CK: So, right, a lattice-ordered group is a group, to begin with, in which any two elements have a sup and an inf, so that gives you your lattice order. They’re torsion-free, so they’re, once you get past countable ones, they’re enormous groups to work with. So my favorite theorems are the representation theorems that allow you to prove stuff because they get unwieldy due to their size. EL: Oh cool. One of my favorite classes in grad school was a representation class. I mean, I had a lot of trouble with it. It was just representations of finite groups, and those were still really out there, but it was a lot of fun. Really algebraic thinking. CK: Well actually these representations allow you to translate problems from algebra to topology, so it’s pretty cool. The classical theorem is by Hahn in 1909. He proved the special cases that any totally ordered Archimedean group can be embedded as a subgroup of the reals, and it kind of makes sense that you should be able to do that. KK: Sure. CK: And then he said that any ordered abelian group, so not necessarily lattice-ordered, can be embedded in what’s called a lexicographical product of the reals. So we could get into what that is, but those are called Hahn groups. They’re just huge products of the reals that are ordered in dictionary order that only live on well-ordered sets. So this conjecture, it’s actually a theorem, but then there’s a conjecture that that theorem is actually equivalent to the axiom of choice. KK: Wow. EL: Oh wow. CK: Right? EL: Can we maybe back up a little bit, is it poss...
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