Category Archives: Research

New Research Area: Algebraic Coding Theory

Over the past year or so, I’ve been transitioning my research to a more applied area, Algebraic Coding Theory; as a youngish PhD staring down the barrel of another job search, this was partially strategic. Another part of it was that this area just seems interesting; in school I had a few graduate classes in […]

On the Absoluteness of Almost-Free Groups

This past year has been my first as faculty at Baylor University. While here I have had the chance to work and collaborate with a remarkable graduate student, Alexandra Pasi-Fitzmaurice. She and her advisor, Daniel Herden, have been investigating a certain family of infinite abelian groups called -free groups, specifically looking at how these groups […]

Automorphisms of Infinite Dimensional Algebras–Part 3

In this post we’ll show that for any algebra with a faithful ideal isomorphic to , every automorphism is inner. Buckle up; this’ll be a long one. It’ll build on my previous two posts, so make sure to read them if you haven’t already: Part 1, Part 2 Let’s gather together some facts which will […]

Fredholm Endomorphisms of Index 0

A while ago I posted a note on the ArXiV about my attempt at applying the theory of Fredholm operators from functional analysis to more general context of -algebras. I wanted to work through an argument here because it seemed like it has a suspiciously nice proof; I’ve learned to be skeptical of such proofs. […]

Automorphisms of Infinite Dimensional Algebras — Part 2

This post is devoted to determining the set of automorphisms of the Toeplitz-Jacobson -algebra . There are two general approaches to finding the automorphisms of these algebras. One could take the “outside-in” approach and look for bijective endomorphisms of . This might be an interesting approach since one can, in theory, leverage the decomposition of […]

Automorphisms of Infinite Dimensional Algebras–Part 1

I’m going to be using this, and most likely the next post or two to organize my thoughts about a fun little problem I’ve been toying around with. Recently, I’ve become interested in directly infinite algebras. Gien a field of characteristic zero , these are -algebras which have elements and such that but . Whenever […]