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 […]

Why Derivations?

This week, I’m going to take a break from the topic that I’ve been writing about for most of this blog because it was a fun excursion into a topic which I found interesting and which seats my research firmly into the bigger picture of Algebra and mathematics in general. In past months I’ve been […]

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 […]

Is Reading Math Hard?

In my 9AM calculus class, students don’t have desks, but have tables where they sit in sets of four. I like this setup because it makes it easy to assign group work and the students already have a pre-made small group for them to work with. One of the pods has a bunch of guys […]

The Gelfand-Naimark-Segal Construction — Part 1

In this series of blog posts I’ll work through the proof that every -algebra on a may be isomorphically embedded into , the -algebra of bounded linear operators on some Hilbert space . This post is meant to give definitions of the objects that we will be studying and possibly some motivation for why this […]

Necessary Intro

Hey all. I’m Dan, and I’m a newly minted PhD in Mathematics and one of the lucky people to then get a job in academia. I’m not quite sure how I’m going to use this blog, but I have a couple ideas. First, I think I’ll use this as a way to teach myself things […]