### September 2007

Recently, a lot of blogs (incited by the Everything Seminar) have been talking about the Axiom of Choice and it’s more horrifying implications. Well, I’m going to jump in on the bandwagon and present deGroot’s problem. This is essentially a continuous version of the Banach-Tarski paradox, and is therefore somewhat more horrifying than the standard version. The solution is due to Trevor Wilson, and my understanding is that it came out of an REU. This post is mostly taken from having to present his paper to one of my professors as part of an independent study on Banach-Tarski. (more…)

Hi, I’m Joe, and I’m a second-year graduate student at Stony Brook University. I’ve just passed the comprehesive exams (qualifying exams at most other schools) at the beginning of the semester and will soon start preparing for orals. I’ve known Charlie for a few years now, and he has been trying to get me to start posting on this site since its inception. I finally found a free moment to make my introduction to the site.

My mathematical interests primarily lie in various parts of geometry and analysis, but I will try my hand at anything to do with physics, so you can expect most of my posts to be physically motivated. My posts will probably be more sporadic than either of my counterparts on the site, but I’m sure Charlie will try to keep me on some semblance of a schedule. That’s all I have for now; I’ll see you again soon.

- Joe

In this post, I defined Grothendieck Topologies and gave some examples. Look there and in related links for more on Grothendieck topologies. We’re going to move on to Sheaves today. This is going to be a minipost, because this does deserve it’s own posting and it’s going to take me some time to work out the next in this series.

I should probably introduce myself (and start writing as well!). I’m Matt, a first-year grad student at the University of Chicago. I’m interested in algebra, geometry, and topology, but I also find various topics in theoretical CS and in logic and set theory really neat — so I doubt there will be much uniformity in the topics I write about!

Topology. It’s something that every math student has to become comfortable with. First you get to learn metric topology (usually on $\mathbb{R}^1$), followed by point set topology. Then in graduate school, pretty much everyone gets an introduction to algebraic topology. However, it all really just studies sets with distinguished collections of subsets. There is, however, a more general version of topology, in fact, a way to put a topology onto a category due to, of course, Grothendieck.

So, I WAS going to give a classical proof of a classical algebraic geometry theorem: that there are exactly twenty-seven lines on any smooth cubic surface. I wanted to avoid using the machinery of divisors and all the attached technicality, but the proof I came up with was rather nasty, needs a lot of lemmas, almost all of which are technical. So instead, I’m going to talk about my understanding of the development of the subject.

I’m posting late yet again. I seem to be making a habit of it, hopefully I’ll pull myself together eventually, but I’m going to back off on my goal of a weekly post for a bit. For now, I’m going to talk group theory and physics, as the title suggests. We’ll begin with the math.