### August 2007

You know you’ve been asked “what’s math research good for?”  I’m talking about at that party that wasn’t full of other math grad students.  You know, right after, “you do new mathematics?  I thought it was all finished!”

I’ve made a stab at writing an answer to this for the general public, and I’d love input from other mathematicians.  Do you think I’m right?  So what is it good for?

Many of the math blogs that I read (and you can find in the blogroll) are writing about the plight of Ali Nesin and his Summer School. Apparently, Prof. Nesin was arrested for running his yearly Summer School in Turkey since he was “teaching without permission”! You can even see pictures of black boards behind police tape on Mathematics under the Microscope. A petition was set up to protest and I urge you to sign it!

The paper Nonisomorphic Verdier Octahedra on the same Base, by Matthias Künzer, popped up on the Arxiv today. It answers the following question: Given a morphism $f : A \to B$ in a triangulated category the cone on $f$ is determined up to a (non-unique) isomorphism; is the same true for a Verdier octahedron constructed from a commutative triangle? The answer is: No, there exist non-isomorphic Verdier octahedra constructed from the same base! The non-uniqueness mentioned above has been a thorn in the side of the standard axioms for triangulated categories and this result doesn’t exactly make their life easier!

To prove this result Künzer uses something called Heller triangulated categories which he describes in detail in the paper Heller Triangulated Categories. These two papers are definitely being added to my list of things that I want to understand!

On a related note, J.P. May has proved that the standard axioms for triangulated categories are over-determined in the sense that one of the axioms can be proved using the other three. This is explained in his paper The Axioms for Triangulated Categories.

I use an RSS thread to follow what is new on the arXiv and recently I came across a paper that claims that the theory of mathematics isn’t consistent. That sounded intriguing so I downloaded the paper and read it. I discovered, what I think is, a serious error and notified the authors. Every night since then I have been in email contact with the authors. We are in complete disagreement about whether there is an error or not but our discussions are (still) very polite and it doesn’t look like either one of us will convince the other(s).

I would appreciate any comments or advice from anyone reading this post that has similar experiences of conversing/arguing with the author(s) of a paper about errors and how such conversations/arguments have turned out.

I don’t want to post a link to the paper, since I feel like that would be like trying to make fun of other people’s work. I can email a link to anyone who is interested in taking a look at the paper, just drop a comment.