The Witt group, or the cohomology of the periodic table of n-categories March 30, 2009
Posted by Noah Snyder in Category Theory, conferences, homological algebra, quantum groups, talks.10 comments
A very popular topic at the Modular Categories conference was the a generalization of the Witt group which is being developed by Davydov, Mueger, Nikshych, and Ostrik. What is this Witt group? Well it’s the simplest case of the cohomology of the periodic table of n-categories!
In this post I want to explain the definition of this cohomology theory and explain why it generalizes the classical Witt group.
First recall the Baez-Dolan periodic table.
Periodic Table
More stimulating mathematics? March 29, 2009
Posted by Ben Webster in jobs.7 comments
It seems that there is a second wave of postdoc positions opening now, administered through the NSF’s Math Institutes. It sounds like they will be some kind of weird mix of the usual NSF postdoc and a position at one of the institutes, and specifically aimed at those who have struck out this year. As we’ve discussed, I’m a little skeptical about inflating the number of temporary positions without creating permanent ones to match, but, in the short term, good news for anyone still looking.
A suggestion: Good refereeing certificates March 29, 2009
Posted by David Speyer in crazy ideas.13 comments
Some one tell me what’s wrong with this idea: Journal editors should publicly acknowledge their best referees. Obviously, they can’t say which papers the referee worked on, but they could write a note saying
To whom it may concern: Jane Doe has refereed more than twenty papers for the Journal of Isotropic Widgets, and she has always done a through and careful job. Our journal is greatly improved by her efforts.
Professor Doe could then list this on her webpage and CV, and hopefully it would help her tenure case and her professional reputation. As things currently stand, referees can list the journals we have worked for on our CV’s, but there is no way to indicate the quality of that work.
The only argument I see against this is that the process of writing these letters would be very subjective. But that’s also true of the writing of recommendation letters and of the acceptance of papers to journals. Am I missing something?
Hall algebras and Donaldson-Thomas invariants I March 25, 2009
Posted by Joel Kamnitzer in Algebraic Geometry, conferences, homological algebra, quantum groups, things I don't understand.4 comments
I would like to tell you about recent work of Dominic Joyce and others (Bridgeland, Kontsevich-Soibelman, Behrend, Pandaripande-Thomas, etc) on Hall algebras and Donaldson-Thomas invariants. I don’t completely understand this work, but it seems very exciting to me. This post will largely be based on talks by Bridgeland and Joyce that I heard last month at MSRI.
In this post, I will concentrate on different versions of Hall algebras. Let us start with the most elementary one. Suppose I have an abelian category 
![c_{[M], [N]}^{[P]}](http://s0.wp.com/latex.php?latex=c_%7B%5BM%5D%2C+%5BN%5D%7D%5E%7B%5BP%5D%7D+&bg=ffffff&fg=666666&s=0 "c_{[M], [N]}^{[P]}")
The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver 
The obvious question concerning Hall algebras is to come up with a framework for understanding them when the Hom and Ext sets are not finite. This is what Joyce has done and he has applied it where 
Extended Haagerup Exists! March 25, 2009
Posted by Scott Morrison in conferences, planar algebras, small examples, subfactors, talks.16 comments
Following on from Noah’s post about the great Modular Categories conference last weekend in Bloomington, I’ll say a little about the talk I gave: Extended Haagerup exists!
The classification of low index, finite-depth subfactor planar algebras seems to be a difficult problem. Below index 4, there’s a wonderful ADE classification. The type A planar algebras are just Temperley-Lieb at various roots of unity (and so the same as 
But what happens as we go above index 4? In 1994 Haagerup gave a partial classification up to index 
and
(in both cases here the initial arm increases in steps of length 4) and another possibility
This result really opened a can of worms. Which of these graphs are actually realised? (Hint, they’re nicely colour-coded :-) What about higher index? What does it all mean? Are these graphs part of some quantum analogue of the classification of finite simple groups? Read one for the answer to the first question, at least.
SF&PA: Subfactors = finite dimensional simple algebras March 23, 2009
Posted by Noah Snyder in Category Theory, representation theory, subfactors.2 comments
Since my next post on Scott’s talk concerns the construction of a new subfactor, I wanted to give another attempt at explaining what a subfactor is. In particular, a subfactor is just a finite-dimensional simple algebra over C!
Now, I know what you’re thinking, doesn’t Artin-Wedderburn say that finite dimensional algebras over C are just matrix algebras? Yes, but those are just the finite dimensional algebras in the category of vector spaces! What if you had some other C-linear tensor category and a finite dimensional simple algebra object in that category?
Let me start with an example (very closely related to Scott Carnahan’s pirate post).
(more…)
Coincidences of tensor categories March 22, 2009
Posted by Noah Snyder in Category Theory, quantum groups, small examples, talks.5 comments
This week Scott and I were at a wonderful conference on Modular Categories at Indiana University. I find that I generally enjoy conferences on more specific subjects, especially in algebra. Otherwise you run the danger of every talk starting by defining some algebra you’ve never heard of (and won’t hear of again the rest of the conference) and then spend a while proving some properties of this random algebra that you still don’t know why you care about (let alone why you should learn about its projective modules). With more specific conferences if you don’t quite get something the first time you have a good change of seeing it again and it slowly sinking in. The organizers (Michael Larsen, Eric Rowell, and Zhenghan Wang) did an excellent job putting together and interesting, topically coherent, and fun conference. I was also pleasantly surprised by Bloomington, which turned out to actually be kind of cute. I have several posts I’d like to give on other people’s talks, in particular there were several talks (by Davydov, Mueger, and Ostrik) on the “Witt group” which involves the simplest case of a kind of cohomology of the periodic table of n-categories and thus should appeal to all of you over at the n-category theory cafe. But I think I’ll start out with our talks (which Scott and I prepared jointly based on our joint work with (Emily Peters)^2 and Stephen Bigelow).
The first of these talks (click for beamer slides) was on coincidences of small tensor categories. The strangest thing about this talk was that I was introduced as a “celebrity math blogger.”
Please note that in the slides I’ve completely elided the distinctions between a quantum group, its category of representations, and (when q is a root of unity) its semisimplified category of representations (where you quotient out by the kernel of the inner product as in David’s post).
More shiny Web 2.0, please March 19, 2009
Posted by Ben Webster in websites.11 comments
A friend of mine from grad school recently suggested a really good idea for a webtool on his blog (it’s a personal blog, so I won’t link there, but he can identify himself in comments if the urge strikes): one should be able to search for theorems based on hypotheses and/or conclusions. So that if you wanted to prove a representation was indecomposible, you could search for theorems where that was the conclusion.
Of course, such a thing would only be functional if it could produce this automatically, which would be a tricky business. On the other hand, computers aren’t as stupid as they sometimes seem. If they can do machine translation, they should be able to do this (eventually). Any Googlers out there looking for something to do in their 20% time?
Interested in mentoring at Mathcamp? March 12, 2009
Posted by Noah Snyder in blegs, Mathcamp.10 comments
The past several summers I’ve worked at Canada/USA Mathcamp, which is a summer program for advanced highschool students. It’s a really exciting work environment with lots of great students and an unparalleled opportunity to teach classes on interesting unusual mathematics. Mathcamp classes have a lot in common with blog posts in a way, as you can see from these posts where I’ve shamelessly mined mathcamp material for posts. So it occured to me that graduate students who read the blog might be interested in applying to work as mentors!
I’ll put the application after the jump. Technically it’s past the deadline, but it turns out we need to hire a few more mentors than we’d expected (fewer returning mentors than we’d hoped), so even though we have some very promising applicants we’d love to have a few more good applicants.
The Conyers Bill Disaster March 7, 2009
Posted by Ben Webster in evil journals.3 comments
So, perhaps you’ve heard about the Conyers bill (“Fair Copyright in Research Works Act”). There are some emails circulating about this online (I got a forward of one originating with David Yetter), mostly leading back to a post of Lawrence Lessig’s at the Huffington Post.
You can read more details at the Lessig post, but the short version is this: this bill would make it illegal for any federal agency to require you to cede any copyright privileges to the government in exchange for research funding. The only current instance of this I know of is that all NIH funded research must be posted on PubMed. This bill would make it illegal for the NIH to require that, which would be a terrible thing to do. John Conyers should be ashamed of himself for introducing such a bill, and you should write you congresscritter about it, especially if they’re on the Judiciary Committee (it would be much better if this died in committee).
On the other hand, certain email forwards seem to have gotten it in their heads that this bill would forbid people who get NSF grants from posting on the arXiv, but it sure doesn’t look like the text of the bill could possibly imply that. Am I right here, or am I somehow misreading things?
On the other other hand, this is something of a big deal, in that it is a big middle finger from all the publishers to even the most mild-mannered open-access advocate. The message is roughly that the publishers think their profit margins are more important that people dying. Which is fine; they’re corporations, that’s how corporations are required to function by law. It just should make you think twice about whether corporations are a good vehicle for scientific publishing.
Anyways, everybody chill out. Then call your congresscritter.
EDIT: Right after posting this, I got another forward of an email by David Yetter, saying that he had misunderstood the legislation and that my (still displeasing) interpretation is correct. (Just so we’re clear, that had nothing to do with the post. It’s just a coincidence.)
