When confusions annihilate

As mathematicians we spend most of our lives confused about something or other. Of course, this is occasionally interrupted by moments of clarity that make it worth it. I wanted to discuss a particularly pleasant circumstance: when two confusions annihilate each other. I’ll give two examples of times that this happened to me, but people are encouraged to provide similar examples in the comments.

In both cases what happened was that I had:

• A question to which I didn’t know the answer
• An answer to which I didn’t know the question

Upcoming conference: Workshop on operator algebras and conformal field theory in Eugene

I wanted to take a moment to plug a conference in my soon-to-be hometown Eugene, OR organized by my once and future colleague Nick Proudfoot.

Aside from Eugene being lovely in August, I felt this conference was worth a post because it’s something of a unique format. Rather than being a bunch of experts on the subject (as it says in the title, the subject is the conjunction of operator algebras and CFT) getting together and giving talks that only they understand, it will be aimed at being educational for graduate students and interested non-experts (such as myself). The format is a bit similar to that of Talbot. In particular, in addition to an organizer (Nick) it has a “leader” who is in charge of mathematical content (but will delegate quite a few of the lectures); that will be the incomparable Andre Henriques.

MSC vs. ArXiv (and some interesting info on mathjobs)

One of my pet peeves is how annoyingly the AMS’s math subject classification is for people working in quantum algebra and quantum topology. The MSC has 97 different major subjects and my field is not one of them, and instead appears many times a subheading. In the new 2009 classification there’s at least the following: 16T, 17B37, 18D10, 20G42, 33D80, 57R56, 58B32, 81R50, and 81T45. Here I’m only counting things that are obviously quantum algebra and quantum topology (for example I didn’t list subfactors, quantum computation, knot invariants, etc.) By way of contrast, on the ArXiv there are only 32 categories, yet one of them (math.QA) contains the vast majority of work in my field (of course, many of those are cross-posted).

This mini-rant of mine came up at dinner at an AMS meeting in Waco (more on the excellent “fusion categories” special session later). Someone pointed out an interesting side-effect of this issue that I hadn’t thought of. One of the awesome things about mathjobs is that rather than simply having a large paper stack of applications, the people on hiring committees can instead sort the applications automatically in many different ways. It makes a lot of sense that mathjobs has this feature, but none of us who were on the applying side of things had ever considered it. Here are a few examples of things you might want to search for: look at people applying from a specific school, find everyone who has a recommendation letter from Prof. X, and (relevant to this post) sort by AMS subject classification.

This means that choosing the right AMS subject classifications is actually somewhat important. If you choose poorly then someone who might be interested in hiring you might never actually find your application among the hundreds they’re looking through. So if you’re in a situation like mine it’s worth asking a professor or two which AMS subject classifications they’d be most likely to look through.

Since then I’ve been wondering whether it might be a useful for mathjobs that the data they ask for also include which arxiv classifications applicants have posted preprints under, as that’s the search that I would want to use if I were on a hiring committee. What do people think? Mathjobs is very responsive to requests, so if people think this makes sense I may send them an email.

The First National Forum of Young Topologists

So, I’ve added to the sidebar a new conference: The National Forum of Young Topologists in new Orleans next month. This actually sounds like a somewhat intriguing conference, concentrating on a mix of math and what I would call “professional development,” for lack of better word. Rather than a laser-like focus on the most recent stuff, it sounds like the math talks are supposed to be more historical in nature, and apparently the lectures by junior people are supposed to also “have the effect of showing examples of success stories starting out as graduate students through obtaining a tenure track position.”

While this all sounds very nice (I’ll admit that as a graduate student I never felt “isolated with [my] entire mathematical universe being [my] Ph.D. adviser,” and thus never felt the need for a conference to correct this, I know some people who’ve had that experience), I just can’t help but mock the name. I mean, I cannot imagine what on Earth the organizers were thinking when they came up with that one (and this comes from a guy who freely admits that the name of this blog was arrived at under the influence of spirituous beverages). I mean, did they put that in the grant application? It sounds like the world’s oddest Stalinist youth organization. The website could really use a couple of posters of square-jawed youths with sickles (or maybe just this guy) with a diagram of the Kirby calculus on the flag. I’m glad they want to have more such conferences (or at least, so they claim on the website), but I hope they don’t feel obligated to stick with the name.

Why the ICM is not as good an idea as it might sound

Jim Humphreys’s comment reminded me of one of my rants that has yet to be bloggified. The non-hyperbolic title is above; those who are amused rather than annoyed by my hyperbolic titles can imagine it was called “why the ICM is a scam.”

Now, as usual, you should take this with big fat grain of salt; I’ve only been to one ICM, during one of the unhappiest periods of my recent life (if you don’t believe me, just wait for the post on why Denmark is a scam), so I’m doing a pretty unscientific extrapolation. That said, I’ve been to one ICM, and it didn’t make me all that eager to go back.

Conference Networking

Early in my graduate student career, I was told by several people that I should go to conferences and talk to professors. If you work in mathematics, you’ve probably heard this piece of advice before, and it’s hard to see how you could damage your career by following it (given reasonable assumptions on your behavior). I encountered two problems:

1. What sort of talking am I supposed to do with a professor if I don’t know anything?
2. How do I make my way into one of those small circles of people that inevitably form between talks?

I’ve heard that some advisors actually go to conferences with their students and introduce them to colleagues, and this pretty much solves both problems, but I’d like to focus on the case that this doesn’t happen, since I imagine it will be the norm for a while. This isn’t meant to be a definitive guide, and I’d really appreciate further suggestions and anecdotes.

Mike on Topological Quantum Computing, at Georgia

I’m here at the 2009 Georgia Topology Conference and Mike Freedman is about to start talking about the current proposal for building a topological quantum computer. I’ll try liveblogging his talk; there’s a copy of the slides at http://stationq.ucsb.edu/docs/Georgia-20090518.pptx (PowerPoint only, sorry!) if you want to see the real thing. I think he recently gave a version of this talk in Berkeley recently, so some of you may have already heard it. I’ll fail miserably at explaining everything he talked about, but ask questions in the comments!

Mike says that the point of the talk will be to explain how it is that there’s a “topological” approach to building a computer, and try to give an idea of the mathematics, physics and engineering problems involved.

The Witt group, or the cohomology of the periodic table of n-categories

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

Hall algebras and Donaldson-Thomas invariants I

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 which has the following strong finiteness properties: namely and are finite for any objects . Then one can define an algebra, called the Hall algebra of , which has a basis given by isomorphism classes of objects of and whose structure constants are the number of subobjects of which are isomorphic to and whose quotient is isomorphic to .

The main source of interest of Hall algebras for me is the Ringel-Green theorem which states that if you start with a quiver , then the Hall algebra of the category of representation of over a finite field is isomorphic to the upper half of the quantum group corresponding to at the parameter .

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 is the category of coherent sheaves on a Calabi-Yau 3-fold.

Extended Haagerup Exists!

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 , as long as you change the pivotal structure). The type D planar algebras (with principal graphs the Dynkin diagrams ) were the subject of Noah’s talk at the conference, and the and planar algebras are nicely described in Stephen Bigelow’s recent paper.

But what happens as we go above index 4? In 1994 Haagerup gave a partial classification up to index . He showed that the only possible principal graphs come in two infinite families

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.