Cyclotomic integers, fusion categories, and subfactors (March)

Frank Calegari, Scott Morrison, and I recently uploaded to the arxiv our paper Cyclotomic integers, fusion categories, and subfactors. In this paper we give two applications of cyclotomic number theory to quantum algebra.

  1. A complete list of possible Frobenius-Perron dimensions in the interval (2, 76/33) for an object in a fusion category.
  2. Given a family of graphs G_n obtained from a graph G by attaching a chain of n edges to a chosen vertex, an effective bound on the greatest n so that G_n can be the principal graph of a subfactor.

Neither of these results look like they involve number theory. The connection comes from a result of Etingof, Nikshych, and Ostrik which says that the dimension of every object in a fusion category is a cyclotomic integer.

A possible subtitle to this paper is

What’s so special about (\sqrt{3} + \sqrt{7})/2?

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More math news

In more parochial math news, my alma mater has hired Sophie Morel. This means that after 374 years we finally have a woman as a full professor. I don’t believe in congratulating a school for doing something that’s at least 75 years overdue (I hear that Emmy Noether was on the market in 1933), but as an alum I’m certainly relieved that this is no longer a continued source of embarrassment. This is just the first step, and I look forward to the day when Harvard has at least two women on faculty, like both of the other institutions I’ve been affiliated with.

Speaking of Prof. Morel, her advisor Gerard Laumon has quite the string of superstar students. Anyone know what the current record is for who has the most students with Fields medals?

Wanted: page of tax info for NSF fellows

So I think something that the world needs is a page with important bits of knowledge and tricks concerning having an NSF graduate or postdoctoral fellowship and paying taxes. The somewhat tricky thing is that you’d need an actual lawyer involved at some point in the process. Is this something one could try to convince the NSF’s accounting department to do? Presumably it’d be easy for the right person to do, and would save a lot of time for NSF Fellows who could then do math instead of trying to figure out taxes.

Here’s some examples of the sort of questions this page could answer:

  • In what ways is NSF income taxable? (My understanding, which is not legal tax advice, is that you must pay income tax on this money by writing “SCH $$$” on the dotted line next to box 7, but you do not need to file a Schedule C nor a Schedule SE nor pay FICA/self-employment tax.)
  • Is it possible to efile when you have taxable scholarship and fellowship income? Or do you have to print out the form in order to write “SCH $$$” on the dotted line next to box 7? (The rumor in the dept. today was that someone knew how to get TurboTax to enter the SCH thing, but none of us actually there knew.)
  • How is the “research allowance” treated? My understanding, which again is not tax advice, is that since this is only for reimbursement it is not income.
  • How are health care costs treated? At Columbia they apparently treat your health care as taxable income issuing you a 1099-MISC for non-employee compensation. This seems contrary to my readings of both the 1099-MISC instructions (where one of the criteria for issuing a 1099-MISC is that there were services rendered) and to section III.B.3. of the NSF Postdoctoral Fellowship Solicitation. But presumably Columbia’s accountants know what they’re doing. Nonetheless it’s extremely difficult to figure out what this money means. Is it FICA taxable? Do I have to file a Schedule C? What are my business expenses if it’s a business that didn’t actually do anything but yet was given money for no reason and then spent all its money on my healthcare? Why didn’t Berkeley issue me a 1099-MISC for the portion of my fees that went to medical care in graduate school? Is it possible to know in advance which schools treat health insurance money this way? If it’s self-employment money then I could wind up paying 15% FICA + 9% State and City + 25% Federal of $6K because of Columbia’s accounting.

Anyone have other good questions? Know anywhere to find answers to these questions?

Knot polynomial identities and quantum group coincidences (February)

Scott Morrison, Emily Peters and I have just uploaded to the arxiv our paper Knot polynomial identities and quantum group coincidences. In this paper we prove several new strange identities between certain specializations of colored Jones polynomials and other classical knot polynomials. For example, we prove that for any knot (but not for links!) the 6th colored Jones polynomial at a 28th root of unity is twice the value of a certain specialization of the HOMFLYPT polynomial (for the exact formulas see the first page of the paper).

Each of these identities of knot polynomials comes from a coincidence of small tensor categories involving the even part of one of the D_{2n} subfactors. For example, the above identity comes from an equivalence

\frac{1}{2} \mathcal{D}_{8} \cong \text{Rep}^{uni}{U_{s=\exp({2 \pi i \frac{5}{14} })}(\mathfrak{sl}_4)}^{modularize}

To recover the knot polynomial identity, one computes the Reshetikhin-Turaev invariant for a particular object on the left (getting half the relevant colored Jones polynomial for knots, but something worse for links) and for the corresponding object on the right (getting the specialization of HOMFLYPT).

(In that equation above there’s a lot of technical terms on the right side. “Uni” means we’re using Turaev’s unimodal pivotal structure instead of the usual pivotal structure, and “modularize” means take the Bruguieres-Mueger modularization where you add isomorphisms between the trivial object and all simple objects that “behave like the trivial object.”)

For most of these coincidences we give no fewer than three separate proofs, as well as an exciting diagram which explains the proof in pictures. After the jump I’ll sketch the flavor of these different arguments.

One of the biggest difficulties in writing this paper was getting all our conventions straight because we constantly need to jump between different quantum groups, and knot polynomials. Life becomes very messy if you’re using one convention for q in one place and a different one somewhere else. So one goal of this paper is to be “full of correctness.” We hope that if you ever wanted to know about the relationship between quantum groups and diagramatics you can just look at this paper and have nice consistent conventions for everything. In particular, if you notice even small errors we’d really love to hear about them so that they can be fixed.
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“Et al” is unethical

So apparently the AMS has a document on Ethical Guidelines. It’s actually remarkably well done. It has lots of tips that can help young mathematicians learn how to behave professionally. I was also impressed by the way that the guidelines avoid making too controversial of stands (which would go beyond the basics of ethics) while still not being milquetoast. For example, “No one should be exploited by the offer of a temporary position at an unreasonably low salary and/or an unreasonably heavy work load” is certainly an ethical obligation, but one that may be difficult to live up to.

I also thought that the guidelines about correct attribution were well phrased. For example:

To give appropriate credit, even to unpublished materials and announced results (because the knowledge that something is true or false is valuable, however it is obtained);

I have my own suggestion for a guideline on ethical use of citations: you should never ever use “et. al.” citations. Furthermore, if journal typesetters add them you should ask them to replace them with full citations.

If a bibliography just says “et al.” many readers are never going to get around to looking at the other names thereby effectively failing to properly attribute everyone. People at the end of the alphabet are already at enough of a professional disadvantage (see What’s in a Surname? The Effects of Surname Initials on Academic Success by Liran Einav and Leeat Yariv,), the use of “et al” just exacerbates this.

Hat tip: I learned about this document in a MathOverflow comment by Bill Johnson

Why is physical intuition possible?

This post is based on a conversation I had with Allan Adams at Mathcamp a few summers ago, and I was reminded of it by an aside in Mike Freedman’s talk in Scott’s backyard on Friday. As usual with blog posts based on other people’s talks, all good ideas in this post should be attributed to Allan and Mike and all mistakes to me. Furthermore I think everything I say here is obvious to people who actually know physics.

My basic confusion was how physical intuition (in particular in quantum field theory) could be applied to so many mathematical settings when there’s only one physical world so there’s no reason to think any intuition built up within that single example would apply any more generally than that one example. What Allan pointed out to me is that it’s not true that physicists are only studying one example. Although there may only be one fundamental theory of physics, by looking at various particular physical systems the limiting behavior becomes its own theory. The physics at the surface of a black hole can be thought of as its own example; the physics of superconductors is its own example; etc. Because all of these examples are physical (they involve minimizing actions, they’re quantum, etc.) they have a lot of attributes in common, so intuition and general techniques can be developed by understanding their commonalities.

Mike made two comments in his talk (on K-theory and superconductors) that flesh out this idea further. He was discussing the BCS superconductor and explained that when physicists refer to a theory by initials they’re not just being polite, what they mean is that you’re studying the mathematical model rather than any particularly instantation of it. In particular, the model doesn’t care if there are exactly 10^9 electron pairs or the exact composition of the material, it is studying the abstract setting that appears in the limit. By calling it the “BCS superconductor” they mean that in some sense they’re studying the physics of a different world. In particular, in the BCS setting since you’re assuming that there’s a huge sea of electron pairs the “vacuum” consists of this huge sea. This explains how physicists can develop intuition for more general notions of vacuum: they’re not always studying the absolute vacuum, they’re also studying other systems with states that have the properties of being a “vacuum.” This particular vacuum has a delightfully strange property. Since a new electron pair doesn’t change the underlying vacuum, in this “world” electric charge isn’t preserved!

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.

How to almost prove the 4-color theorem

Vaughan Jones often quips at the beginning of talks on Planar Algebras (see these lectures, for example) that the worst thing you can say about Planar Algebras is that they have not yet yielded a proof of the 4-color theorem. In this post I’ll sketch how a common “evaluation algorithm” (used by Greg Kuperberg and by Emily Peters, for example) almost proves the 4-color theorem. I believe this (failed) argument is due to Penrose, though I’m taking it from an article of Chmutov, Duzhin, and Kaishev and some notes of John Baez’s. There are some more elaborate attacks (by Kauffman, Saleur, Bar Natan, and probably others) that I won’t discuss at all. This is the second of what hopefully will be a short series of posts on “evaluation algorithms” (the first was on the Jellyfish algorithm).

The outline of the post is as follows. First I’ll explain a standard reduction of the 4-color theorem to a question about 3-coloring edges of trivalent graphs. Second I’ll explain why 3-colorings of edges is a question about finding a positive evaluation algorithm for a certain planar algebra. Third, I’ll discuss “Euler characteristic” evaluation algorithms. Fourth I’ll explain how this technique almost answers the 4-color theorem.
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New developments in the blnosphere

Let me start out by apologizing for two things, first the horrible pun in the title, and second my absence from the blog for the summer. Between moving twice (once cross-country), graduating, getting set up at a new job, buying furniture, trying to finish some papers, and being academic coordinator at Mathcamp I was pretty swamped. As a result I missed out on some developments in the math blogging.

Frequent commenter Danny Calegari started a blog in May. It pays to occasionally click on the links in comments here as sometimes you’ll find brand new blogs. My mathcamp friend, Matt Kahle, who is a postdoc at Stanford also started a blog. It has a fun mix of some elementary stuff (like the Rubik’s cube) and some of his research (which as an interesting mix of topology, combinatorics, and statistical mechanics, it definitely involves a lot of sending n to infinity in ways that would make my advisor happy). I’ve been meaning to link to both of those since sometime in June but just haven’t gotten around to it (though I did manage to add them both to the blogroll). It’s been that sort of summer, just ask me about my passport. Also, low dimensional topology has become a group blog. I find group blogging a great model both as a reader and blogger because it promotes conversations and allows one to maintain a reasonably updated blog even when someone disappears a whole summer.

Finally, over the summer there was a great conversation about what mathematicians need to know about blogging. Here’s my two cents. One thing incredibly valuable thing about blogging is the opportunity to have discussions and get advice about how to be a mathematician. It’s often hard in real life to have a discussion involving people at many different places in their careers about professional questions. In that spirit, here’s a question I’ve been wrestling with lately. How do you balance your research time between the following three activities: working on problems you basically know how to solve, working on problems you don’t know how to solve but are important problems, and learning new tools. When I was in graduate school I felt like it was pretty easy to balance things because any time I had any idea that was at all worthwhile I just worked on it and when I didn’t, I learned new things. I had few enough research-worthy ideas that it was feasible to think about all of them. Now that I know more I can’t keep doing that because I simply don’t have time to work on all the easy problems that I could solve. So the need comes to prioritize. I was wondering how other people strike this balance.

The Jellyfish Algorithm

Stephen Bigelow, Scott Morrison, Emily Peters and I have a preprint up on the arxiv today about the extended Haagerup subfactor and its planar algebra. Scott already explained nicely the story that this paper fits into. Today I wanted to tell you about one of the key elements of this paper, Stephen’s “Jellyfish algorithm.” This algorithm is a kind of “evaluation algorithm” and I hope to get around to putting up some more posts on some other evaluation algorithms as well as some open questions concerning evaluation algorithms (one would prove the 4-color theorem and another is related to whether the exceptional lie algebras aren’t so exceptional after all).

What is an evaluation algorithm? Well suppose you have a bunch of diagrams (think formal sums of links) constructed out of a few generators (think crossings) modulo some relations (think the HOMFLY relations). There are two obvious questions. Are these rules enough to reduce every link to a constant times the empty diagram? Does any way of reducing a given diagram give the same answer? Usually the former question is substantially easier than the latter. Often the first question is answered by giving an explicit algorithm for simplifying diagrams.

One simple example is the Kauffman bracket description of the Jones polynomial. Here the evaluation algorithm is very easy: find a crossing and resolve it, wash, rinse, repeat. This clearly terminates because at every step you get fewer crossings.
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