Concrete Categories October 26, 2009
Posted by David Speyer in Algebraic Topology, Category Theory.12 comments
In many introductions to category theory, you first learn the notion of a concrete category: A concrete category is a collection of sets, called the objects of the category and, for each pair 
Using terminology from a discussion at MO, I’ll call a category concretizable if it is isomorphic to a concrete category. For example, 
At one point, I learned of a result of Freyd: The category of topological spaces, with maps up to homotopy, is not concretizable. I thought this was an amazing reflection of how subtle homotopy is. But now I think this result is sort of a cheat. As I’ll explain in this post, if you are the sort of person who ignores details of set theory, then you might as well treat all categories as concrete. My view now is that specific concretizations are very interesting; but the question of whether a category has a concretization is not. I’ll also say a few words about small concretizations, and Freyd’s proof.
MSC vs. ArXiv (and some interesting info on mathjobs) October 25, 2009
Posted by Noah Snyder in conferences, inside baseball, jobs, math life, the arXiv.19 comments
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.
Rhombus tilings and an over-constrained recurrence October 21, 2009
Posted by David Speyer in combinatorics, things I don't understand.13 comments
I recently visited Robin Pemantle and his student Peter Du at UPenn. We talked about tilings of planar regions, generating functions and asymptotics. Towards the end, we talked about a bit about a very classical example, which is what I want to tell you about today.
In most planar tiling problems, the goal is an asymptotic analysis for tilings of large regions, because there isn’t enough structure to do better. This is the approach taken in the beautiful work of Kenyon, Okounkov, and collaborators.1, 2, 3 Sometimes, there is enough structure to give exact solutions with explicit generating functions. This is the situation with Aztec Diamonds, fortresses, and other several other examples.4,5 The central name here is Jim Propp 6, 7, who has developed this theory together with many undergraduate and graduate students (including me).
And then there is one case: rhombus tilings of a hexagon. These have almost too much structure; more structures than one would expect to be compatibly possible. In this post, I want to talk about this example. In particular, I want to ask you a question which I thought about a bit on the train ride back and see whether any of you have some thoughts.
Polymath projects on StackExchange/mathoverflow? October 20, 2009
Posted by Scott Morrison in Math Overflow, polymath.8 comments
I’ve been thinking a bit about whether the StackExchange software (which mathoverflow is running on) could be used to host a polymath project.
I’d imagine it involving many many questions and answers, with links between them, modelling the division of the “big question” into its constituent chunks.
There are some big advantages — in particular, it’s easier to pay attention to individual parts, because there’s more structure than in blog comments.
As a first approximation, you might start out like this: Terry asks the polymath7 problem, linking elsewhere for motivation and background. Tim posts a first ‘answer’: “Could we attack this by proving Lemmas X, Y and then generalising the approach of Theorem Z?” and at the same time creates questions corresponding the Lemmas X and Y and a more open question about Theorem Z. Other participants can then go to those questions to give their thoughts. Answers don’t have to be “answers” in the convention sense — they’re just meant to correspond to “ideas”, and should often link to a new question if it’s obvious that the idea needs further development. The StackExchange software allows for comments on answers, which would allow short responses to previous answers.
The big disadvantages of StackExchange are that
* at this point, there’s no LaTeX support, although this will hopefully change.
* the reputation system inhibits new participants, at least at first (they can still ask and answer questions, but commenting and upvoting are limited).
* it may end up harder to understand the “big picture” than in a blog thread.
The solution to the first two of these may be to try a polymath project at mathoverflow.net itself, rather than a new installation. Many participants will already have reputation (and on an established site it’s very easy to gain enough reputation to comment and upvote, because any decent question will quickly garner reputation). It’s easy to filter questions by tags, so I think you could ignore everything else happening on mathoverflow.net if you wanted to.
The last problem might be addressed by having a “community wiki” answer at the top of each question, summarising progress so far, as well as regular progress reports on blogs.
Why do I find MathOverflow fun and nLab not? October 15, 2009
Posted by Ben Webster in websites, wiki.32 comments
There’s been a very interesting dicussion in comments on Scott and Anton’s post about the strengths and weaknesses of MathOverflow and nLab, and I thought it might be good to divert to a new post (and use my position as blogger to top-post my thoughts). (edited since the original post. Look below for more) (more…)
Math Overflow October 14, 2009
Posted by Scott Morrison in Math Overflow.120 comments
(written collaboratively by Anton Geraschenko and Scott Morrison, in google wave)
Math Overflow (MO) is a brand new mathematics questions and answers site. You should go give it a try! Several of the regular readers of this blog are already there. It’s much more fun if you actually ask a question, and the only way to get the full experience.
20 Questions: October 6th October 8, 2009
Posted by Critch in 20 questions.20 comments
Hi folks,
Here are the latest problems from the 20 Questions seminar for the query-hungry :)
How to almost prove the 4-color theorem October 7, 2009
Posted by Noah Snyder in planar algebras, quantum topology.19 comments
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.
(more…)
REUs October 6, 2009
Posted by Ben Webster in math life, Mathcamp, Undergraduate.15 comments
Rereading Noah’s graduate school advice post, I realized I’d forgotten to stir up trouble at the time about his comments on REUs. In part, one should understand this post as an attempt to goad him into explaining.
First, a little personal history. I’m basically the poster-child for REUs; doing an REU in the year between my junior and senior years was the only organized math outside of school I participated in before grad school, and was the only experience with research I had before grad school, essentially. I had an excellent relationship with my advisor from the REU, met several people I liked a lot, did good enough research to turn it into a solo paper (admittedly, several years later), and generally had a really excellent experience. I’ve always recommended REUs to students, especially if they were considering grad school.
This has always seemed like a no brainer to me. If nothing else, since an REU is essentially the only real chance that an undergrad has to test drive grad school before committing years of their life to it. So, I’ll admit, I was a little surprised to find out that “REU-hater” is a category of person that exists. And now I’m curious; are there any more of you out there?
Israel Gelfand (1913-2009) October 5, 2009
Posted by Ben Webster in Uncategorized.12 comments
News is circulating on the internet that Israel Gelfand has died. My ultimate source for this is LiveJournal, so take it with a grain of salt, but it’s not hard to believe, given that the guy was 96. Anyways, seminars will never be the same again.
