Orders on Coxeter Groups — with a Problem Set! January 30, 2009
Posted by David Speyer in Uncategorized.10 comments
John Mangual asks:
So, David, how do weak order and bruhat order work on the permutation group, S_n ? As I understand, Coxeter groups are (finitely generated?) reflection groups and your description of a particular element depends on your choice of presentation. These two orders have to do with the length of the “reduced” word representing a particular element. I still don’t totally get how this works. Also, do they form a distributive lattice under these orders?
That’s a great question, but I have a talk and two papers to work on (and at least one of my coauthors is reading this blog!) So I’ll make it a problem set. If you want to work out the case of weak order on 
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Frank Morgan has a blog January 30, 2009
Posted by Ben Webster in Uncategorized.add a comment
Here. When he becomes AMS vice-president on January 31 (I’ll just note, the AMS has 3 vice-presidents; presumably they have to do “Rock, Paper, Scissors” to figure out who takes over if the current president resigns), that will probably make the first AMS officer to also be a Blogger-American. Not to mention probably the first AMS officer who has been filmed doing the “Where the Hell is Matt? dance.
Definitions: Weak or Strong? January 29, 2009
Posted by David Speyer in Uncategorized.20 comments
I am currently writing a paper where I need to define an antimatroid, which is a combinatorial structure that can be axiomatized in several equivalent ways.
In the axiom system I am using, an antimatroid consists of a finite set 
Of course, I could equivalently say
My preference is for 
The (lack of) effect of birthdates on Fields Medals January 28, 2009
Posted by Ben Webster in fields medals.9 comments
I’ll just note, in response to Michael’s post on how birthdates affect Fields Medals: If you actually look at this data, there’s very little support for this idea, given that most people who’ve received Fields Medals in the last 40 years have not only not been just barely elligible, but a small majority of them would have been eligible at the next ICM! In both 1978 and 1986, every single person who the Fields Medal would been eligible in the next go around. 7 of them (Tao, Baker, Novikov, Margulis, Fefferman, Donaldson and Faltings) had 2 more cycles of eligibility.
I’ve got to admit, I was a little surprised by this data, though interestingly young Fields Medalists seemed to peak around the early 80′s, exactly when the small wartime generation would have been turning 40. In recent years, this has been less pronounced (though 3 out of 6 in 2002 and 2006 would have been eligible 4 years later, so this hasn’t entirely changed).
Incidentally, by this logic, I am horrible positioned to get a Fields Medal. I’m 3 weeks too old to get one in 2022. If I do something really amazing in 2019, I’m going to be annoyed (not really. After all, awesome things are worth doing for their own sakes. On some level, being “the guy who would have gotten a Fields Medal if he’d been 3 weeks younger” is more memorable and unique than being “the guy who won a Fields Medal.” After all, there are lots of those).
A couple of links January 27, 2009
Posted by Ben Webster in Uncategorized.3 comments
I’m just pointing this out since it’s a new comment at the end of an old thread, but those interested in the el Naschie situation may want to read this comment. If this is true, then it is going to make an excellent TV movie.
In other news, Tim Gowers asks “Is massively collaborative mathematics possible?” His opinion seems to be, and I agree, that the answer is that it is possible, becoming more so every day, and in fact, quite desirable. I mostly agree; it reminds me of some of the things I wrote previously about sharing ones ideas.
Welcome (welcome) welcome! To the MIT Mystery Hunt! January 23, 2009
Posted by David Speyer in Uncategorized.4 comments
Noah and I spent the weekend at the MIT Mystery Hunt, a weekend long puzzle competition held at MIT each Martin Luther King day weekend. Teams of 20-120 people work to solve an astonishing collection of puzzles, starting at noon Friday and going until sometime on Sunday. There are puzzles of every sort — wordplay, crosswords, logic, bizarre images, scavenger hunts and new puzzle forms invented for the day — many of whose answers feed into other puzzles. The puzzles and solutions are now available: this page explains the plot of the hunt, while this page shows you all of the puzzles as we saw them.
I thought I’d single out some of the puzzles that might be particularly enjoyable to the mathematically minded. In each case, the puzzle should output a word or short phrase. My descriptions below contain slight spoilers, in order to be friendlier to people who are not familiar with the Hunt, but I never spoil the interesting part of the puzzle.
arXiv-stalking January 23, 2009
Posted by Ben Webster in blegs, the arXiv.10 comments
At some point in my graduate career, some kind person told me that I should be subscribing to arXiv emails. This was, of course, a big improvement over not paying attention to what was on the arXiv at all, but still more trouble than it should be. the arXiv classification is way too crude for my purposes, and I just get overwhelmed.
What would be much better is if I could get emails compiling submissions of particular authors (effectively, arXiv-stalking them). Anyone have ideas on how to do this?
EDIT: I ended up writing a Yahoo Pipe that does this. You can use it yourself here, and if you clone it, you can use your own list instead of mine. I’ve also included some keyword sorting, to address the concerns of David and Allen.
Stimulating mathematics January 16, 2009
Posted by Ben Webster in inside baseball, jobs.10 comments
So, a draft of the stimulus package is making its way around the internets.
Notable for academic mathematicians:
- 3 billion to the NSF; 2 billion of that specifically for “employment opportunities.” Under current allocations, math’s share would be between $100 million and $50 million.
- A $500 increase in the Pell Grant.
- Billions in direct aid to state universities (couldn’t find a specific number on that one; $39 billion to all levels of education).
The real question is: will any of this money make it into the job market this year? If it does, in what form? Does giving out more grants count as “employment opportunities” or does that mean a lot of people being directly employed by the NSF (a second round of postdoctoral and graduate fellowships when the stimulus comes through, perhaps)?
Does anyone more informed in the ways of the NSF than me have ideas?
Lurie on TFTs January 15, 2009
Posted by Chris Schommer-Pries in Algebraic Topology, groupoids, QFT, tqft.2 comments
This morning Jacob Lurie posted a draft of an expository paper on his work (with Mike Hopkins) classifying extended (infinity, n)-categorical topological field theories and their relation to the Baez-Dolan cobordism hypothesis.
Should make for some intersting bedtime reading…
Generalized moonshine I: Genus zero functions January 8, 2009
Posted by Scott Carnahan in group theory, mathematical physics, Number theory, Paper Advertisement, representation theory.21 comments
This is a plug for my first arXiv preprint, 0812.3440. It didn’t really exist as an independent entity until about a month ago, when I got a little frustrated writing a larger paper and decided to package some results separately. It is the first in a series of n (where n is about five right now), attacking the generalized moonshine conjecture. Perhaps the most significant result is that nontrivial replicable functions of finite order with algebraic integer coefficients are genus zero modular functions. This answers a question that has been floating around the moonshine community for about 30 years.
Moonshine originated in the 1970s, when some mathematicians noticed apparent numerical coincidences between the theory of modular functions and the theory of finite simple groups. Most notable was McKay’s observation that 196884=196883+1, where the number on the left is the first nontrivial Fourier coefficient of the modular function j, which classifies complex elliptic curves, and the numbers on the right are the dimensions of the smallest irreducible representations of the largest sporadic finite simple group, called the monster. Modular functions and finite group theory were two areas of mathematics that were not previously thought to be deeply related, so this came as a bit of a surprise. Conway and Norton encoded the above equation together with other calculations by Thompson and themselves in the Monstrous Moonshine Conjecture, which was proved by Borcherds around 1992.
I was curious about the use of the word “moonshine” here, so I looked it up in the Oxford English Dictionary. There are essentially four definitions:
- Light from the moon, presumably reflected from the sun (1425)
- Appearance without substance, foolish talk (1468 – originally “moonshine in the water”)
- A base of rosewater and sugar, or a sweet pudding (1558 cookbook!)
- Smuggled or illegally distilled alcoholic liquor (1782)
The fourth and most recent definition seems to be the most commonly used among people I know. The second definition is what gets applied to the monster, and as far as I can tell, its use is confined to English people over 60. It seems to be most popularly known among scientists through a quote by Rutherford concerning the viability of atomic power.
I’ll give a brief explanation of monstrous moonshine, generalized moonshine, and my paper below the fold. There is a question at the bottom, so if you get tired, you should skip to that.
