Monthly Archives: November 2009

Airport barbecue

You can never get good pizza in the airport, or deli, or burritos, or Chinese. But I just ate yet another satisfying airport meat-and-two-sides, this one at the Speedway Grill in Charlotte-Douglas, where we’re paused on our long way home from Thanksgiving. It was good enough that CJ, to my great surprise, demanded more than his share of green beans; and it was only the second best airport BBQ I’ve had this month. (Brookwood Farms BBQ in Raleigh-Durham takes the crown here.)

Why does barbecue translate so well to the airport setting?

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Local points on quadratic twists of X_0(N)

A new paper on the subject by my Ph.D. student, Ekin Ozman, is up on the arXiv today.

The twists in question are isomorphic to X_0(N) over a quadratic field K = Q(sqrt(d)), but not over Q; the twist is via the Atkin-Lehner involution w_N, which is to say that the rational points on such a twist are in bijection with the points P of X_0(N)(K) satisfying

P^\sigma = w_N P

where \sigma is the generator of Gal(K/Q).  Call this twist X^d(N).

Why do we care about these twists?  For one thing, X^d(N) parametrizes certain classes of Q-curves, elliptic curves over Qbar which are isogenous to all their Galois conjugates.  These guys turn out to be more flexible than elliptic curves over Q for constructing “Frey curves” attached to Diophantine equations, but have all the same handsome modularity properties as elliptic curves over Q, allowing the usual Mazur-Frey-Serre-Ribet-Wiles style argument to go through.

More generally, though, the X^d(N) form a very natural class of modular curves whose arithmetic hasn’t been much thought about.  For instance:  is there an analogue of Mazur’s theorem?  That is, do these curves have rational points?  Here one immediately encounters a divergence from the untwisted case — X^d(N) might not even have local points!  The cusps, which provide Q_p-points on X_0(N) for every N and p, are not rational points on X^d(N); without them, there’s no reason for X_0(N)(Q_p) to be nonempty.  In a 2004 survey paper about Q-curves, I asked (Problem A in the linked paper) for which d and N the curve X^d(N) had local points everywhere; this certainly has to be settled before any investigation of the global points can start.  Pete Clark (see the appendix to this paper), Jordi Quer, and Josep Gonzalez got partial results on the problem; now Ekin has almost entirely solved it, getting an exact criterion for the existence of local points whenever K and Q(sqrt(-N)) have no common primes of ramification.

I was never able to see a simple way to do this — and it turns out that’s because the answer, as Ekin works it out, is actually pretty complicated!  So I won’t state her theorem here; I’ll just say that it’s competely explicit and it allows you to compute just about whatever you want.  For example:  the number of squarefree d < X such that X^d(17) has local points everywhere is on order of X / (\log X)^{5/8}.  (She could compute the constant, too, if for some terrifying reason you needed it…)

In the end, a lot of the X^d(N) have local points everywhere; and because they are all covers of the quotient X_0(N)/w_N, which has finitely many rational points once N is big enough, many of them don’t have any global rational points.  In other words, you have a healthy population of curves violating the Hasse Principle.  (This observation is due to Clark.)  Are these failures of the Hasse principle always due, as some people expect, to the Brauer-Manin obstruction?  In the last section of the paper,  Ekin works out one case in full — namely, X^{17}(23) — and shows that it is indeed so.

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well, did we, okay did we determine that Moebius maps were like isometries or whatever?

From the Michigan Corpus of Academic Spoken English, a 16,000 word transcript of an undergraduate math study session.  In case you ever wanted to know what it really sounds like when students work on our homework.

S1: what if- what if A plus B, equals two times Y and C plus D equals two? [S3: yeah. ] it just has to be proportional so you can’t break it up… but if we have A and C being whatever, then let’s make them something that works.

S2: like one?

S1: let’s… like what if you made, A equal Z and C equal one or something.

S2: but they can’t equal whatever because in the bottom A over C has to equal Z.

S1: i know. [S2: okay ] you make it so that it works.

S2: so you want A to be equal to Z, and C to be equal to one.

S1: okay, so what if we do that…? well no then that gives us uh, Z in the Y equation. unless B equals like Y minus Z or something well it could be done… it’s gonna get complicated though… so if A equals Z,

S2: i think this sucks.

Continue reading

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Cinnamon garlic eggs

CJ wanted scrambled eggs for lunch today.  While we were making them he said he thought they needed some spices.  But which ones?  He went over to the cabinet and picked out some cinnamon (“because that will make the eggs really sweet.”)  Then he dithered a bit between dried dill and garlic powder, but decided on the latter (“because that will give the eggs a nice spice.”)  A little black pepper, too, as usual.

And you know what?  Cinnamon garlic eggs are awesome. Per Google we are the first people ever to make them.  Don’t let us be the last!

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Mihailescu on the Leopoldt conjecture

You might have heard that Preda Mihailescu claimed a proof of Leopoldt’s conjecture earlier this year.  He just gave a series of lectures on the paper at Cambridge, and Minhyong Kim was there to blog about it.  Mihailescu has provided an “executive summary” of the argument.

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Things I now know how to do: signal apology from inside a car

I mentioned a couple of months ago I didn’t know how to convey the sentiment “I’m sorry” while driving.  Why didn’t I think of this? (via MetaFilter.)

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Disks in a box, update

In April, I blogged about the space of small disks in a box.  One question I mentioned there was the following:  if C(n,r) is the space of configurations of n non-overlapping disks of radius r in a box of sidelength 1, what kind of upper bounds on r assure that C(n,r) is connected?  A recent preprint of Matthew Kahle gives some insight into this question: he produces configurations of disks which are stable (each disk is hemmed in by its neighbors) with r on order of 1/n.  (In particular, the density of such configurations goes to 0 as n goes to infinity.)  Note that Kahle’s configurations are not obviously isolated points in C(n,r); it could be, and Kahle suggests it is likely to be, that his configurations can be deformed by moving several disks at once.

Also appearing in Kahle’s paper is the stable 5-disk configuration at left; this one is in fact an isolated point in C(5,r).

More Kahle: another recent paper, with Babson and Hoffman, features the theorem that a random 2-complex on n vertices, where the edges are all present and each 2-face appears with probability p, transitions from non-simply-connected to simply connected when p crosses n^{-1/2}.  This is in sharp contrast with the H_1 of the complex with Z/ ell Z coefficients, which disappears almost surely once p exceeds 2 log n / n, by a result of Meshulam and Wallach.  So in some huge range, the fundamental group is almost surely a big group with no nontrivial abelian quotient!  (I guess this doesn’t formally follow from Meshulam-Wallach unless you have some reasonable uniformity in ell…)

One naturally wonders:  Let pi_1(n,p) be the fundamental group of a random 2-complex on n vertices with facial probability p.   If G is a finite simple group, what is the expected number of surjections from pi_1(n,p) to G?  Does it sharply transition from nonzero to zero?  Is there a range of p in which pi_1(n,p) is almost certainly an infinite group with no finite quotients?

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What to do in talks

Jason Starr, in comments, makes the excellent point that listing good things to do in talks helps society more than listing bad things.  Here’s his list:

  • Tell a joke.
  • Answer good questions from the audience.
  • Give a simple example before giving a difficult one.
  • Explain some history.
  • Explain why some famous problem is hard.
  • Break the chalk so it doesn’t squeak.
  • Repeat a soft-spoken audience member’s question / remark so everybody hears it.

Good stuff!  In the same spirit, here’s a tip sheet I wrote I few years ago for grad students giving talks at the Graduate Student Conference in Number Theory.  The first tip on the list is “Tell a story.” and I stand by that placement.

Contribute your own must-do’s in comments!

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What not to do in talks

Fearing Tavern‘s comment on the previous post links to his exhaustive list of “things not to do in a conference talk”:

use cliche’ expressions
include unecessary equations
read paragraphs from slides
animate unecessarily (vis-a-vis powerpoint)
lie
get caught lying
read formula’s out loud
ignore your time requirement
cry (or sound as though you will)
assume nontrivial background knowledge
mispronounce people’s names
mumble
use a separate laptop from previous speaker (often causes technical difficulties)
forget to conclude
change notation
use .AVI movies (or anything else specifically Windows)
pander to famous audience members
be afraid to ask for clarification on audience member questions

I think I’ve done seven of these, though I won’t reveal which.  What about you?

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In which Athens, GA is great

Blogging from the airport shuttle on my way from Athens to ATL. “Radio Free Europe” on the radio, on a station whose bumpers bill it as “the college football voice of the south.”

Things I liked about Athens:

Grits at Five Star Cafe.

Steak and gravy (roughly equivalent to chicken fried steak) at Weaver D’s (“Automatic For The People.”) White, very peppery gravy, as it should be. After lunch, coffee next door at Jittery Joe’s, a coffee roaster/bike shop/used book store in a converted barn.

The great young group in number theory and algebraic geometry at UGA, almost all hired in the last three years.

Cheesecake with green tomato relish at Five and Ten. I am less enthusiastic about dessert than most people but this was the best thing I ate yesterday.

R.E.M tourism. Had the shuttle driver take us by the abandoned church where they used to practice in 1981. And I went to the 40 Watt to see an Athens band, Twin Tigers, play. It’s true what they say about music in Athens; the place was much bigger than the High Noon and was packed, even for the first band. And not just with tragic indie kids with ironic mustaches. I think regular kids in Athens go to indie rock shows, kids in fraternities who listen to the college football voice of the south. Maybe there’s a counterculture of tragic country and western kids who run the college radio station!

As for Twin Tigers: loud washes of sound which was boring at first but which won me over. Given the style of guitar-playing, surprisingly non-deadpan vocals, heavily reverbed a la Simple Minds. “Automatic” was the standout track. Don’t know how to include links when blogging by phone, but I think you can download this free.

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