UCLA mathematician Terry Tao is one of the recipients of a 2006 Fields Medal. Among other things, he proved that the set of prime numbers contains arithmetic progressions of arbitrarily long length.

The other winners are Grisha Perelman, for his work on the Poincare conjecture; Andrei Okounkov, with whom I played soccer once at a conference; and Wendelin Werner.


Tito, you are officially killing me

Terry Francona again bats Coco Crisp in the leadoff, and Coco goes 0-for-6. (Nice bid on the Giambi homer though.)

Instead of bringing in Jon Papelbon coming on in the beginning of the 8th, Tito brings in Mike Timlin to give up a hit to Johnny Damon and hit Derek Jeter. Tito then brings in Javier Lopez to walk Bobby Abreu. So instead of Papelbon getting 6 outs with the bases empty, he has to get 6 outs with the bases loaded.

With Manny Ramirez and David Ortiz on 1st and 2nd in the bottom of the 9th, tie game, none out, Tito orders Kevin Youkilis to bunt instead of either letting him hit or bringing in a sub to bunt. Predictably, the guy with the .420 OBP and no real bunting expertise bunts into a force-out at 3rd, eventually allowing the Yankees to wriggle out of the 9th unscathed.



Lousy international governing bodies

The International Astronomical Union has decided that the asteroid Ceres, the recently discovered rock Xena, and Pluto's moon Charon are now planets. This ensures a huge profit for publishers of science texts over the next couple years and many more headaches for helpless fourth-grade kids. WON'T SOMEBODY PLEASE THINK OF THE CHILDREN????


You can't say that in Germany

Germany has some of the most restrictive (ridiculous) hate speech laws in the known universe. Apparently the government is mulling the question of whether the mock-crucifixion act in Madonna's concerts violates these laws.

If so, I'd imagine that the following songs should also be banned in Germany:

The Beatles, The Ballad of John and Yoko
Live, Selling the Drama
Billy Joel, Only The Good Die Young

Feel free to add to this list in the comments. This could be almost as fun as the list of songs that were declared inappropriate by ClearChannel in the wake of 9/11.

The Poincare conjecture

There's an article in today's New York Times about the Poincare conjecture, and how it was solved (or at least thought to have been solved) by Grisha Perelman.

For the non-geometers in the audience, the Poincare conjecture asserts that if X is a smooth orientable 3-dimensional space that has the property that any closed non-self-intersecting loop can be shrunk to a point, then X is a 3-sphere. (To see that this is a nontrivial statement, think about drawing a loop on the surface of a donut.) Such a space is said to be "simply connected".

The basic technique that Perelman uses is Ricci curvature flow. Curvature is fairly easy to explain for 2-dimensional spaces sitting in our ordinary 3-space. Pick up a ball and pick any point on it. Draw a curve on the ball going through the point. The curve bends in a certain direction. Now draw another curve going through the point, but perpendicular to the first curve. The second curve bends in the same direction as the first one. The curvature of the sphere at that point is positive; in fact, the curvature of a sphere at any point is positive.

Now pick up a Pringle, and pick a point on it. If you draw two curves through the point, they'll bend in opposite directions. The curvature of the Pringle at any point is negative.

The neat thing is that the curvature of a surface, which is a local property of each point on the surface, tells you something about the global topology of the surface. If you have a closed, orientable 2-dimensional surface and its average curvature is positive, it must be topologically equivalent (smoothly deformable) to a sphere. ("Average" here means "integrate a 2-form and divide by a constant", but that's a mouthful.) If its average curvature is zero, it must be topologically equivalent to a donut with one hole. (Pick up a donut; the points on the outside rim have positive curvature, while the points on the inside rim have negative curvature.) If its average curvature is negative, it must be topologically equivalent to a donut with more than one hole. This is essentially the Gauss-Bonnet theorem.

Sadly, the previous paragraph's formulation of curvature is insufficient for more general situations -- specifically, we need a notion of curvature for higher-dimensional manifolds, and this notion of curvature should be intrinsic to the manifold instead of depending on the way it is situated in an ambient space. The modern formula of curvature is as a (1,3)-tensor, meaning that it is a function which takes three tangent vectors and spits out another tangent vector. Here's how it works: pick a point on the manifold, pick two tangent vectors, and draw a little square anchored at the point using the two vectors. Take a third vector, and transport it around the square. The curvature tensor tells you how much the third vector changes when you transport it. To see that this is not trivial, think about sitting at the Equator of the Earth carrying a vector that points north. Walk east around the Equator 90 degrees, then walk up to the North Pole, and finally walk back down to your starting point, all the while not changing the direction of the arrow.

The curvature tensor is a mouthful, so Ricci invented a contraction of it called the Ricci tensor. It's a (0,2) tensor, so it takes as input two tangent vectors and spits out a number. You lose some information this way, but it's a more managable object to deal with.

Oh, I need to tell you about the metric tensor on a general manifold. It too is a (0,2) tensor, meaning that you take two tangent vectors at a point, and it gives you a number. In our ordinary geometry this number would just be the product of the lengths of the vectors times the cosine of the angle between them. In abstract differential geometry, we need the metric tensor to define the notions of "length" and "angle" on our manifold, and in fact we need to have the metric tensor first in order to properly define the curvature tensor, but whatever.

Okay, so now we can sort of say what Ricci flow is. The Ricci flow simply causes the metric to change with time, and the rate of change of the metric tensor is given by the (negative of the) Ricci tensor. They're both (0,2)-tensors, so this makes sense mathematically. However, as the metric tensor changes, so does the Ricci tensor, so the Ricci flow equation becomes highly nonlinear and difficult to solve. In our previous discussion of 2-dimensional surfaces, what would happen is that the positive-curvature regions would shrink and the negative-curvature regions would grow. If you took a sphere, squished it so it had more of a dumbbell shape and then ran the Ricci flow, the thin neck of the dumbell would grow, while the round ends would shrink. Eventually the kinks would smooth out and the dumbbell would turn into a sphere, then start shrinking to a point.

Naturally, the Ricci flow is a bit more complicated in higher dimensions, but what Perelman essentially does is take the simply connected 3-manifold X, run the Ricci flow on it, perform surgery on the singularities that appear and resume the Ricci flow. Only finitely many surgeries are required in any finite time interval, and Perelman ends up studying the limit as t goes to infinity, then infers from the geometry of the limiting manifold that the original X was topologically a 3-sphere.


Daft sports rules

Apparently in Little League, every player on a team has to play 3 consecutive defensive outs and get one at-bat. If that doesn't happen, the team forfeits the game

Of course, what happens is that in a Little League World Series semifinal game, Vermont goes up 9-7 against New Hampshire in the bottom of the penultimate (fifth) inning but still has one player who hasn't entered the game.

New Hampshire gets a run back in the top of the sixth, but then Vermont figures out that they need to give up another run in order to give their last player an at-bat in the bottom of the inning. So the Vermont pitcher starts throwing the ball in the dirt, to the backstop, etc. However, the next three New Hampshire batters strike out intentionally in order to end the game and force Vermont to forfeit.

This is even worse than the soccer game between Barbados and Grenada. I wonder what happens if the Vermont pitcher simply refuses to pitch, thus preventing them from recording any outs?



Michigan Stadium is widely recognized as one of the least intimidating large stadiums in the US. Part of it is due to the design of the stadium -- the seating is very flat and shallow, allowing most of the noise that the crowd makes to escape. Part of it is also that Michigan alums are all wealthier and better-behaved than their counterparts at Ohio State, Tennessee, Florida, Florida State, LSU and other places.

Now, it appears that Michigan is going to add luxury boxes to its stadium. Le sigh.


How to defeat an airplane hijacker

Here are Steven den Beste's suggestions, from a few years ago. 'Cause you know, you can't use a broken beer bottle right now.


When will the PC era finally die?

There's a really long thread on the usage of politically correct terms over at the Volokh Conspiracy.

I find it really embarrassing when people of East Asian descent object to the use of the term "Oriental" when describing a person. If I recall correctly, the basic reason "Oriental" is considered offensive is basically because this bozo said so. Now given that "Oriental" was in fairly common usage up to, say, around 10 years ago, it seems rather insulting to imply that people who use the term today are subconsciously racist.

However, suppose for the moment that one accepts Said's argument that "Oriental" inherently means "the West is superior to the East". Even if this is the case, the term "Asian" doesn't solve the problem, as the term "Asia" is also thought to mean "from the East".

The lifespan of PC terms seems to be around 20-40 years (e.g. "handicapped"-"disabled"-"differently abled", or "Negro"-"African-American"-"black"). Maybe in 20 years "yellow" will become the preferred term. Who knows.