Friday, April 18, 2003

Two From the Times Check out this article from The New York Times today. It describes the effects of the recent hostilities in the Middle East on the work of American scholars. Iraq and Pakistan, for example, house some of the richest fossil beds on Earth; an invaluable resource for paleontologists. The best examples of transitional forms linking whales to land animals (fossils which provide a convenient counterexample to creationist claims that there are no transitional forms) were found in the Middle East. And, obviously, the entire area is a treasure trove for archaeologists. On top of this, many of Iraq's historical treasures have been looted in the days following the war. Of course, the inability of a few scholars to carry out their research does not compare to the suffering of the Iraqi people, first under Saddam Hussein and now under the current wave of anarchy. But it does provide another perspective on the goings-on in that part of the world.

And while you're at the Times website, don't miss Paul Krugman's latest column here. He discusses the bogus environmental efforts of the Bush administration.

Thursday, April 17, 2003

Love and Diff-E-Q's A new book called The Mathematics of Marriage argues that the success or failure of a marriage is determined by a few differential equations. Mathematician Jordan Ellenberg provides an interesting review, along with a nice introduction to differential equations, in this Slate article.

Wednesday, April 16, 2003

Profiles in Right-Wing Courage Shameless congressional demagogue J.D. Hayworth (R-AZ), together with more than one hundred of his colleagues, has urged Columbia University president Lee Bollinger to fire Nicholas De Genova, an assistant professor of anthropology at the college. De Genova raised hackles when, during an impassioned anti-war statement at a recent teach-in, he openly wished for "a hundred Mogadishus". This was part of a larger argument against U.S. imperialism and our intervention in Iraq.

Of course, Hayworth doesn't really care too much about the hiring practices of Columbia University. He simply knows a good chest-pounding moment when he sees one. Harrumphing about the poorly reasoned musings of one obscure academic is a good way to impress his constituents, while simultaneously distracting the increasingly juvenile media from more important issues.

One thing the right does very well, aided by the media, is to elevate any ridiculous statement made by any leftist, no matter how obscure, to the level of a national scandal. They will holler about it until the media, cowed by incessant charges of liberal bias, feel compelled to cover it as actual news. Similarly vile statements on the right don't rate a mention.

A pro-Hayworth statement can be found at the Christian news source Agape Press, available here. Columbia's Department of Anthropology has issued a statement defending De Genova here.

Tuesday, April 15, 2003

The Poincare Conjecture The New York Times is reporting that a Russian mathematician named Grigori Perelman is announcing a proof of the Poincare conjecture. The article can be found here.

At the turn of the milennium, a panel of mathematicians was asked to identify the most important unsolved problems in mathematics. The Poincare conjecture made the list. A one million dollar prize has been offered by the Clay Mathematics Institute in Cambridge, Massachusetts to anyone who could solve it. If Perelman's work holds up (a very big if, given the number of false solutions that have been offered over the years), he will receive the prize (together with another mathematician, Richard Hamilton, a professor at Columbia University, on whose work Perelman has based his proof).

To understand what the Poincare conjecture asks, begin by considering the surface of a sphere. Draw a loop on this surface, and imagine that you gradually shrink the loop over the sphere's surface. The loop can be shrunk indefinitely, until it gets down to a single point. Contrast this with the surface of a doughnut. Here it is possible to draw a closed loop on the surface for which it is impossible to shrink it down to a point. For example, imagine that you grab the doughnut with your thumb and index finger, so that your fingers meet through the hole of the doughnut. The loop formed by your thumb and index finger can not be continuously deformed to a single point. Not if we're confined to moving around on the doughnut's surface, anyway. Similarly, if you tie a loop of string around the doughnut like a belt, then this loop also can not be deformed.

The surface of a sphere and the surface of a torus are examples of two-dimensional surfaces. It turns out that the sphere is essentially the only two-dimensional surface on which it is possible to shrink any loop to a point. But wait! What about a cube, or a tetrahedron? Well, it is indeed possible to shrink any loop to a point on the surface of a cube or a tetrahedron. However, mathematicians regard such surfaces as equivalent to a sphere. Imagine that your cube was made of rubber. It would then be possible to smooth out the corners of the cube, and gradually mold the whole surface until a sphere emerged. Any two surfaces that can be deformed into one another by such a "rubber-sheet" transformation are viewed as equivalent. The rubber can be pulled and stretched and molded at will, with the sole restriction being that the rubber must never be torn or cut. A sphere can not be deformed into a torus by such a transformation, so they are not equivalent.

But why should we stop at two-dimensions? Though they are rather hard to picture, it is a simple matter to define higher-dimensional surfaces, and in each dimension there is a surface that is clearly analagous to a sphere. Call that analog the n-dimensional sphere. The question is, if you are given an n-dimensional surface on which it is possible to shrink any closed loop to a point, is your surface necessarily equivalent to the n-dimensional sphere?

If the dimension is five or higher the answer is yes. That was shown by Stephen Smale in 1960. In 1981, Michael Freedman demonstrated that the answer is still yes in dimension four. Smale and Freedman both received Field's Medals (the mathematical equivalent of the Nobel Prize) for their troubles.

All of which should make the attentive reader wonder about dimension three. That's the Poincare conjecture. Specifically, the conjecture is that the answer remains yes in the incredibly complicated case of three-dimensional manifolds, and that is what Dr. Perelman believes he has shown.

Many proposed proofs of the conjecture have come and gone, all of them suffering from subtle flaws. The history of the conjecture is replete with people who devoted huge chunks of their lives to it, only to wind up defeated and insane. Hopefully Dr. Perelman will avoid that fate.

Monday, April 14, 2003

Life on Mars? Once again The Onion gets it exactly right. Check out this article entitled "Mean Scientists Dash Hopes of Life on Mars. This precisely the sort of crap real scientists put up with on a daily basis...

The Human Genome Project The Los Angeles Times is reporting that the human genome has now been sequenced to a 99.9% level of accuracy. The article can be found here.

Sequencing the genome will undoubtedly affect the way we view medicine, genetics and evolution. Precisely what that effect will be is an open question, however.

Sunday, April 13, 2003

Conventional Wisdom Fails Again... I dimly recall learning to tie my shoe as a child. For the longest time it seemed to me that the second loop appeared magically from out of the ether, and it was quite some time before I mastered the techinique. I had to settle for making two loops and crossing them. This difficulty with shoe-tying was matched many years later by my even greater frustration over learning how to tie a necktie...

Well, guess what? Turns out the conventional methods of shoe-tying are hardly the most efficient. An Australian mathematician has proven that more efficient methods exist, though they seem rather more complicated than the conventional way. Discover magazine has a short description of this work here. Follow the link and proceed to the R&D section.

Sequencing SARS According to this New York Times article, scientists working in the United States and Canada have successfully sequenced the genome of the virus believed to cause SARS, the flu-like disease that has been spreading at an alarming rate over the last three weeks. The bug in question is apparently a new kind of coronavirus, one previously unknown either in plants or animals. It is hoped that understanding the virus' genome could lead to effective treatments against it, or at least make it possible to control the vector spreading it. The article notes the relationship between the present virus and the hantavirus that afflicted the American Southwest a decade ago. In that case knowing the structure of the virus made it clear that it was being spread by rodents, and deer mice were quickly identified as the culprits. Controlling the deer mice population brought the disease under control.

It is worth pointing out that all of this research has taken place in government-funded laboratories in the United States and Canada. Looks like government can do something right after all.