Poincare's Prize

I recently read Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles by George C. Szpiro. I recommend it highly. Some time back I recommended another book on the same topic, The Poincaré Conjecture: In Search of the Shape of the Universe by Donal O'Shea. If you can only read one book on the topic, I recommend the Szpiro book.

Both authors are fine writers. The books are of similar length. O'Shea's book is 200 pages followed by 72 pages of supplementary material: endnotes, two glossaries, a timeline, and an 11-page bibliography. Szpiro's book is 262 pages followed by 32 pages of endnotes and bibliography. Each book provides a different interesting aspect of Poincarés life: Szpiro's book relates Poincarés career as a mining engineer, in the course of which he exhibited great personal courage and deductive ability worthy of Sherlock Holmes to investigate a mining disaster. O'Shea spends a fairly lengthy chapter on the Klein-Poincaré correspondence which has been put forth as an example of the way academics can cooperate even when their countries are mortal enemies. O'Shea's careful reading shows the antagonism simmering beneath the surface of their "polite" academic discussion.

Szpiro introduces a great deal of the mathematics that led to the proof of the conjecture by Grigori Perelman in 2002, almost 100 years after Poincaré made the conjecture. He illustrates the math by very clever analogies, avoiding any attempt to go to deeply into the mathematics, which it seems to me is the only way to present material of such awesome complexity and abstraction to a lay audience (in which group I include myself.)

Like O'Shea, Szpiro shows mathematicians warts and all, as he discusses priority disputes such as the Smale-Stallings-Zeeman controversy of the proof of the Poincaré conjecture in higher dimensions (which preceded the proof of the original three-dimensional conjecure). He is not afraid of picking sides: He argues that Smale deserves credit for the proof, but that his abrasive personality made it difficult for him to get help in establishing priority.

Nor is Szpiro shy in assigning full credit for the final proof to Perelman, though standing on the shoulders of many giants, especially William Thurston and Richard Hamilton. Perelman has been pictured as an eccentric loner, refusing the Fields Medal and the $1,000,000 Millennium prize for no good reason. Szpiro sees him as a man of utmost integrity and great friendliness to those who share his seriousness. It is not surprising, then, that Szpiro takes the great Chinese mathematician Yau Shing-Tung to task for pushing the claims of his students Cao and Zhu, who wrote a paper in which they claimed to given the first real proof of the conjecture, based on Perelman's "outline".

If you are interested in mathematics, you owe it to yourself to read either Szpiro's or O'Shea's book on the Poincaré conjecture.

Photographs of mathematicians

In his review of Mariana Cook’s new book, Mathematicians: An Outer View of the Inner World, Boston Globe writer Mark Feeney writes "There has yet to be a mathematician maudit, or a Byronic mathematician (other, that is, than Byron’s daughter, Ada)." To which I reply, "What about Evariste Galois?"

The book is 92 black-and-white portraits of mathematicians, and looks quite interesting.

My Want Ad

I've decided to move ahead with some ideas I've been developing on teaching mathematics to individuals with mathematics anxiety. The object is to develop and sell a software-based product that can be used by adults in their own home. I believe that much mathematics anxiety in adults is a form of PTSD (post traumatic stress disorder) and needs to be addressed before mathematics content can be mastered. My program would teach students relaxation techniques they need to use before attempting mathematics lessons. The lessons themselves would be tailored for adults likely to experience stress in learning mathematics.

I have placed an advertisement looking for help (no pay yet) with a start-up business to bring this all about. So far, this advertisement has been sent out to Acton Networkers, a local group of mostly technically savvy job seekers. For a few more details, see http://www.scribd.com/doc/17717501/Advertisement-for-StartUp-Workers.

Learning Theory and Mathematics

I am starting work on a project to develop self-study materials to teach mathematics to adults with math anxiety, phobias, or just plain stress. I've recently been looking into methods that involve getting the student into a state of relaxed awareness prior to a session of math awareness. These use music (at about 1 beat/second, such as Baroque music), yogic breathing techniques, or other methods to teach the student to synchronize body and mind and facilitate communication between the two brain hemispheres. Claims for these techniques are astounding. They have mostly been used in teaching language, or other subjects where memory is paramount.

These approaches have been dismissed as pseudoscience by some, but there seems to be quite a lot of evidence that they work. See the "suggestopedia" method of Georgi Lozanov (http://en.wikipedia.org/wiki/Suggestopedia) or the Institute of HeartMath (http://www.heartmath.org/education/overview.html).

I would love to hear from anyone who has experience in these methods, or in related ones, particularly as applied to mathematics learning.

E13. The Very Bad Key Drive

The following problem was a joint effort. I don’t know the original author. The original version, involving a poisoned keg of wine, was passed on to me by Arshag Hashian of Northeastern via Sandy Blank. My nephew, Michael Ash, modified the statement of the problem to make it politically correct. My sister, Arlene Ash, improved the exposition of my solution.

Here it is:

You have five expendable computers and 240 key drives. Exactly one of the drives has a very bad problem. Any computer that has mounted the bad drive during the previous day will be destroyed when the cron system maintenance runs on the computer at midnight.

You need to use the data on the 239 good drives on a nonexpendable computer in 48 hours and so you can have only two rounds of testing. How can you determine which is the bad drive?

For the solution, click here.

E12. Ladder against a wall (Part II)

Here is another ladder against a wall problem, from Coxeter's classic Introduction to Geometry. It is somewhat atypical of the book in that the interesting part seems to be the algebra, rather than the geometry. To make this more of a challenge, try to do it without using a CAS.

A 24-foot long ladder rests against the horizontal ground and a vertical wall in such a way that it touches a cube. The cube is 7 feet on a side and is placed flat on the ground, touching the wall. Find the height of the top of the ladder.

This is a two-dimensional problem which could be stated a little less colorfully in terms of a square and a line segment. It's easy to see that there must be (at least) two answers because of the symmetry of the problem; if a line segment of length 24 passes through (7,7) with endpoints on the positive coordinate axes, the reflection of that line segment in the line y = x will satisfy the same conditions. I found several ways of setting up the problem, all of which result in a 4th degree equation. However, the solutions are quadratic irrationals. In one approach, the biquadratic factors into two quadratics with integer coefficients. In another, a somewhat obvious substitution does the trick.

E11. Ladder against a wall

A ladder is placed against a (vertical) wall and the bottom of the ladder is moved away along the (horizontal) ground. What is the shape of the curve traced by the midpoint of the ladder?

It is very easy to work out the answer to this problem, and I won't bother to do that here. If you haven't seen the problem before, test your intuition. Try to sketch what you think curve looks like before solving the problem. (In particular, is the curve concave up or concave down?) The first time I saw this, my intuition was wrong.