This month's topics:

Too good at math?

According to Slate Magazine's Fred Kaplan (posting updated May 18, 2007), Paul Wolfowitz' problem is that "He's too good at math." Wolfowitz, the former Deputy Secretary of Defense and the soon-to-be former President of the World Bank, majored in mathematics and chemistry at Cornell (his Ph.D. is in political science). His excellence in math is in fact a matter of record: according to Anil Nerode "Paul was one of the two or three smartest math students I've ever seen." (Quoted by David Dudley in the Cornell Alumni Magazine Online). How this talent is a flaw is not clear from Kaplan's analysis, which begins by positing "... judgment and character trump dedication and belief ..." as the root explanation for Wolfowitz's downfall. Judgment and character, dedication and belief, both pairs of traits are quite independent of mathematical ability. What the Slate piece shows in fact is that Kaplan himself has had some disagreeable experiences with mathematicians: "In math, methodologies and answers are right or wrong, and those who choose the wrong ones are properly ignored or savagely dismissed. Mathematicians who enter the political realm tend to retain this attitude." Perhaps at the undergraduate level, where answers can be checked at the back of the book, one can reduce everything to "methodologies and answers are right or wrong." Kaplan does not seem to realize that the universe of a working mathematician is much more like real life, where one struggles to disentangle what is true from what one would like to be true.

Mathematical patterns in songs

One of the videos generated by the the 2007 New Yorker Conference has their staff writer Malcolm Gladwell interviewing Mike McCready, whose company, Platinum Blue, has developed computer algorithms for analyzing songs. In the interview, McCready describes (in rather non-specific terms) how Platinum Blue's software has identified 30 quantifiable elements in the makeup of a song which are significant enough for neighborhoods in this 30-dimensional space to be commercially exploitable. For example, a pop hit will fall with very high probability into one of some 60 clusters. Taking the points corresponding to the songs in an album and overlaying them on the display of hit clusters can help a producer identify which track should be released as a single. The ultimate commercial application, McCready believes, will be a personal recommendation service, where Platinum Blue takes a set of your favorite songs (which can include classical items or music from exotic cultures) and generates a list of music titles which you are mathematically guaranteed to like, even though you may never have heard of them.

Curvature and the growth of cells

A mathematics article was published, April 26, 2007, in the general science journal Nature. This unusual occurrence is due to the prominence and wide applicability of the result. Robert MacPherson and David Srolovitz solved the 50-year old problem of generalizing to three dimensions John Von Neumann's work on the growth of cells in planar tesselations. The hypotheses in both cases are that cell walls move with a velocity proportional to their mean curvature, and that domain walls meet at 120°, hypotheses which are realized in many physical and biological contexts.
    Von Neumann showed that the rate of change dA/dt of the area A of such a cell can be expresed in terms of γ the surface tension of a domain wall, M a kinetic coefficient describing the walls' mobility and n the number of vertices where distinct walls intersect, by

dA/dt = –2πMγ(1 – n/6).

So for example in the tesselation portion shown in Fig. 1, the 8-vertex regions A and B will grow at the expense of the 2-vertex region C.

tesselation example

Fig. 1. With the common factor 2πMγ set to 1, Von Neumann's formula tells us that dA/dt = dB/dt = 1/3, while dC/dt = – 2/3.

   MacPherson and Srolovitz's formula for the rate of change of the volume of a domain D in a 3-dimensional tesselation is formally analogous but requires the new and ingeniously defined mean width L(D), which they describe as "a natural measure of the linear size" of D. In terms of L(D), their formula reads

\frac{dV}{dt} = -2\pi M \gamma ({\cal L} D) - \frac{1}{6}\sum_i e_i),
where ei is the length of the i-th 1-dimensional edge of D, and the sum is taken over all the edges. Note that following our initial requirement, faces meet 3 by 3 along an edge with dihedral angles 120°.

The mean width L(D) is computed in two steps. First, for each line through the origin, the Euler width of D along is the integral along of the Euler characteristic of the intersection of D with the plane perpendicular to (see Fig. 2):

.

So if D is convex (χ always = 1), is exactly the length of the projection of D on .

image from Nature

Fig. 2. For D a 3-dimensional domain, and a line through the origin, the Euler width of D along is calculated by measuring, for each point p on , the Euler characteristic of the intersection of D with the plane through p perpendicular to , and integrating along . Image from Nature 446, 1053-1055, used with permission.

Then L(D) is computed as twice , averaged over the space RP2 of lines through the origin:

,
where d is normalized to have total integral 1.
   The authors state that their formula and Von Neumann's are both special cases of a general n-dimensional formula, which they give. The Supplementary Information for their article (entitled "The von Neumann relation generalized to coarsening of three-dimensional microstructures") gives the proof of their 3-dimensional formula and rules for computing L(D); for example the cube of side length a has mean width 3a.