This month's topics:
Mathematicians, the NSA and domestic spying (cont.)The Chronicle of Higher Education has taken up the story with some useful background information. "Shunned as NSA Advisers, Academics Question Their Ties to the Agency," by Paul Basken, appeared on February 10, 2014. As Basken tells the story, the comfortable and productive relationship between the National Security Agency and mathematical academe, which had evolved during the Cold War, "began shifting during the 1990s, as the Internet rose in popularity. A series of intelligence failures exposed the NSA as too focused on intercepting satellite and radio signals, and insufficiently attuned to the growth of computerbased communications traveling in cables. To catch up, the NSA turned to technology companies and other corporate experts, and to even greater levels of secrecy." In particular the names of the members of the Advisory Board, now called the Emerging Technologies Panel, are secret; but "those inside and outside the NSA have said that academics have been left with a diminished role on the board." Baskin cites recent allegations that lead some to believe that the Agency has embarked on security strategies that are "selfdefeating in the long run" for "scientists and for the country as a whole." He refers to comments by Matthew M. Aid, a military historian specializing in the NSA, and suggests: "With fewer university advisers, the board may have lost some of the moderating influence that derives from a more holistic approach." Then there's this: "The NSA has extensive ties to the American Mathematical Society, using it to directly recruit workers for sabbaticals at the agency, and to help the agency finance college math programs and researchers in general." And "The mathematical society also administers the distribution of grants financed by both the NSA and the National Science Foundation" (a "correction" at the end of the piece, dated February 21, states that the grants in question "are travel grants, not research grants.") Homer Simpson, Math WizardSimon Singh's "The Simpsons and Their Mathematical Secrets" (Bloomsbury, 2013) was reviewed by Amir Alexander in the New York Times on January 28, 2014. The review ("Examining the Square Root of D'oh") starts with "Did you know that Homer Simpson disproved Fermat's last theorem? He did, or so it seemed, when he scribbled $3987^{12}+4365^{12}=4472^{12}$ on a blackboard in a 1998 episode of 'The Simpsons.'" The equations and diagrams on Homer's blackboard in Season 10, Episode 1 of "The Simpsons" span mathematical physics, number theory, cosmology and topology. Singh's book documents how references to highlevel mathematics have been woven into "The Simpsons" from the start (1989) of that popular cartoon sitcom, and tells us why. As Alexander puts it: "Perhaps the most surprising revelation is the composition of 'The Simpsons' creative team, which over the years has included J. Stewart Burns, who has a master's degree in math from Harvard; David X. Cohen (master's in computer science, University of California, Berkeley); and Ken Keeler (Ph.D. in applied math, Harvard). Most astonishing is Jeff Westbrook (Ph.D., computer science, Princeton), who was an associate professor at Yale before he joined the team." Alexander wonders: "Why would so many talented mathematicians forsake academia to write outrageous stories and gags for an animated TV show?" And comes up with an answer: "Think of it this way: To write an episode of 'The Simpsons,' one begins with a known set of characters Homer, Bart, Lisa, Marge and confronts them with a problem. The rest of the episode follows the characters through a complicated series of moves until the problem is resolved." "In their basic structure, then, mathematical reasoning and storytelling might not be so different after all. " As for the problematic equation $3987^{12}+4365^{12}=4472^{12}$ on Homer's blackboard, Alexander links us to Wordplay, the Times' crossword blog, where Gary Antonick checks it out on a calculator. The left and righthand sides both give $6.3976656\times 10^{43}$. "So the equation appears to be valid. But  it can't be, according to Fermat's famous last theorem, which says there are no three integers that can satisfy the following equation when $n$ is an integer greater than $2$. $$a^n + b^n = c^n.$$ Homer's values do seem to check out. But Fermat says they can't. What's going on?" Antonick left this as a challenge to his readers, and "Many commenters (e.g., Ravi in Hyderabad) correctly showed why the solution must be false, and suggested that it must be a nearmiss solution. In other words, the numbers are close enough to fool the limited display of the Google display, but a closer examination reveals a discrepancy." [Perhaps Ravi noticed that the lefthand side is divisible by 3, but the righthand side is not.] An exhaustive compilation of references to mathematics in "The Simpsons" has been maintained since about 2000 by Andrew Nestler (Santa Monica College) and Sarah Greenwald (Appalachian State University). Singh gives his own views on the phenomenon in an article in The Observer, September 21, 2013: "The Simpsons' secret formula: it's written by maths geeks." Flipping pancakes with mathematicsMore Simon Singh. In his blog on The Guardian's website he posted "Flipping pancakes with mathematics. Mathematical minds love a problem that's easy to pose but tough to solve" (November 14, 2013). The story is about the CCNY mathematician Jacob Goodman, and his analysis of the problem: how many flips does it take to arrange a stack of (differentsized, circular) pancakes according to size? For three pancakes, as many as three flips may be required to get them restacked by size. Image adapted from Singh's blog posting. According to Singh, when Goodman first published his work on the topic he used a pseudonym so as not to derail his ascension to tenure. In fact the maximum number $F(n)$ that can be required for a stack of $n$ pancakes is not known past $n=19$, although there are bounds: Bill Gates proved, in a 1979 paper with Christos Papadimitriou (EE, Berkeley) that $F(n) \leq (5n+5)/3$. Singh tells us that he learned about this problem during his research on "The Simpsons." David X. Cohen "casually mentioned that he had coauthored a paper titled, On the Problem of Sorting Burnt Pancakes." The difference is that each pancake must end up burnt side down. The neurophysiology of the perception of mathematical beauty"The experience of mathematical beauty and its neural correlates" appears in Frontiers of Human Neuroscience, February 13, 2014; among the authors is Michael Atiyah. From the Abstract: "... we used functional magnetic resonance imaging (fMRI) to image the activity in the brains of 15 mathematicians when they viewed mathematical formulae which they had individually rated as beautiful, indifferent or ugly. Results showed that the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbitofrontal cortex (mOFC), as the experience of beauty derived from other sources." A byproduct of the research was a kind of beauty contest among equations, since some produced higher "parameter" values than others. "The formula most consistently rated as beautiful ... was Leonhard Euler's identity $$1+e^{i\pi}=0;$$ the one most consistently rated as ugly was Srinivasa Ramanujan's infinite series for $1/\pi$, $$\frac{1}{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^{\infty}\frac{\displaystyle (4k)!(1103+26390k)} {\displaystyle (k!)^4 396^{4k}}.$$ Other highly rated equations included the Pythagorean identity, the identity between exponential and trigonometric functions derivable from Euler's formula for complex analysis, and the CauchyRiemann equations. Formulae commonly rated as neutral included Euler's formula for polyhedral triangulation, the Gauss Bonnet theorem and a formulation of the Spectral theorem. Low rated equations included Riemann's functional equation, the smallest number expressible as the sum of two cubes in two different ways, and an example of an exact sequence where the image of one morphism equals the kernel of the next." The complete set of equations presented to the subjects is Data Sheet 1.PDF. Beauty and Understanding. The authors attempt to disentangle a subject's perception of the beauty of an equation from the subject's evaluation of his or her understanding of the equation. There was, in fact, "an imperfect correlation between understanding and the experience of beauty." This leads them into wooly territory, and ".. to the capital question of whether beauty, even in so abstract an area as mathematics, is a pointer to what is true in nature, both within our nature and in the world in which we have evolved." They give as an example Hermann Weyl's work on gauge theories. "Rejected at first (by Einstein) because it was thought to conflict with experimental evidence, it came subsequently to be accepted but only after the advent of quantum mechanics, which led to a new interpretation of Weyl's equations. Hence the perceived beauty of his mathematical formulations ultimately predicted truths even before the full facts were known." This research was advertised in a press release, "Mathematical beauty activates the same brain region as great art or music," from University College, London. It was picked up in a BBC broadcast by James Gallagher on February 12: "Mathematics: Why the brain sees maths as beauty." The subtitle: "Brain scans show a complex string of numbers and letters in mathematical formulae can evoke the same sense of beauty as artistic masterpieces and music from the greatest composers."
Tony Phillips 
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