This month's topics:
Visualizing the curvature tensorA special section in Science for August 3, 2012 covered black holes, and included the "perspective" piece by Kip Thorne, "Classical Black Holes: The Nonlinear Dynamics of Curved Spacetime." Thorne begins by describing a way that he, Robert Owen and collaborators (Physical Review Letters 106 151101) have devised "to visualize the Riemann curvature tensor, which embodies the curvature of spacetime. Just as the electromagnetic field can be split into an electric field and a magnetic field, so the Riemann tensor can be split into a tidal field ${\cal E}$ that stretches and squeezes anything it encounters, and a framedrag ${\cal B}$ that twists adjacent inertial frames with respect to each other. ... Just as electric and magnetic fields can be visualized using field lines, ... so ${\cal E}$ and ${\cal B}$ can each be described by three orthogonal sets of field lines, called (tidal) tendex lines for ${\cal E}$ and (framedrag) vortex lines for ${\cal B}$." The tendex lines for a stationary (nonrotating) spherical object "which could be the Earth, Moon, Sun or a nonspinning black hole." A test object falling towards the object is stretched in the red directions and compressed in the blue. Along with its own pattern of tendex lines, a spinning mass has a framedragging effect on spacetime. This image shows the vortex lines around a black hole spinning at 95% of its maximum possible rate (black arrow: spin axis). For an object falling along a red line, gyroscopes at top and bottom would precess counterclockwise with respect to each other. Along a blue line, the relative precession would be clockwise. Images adapted from Science 337 536. Thorne uses this imagery to describe the results of simulations of blackhole collisions and close encounters. He remarks that the resulting patterns of gravitational waves should be observable by a new generation of detectors coming on line in 2017. Math in the plant physiology curriculumAn article in the online journal Bioscience Education addresses the perception that "Biology has often been considered the ideal career for students inclined to science but mathematically challenged" and the difficulty that Biology students face with test questions involving a formula, a graph, or a table. The authors (A. LLamas, F. Vila and A. Sanz) based their study on ten years of instruction in Plant Physiology at the University of Valencia, in Spain. They report that "the percentage of correct answers for questions requiring mathematical skills is 16% lower than for the corresponding nonmathematical questions." In particular, questions involving math skills are almost twice as likely to be left unanswered. The authors' interpretation is that students lack what they call "selfefficacy:" a familiarity with this kind of problem and some confidence that it can be mastered. They recommend strengthening the curriculum, "less in the sense of adding more subjects of mathematics, but rather in increasing their practical use in the various experimental disciplines that use them as a tool." Llamas, Vila and Sanz' work was picked up in the "Editor's Choice" section of Science, August 3, 2012, with the heading: "Math $+$ Science $=$ Success." Is Algebra Necessary?That was the title of an "opinion" piece published in the Sunday Review of the New York Times, July 29, 2012. The author, Andrew Hacker, is emeritus professor of political science at Queens College, CUNY. The lead illustration shows desperate hands emerging from a sea of formulas, as the text begins: "A typical American school day finds some six million high school students and two million college freshmen struggling with algebra. In both high school and college, all too many students are expected to fail. Why do we subject American students to this ordeal?" Hacker examines and dismisses some of the standard reasons; he suspects that "that institutions and occupations often install prerequisites just to look rigorous ... Mathematics is used as a hoop, a badge, a totem to impress outsiders and elevate a profession's status." Hacker distinguishes proficiency in algebra, which he exemplifies as the ability to prove $(x^2+y^2)^2 = (x^2y^2)^2 + (2xy)^2$, from quantitative literacy, which "clearly is useful in weighing all manner of public policies." "What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey." Along that line, he proposes that "mathematics teachers at every level could create exciting courses in ... 'citizen statistics'." These "would familiarize students with the kinds of numbers that describe and delineate our personal and public lives." He remarks, "More and more colleges are requiring courses in 'quantitative reasoning.' In fact, we should be starting that in kindergarten." At the same time, "mathematics departments can also create courses in the history and philosophy of their discipline, as well as its applications in early cultures. Why not mathematics in art and musiceven poetryalong with its role in assorted sciences? The aim would be to treat mathematics as a liberal art, making it as accessible and welcoming as sculpture or ballet." New strategies for the Prisoner's DilemmaAccording to an August 16, 2012 posting in the Physics Arxiv Blog of Technology Review ("The Emerging Revolution in Game Theory"), "The world of game theory is currently on fire." William Press (Computer Science, UTA) and Freeman Dyson (IAS) have discovered "a previously unknown strategy for the game of prisoner's dilemma which guarantees one player a better outcome than the other. That's a monumental surprise. Theorists have studied Prisoner's Dilemma for decades, using it as a model for the emergence of cooperation in nature. This work has had a profound impact on disciplines such as economics, evolutionary biology and, of course, game theory itself. The new result will have impact in all these areas and more." The blog author sketches the rules of the game: "Alice and Bob have committed a crime and are arrested. The police offer each one a dealsnitch and you go free while your friend does 6 months in jail. If both Alice and Bob snitch, they both get 3 months in jail. If they both remain silent, they both get one month in jail for a lesser offence." "In a single game. the best strategy is to snitch because it guarantees that you don't get the maximum jail term. However, the game gets more interesting when played in repeated rounds ... ." Until now, received wisdom ("based on decades of computer simulations and a certain blind faith in the symmetry of the solution") has been that a titfortat approach, in which each player copies the opponent's behavior in the previous round, was the best strategy; both opponents spend the same time in jail. But it turns out that titfortat is only one member of the family of "zero determinant strategies" discovered by Press and Dyson, that can make the other player spend "far more time in jail (or far less if you're feeling generous)." See the Press and Dyson article, "Iterated Prisoner's Dilemma contains strategies that dominate any evolutionary opponent."
Tony Phillips 
Comments: Email Webmaster 
© Copyright
, American Mathematical Society

