# Math Digest

## Summaries of Media Coverage of Math

Edited by Allyn Jackson, AMS
Contributors:
Mike Breen (AMS), Claudia Clark (freelance science writer), Lisa DeKeukelaere (Brown University), Annette Emerson (AMS)

### January 2006

"Unwed Numbers" by Brian Hayes. American Scientist, January-February 2006, pages 12-15.

 They first appeared in the US in 1979. Five years later they showed up in Japan. After appearing in London in late 2004, they became wildly popular in the UK. Subsequently they returned to the US this past spring. No, it's not a rock group: It's Sudoku. And, in the latest issue of American Scientist, senior writer (and self-proclaimed Sudoku aficionado) Brian Hayes writes about these popular puzzles, beginning with their history and some basic techniques for solving them. The balance of this article considers the question of puzzle hardness and whether or not there are "clear-cut criteria for ranking or classifying the puzzles." Hayes starts by exploring the number of solutions there are to Sudoku puzzles of varying sizes, including a well-known subset of these puzzles: Latin squares. He then discusses some proposed ways of identifying the hardness of the puzzles by presenting some of the thinking about their membership in the class NP, as well as considering the relationship between the uniqueness of solutions and the number of "givens" that appear in a Sudoku puzzle. Hayes concludes the article by discussing another proposed way to classify the puzzles: "those that can be solved by logic alone' and those that require trial and error'" or backtracking. But he finds this criterion for ranking puzzle difficulty less than promising or clear-cut. Still, solving the puzzles remains a satisfying exercise, even if they remain difficult to classify. --- Claudia Clark

"By the Numb3rs," by Frank Roylance. Baltimore Sun, 27 January 2006.

 Sommer Gentry and Dorry Segev. Baltimore Sun writer Frank Roylance discusses the popular TV show, NUMB3RS, and a husband-and-wife team whose research inspired the topic of a recent episode. In the episode, investigators used a mathematical model to find the most likely recipient of a black-market kidney. The story was inspired by the real-life work done by transplant surgeon Dorry Segev and mathematics professor Sommer Gentry (pictured). Sometimes an ailing patient and potential donor find their tissues are not compatible. However, if another mismatched pair of patient and donor can be found whose tissues match the first pair, a paired organ donation can be done and both patients will receive the much-needed transplant. Working at the John Hopkins University School of Medicine, Segev and Gentry used optimization theory to develop a model for finding and matching up such pairs. In fact, Roylance notes, thousands of additional matches could be made each year through an optimized, national system for paired organ donations. (Currently no such system exists.) The fact that the NUMB3RS series producers "have worked hard to get the math right," and have made mathematics more engaging to the public by drawing upon real-world mathematics, has generated a lot of enthusiasm among math teachers: more than 25,000 teachers currently download materials linked to each NUMB3RS episode, Roylance reports. --- Claudia Clark

"Scientists use Web site to track money and predict diseases," by Alicia Chang. Associated Press, www.SignOnSanDiego.com, 26 January 2006;
"Fitting the bill," by Rory Howlett. Nature, 26 January 2006, page 402.

Understanding human travel patterns may help scientists' efforts to predict the spread of viruses, which travelers can sometimes carry. However, a lack of solid research on travel patterns has led researchers to instead investigate the movement of something travelers always carry: money. Using the website www.wheresgeorge.com, which tracks the locations of bills throughout the United States, researchers found some unexpected patterns. In previous models, the probability that a person would travel a given distance depended on the distance in question, but tracking the bills revealed that money stayed within 50 kilometers of its starting point for much longer than expected. The researchers' new model ("The scaling laws of human travel," by D. Brockmann, L. Hufnagel and T. Geisel, Nature, 26 January 2006, page 462) matches well with two different surveys on long-distance American travel, bolstering the strength of the hypothesized claim that people and money travel with similar behavior. These findings are especially important given the current efforts to contain the avian flu virus.

--- Lisa DeKeukelaere

Review of Beyond Coincidence: Amazing Stores of Coincidence and the Mystery and Mathematics Behind Them, by Martin Plimmer and Brian King. Reviewed by William Grimes. International Herald Tribune, 26 January 2006, page 11.

The reviewer calls this book "a collection of stranger-than-fiction anecdotes wrapped loosely in colorful intellectual tissue paper," though he also concedes that it is a "superior example" of its genre. The book describes many weird coincidences (such as an ice dealer named I. C. Shivers and a Canadian farmer named McDonald whose postal code is EIEIO), while dispelling the notion that such coincidences really are all that weird. The mathematical laws of chance show that coincidences are bound to occur, and the tendency to read meaning into them simply means that "most people have a poor grasp of statistics." Also, because humans depend for their survival on their ability to perceive order and patterns, they may, when confronted with strange coincidences, tend to see order and patterns that are not really there.

--- Allyn Jackson

"Edge.org: What is your dangerous idea?" Open Source: a public radio show with Christopher Lydon, National Public Radio, 24 January 2006.

Open Source is advertised as being a blog on live radio. Among the guests on the "Edge.org: What is your dangerous idea?" program was mathematician Steven Strogatz (Cornell University). His "dangerious idea" is: "Mathematicians have figured out almost everything that's humanly possible. (Computers will take it from here.)" He cites Brian Davies' recent article, "Whither Mathematics?" in the Notices of the American Mathematical Society: "[Davies] mentions, for example, that the four-color map theorem in topology was proven in 1976 with the help of computers, which exhaustively checked a huge but finite number of possibilities. No human mathematician could ever verify all the intermediate steps in this brutal proof, and even if someone claimed to, should we trust them? To this day, no one has come up with a more elegant, insightful proof. So we're left in the unsettling position of knowing that the four-color theorem is true but still not knowing why." Strogatz also notes human vs. computer chess games in which the computer wins but we don't know exactly how. He says that the teaching of elementary geometry won't likely change; teachers will still instill the concept of making a series of steps to attain logical explanations and conclusions. But host Lydon wonders if the old adage of his and other mathematics teachers ("I don't care what your answer is as much as how you got it") will apply less as students use computers and don't understand the process or the complex algorithms making the computer work. The interview with Strogatz and related online blogs may be found as links off the Open Source website, and the Edge.org website includes "dangerous ideas" from many scientists and journalists from the world of mathematics, mathematical physics, and computer science, including Lawrence Krauss, Freeman Dyson, Brian Greene, Frank Tipler, Rudy Rucker, John Allen Paulos and Keith Devlin.

--- Annette Emerson

"Math will rock your world," by Stephen Baker with Bremen Leak. Business Week, 23 January 2006, pages 54-60.

With so many people using the Internet to shop, do business, and post articles and blogs, a tremendous amount of data is there for the taking. But with this vast amount of data comes the challenge, for businesses that wish to use this information, of intelligently analyzing this data. This is where mathematics comes in, according to writer Stephen Baker in his cover story in Business Week. Baker describes how mathematicians, or "quants," are "helping map out advertising campaigns ... changing the nature of research in newsrooms and biology labs, and ... enabling marketers to forge new one-on-one relationships with customers." Baker also describes how companies are using data about themselves to increase productivity and "shake up the workplace." For example, IBM senior manager of stochastic analysis Samer Takrite and his team are using mathematics to model IBM's workforce. "Of course," Takriti acknowledges, "people are complicated." But, if this effort is successful, IBM will offer these services to other companies, anticipates Baker.

Doing this work without sacrificing individual privacy will be a challenge: Microsoft cryptographer Cynthia Dwork acknowledges that, in Baker's words, "mathematically gifted hackers can continue to pry open doors that she and her team slam shut." Another challenge will be for managers, entrepreneurs, and the public to become mathematically savvy and able to "question the assumptions behind the numbers." Mathematicians, too, must be wary of placing too much faith in the numbers: in modeling people and their behavior, they would benefit from the expertise of others, including those from the social sciences.

--- Claudia Clark

"Hunter-Gatherers Grasp Geometry," by Constance Holden. Science, 20 January 2006, page 317.

Although some students of high school geometry might disagree, basic geometric concepts may be "a part of basic human cognitive equipment," according to writer Constance Holden, reporting on a research article appearing in the same issue of Science. Using two non-verbal tests, anthropologist Pierre Pica, one of a team of researchers headed by Stanislas Dehaene, tested 14 children and 30 adults belonging to a group known as the Mundurukú, who live in a remote part of the Amazon. One test required subjects to pick the one item that didn't fit in with a group of six geometric figures, some of which illustrated basic concepts such as symmetry. For the second test, subjects performed a test requiring them to use a map to locate a hidden object.

The results? Holden reports that subjects correctly answered about two-thirds of the first test questions and about 71 percent of the questions on the second test. When compared with a control group of 26 U.S. children and 28 U.S. adults, the Mundurukú performed at about the same level as the U.S. children. Dehaene concludes that "even without education, and living in isolation without artifacts such as maps, you can have a developed geometrical intuition." Not all scientists agree with the findings: Rosalind Arden, a doctoral student at King's College in London, argues that the tests are more a measure of "general reasoning ability" than a "shared core of geometric knowledge." But others do: Steven Pinker of Harvard University calls the research "a nice addition to the literature on cognitive universals."

--- Claudia Clark

"Hip 2B2," by Amy Dorsett. San Antonio Express-News, 14 January 2006, page 1;
"Educators show ways to climb math mountain," by Michelle M. Martinez. San Antonio Express-News, 15 January 2006, page 4B.

 These two articles are about the Joint Mathematics Meetings, which were held January 12-15, 2006, in San Antonio, Texas. The earlier article, a general one about the meeting, ran on the front page of the paper. The later article is about some talks in the Mathematical Association of America session Countering "I Can't Do Math": Strategies for Teaching Underprepared, Math-Anxious Students. Martinez gives quotes from Debasree Raychaudhuri (California State University in Los Angeles) and Ann Hanson (Columbia College), two of the presenters at the session, about how they try to overcome students' anxieties about math. --- Mike Breen

"Taking Anxiety Out of the Equation," by Elizabeth F. Farrell. Chronicle of Higher Education, 13 January 2006, page A41.

Farrell writes about efforts in colleges and universities to improve the teaching of mathematics. For many students the difference between high school math courses and those in post-secondary institutions is too great for them so they change majors to avoid math. The article states that one of the main causes of math anxiety is a "dropped stitch": a gap in a student's education that stops him or her from learning other concepts. The article also states that "Experts say students can overcome math anxiety by using two strategies: telling their professors when they are confused, and staying on top of their homework."

--- Mike Breen

"Twin Prime Conjecture." NOVA scienceNOW, 10 January 2006.

The January 10, 2006, PBS broadcast NOVA scienceNOW described some of the top science stories of 2005, including the proof by Goldston, Yildrim, and Pintz about gaps between prime numbers. The segment included an original song about prime numbers and cameos by mathematicians. The day after the program was aired PBS posted a web page that includes "Seven Prime Questions"", in which mathematicians (Mel Nathanson, City University of New York; Kevin O'Bryant, City University of New York; David Chudnovsky, Brooklyn Polytechnic University; and Carlos Moreno, City University of New York) explain what prime numbers are. The web page also includes various options to download the twin prime conjecture song and/or the complete video segment, and links to additional resources, "Send Feedback," bios and more.

--- Annette Emerson

"Doing the maths," by Ehsan Masood. openDemocracy, 10 January 2006.

The occasion for this article is the awarding of the first ICTP Ramanujan Prize to Marcelo Viana, an outstanding mathematician and native of Brazil who is on the faculty of the Instituto de Matem\'atica Pura e Aplicada in Rio de Janeiro. The prize is given by the International Center for Theoretical Physics in Trieste (ICTP). The article discusses the problem of "brain drain", in which talented scientists and mathematicians from poorer countries emigrate to richer countries. Honors like the Ramanujan Prize, which recognizes young mathematicians working in developing countries, may help to counteract this trend. The article also describes activities of the ICTP, which was founded by the Pakistani theoretical physicist and Nobel Laureate Abdus Salam. ICTP aims to develop scientific and mathematical talent in developing countries.

--- Allyn Jackson

"Irrationales bei Airlines und Passagieren: Das Braess-Paradoxon am Beispiel der Flugroutenwahl (Irrationality among airlines and passengers: The Braess Paradox on the example of an airline network)", by George Szpiro. Neue Zuercher Zeitung, 9 January 2006.

 This article deals with the so-called Braess Paradox as applied to an airline network. This result shows that adding an edge to a network can increase the pressure on the other edges, rather than decreasing it. Dietrich Braess published a paper on the paradox in 1969. An English translation appeared only very recently, in the November 2005 issue of Transportation Science. --- Allyn Jackson

"The torturer's dilemma: the math on fire with fire," by Jonathan David Farley. San Francisco Chronicle, 8 January 2006.

Jonathan David Farley, a science fellow at Stanford University, is a mathematician who has used mathematics to model terrorist networks. In this article, he describes "reflexive theory", which was developed during the Cold War by a Soviet mathematical psychologist named Vladimir Lefebvre. Lefebvre's theory provides a mathematical framework for modeling moral decisions. It was used extensively by the Soviet defense establishment but was unknown in the West. With very simple assumptions, Farley writes, "Lefebvre showed that in a society that accepted the compromise of good with evil, individuals would more often seek the path of confrontation with each other." Lefebvre's theory can be used to examine the question of whether the United States should use torture in its fight against terrorism. The theory shows, Farley says, that "If Americans begin to accept the use of torture, American society might turn into a society of individuals in conflict."

--- Allyn Jackson

"A leap into hyperspace," by Haiko Lietz. New Scientist, 7 January 2006, pages 24-27.

This article discusses the little-known work of a German physicist named Burkhard Heim. Heim, who was born in 1925 and died in 2001, formulated a physical theory that he hoped could be used to build a new kind motor that would propel spacecraft at enormous speeds. As the article puts it, the spacecraft "could leave Earth at lunchtime and get to the moon in time for dinner." Heim came up with his ideas in attempting to formulate a theory that would unify quantum mechanics and Einstein's theory of relativity. One of his results "was a theorem that led to a series of formulae for calculating the masses of the fundamental particles---something conventional theories have conspicuously failed to achieve," the article says. This led to a mathematical description of an eight-dimensional universe in which conversion between electromagnetic and gravitational energy is effected by pairs of particles called "gravitophotons". Heim's work has remained relatively obscure: What little he published appeared only in German, and his writings are difficult to understand. The article notes that the majority of physicists have never heard of Heim's theory. The reason it is receiving attention now is that the American Institute of Aeronautics and Astronautics last year awarded its prize in future flight to proposed experimental tests of a spacecraft engine based on Heim's ideas.

--- Allyn Jackson

"Bayes Rules." The Economist, 5 January 2006.

 Thomas Bayes (1702-1761). Bayesian statistics involves deriving a conclusion from a small set of data points by assuming certain properties about the behavior of the entire system. Developed by Bayes in the 1700s, the method has often been eclipsed by "frequentism," an approach that uses more data points and fewer assumptions. But a recent study by researchers at Brown University and MIT indicates that Bayes' idea emulates the human thought process: study participants correctly estimated quantities such as the length of time a congressman would serve when given only a single fact, the amount of time he had currently been in office. The participants drew conclusions by making assumptions based on personal previous experience, yet their predictions matched well-known statistical event probabilities. These findings might explain the prevalence of superstition, as people make assumptions in an attempt to link a few random events to a larger picture. --- Lisa DeKeukelaere

"Mathematik zum Kugeln": Review of The Pea and the Sun: A Mathematical Paradox, by Leonard M. Wapner. Reviewed by George Szpiro. Neue Zuercher Zeitung, 1 January 2006.

Szpiro recommends this book, which discusses the so-called Banach-Tarski paradox. This counter-intuitive result shows how one can cut a sphere into pieces and rearrange the pieces to produce two spheres of the same size as the original one.

--- Allyn Jackson

"Raoul Bott; Top Explorer of the Math Behind Surfaces and Spaces," by Bryan Marquard. Boston Globe, 4 January 2006, page A20;
"Raoul Bott, an Innovator in Mathematics, Dies at 82," by Jeremy Pearce. New York Times, 8 January 2006;
"Mathematics innovator Raoul Bott dies." United Press International, 9 January 2006.

 Raoul Bott. Photograph courtesy of Harvard University Mathematics Department. These obituaries describe the life and work of Raoul Bott, one of the outstanding mathematicians of the twentieth century, who made deep contributions to differential geometry and topology. His name is a household word among mathematicians and has been attached to such important results as the Bott Periodicity Theorem and the Atiyah-Bott Fixed Point Theorem. Not only was Bott a great mathematician, he had a warm and gregarious personality and was deeply beloved by many in the mathematical community. His Harvard colleague Clifford Taubes is quoted in the Boston Globe obituary as saying: "His theorems were fantastic, but there are people with fantastic theorems who are not loved the way he was loved. Everyone considered him a father figure. He was just such a gentleman and gregarious. He loved to laugh, he loved life. He taught us to look for beauty and art in everything." --- Allyn Jackson

"Mysterious Death of a Mathematician Finally Solved?," by Susan Kruglinksi. Discover, January 2006, page 10.

In the course of writing the book The Equation That Couldn't Be Solved, Mario Livio's research led him to conclude that mathematician Évariste Galois died in a fight over a woman. Galois was found in a field near Paris with one shot to his stomach. He died in a hospital the next day, 31 May 1832, at the age of 20. Though historians have concluded he was shot in a duel, no one knew who shot him. Livio is one of many scientists fascinated by the case and describes Galois as "a romantic character and truly one of the most original thinkers in the history of science." Kruglinski quotes Livio as declaring that "group theory, the study of symmetries, is the `bread and butter' of modern physics," and that investigation is one of the enjoyable parts of his job.

--- Annette Emerson