# Math Digest

## Summaries of Articles about Math in the Popular Press

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

### March 2005

 Davis traces the attempts to make logic itself the subject of inquiry, from Aristotle, to George Boole, to Gottlob Frege, to Bertrand Russell, and states that "during the 20th century, as logic developed into a vibrant branch of mathematics, tension remained between those two aspects of the field: logic as part of mathematics and logic as the foundation of mathematics." The book under review is never actually reviewed, but this piece provides a nice introduction to Alfred Tarski, one of the great logicians of the 20th century, who proved his famous theorem on the "nondefinability of truth," who "always seemed to work on formulations of logical matters that could hope to have the same elegance found elsewhere in mathematics," and who was a great teacher. --- Annette Emerson

"Why W?" by Brian Hayes. American Scientist, March-April 2005, pages 104-108.

 The Lambert W function, defined as "the inverse function associated with the equation WeW = x," is a rising star in the cluttered sky of mathematical equations. With applications ranging from ballistic projectiles to delay differential equations and the omega constant, some say it should join logarithms and exponents in the fundamental mathematical toolbox. The function is difficult to evaluate for two reasons. First, there is no closed form of the solution: One must first guess W based on x, then adjust the guess accordingly and iterate. Second, for certain x there are multiple W which solve the equation. Though it was originally postulated by Euler, the function bears the name of the relatively obscure Lambert because "naming yet another function after Euler would not have been useful." Its presence in the program Maple played an important role in its path to stardom. --- Lisa DeKeukelaere

There's more to cultural craft patterns than textbook symmetry, expounds Symmetry Comes of Age: The Role of Pattern in Culture, a collection of articles described as "anthropology for mathematicians." Patterns woven into baskets and belts often demonstrate societal ideals and intentional asymmetries that mathematical scrutiny would cast aside as being imperfections. Sometimes these patterns relate to cultural norms that can't be easily explained, similar to our innate knowledge of which necktie motifs and color-schemes are acceptable in American business settings. Trying to manufacture an "acceptable" pattern based solely on the reproduction of symmetries will often fail because it lacks the necessary anthropological underpinnings.

--- Lisa DeKeukelaere

"NYU's Peter Lax Wins 'Nobel Prize of Mathematics'," by Gary Shapiro. The New York Sun 23 March 2005.;
"From Budapest to Los Alamos, a Life in Mathematics," by Claudia Dreifus. The New York Times, 29 March 2005.;
"Abel-Preis 2005 an Peter D. Lax: Ein Vertreter der angewandten und der reinen Mathematik," by George Szpiro. Neue Zuercher Zeitung, 23 March 2005;
"Abel Prize." Random Samples, Science, 1 April 2005, page 47.

 The New York Sunarticle quotes Sylvain Cappell, professor at the Courant Institute of Mathematical Sciences, New York University, on Peter D. Lax's achievements: "Besides being the dominant figure in applied mathematics in his time, he's also one of the world's central figures in pure mathematics." Lax, who is also a professor at Courant, is to receive the prestigious 2005 Abel Prize "for his groundbreaking contributions to the theory and application of partial differential equations and to the computation of their solutions" (The Norwegian Academy of Science and Letters announcement). The article captures some personal background about Lax, along with quotes from colleagues and from Lax himself. The New York Times piece--a conversation with Lax--includes his account of his family coming to America to escape the Nazis when he was 15, going to Stuyvesant high school, working at Los Alamos National Laboratory, speculating on what his mentor John von Neumann would think of today's computerized world, and citing what his own most significant contributions have been: work on shock waves, the Lax-Phillips semigroup in scattering theory, and analysis with his student Dave Levermore of "what happens to solutions of dispersive systems when dispersion tends to zero." When asked "has mathematics become too complex for anyone to understand all of it?" Lax says yes, noting that mathematics is a broad subject. But he notes that "it is also true that as mathematics develops, things are simplified and unusual connections appear. Geometry and algebra, for instance, which were so very different 100 years ago, are intricately connected today." --- Annette Emerson

"Proof and Beauty." The Economist, 31 March 2005.

How do you know that a proof is correct? Because a computer says so? Because someone smarter than you believes it? These questions poke at the heart of the debate over the strength of computer-dependent proofs, and the answers affect the perceptions of both validity and beauty in the field of mathematics. Recent proofs, such as that of the four-color problem and the Kepler conjecture, have used computers to sort through thousands of possible case situations in order to solve a problem. Computers are capable of formal logic proofs, which eschew inference and proceed based on a strict set of rules, yet few are enthusiastic about checking the gargantuan number of required steps. This dilemma has left journals with the difficult task of assessing and presenting these important proofs in a timely fashion without misleading readers as to their accuracy. Changing the criteria for popular acceptance from detailed verification to trust may alter the idea of proof as we know it.

--- Lisa DeKeukelaere

"Dot.compass," by Emily Rooney. Greater Boston, 28 March 2005.

Each Monday Greater Boston (on Boston's PBS TV station WGBH) highlights websites of interest in a Dot.compass segment. The March 28 program on "Fantasy Hockey Sites" highlighted WhatIfSports.com. Program host Emily Rooney explains that "mathematical algorithms" are used to (as promoted on the WhatIfSports website) "allow users to create teams of their all-time favorite players and pit them against other user-created teams," while the site simulates games (baseball, basketball, football and hockey) and produces scores based on various data.

--- Annette Emerson

"Student Scientists to Watch," by Ben Harder. Science News, 19 March 2005, page 181.
"Meet the Best and the Brightest," by Otis Port. Business Week, 28 March 2005.

These articles list the top 10 winners in the 2005 Intel Science Talent Search. First place and a US$100,000 scholarship went to David Vigliarolo Bauer of Hunter College High School in New York City for his design of a device that helps detect neurotoxins. Two mathematics projects that finished in the top ten were from Robert Thomas Cordwell of Manzano High School in Albuquerque, who won fourth place for "his study of the mathematical properties of sets of points that lie along a circle and can be connected into polygons," and from Po-Ling Loh of James Madison Memorial High School in Madison (WI) who won tenth place for her project studying closure properties of certain dihedral groups. Cordwell will receive a US$25,000 scholarship and Loh will receive a US$20,000 scholarship. The Science Service website has more details on the winners and the competition. --- Mike Breen Return to Top  Thinking about infinity can be like trying to interpret abstract art: You are looking at a conglomeration of tangible pieces, but you're never quite sure what the whole thing amounts to. Singh reports that in The Infinite Book, John D. Barrow helps pin down the facts about this elusive concept and that his writing gives new life to well-worn examples, addressing topics from the Schwarzschild radius of a black hole to connections between Fawlty Towers and the Infinite Hotel. Barrow also keeps readers entertained with a collection of shocking and amusing anecdotes. Infinity may always be a bit abstract to most people, but Singh says this book provides readers with some of the principles and motivation needed to appreciate the myriad exhibits in the museum of mathematics. ---Lisa DeKeukelaere Return to Top  Scientific American online reports on research by Bruce Malamud (King's College London) and colleagues in which they analyzed over 88,000 wildfires in the U.S. between 1970 and 2000. They found that "small fires occur most frequently and the largest ones are the least likely to occur" and that "going from east to west, the ratio of small fires to large ones decreases." The summary states that their results allow "for the classification of wildfire regimes for probablistic hazard estimation in the same vein as is now used for earthquakes." The findings---categorized "frequency-area statistics, power-law distribution, Bailey ecoregion divisions, U.S. Department of Agriculture Forest Service, and probabilistic hazard"---were published online before print in the Proceedings of the National Academy of Sciences ("Characterizing wildfire regimes in the United States," 21 March 2005). --- Annette Emerson Return to Top "UW-Madison grad student makes math history," by Ron Seely. Wisconsin State Journal, 18 March 2005. "Classic maths puzzle cracked at last," by Magie McKee. New Scientist, 21 March 2005. "'Cranky' Proof Reveals Hidden Regularities," by Dana Mackenzie. Science, 1 April 2005, pages 36-37. "Greeley math whiz cracks old equation," by Doyle Murphy. The Greeley Tribune, 17 April 2005.  These articles concern a result by Karl Mahlburg, a graduate student of Ken Ono at the University of Wisconsin, regarding partitions. Freeman Dyson characterized the result as, 'Beautiful and totally unexpected.' The historical background of the result traces to Ramanujan and goes through Dyson, George Andrews, Frank Garvan, Oliver Atkin, and Ken Ono. Ramanujan made an original discovery (1919) about partitions of numbers dealing with the primes 5, 7, and 11, which Dyson later thought could be explained by a function he called the "crank" (a function that was yet to be discovered). Later George Andrews and Frank Garvan found the crank function which did indeed imply Ramanujan's discovery. In 2000, after looking at Ramanujan's workbooks, Ken Ono further generalized Ramanujan's discovery to primes bigger than three, using modular functions. His was an analytic proof of a deep generalization. Now his graduate student has provided a combinatorial proof that ties Ono's generalization with the crank function. Mahlburg's work has been submitted to the Annals of Mathematics. The Wisconsin State Journal ran the story on its front page. See also: "Pieces of Numbers," by Erica Klarreich. Science News, 18 June 2005, pages 392-393. --- Mike Breen Return to Top "New largest prime discovered," by Erica Klarreich. Science News, 19 March 2005, page 188.  The largest known prime number is now 225,964,951-1, a number that is 7,816,230 digits long. (According to the article the number would completely fill 58 issues of the Science News.) The number was found through the Great Internet Mersenne Prime Search (GIMPS), a distributed computing project, on the computer of Martin Nowack, a German eye surgeon. A Mersenne prime is one of the form 2p-1. For such an expression to be prime, p must be prime, but p being prime is not enough to ensure that 2p-1 is prime (for example, p = 11). The previous largest known prime was discovered through GIMPS in May. Over 2000 years ago Euclid gave an elegant proof that there are an infinite number of prime numbers. --- Mike Breen Return to Top This book is one of several popular books on the Riemann Hypothesis (RH) to come out in recent years. RH is one of the biggest unsolved questions in mathematics today. A proof of the hypothesis would yield a wealth of information about the nature of prime numbers, which are the basic building blocks of arithmetic. Calling this book a "fine addition" to the RH literature, the reviewer writes, "Rockmore is an excellent guide to take you right to the edge of the mathematical map." --- Allyn Jackson Return to Top "Artists on science: scientists on art." Special section in Nature, 17 March 2005, pages 293-323.  This issue of Nature includes a special series of articles by writers, visual artists, a composer, and two scientists on their connections to art and science and their thoughts on these connections. Three of the articles specifically mention mathematics as inspiration: In "Fiction informed by science," A.S. Byatt lets the reader know how mathematics enlightened her and plays a role in her four-part series of novels. She recalls her discovery that in snails the "Fibonacci spiral was an example of a Platonic order---a sense that an invisible mathematical order informed all our physical accidental world." In "A tale of two loves," Alan Lightman (physicist and novelist) notes that as a child he was drawn to mathematics: "When my maths teachers assigned homework, I relished the job. I would save my maths problems for last, right before bedtime--like bites of chocolate cake awaiting me after a long and dutiful meal of history and Latin....I loved the abstractions, letting x's and y's stand for the number of nickels in a jar or the height of a building in the distance...And when you had found that answer, you were right, unquestionably right." In later years and as a novelist he "realized that the ambiguities and complexities of the human psyche are what give fiction, and perhaps all art, its power," and now thinks that the sciences and arts contribute to each other, are always in tension, and offer humans different---sometimes synthesized---ways of looking at our world. In "From science in art to the art of science," Martin Kemp writes that "shared intuitions about the natural world drive the pursuits of artists and scientists," and includes photographs and descriptions of mobiles, kinetic and geometric sculptures, etc. of some 20th century artists. --- Annette Emerson Return to Top "Math confusion setting in," by John Powers. The Boston Globe, 16 March 2005.  Without really getting into the mathematics of the new scoring system for competitive figure skating (used at the World Figure Skating Championships in Moscow in March 2005), Powers reports that the skaters and audiences find the new scoring system "numbing" and "bewildering." In an effort to avoid the corruption and collusion of the 2002 Winter Olympic judges, the new system does not link marks to judges and breaks down the old technical and artistic scores into scores for separate sequences and elements for each. The new scoring may force skaters to change their programs to fit the new method. While admitting that the new scoring system may lessen subjective influences, Powers concludes that "the new system is numbers stacked upon numbers, with base values and grades of execution and factored program components and deductions, all of which adds up to scores like 113.93. It's a system in which a key-punch error can cost you a medal if nobody catches it. Compared with the old 6.0 system, this is a quadratic equation." --- Annette Emerson Return to Top This article reports on the math game show "Who Wants to Be a Mathematician?", which is sponsored by the AMS. The show, created by AMS public awareness officer Mike Breen, gives high school students a chance to show off their math skills, win some prizes, and have a lot of fun in the process. The show on which the Providence Journal reported was held on the campus of Providence College. William Shore, a student at Rocky Hill School in Warwick, Rhode Island, took home the grand prize of US$2,000.

--- Allyn Jackson

 Knuth leads the interviewer around his home, pointing out that mathematics was used to design the kitchen and still informs nearly every aspect of his life and lifetime of work in computer science. He notes that computer programming is not set in stone, that mathematical analysis and refinements---and the ideas and work of many individuals---are constantly improving programs. The NPR web page describes Knuth as "legendary in the computer science world for writing a series of must-have reference books called The Art of Computer Programming. Part cookbook, part textbook, part encyclopedia, these books are also considered by many to be technical and personal works of art." Knuth provides some gems of personal philosophy, including his remark that "efficiency" is fun to think about. Mathematicians and math publishers are of course familiar with his development of TeX, the program that revolutionized typesetting of mathematics. Listen to the informal tour of Knuth's home and mind on the NPR website. --- Annette Emerson

 Hyperbolic plane crocheted by and photographed by Daina Taimina National Public Radio's Jacki Lyden interviews mathematician Daina Taimina (Cornell University) about the hyperbolic planes she crochets. She explains how she started modeling the shapes with paper, then tried knitting them, but found that crochet retains a more rigid structure that students and others can see, touch and turn to understand the geometrical concept of a hyperbolic plane. Taimina's models are on view on the Institute for Figuring website, and a detailed explanation of her works is in an article by Taimina and her husband David Henderson, "Crocheting the Hyperbolic Plane." See also:"Professor Lets Her Fingers Do the Walking," by Michelle York, The New York Times, 11 July 2005. --- Annette Emerson

"Crunch time": Review of Electronic Brains, by Mike Hally. Reviewed by Barry Fox. New Scientist, 12 March 2005, page 51.

According to the review the author of this book "traveled the world recording interviews with people who knew first-hand how the first electronic computers were built in the 1940s and 1950s." Among the topics discussed in the book is the work of Alan Turing, and the development of the ENIAC and UNIVAC computers. "Well researched and a rattling good read, [the book] is a terrific value," the reviewer writes.

--- Allyn Jackson

"QuickStudy: Wavelets," by Russell Kay. Computer World, 7 March 2005.

The author provides a brief explanation of Fourier analysis and wavelets, used to analyze data to discover underlying patterns and information. He highly recommends Dana MacKenzie's article "Wavelets: Seeing the Forest and the Trees", noting that "wavelet algorithms allow us to record or process different areas of a scene at different levels of detail (resolution) and using greater amounts of compression (scale)." Kay introduces wavelets: "Unlike the sinusoidal, endlessly repeating waves used in Fourier analysis, wavelets are often irregular and asymmetric, with values that die out to zero as they move farther from a central point." Among the applications of this area of study he includes the compression algorithm used by the FBI to store its fingerprint database, wavelet compression to create MPEGs on the web and reduce noise, and---being explored now---for medical diagnosis and weather forecasting. There's a nice illustration of an apple as a JPEG compression compared with a wavelet algorithm.

--- Annette Emerson

"Who Says a Woman Can't Be Einstein?," by Amanda Ripley. Time, 7 March 2005, pages 51-60.

There isn't much math in the cover story from Time's "The Math Myth" issue. The article is due to Harvard president Larry Summers' remarks on gender disparities in math and science, and examined whether there was evidence to support a biological basis for gender differences in performance in science and math. Psychologists and brain researchers agreed that there are differences between male and female brains, but whether those differences affect behavior or outweigh environmental factors is not known. Indeed, "most scientists still cannot tell male and female brains apart just by looking at them." The article points out that without taking into account the complex interaction of biology and environment, "it would be all too easy to look at the latest research on the brain and conclude, say, that men may not in fact make the best university presidents. For example, studies show that men are slightly more likely to say things without realizing how their actions will affect others."

--- Mike Breen

"Primed for Numbers," by Rich Monastersky. The Chronicle of Higher Education, 4 March 2005, page A1.

In the aftermath of Harvard University president Lawrence Summer's remarks regarding the differences between men's and women's participation in science and engineering careers, Rich Monastersky examines the findings of a number of experts regarding the "nature" of mathematical ability in boys vs. girls. On the one hand, some researchers point to biological differences in the brains of males and females, or the effect of hormones, that give boys an advantage in performing certain mathematical tasks, or that cause more males than females to pursue math and science careers. However, other researchers point to cultural influences to which girls are subject, including an emphasis on relationships, and the lack of female role models in math and science fields, both of which can lead girls to choose careers outside of math and science. But whether differences are inborn or learned, changes such as allowing students in engineering majors to choose more electives, educating students better about math and science careers, or providing additional training in areas such as spacial thinking may make a difference in the numbers of girls pursuing careers in science, math, and engineering.

Two sidebars appear with this article. One focuses on reasons that Chinese children may perform better at mathematics than American children, according to education and psychology professor Kevin Miller. These reasons include the following: The names of certain numbers in Mandarin Chinese are "more informative" than their English counterparts, Chinese teachers specialize in, and teach, either math or reading rather than teach all subjects, and, in China, success is believed to come from hard work as opposed to ability, a more American idea. In the second sidebar, Monastersky considers an effect of stereotyping described by psychology professor Claude Steele as "stereotype threat": When a person faces a detrimental stereotype, according to Steele, his or her performance seems to suffer. Ironically, Steele notes that this type of threat "harms people most when they excel in a given subject."

--- Claudia Clark

Visualizing proteins, studying environmental issues, or learning how to contain epidemics: these are just a few of the ways that mathematical modeling is becoming a critical tool in biology. At the same time, the mathematics involved in the study of biological questions is "as deep, elegant, and beautiful as the mathematics of relativity, quantum mechanics, and subatomic particles," notes St. Olaf College mathematics professor Lynn Arthur Steen. In this article, Steen discusses the problems that the "new biology" faces in undergraduate institutions. These problems include a mathematics and biology faculty trained in a "monodisciplinary culture," departmental and institutional policies that discourage interdisciplinary work, and a lack of resources and courses reflecting the latest work in biology. Steen offers some solutions to these problems, such as developing new curricula in the interface between mathematics and biology, and getting math and computer science undergraduates "hooked on mathematically fascinating biological problems early in their college careers." But it doesn't stop there. He argues that "sound, quantitative understanding of 21st-century biology" is not a luxury required by only a few students, but by all citizens.

--- Claudia Clark

"Flaw Found in Data-Protection Method," by Charles Seife. Science, 4 March 2005, page 1389;
"Chinese mathematician decodes two int'l cipher systems". People's Daily Online English, 29 March 2005.

Three code breakers have found a way to break the Secure Hash Algorithm (SHA-1), a standard cryptographic function introduced by the National Institute of Standards and Technology in the early 1990s. The discovery does not render SHA-1 unusable, however, because even with this discovery, breaking SHA-1 is still beyond the capabilities of modern supercomputers. What the code breakers have done is establish an attack which reduces by a factor of 1000 the number of attempts necessary to exploit SHA-1.

--- Mike Breen