Math Digest

Summaries of Media Coverage of Math

Edited by Allyn Jackson, AMS
Contributors: Mike Breen (AMS), Claudia Clark (freelance science writer), Lisa DeKeukelaere (2004 AMS Media Fellow), Annette Emerson (AMS), Brie Finegold (University of California, Santa Barbara), Adriana Salerno (University of Texas, Austin)

Elena Naumova (pictured at left) is a professor at the Tufts University School of Medicine who uses mathematics to try to find patterns in the spread of infectious diseases (thus the article title), especially as they relate to seasons and weather. She has found that in Massachusetts, salmonella peaks at the end of July, while populations of giardia (a parasite) and shigella (a bacterium) spike one month later. Naumova hopes eventually to be able to predict outbreaks, just as weathermen do. Nina Fefferman (Rutgers University) says of Naumova's work: "While her research is on the numbers side, she pulls in an understanding of the biology really well. Her most frequent collaborator is her husband, who is an immunologist, so a lot of the modeling is not just about the mathematics, but also how the biological systems work. She approaches it from both ends." (Photo: Pat Greenhouse/Boston Globe/Landov.)

--- Mike Breen

The National Council of Teacher Quality (NCTQ), a nonpartisan research and advocacy group, has issued a report that says that elementary school teachers are poorly prepared in math. NCTQ president, Kate Walsh, said, "As a nation, our dislike and discomfort with math is so endemic that we do not even find it troubling when elementary teachers admit to their own weaknesses in basic mathematics. Not only are our education schools not tackling these weaknesses, they accommodate them with low expectations and insufficient content." The report looked at 77 elementary education programs across the country and found

• Low expectations for math knowledge
• Low math standards for teacher certification
• Lack of national consensus on what teachers should learn
• Irrelevant college course work with insufficient focus on math content
• Haphazard state guidance
• Very little, if any, perparation in the foundations of algebra

The report also found that it is possible in some states to pass the teacher licensing exam without answering a single math question on the exam correctly. The NCTQ recommends higher admission standards for education programs and more rigorous state licensing exams. MacMurray College, in Illinois, was one the schools in the study. Its president, Colleen Hester, faulted the study's methodology: "They just looked at syllabi," and not at measures of teacher competency or of elementary students' knowledge. Said a recent MacMurray graduate, "I felt prepared, but I don't think any college class can prepare you for actually teaching. I think experience in classrooms is what gets you ready, but as far as the content and what math skills I had to teach kids, that I was ready for." In response to the question (from the NBC 13 report), "Should someone be to blame---for teachers not knowing all the answers?", John Mayer, mathematics professor at the University of Alabama, Birmingham, says: "There's enough to go around... but ultimately it's up to parents and students to make sure they're getting the math knowledge they need for the future."

--- Mike Breen

This article uses the "postage stamp problem" to discuss the deeper problem known as "P vs. NP". The latter is one of the main outstanding problems in computer science today, and the Clay Mathematics Institute has offered a US$1 million prize for its solution. The postage stamp problem is the following. If you have a supply of, say, 2 cent stamps and 5 cent stamps, then there are only two values of postage you cannot make with these stamps, namely 1 cent and 3 cents. It is a general fact that, given an unlimited supply of stamps of certain denominations, there is always a key value above which any postage total can be put together with the stamps. The million-dollar question, as Stewart has framed it, is, What is that key value? The postage stamp problem is NP-complete, meaing that it appears to be much more difficult than problems solvable by polynomial-time algorithms. But recent work involving large stamp denominations has shown that particular cases of the postage stamp problem can be solved quickly. Perhaps, then, only certain cases of NP-complete problems are really hard to solve and most cases one is likely to encounter are solvable. This could have practical implications, Stewart notes. "Even if the method that your bank uses to encrypt your account information is NP-complete `in general'---which is more than can be proved for most practical encryption systems---the particular version that your bank is using might nevertheless be insecure."

--- Allyn Jackson

The Stonehenge monument has fascinated people for centuries. Many mystical and even outlandish interpretations have been proposed to explain the monument's creation and purpose. This book examines some of the mathematical and engineering aspects exhibited at Stonehenge. "No one denies that those who authorized Stonehenge had access to mathematical and constructional talent," the reviewer, an archeologist, writes. "But archeologists are rightly wary of ancient geometries `hidden' in a monument's layout." He found the book to be "refreshingly free from the lunacies of the past."

--- Allyn Jackson

How common is the incorrect belief that the number of miles a car travels on one gallon of gas (mpg) is directly proportional to the number of gallons it takes to travel a 100 miles (gpm)? Only one in 77 college students correctly identified the non-linear relationship between miles per gallon (mpg) and gallons used per 10,000 miles. This misconception stems from the idea that the mpg rating is synonymous with "how green" a car is. And certainly, the level of "green-ness" of a car is directly proportional to how much fuel it needs to consume to go a certain distance.

Although mpg and gpm ratings contain the same numerical information, the public's perception of the relationship between the terms "fuel efficiency" and "good for the environment" may lead many people to be surprised (at first glance) by the following fact. Upgrading an old clunker from 10 to 12 mpg will save the same amount of gas over a 60 mile trip as trading in a respectable 30mpg compact for a 60 mpg hybrid. Thus the authors of this article argue for changing the current measurement units from miles per gallon to gallons per 100 miles. A study by the Duke University researchers shows that consumers easily and accurately compare different levels of efficiency using the gpm units. Extending the theme of ease of comparison, the point is raised that a common metric is needed to measure the greenhouse gas emissions that result from various activities.

--- Brie Finegold

Enrique Cerda (University of Santiago, Chile) and a team of researchers have found that the inner angles made by rolled-up materials (such as paper) are very close to one another regardless of the thickness of the material or the width of the coil. In all the materials and rolls they tested, they found that the angle between the inner end of the rolled-up material and the circular roll is within a degree of 24.1°. Cerda is surprised that no one had discovered this before. He says that although the mathematics involved was complicated, the result could have been discovered by 18th century mathematicians, but "it seems no one thought to ask."

--- Mike Breen

Statistical analysis of patterns in musical compositions, such as those of Mozart and Bach, may allow computers to classify music by time period and settle disputes over a piece’s true composer. Similar techniques have been used to identify the authors of literature, based on a comparison of how frequently specific words are used in relation to each other. This principle is applied to music by substituting notes, chords, and other musical building blocks in order to measure similar patterns. Using this idea, as well as some information theory, computers can break a piece of music into a group of segments having the largest possible difference in terms of these building blocks, producing results comparable to how a trained human ear would parse the same composition. The author suggests it may someday be successful at measuring complexity, as well.

--- Lisa DeKeukelaere

Breaking the record previously held by a student of Sir Isaac Newton, New Yorker Alia Sabur received her bachelor's degree in mathematics at the age of 14 and is now the youngest college professor in the world according to the Guinness Book of World Records. Her research in nanotechnology and her enthusiasm for teaching and speaking have led her to a job at Konkuk University in Seoul, South Korea. Already having taught math and physics in New Orleans in an effort to advance the recovery from Hurricane Katrina, Sabur hopes to encourage international collaboration and be a role model for young women. By inspiring other girls to go into the sciences she says "they can prove the same thing: that girls are good at math and science and that you don't have to be really nerdy or weird to be successful in them."

--- Brie Finegold

From May 12^th through May 16^th, over 1,500 high school students from around the globe converged on Atlanta, Georgia, for this year's Intel International Science and Engineering Fair. In her article, Rachel Ehrenberg describes the work of some of the top prize-winners, including Intel Foundation scholarship winner Sana Raoof. Raoof "won for her research into the Alexander-Conway polynomial invariant for chord diagrams, work that sheds light on a branch of mathematics known as knot theory," writes Ehrenberg. This research could be applied to biochemical problems. In the second article, Bonnie Elgie interviewed Mate Bezdek, second place winner in the mathematics category. Bezdek won US$1,500 for his project entitled "The Blaschke-Lebesgue Problem Revisited." In this project, "Bezdek proposed a new theorem for computing and estimating volume in two- and three-dimensional polygons for this well-known, 100-year-old problem," writes Elgie.

--- Claudia Clark

Imagine looking through a telescope at a star far, far away. Now imagine that there are huge planets between you and the star. The light emanating from the star would bend around these planets, so that you would see more than one star when in reality there is only one. This is the basic principle behind what astrophysicists call "gravitational lensing". Sun Hong Rhie, while at the University of Notre Dame in Indiana, studied a gravitational lens comprised of a very specific configuration of n coplanar objects and realized that the number of images obtained from one light source was 5n - 5. Gravitational lenses give astronomers details of the distant universe, as Battersby says, but surprisingly they also helped prove a mathematics theorem. Completely independently, mathematicians Dmitry Khavinson and Genevra Neumann, from the University of Northern Iowa in Cedar Falls, were trying to find the zeros of a family of rational harmonic functions, inspired by a conjecture of A. Wilmshurst that attempted to extend the Fundamental Theorem of Algebra. Khavinson and Neumann found that if n is the degree of the rational function an upper bound on the number of zeros is 5n-5, but they couldn't prove that the bound was sharp. It so happens that the function Rhie used to find the number of "lensed" images was a harmonic rational function of the type Khavinson and Neumann were studying, and so when they discovered her research they were able to finish their proof. Thia research is described in "From the Fundamental Theorem of Algebra to Astrophysics: A 'Harmonious' Path", by Dmitry Khavinson and Genevra Neumann, Notices of the American Mathematical Society, June/July 2008.

--- Adriana Salerno

Although our grandmothers may not have envisioned their knitting needles or crochet hooks as mathematical tools, many mathematicians are turning to the handicraft to fashion models of non-orientable surfaces, hyperbolic planes, and Lorenz manifolds. Tactile insight into difficult problems is provided for both the creators of these surfaces and those who observe the finished product.

The impetus for all the articles is the "Hyperbolic Crochet Coral Reef" exhibition held at the Hayward Gallery in London. The project, conceived and curated by sisters---and Institute for Figuring co-directors---Margaret and Christine Wertheim, draws attention to the fragility and endangerment of Australia's Great Barrier Reef. The unique exhibit---actually a compilation of works by many crocheters---first drew media attention in New York before the exhibition traveled to the U.K. The "Woolly thinking" article explains how Margaret Wertheim, a physicist and mathematician, became inspired by mathematician Daina Taimina's hyperbolic crochet work; the "Hyperbolic crochet" article delves into this marriage of mathematics and craft; and "How crochet solved an age-old maths problem" tells the story of how Daina Taimina (Cornell University) started creating hyperbolic shapes with crochet and what they have to do with mathematics.

During the 2009 AMS-MAA Joint Mathematics Meetings, there will be a Special Session on Mathematics in Fiber Art co-organized by Carolyn Yackel of Mercer University and Sarah-Marie Belcastro of Smith College, who were featured in some of the articles. Talks in the session will relate the fiber arts to symmetry groups, computer science, embeddings of graphs in surfaces, and cellular automata.

--- Annette Emerson and Brie Finegold

A doctored photograph can ruin a political campaign, put a person behind bars, or win a photojournalist undeserved accolades. Hany Farid (pictured at left), a computer science professor at Dartmouth College, describes several techniques he and his students have developed to identify such frauds, in what he describes as an arms race between perpetrators and forensic analysts—similar to battles fought by spammers and hackers. Two of Farid’s techniques examine the eyes of the people in the photographs. By performing mathematical analysis based on the shape of the eyes, their relative position on the face, and the location of the "twinkle", Farid can determine whether the relative location of the camera and light source differs between the subjects—which would indicate that multiple photos have been stitched together. His other techniques include a similar method to identify a light source based on the shape and color gradation on inanimate objects, as well as a way to search for cloned regions based on comparisons between blocks of pixels. (Image courtesy of Hany Farid.)

--- Lisa DeKeukelaere

Click here for a list of links to web pages of publications covered in the Digest.