"When Math(s) Turns Out To Be Useful," by Davide Castelvecchi. Degrees of Freedom.
While some deem abstract mathematics as useless until proven otherwise, others argue that the intrinsic beauty of mathematical ideas makes their utility irrelevant. Meanwhile grant-writing mathematicians struggle to convince other scientists that their activities do indeed turn out to be useful. Aside from looking at recent successful grant proposals (as discussed in the Secret Blogging Seminar), we might look for examples of how abstract mathematical ideas influence technology and modern scientific discovery.
In the fourth post of his newly minted math and physics blog, Dr. Castelvecchi (a mathematician-turned-journalist) discusses a Nature article entitled "The unplanned impact of mathematics," by Peter Rowlett. Readers of Rowlett's article are asked to view money spent on mathematics research as a long-term investment. To show that such investments can pay off, Rowlett, an educator and blogger in the United Kingdom, solicited examples of how abstract mathematics has helped advance our understanding of other scientific topics like stock market volatility, genetics, telecommunications, computer graphics, quantum physics, cosmology, and the spread of disease. In an ongoing project of the British Society for the History of Mathematics, Rowlett continues to solicit other examples of the "unplanned impact of mathematics."
Castelvecchi sums up four of the seven topics covered in the Nature article, one of which links Johannes Kepler's ideas about sphere packings and ideas about the exceptional Lie group E8 to the transmission of information over noisy channels. Castelvecchi also comments that while the Nature article does not resolve the question of whether mathematical research is a "waste of time," it might not matter since mathematicians are cheap to fund (in comparison to other scientists).
"Math Contests Kind of Suck," and "What is an earnings surprise?" Mathbabe Catherine O'Neil.
Amongst the new math blogs popping up this summer is Mathbabe, a blog that focuses alternately on mathematics eduction and financial mathematics. As Dr. O'Neil has been both an academic and a quantitative analyst, she has some unusual insights into both worlds, and her frequent and opinionated posts elicit many comments that are themselves worth reading.
So what is the downside to math contests? On the one hand, Dr. O'Neil acknowledges that the process of preparing for them exposes students to non-standard mathematics. On the other hand, the more a student cares about winning, the more alienated from the subject he/she will become after a loss. Also, too much emphasis is put on quickly coming to a correct answer. O'Neil also brings up the possibility that fewer girls win or enjoy contests precisely because of their highly competitive and hurried nature. However, many commenters pointed out the existence of USAMTS a contest in which participants are given many days rather than a few minutes to think about problems. Others agreed wholeheartedly that more emphasis should be placed on collaborative enrichment as opposed to contests. Lastly, there is the question of whether winning contests is a valid predictor of future success in mathematics (as some students may feel that it is).
Borrowing terminology from the financial world, one might use the term "earnings surprise" to denote the difference between a student's predicted success (as measured by contests) and his/her actual success later on as a mathematician. Indeed, an "earnings surprise" in the financial world is the difference between the projected earnings of a company and the actual earnings (as reported by the company on a quarterly basis). To learn more about a model for projecting earnings (and how it is similar to a model for projecting levels of glucose in the body), you can read the post "What is an earnings surprise?"
Currently, Dr. O'Neil is teaching number theory to high school students at Hampshire College Summer Studies in Mathematics.
Math holidays are a relatively new phenomenon, and many bloggers have recently posted commentary on Tau Day (6/28), celebrated by those who see Tau (the ratio of the circumference of a circle to its radius) as a better "circle constant" than Pi (the ratio of the circumference of a circle to its diameter). In his four-year-old blog, Mike Croucher, who works at the University of Manchester, brought to light that not only that June 28th was Tau Day, but it might be called a Perfect Day since both 6 and 28 are perfect numbers. This observation originally came via tweet from mathematician Marcus du Sautoy, who incidentally will be hosting a new show (which first aired on July 27th) on BBC Two called The Code.
In his Tau Day post, blogger Matt Lane asked Tau Day founder Dr. Michael Hartl what foods might be appropriate for celebrating Tau Day. Since Pi Day's popularity largely rests on the consumption of lots of pie, Mr. Lane (who is working towards his PhD in mathematics at UCLA) suggested two pies baked into one cake. On the more serious side, Lane mentions that he hopes that Tau Day will not replicate the Pi Day tradition of reciting as many digits of Pi as one can, an activity lacking mathematical merit.
Carl Bialik's post focuses on the growing number of dates to which people assign mathematical significance. In fact, Bialik interviews Lane who says “simply finding a pattern doesn’t make the pattern significant, and assigning some significance to it is what rings of numerology to me." While Bialik quotes several notable mathematicians who seem not to care if it's Odd day or Square Root Day or Tau Day, most seem to think that such days are harmless and fun. Dr. Kenneth Ribet says, “I do think that it’s better to actually learn mathematics than to organize social events based on mathematical curiosities. If the people doing the social events also learn some math, then this is fine.” In an effort to pin some mathematical heft onto the day of his post, Bialik declares July 16th, 2011 "Square-Prime Number Sandwich Day." Maybe this is the day one eats a square piece of bread between two pieces of prime rib.
--- Brie Finegold
"Crime's digital past," by Bruce Bower. Science News, 30 July 2011, p. 20.
In this article, Bruce Bower describes the findings of a team of researchers who used computer software to probe the "digitized records of the more than 197,000 Old Bailey trials... that took place from 1674 to 1913." For example, when the team calculated trial lengths for guilty and non-guilty verdicts over this time period, they found that, around 1825, trial lengths diverged: "One set of trials maintained a previous drift toward lengthy proceedings, whereas a second set of hearings concluded quickly." At around the same, guilty pleas also increased. In fact, prior to 1825, most defendants pled innocent and sought a trial by their peers while, after 1825, "the number of guilty pleas soared" as lawyers "increasingly encouraged clients to plead guilty." For researcher Tim Hitchcock, "Finding a revolution in legal practice at that time came as a complete surprise."
Another set of data the team looked at were record from bigamy trials during the same 239 years. They found that "the number of cases of female bigamy rose from virtually nil to an average of six or seven annually after 1880," which, along with other data, provides suggestive evidence of a "new strain of female independence" in Victorian England.
--- Claudia Clark
"Mathematical medallist: Seducing CEOs and socialists": Interview with Cédric Villani. Interviewed by Jacob Aron. New Scientist, 28 July 2011.
Cédric Villani received a Fields Medal in 2010 for work that shed new mathematical light on longstanding questions from physics. Around that same time he became director of the Institut Henri Poincaré in Paris. This interview was done in London, where Villani was giving a public lecture hosted by the London Mathematical Society. In the interview, Villani said that he has basically done no research in the preceding year and has instead focused his energies on reaching out to the general public. He has given more than 100 lectures to "politicians, journalists, school kids, university students, all kinds of people". For a lecture to a rally of the extreme left, "I emphasized the values of internationalism and idealism, which are the basis of science," he said. "With the CEOs, I focused on the daring part of my work, the fact that as researchers we are adventurers and try to explore and conquer, and the fact that as the director of an institute I am a bit like a CEO. I discovered that in all of these worlds that I was not familiar with, I could always find a way to connect." Asked about the biggest problem facing mathematics right now, Villani pointed to the Riemann Hypothesis (see the Clay Mathematics Institute's page on this problem) as perhaps the greatest outstanding challenge in the field. He also briefly touched on the highly fruitful interactions with computer science and with physics. (Photo of Villani in the AMS exhibit at ICM 2010, by Dana Chyung.)
--- Allyn Jackson
"Santa Clara University math scholar fights crime with mad math skills," by Mike Cassidy. San Jose Mercury News, 28 July 2011.
Santa Clara University Assistant Professor George Mohler has developed a statistical model, currently in use by the Santa Cruz police department, that predicts the time and location of crimes and has earned him the label "crime-fighting math professor superhero." Mohler's model is based on the principles of earthquake prediction, specifically that a big quake will have aftershocks. By sifting through mounds of historical crime data, Mohler's program identifies the probable "aftershock" locations of other crimes, which are often near the location of the initial crimes because "burglars, like most of us, hate long commutes." A Santa Cruz PD crime analyst says Mohler's model is highly correlated to actual crime reports, and Mohler's model has arrived--fortuitously--at a time when decreases in funding for police officers have increased the imperative for smart technology to assist police work.
--- Lisa DeKeukelaere
"A Sleepaway Camp Where Math is the Main Sport," by Rachel Cromidas. The New York Times, 27 July 2011.
The camp in question is the Summer Program in Mathematical Problem Solving (SMPMS) which took place at Bard College in Annandale-on-Hudson, NY for three weeks this summer. Seventeen low-income seventh-graders were able to attend free of charge, thanks to financing from the Art of Problem Solving Foundation. The camp's purpose was to challenge students who already excel at math. Each day students received six hours of instruction, and participated in some non-mathematical activities. The director of the camp, Daniel Zaharopol, said, "These are students who have a tremendous amount of potential and are really ready for a lot more then they're able to get in schools." The article concludes with a quote from Mattie Williams, summarizing her initial experience at the camp: "The first night we all sat in each other's rooms and talked about what we wanted to do, and how, oh, I miss my mom, I miss my dad....Then we had a pillow fight." (Photo by Ana Portnoy.)
--- Mike Breen
"Perspective in Math and Art," by Annalisa Crannell. Inside Higher Ed, 18 July 2011.
"The enduring myth of music and maths," by Timothy Gowers. The Independent, 6 July 2011.
Mathematician Timothy Gowers says that one way he recovers from the conversation-killing admission that he is a mathematician is to note that he is also a musician. This prompts many people to repeat the often touted connection between mathematical and musical aptitude, and how one can be good at one discipline by doing well in the other. While granting that both mathematics and music have similar patterns, symbols and counting, Gowers goes on to debunk the Mozart Effect, "where children who have been played music by Mozart are supposedly more intelligent, including at mathematics, than children from a control group." He notes that the connection is more anecdotal than statistically proven, and that "there are plenty of innumerate musicians and tone-deaf mathematicians," so is probably more related to both involving hard work and discipline. Gowers brings up the interesting point that "abstract structures are not confined to mathematics and music. If you are learning a foreign language then you need to understand its grammar and syntax, which are prime examples of abstract structures. And yet we don't hear people asking about a mysterious connection between mathematical ability and linguistic ability. My guess is that that is because the connection exists but not the mystery: grammar feels mathematical, so it would hardly be a surprise to learn that mathematicians are better than average at learning grammar." In the end, he is far more interested in wanting to know which instruments and composers mathematicians are drawn to, and how mathematicians listen to music. Meanwhile, "this is uncharted territory and all we can do is speculate."
--- Annette Emerson
"Maths powers Google bid strategy." BBC News, 4 July 2011.
In a recent auction to acquire wireless technology patents from the bankrupt firm Nortel, Google reportedly placed bids in the form of non-whole-number constants such as Meissel-Merten’s constant and Brun’s constant. At one point, Google met a competitor’s bid of $3 billion with a counter bid of $pi billion. Outsiders were unsure whether Google’s choices were born of confidence or boredom, but the company ultimately lost out to a consortium that included Apple and Microsoft. Intellectual property analyst Florian Mueller notes that the sale was a prime opportunity for Google to acquire patents that would pave the way for the success of its Android product, and he was surprised that the search giant pulled out of the race.
--- Lisa DeKeukelaere
"More than sum of its parts": Review of The Man of Numbers by Keith Devlin, Cosmic Numbers by James D. Stein, and The Mathematics of Life by Ian Stewart. Reviewed by Jesse Singal. The Boston Globe, 3 July 2011.
In this article, Singal reviews three recently published books about mathematics. First up is The Man of Numbers: Fibonacci’s Arithmetic Revolution, in which Keith Devlin describes Fibonnaci’s book, Liber Abaci, or Book of Calculations, published in 1202. At a time when Western Europeans were still working with Roman numerals, Fibonacci’s book demonstrated “just how powerful the Hindu-Arabic number system and arithmetic were.” Next up, we learn about James Stein’s book, Cosmic Numbers: The Numbers that Define Our Universe. As Martin Rees did in his 1999 book Just Six Numbers: The Deep Forces that Shape the Universe, Stein “examines the profound way the values of physical constants affect our universe,” while, at the same time, “moving beyond cosmology to look at numbers like absolute zero and the ideal gas constant.” Finally, Singal turns to Ian Stewart’s The Mathematics of Life, which “seeks to show readers how numbers underpin biology’s endless mysteries.” Singal appreciates Stewart’s ability to “communicate wonder,” as well as his skepticism in the face of the misuse of numbers in “mainstream treatments of science.”
--- Claudia Clark
"Africa AIMS high," by Neil Turok. Nature, 30 June 2011, pages 567-569.
In 2003, physicist Neil Turok, currently director of the Perimeter Institute at the University of Waterloo in Canada, launched the African Institute for Mathematical Sciences to help prepare Africa's top mathematics and science graduates for careers in research, industry, and government. A native South African and son of anti-apartheid activists, Turok wanted to do something to help his continent prosper. He chose to focus his energies on the mathematical sciences for three main reasons: mathematics is the backbone of all science and technology, it is a universal language that can unite people, and it is cheap: "All you need is a library, a computer lab and a lecture hall." To date AIMS has graduated 361 students, with 224 master's degrees and 125 PhDs either completed or under way. One-third are women, whom Turok notes are "a particular success." "Away from family responsibilities they can focus full time on developing their minds," he writes. "Many undergo a remarkable transformation, from shy graduates leaving home for the first time, into impressive young scientists and leaders." The article describes the unique learning environment at AIMS, noting the important role played by guest lecturers who come from abroad. "There is no shortage of volunteer lecturers---we have 500 offers, and only 25 are needed each year." Turok also describes his goal of expanding AIMS across the African continent and his efforts to raise the necessary funds.
--- Allyn Jackson
"'Tau day' marked with opponents of maths constant pi," by Jason Palmer. BBC News, 28 June 2011.
The mathematical constant pi is a popular character. British pop singer Kate Bush wrote a song about the number and Darren Aronofsky directed the movie Pi. Many even celebrate March 14th (3.14) as 'pi-day'. Still the number has its critics and a group of mathematicians wants to replace pi with a new constant, tau. Tau is the ratio of a circle's circumference to radius, while pi is the ratio of circumference to diameter. Tau is twice as large as pi (about 6.28) and makes some calculations in geometry easier. However, given how we all know and love pi, replacing it is going to be a hard task [Ed. note: Not as easy as ...].
--- Baldur Hedinsson
"Listen by numbers: music and maths," by Marcus du Sautoy. Guardian, 27 June 2011.
What is it, mathematician and sometime Guardian columnist Marcus du Sautoy asks, that links music and mathematics in the minds of so many of their practitioners? After examining how arithmetic underlies the musical vocabulary--with counting essential for rhythm, and division defining the octaves--du Sautoy considers the role of aesthetics in both practices. In math, as in music, he explains to the uninitiated, there is surprise and excitement to be found. The role of the mathematician, like that of the composer, is to pick the beautiful and interesting melodies out of the noise of the world. A mathematical proof, like a piece of music, can be repeated and savored, as the listener comes to understand "how themes are established, mutated, interwoven and transformed." But after humming these well-known tunes, du Sautoy concludes that it is for love of certain kinds of structures and patterns--symmetries and regularities both simple and complex, from the Fibonacci sequence to geometry--that mathematicians are drawn to music, and musicians drawn to mathematics.
--- Ben Polletta
"Mathematics Teachers' Subtle, Complex Disciplinary Knowledge," by Brent Davis. Science, 24 June 2011, pages 1506-1507.
When it comes to teacher subject knowledge, most people focus on explicit knowledge. Davis writes that the most important skills for math teachers are "tacit, like skills involved in playing concert piano, learned but not necessarily available to the consciousness." An expert's skill in understanding and using different interpretations is a subtle complexity that Davis says is not measured by typical tests or addressed in teachers' courses. He writes that recent research suggests that there is little or no correlation between math courses taken by teachers and performance by their students on standardized tests but that new approaches to preparing math teachers could improve their students' understanding of, and attitudes towards, mathematics. The article concludes with: "In a knowledge-based economy, the development of conceptual fluency is of increased importance and has been the focus of major initatives in school mathematics. Emerging research into the subtlety and complexity of teachers' knowledge not only reveals that these initiatives have fallen far short of their lofty goals, it may offer an important route to achieving them."
--- Mike Breen
'Knot theory', Osaka University, bring mathematical knowledge to a new game, (in Japanese) " Asahi Shimbun, 17 June 2011.
This article (which seems to be available only in Japanese) discusses a computer game based on a new result in knot theory. Daniel Moskovich described the article in an entry in the Low Dimensional Topology blog: "The story is occasioned by the release of a fun computer game based on a theorem of Ayaka Shimizu. Ayaka is a member of Akio Kawauchi's Knot Theory group at OCAMI [Osaka City University Advanced Mathematical Institute]. The theorem appears in her preprint `Region crossing change is an unknotting operation'... Kawauchi, Shimizu, and Kishimoto [another member of the group at OCAMI] turned it into an addictive computer game! I must say, their game is the most fun game I've seen come out of topology... and you don't need to know any topology in order to play it... Asahi Shimbun picked up that this was an amazing new gaming idea, and took the opportunity to run a full-page spread on Knot Theory. I vote this article as one of the best media exposures mathematics has ever had." Click here to play the game.
--- Allyn Jackson
"'Origami Engineer' Flexes to Create Stronger, More Agile Materials," by Zeeya Merali. Science, 17 June 2011, pages 1376-1377.
Surprise guests from out of town? No problem, fold out an extra bedroom! Surprise heat wave? Fold out a pool! Origami houses--reconfigurable at the push of a button--are a dream of origami engineers like Zhong You of the University of Cambridge. You's talent, says colleague Erik Demaine, is in finding near-term applications that bring origami technology closer to that dream. One example is You's origami stent. Folded, it's narrow enough to travel down a blood vessel; unfolded, it expands against the vessel wall. And unlike stents made of fabric stretched over a metal framework, You's one-piece stent won't fall apart.
Origami engineering first appeared in the 1970s, when origami folds were used to pack solar panels into small volumes for space flight. But only recently, as mathematicians have begun to elucidate the geometry of folding, have scientists like You been able to move beyond folding by trial-and-error to designing and mass-producing origami structures. You and doctoral student Joe Gattas hope to design emergency shelters that can be transported as flat sheets and unfolded at disaster sites. Origami may also have applications to packaging materials--from grocery bags (You has designed a folding steel version) to cardboard boxes. Origami crease patterns might even be applied to strengthen materials and make them more rigid, says You. And if none of this works out, he says, "People keep offering to buy my pieces as stress-relieving toys ... I can set up a business as a toy maker."
--- Ben Polletta
Move over modern art—there’s about to be a new museum in town. Mathematician Glen Whitney and his team have raised US$22 million of the $30 million needed to launch the Museum of Math (MoMath) in Manhattan, and in the mean time they’re taking their show on the road with a travelling “Math Midway” exhibition. The display is intended to serve as a test bed for exhibits in the future museum, and Whitney is convinced that a square-wheeled tricycle—which rides smoothly on a scalloped track—will be as big a hit in midtown as it has been elsewhere. (The tricycle demonstrates a mathematical principle about how specific shapes behave in the real world.) He aims to make the museum a place where people of all ages are comfortable getting excited about math, but he is specifically targeting children in the fourth-to-eighth grade range, where students typically begin to engage less in math. (Image: Artist's rendering of Museum of Mathematics, courtesy of the Museum of Mathematics.)
--- Lisa DeKeukelaere
"Preparing Future Math Teachers," by William H. Schmidt, Richard Houang, and Leland S. Cogan. Science, 10 June 2011, pages 1266-1267.
In this article, Schmidt, Houang, and Cogan of Michigan State University explore the long-standing question of what constitutes a well-qualified middle school mathematics teacher. To do so, they compared the results of the 2010 Teacher Education and Development Study in Mathematics (TEDS-M)—a survey of future math teachers who are in their last semester of training—among the 16 participating countries. Within each country, they also compared the TEDS-M results with that country’s eighth-grade mathematics achievement as measured by international tests such the TIMMS. Finally, they compared the average TEDS score of over 80 U.S. institutions with the average SAT mathematics score “for the institution’s TEDS-participating students.” Among their results, they found that “U.S. future teachers’ TEDS scores straddle the divide between countries whose middle school students do better than the United States on international tests…and those who do not.” In fact, “if Taiwan and Singapore were to select their average eighth graders…to become future middle school mathematics teachers, the United States would have to draw its future teachers from above their 75the percentile to be comparable to those from Taiwan and Singapore in their knowledge of mathematics.” Among their recommendations: “a combination of recruiting those who have strong quantitative backgrounds together with a greater emphases on rigorous mathematics in teacher preparation. The latter needs to be driven by tougher middle school mathematics teacher certification requirements.”
--- Claudia Clark
"Unzipping Zipf's Law," by Lada Adamic. Nature, 9 June 2011, pages 164-165.
This article describes the work of S.K. Baek et al., recently published in the New Journal of Physics, in developing a generic model for systems that behave according to a “power law,” meaning that the odds of observing a group of a given size are inversely proportional to that size raised to a fixed power. Zipf’s law, for example, can describe “populations” from city sizes to word counts in a book and states that the probability of observing a grouping of size k is equal to 1/(kα), where α is a constant between 1 and 3. Scientists have developed models for many systems that generally adhere to a power law, but the models use empirical data to account for each specific system’s slight deviations from the typical power law curve, and therefore the models are not transferrable between systems. Baek and his colleagues set out to find a model that would apply to all power law systems and constructed a “random group formation” (RGF) model , which figures out a distribution of group sizes that minimizes the amount of information needed to locate an item within the population, knowing only the size of the group in which it resides. The author notes that RGF requires further development, but sets a high standard for future efforts to develop general power law models.
--- Lisa DeKeukelaere
"Mathematician's newly discovered letters to be digitised at UCC," by Louise Roseingrave. Irish Times, 2 June 2011.
The English mathematician George Boole is regarded as the founder of modern computer languages. He developed the logic that became the basis of all things digital. Thanks to the field Boole helped ignite, much of his work is now available on the internet. Some 4,000 items, including printed books and papers have been digitized and made available online. (Image: Scarf created by Carmel Creaner, Cork, for University College Cork, photo by John Sheehan Photography.)
--- Baldur Hedinsson
"Game theory study says GPO fees don't hurt hospitals," by Brendon Nafziger. DOTmed News, 2 June 2011.
A group purchasing organization, or GPO, serves its member hospitals and health-care providers by negotiating purchases of medical supplies. It's strength in numbers--the high-volume purchases they arrange save their members money. For their trouble, Congress allows GPOs to collect administrative fees from suppliers, typically about three percent of the sale. But where do these fees really come from? Vendors and other critics of GPOs say that they come out of members' pockets, and that repealing the "safe harbor" provision allowing GPOs to charge such fees would save healthcare providers billions. Not surprisingly, the GPOs point to data indicating that their members save money over negotiating alone with suppliers, while medical suppliers point to data that online auctions yield better prices than GPOs. To get to the bottom of the issue, Purdue University professor of management Leroy B. Schwarz used game theory. In his model, multiple GPO members, a single GPO, and a medical supplier with one competitor interact. The GPO and the supplier seek to maximize their profit, while the GPO members seek to minimize their costs--which include both the price of goods and the cost of negotiating contracts and researching providers. Under these conditions, says Schwarz, the fees that GPOs assess change independently from the prices their members pay--so the fees come out of vendors' pockets, not members'. Which explains why medical suppliers are such staunch critics of GPOs, even if it proves them wrong.
--- Ben Polletta
"These are the biggest numbers in the universe," by Alasdair Wilkins. io9.com, 1 June 2011.
You’ve probably heard of extreme sports…but extreme math? That’s the topic for this article in which Alasdair Wilkins describes for the general reader the biggest meaningful numbers in the world—“meaningful” depending upon your perspective. Wilkins starts by defining googol (10100) and googolplex (1010100), then gives examples of some very large numbers with real-world significance, starting with the world’s current population—about 6.92 billion—all the way up to one estimate of the number of universes in the multiverse: 1010107. At this point, we are already well beyond the amount our brain can literally perceive—roughly 101016—but there are “far larger numbers lurking out there,” which Wilkins goes on to describe. These include Mersenne primes (the largest currently known is 243,112,609 – 1), Skewes’ number 10101034, and mind-blowingly huge Graham’s number, the largest number ever used in a mathematical proof, according to the Guinness Book of World Records.
If this is not big enough for you, check out Wilkins’s follow-up extreme math post in which he discusses infinity.
--- Claudia Clark
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