"Recipe for Disaster: The Formula That Killed Wall Street," by Felix Salmon. Wired Magazine, 23 February 2009;
Models are often cast as dangerous seductresses, causing the downfall of powerful leaders. In the cover story in Wired, the model is mathematical. The article's color-coded breakdown of economist David Li's "fatally flawed function" deems a parameter "all powerful", an equation "irresistible", and the concept of equality a "dangerously precise concept" that leads financial experts to divorce themselves from reality. Even after this exposé of the habitually hidden phi's and gamma's of probability, one of the most striking features of "Gauss's copula function" is that "copula" is Latin for "coupled". But the terminology and tenor of this article may simply be a sign of the times. In a recent letter to his shareholders, Warren Buffet cautioned, "Beware of geeks bearing formulas" and chided his audience for being taken in by "a nerdy-sounding priesthood, using esoteric terms such as beta, gamma, sigma and the like." Buffet betrays his own point by describing letters of the Greek alphabet (which Greek people use every day) as "esoteric", a word deriving from Greek.
For those who see mathematics as a foreign language, the Wired article slowly introduces a more nuanced truth, and numerous hyperlinks help explain unfamiliar terms and concepts. A correlation number (the "irresistible" one) indicates the likelihood that two companies will default simultaneously on paying their bonds. Li's function provided a fairly easy means to calculate correlation numbers, and thus pool bonds together in a way that supposedly minimized or at least quantified risk. Securities other than bonds may be pooled, and these pools are called Credit Default Obligations (CDO's).
Even further removed from any concrete security are Credit Default Swaps (CDS's). A CDS is insurance against default for the owner of a bond or CDO. But the CDS's themselves can be traded independently so that every bond or CDO could give rise to many CDS's. Over the past ten years, the CDS market has ballooned over ten times. It seems that Li's model used CDS performance, which is an indicator of the risk that investors perceive, as a means of determining actual risk. Thus no research into the underlying financial instruments was needed to come up with a correlation number. This is similar to polling your friends to see who they are voting for and then casting your vote according to the majority of your friends' votes. It makes your decision easier, and it works well if your friends are all reading up on the candidates. But if everyone did it, the results might not be ideal since no one would be learning about the candidates directly.
While acknowledging that Li is not personally to blame, the author still paints managers as less culpable than "quants" (quantitative analysts). Salmon explains bankers' blindness to the risk involved in CDO's and CDS's by pointing out that the mathematical model is a "black box" whose output is hard to put to a "smell test". Another excuse for why the risk was not apparent was that despite Salmon's earlier assertion that Li used "relatively simple math---by Wall Street standards", he later writes that financial managers "lacked the math skills to understand what the models were doing". Although those commenting on Salmon's article, in the other coverage cited above, empathize more with the quants, they tend to still focus on the need for better quants as opposed to better managers. Salmon writes, "... too many quants see only the numbers before them and forget about the concrete reality the figures are supposed to represent. They think they can model just a few years' worth of data and come up with probabilities for things that may happen only once every 10,000 years." Salmon's use of a number of years more appropriate to an archeological timeline than a financial one brings home the point that the creators of mathematical tools are not to blame for their misuse. No one has ever accused the knife industry of causing a stabbing.
--- Brie Finegold
Recent columns by Marcus du Sautoy. Times Online, February and March 2009.
"What do we need algebra for?," by Keith Devlin and Scott Simon. NPR Weekend Edition, 28 February 2009.
Keith Devlin, NPR's math expert, discusses algebra, spreadsheets, and why we're still doing math ourselves after the advent of computers. Devlin notes that most people find algebra difficult because they try to approach it like arithmetic, which they have already spent years mastering. Unlike arithmetic, however, algebra requires looking at numbers logically and analytically. Devlin presents the ubiquitous computer spreadsheet as the consummate example of using algebra in everyday life and the best way to show students algebra's value. Spreadsheets take care of the arithmetic, but the user must understand algebra in order to organize the spreadsheet and achieve the desired result. Although some believed that computers and calculators would render human mathematical skills unnecessary, Devlin argues that computers simply changed the type of math humans need to practice from arithmetic to algebra.
--- Lisa DeKeukelaere
"Scientists build computer model for snowflakes," by Andrew Stern (AP). Boston Globe, 25 February 2009.
"Simons's Notion: All In, Then All Out," by Aaron Lucchetti and Jenny Strasburg. the Wall Street Journal, 25 February 2009.
"James Simons helped put Stony Brook University on the map with his work as a mathematician, philanthropist and billionaire hedge-fund manager. He also helped introduce the college to Bernard Madoff." So begins this article about how Simons first encouraged the school to invest with Madoff in 1991, then recommended that the school pull out all of its money in 2004. In the early 2000s Simons and Robert J. Frey (current co-chair of Stony Brook Foundation's investment committee), were concerned "about the lack of transparency in Mr. Madoff's firm and the possibility that his stellar returns might not be sustainable." By 2004 Stony Brook's investments with Madoff had grown to US$8 million. "Mr. Simons urged Stony Brook to pull out all its money, but other members of the foundation's board disagreed" and pulled out US$3.5 million. Stony Brook, one of Madoff's first institutional clients, now faces a US$5.5 million loss in the "alleged Madoff fraud," but it could have been worse without the earlier advice of Simons.
--- Annette Emerson
"? = Beauty," by Sebastian Smee. Boston Globe, 22 February 2009.
A new exhibit at the Museum of Fine Arts, Boston, called "Splendor and Elegance," showcases the decorative arts and drawing collection of Horace "Woody" Brock, who has five degrees in mathematical economics, and political philosophy. He is the founder and president of Strategic Economic Decisions, Inc., "a think tank specializing in applying the economics of uncertainty to forecasting and risk assessment." When it comes to acquiring and judging his collection of clocks, chairs, candlesticks, etc., Brock "believes he has cracked the secret of beautiful design. He even has equations and graphs to prove it." In sum, "Designed works, Brock states, can be broken down into 'themes' and 'transformations.' If a work's theme, or main design motif, is simple, we are most satisfied when its transformations within that work are complex, and vice versa." The lengthy article quotes his strong opinions on visual beauty and collecting, touches on the beauty of Mozart's music, and includes a graph, "Representation of Aesthetic Beauty," depicting "maximal beauty" and "the optimal tradeoff."
--- Allyn Jackson
"Only 3.14 More Shopping Days Until Pi Day," by Holly Fletcher. Indiana Gazette, 18 February 2009.
"Idea of Infinity Stretched Back to Third Century, B.C.," by Robin Lloyd. Live Science, 17 February 2009.
This article, and an Associated Press report ("Uncovering ancient secrets beneath the surface," by Randolph E. Schmid, 16 February 2009), cover a talk by Uwe Bergmann at the annual meeting of the American Association for the Advancement of Science. Physicist Bergmann (Synchrotron Radiation Lightsource at Stanford University) and classicist Reviel Netz (Stanford University) have examined and scanned the ancient parchment page from the 348-page Archimedes Palimpsest using modern x-ray and spectral imaging, and "the new reading reveals that Archimedes was engaged in math that made conceptual use of actual infinity", as Netz describes on the Web site ArchimedesPalimpsest.org. The palimpsest is at the Walters Museum in Baltimore.
--- Annette Emerson
"Editorial: Trying to fathom 13-figure spending." Minneapolis Star-Tribune, 15 February 2009.
With the huge budgetary figures in the news, this editorial admits that one trillion is a big number (enough to buy each American 1,000 boxes of Girl Scout Thin Mint cookies), but not out of line with other budget figures. The paper looked to Doug Arnold (University of Minnesota and president of the Society for Industrial and Applied Mathematics) for comparison. He pointed out that one trillion dollars is about 7 percent of the annual US GDP. As to whether the large amount is enough to revive the US economy, Arnold responds that that is an economics question and says, "The hope is the best economic minds are the people giving advice to the administration. And that those are the ones he's listening to."
--- Mike Breen
"A head 4 figures: How will Marcus du Sautoy make maths fun?" An interview with Deborah Orr. The Independent, 14 February 2009.
Marcus du Sautoy has frequently generated media coverage, but U.K. media have been particularly interested in du Sautoy since he was recently named Simonyi Professor for the Public Understanding of Science at Oxford University. His enthusiasm for mathematics, commitment to promoting awareness of its beauty and applications, and opinions of mathematics education are covered in these two recent interviews. In the Montgomery interview he recalls one of his "eureka" moments: "when I discovered a new symmetrical object which had never been seen before. Mathematicians talk about that flash they get, and it was almost electrical. You could probably look at the neuroscience of it and see that there actually was some sort of electrical surge. In that moment, a new thing was created. Which was mine. Nobody has ever thought of it before and it made a connection between two totally different areas of mathematics. I took my subject in a completely new direction." He is currently working on an online math school "where people can just play games and learn maths through doing it." He is convinced that math is a creative endeavor, and that in addition to its beauty it has many applications.
--- Annette Emerson
"Take me to your mathematician," by Martin Rees. New Scientist, 14 February 2008, pages 37-39.
This article is based on a contribution to a panel discussion about the relationship between mathematics and science. In addition to the author, Martin Rees, president of the Royal Society and former Astronomer Royal, the panel included mathematicians Michael Atiyah and Alain Connes. Rees begins his article by noting that, if humans were ever to have contact with alien beings, mathematics would provide "the surest common culture". The article then ranges widely over intersections between mathematics and science, including the mysteries and promise of string theory, the question of whether the universe is infinite, and efforts to understand highly complex phenomena, which Rees says is "perhaps the most challenging [frontier] of all". In discussing efforts to find a "unified theory" of physics, he notes that, just as a fish is intimately familiar with the properties of water but cannot grasp that water consists of atoms of hydrogen and oxygen, human beings might not be able to grasp a unified theory of physics. If the universe is infinite, Rees writes, the theory of infinity created by 19th century mathematician Georg Cantor could play a central role. Without Cantor's exotic ideas about different sizes of infinity, "cosmologists will not be able to firm up the concept of the multiverse theory and decide, without paradoxes or ambiguities, what is probable and improbable within it." Rees also discusses the "game of life" invented by mathematician John H. Conway, which showed how simple rules can give rise to extremely complex behavior, and muses on the widespread use of computer simulations, which are changing the face of many areas of science. "Maybe in the far future..., post-human intelligence will develop hypercomputers with the processing power to simulate living things---even entire worlds," he writes. "This raises a disconcerting thought: perhaps that is what our universe really is."
--- Allyn Jackson
Cipra describes three events at the 2009 Joint Mathematics Meetings: the session on redistricting, a talk on fractal billiards, and Peter Winkler's Invited Address "Stacking Bricks and Stoning Crows."
Richard Pildes (New York University School of Law) says of redistricting, in which a state's ruling party redraws districts in its favor, "the problem is much worse than it used to be." A social science graduate student at Caltech, Alan Miller, presented a method to quantify the bizarreness of a political district, which involves "the probability that the most direct path between two randomly chosen voters in the district crosses district lines." With this measure, Maryland's Third District is the most bizarre at 0.860. The least bizarre district is Connecticut's Fourth, at 0.023. Rehmeyer writes of Zeph Landau (University of California, Berkeley) who uses the mathematics of fair division to create districts. Another article on the redistricting session and other Joint Meetings events was summarized earlier in the Math Digest.
Robert Niemeyer, a graduate student at the University of California, Riverside, and his adviser Michel Lapidus are researching fractal billiards "in which a point-mass cue ball rattles around inside a shape [such as the Koch snowflake] whose boundary seemingly consists of nothing but corners." Currently Niemeyer is trying to find examples of periodic orbits. The research could apply to sound bouncing off a rough surface and to solutions of wave equations in quantum chaos.
Peter Winkler's Invited Address showed how to extend stacked bricks beyond the usual length associated with the harmonic series and the natural log of the number of bricks. The method, published in the American Mathematical Monthly, shows how to increase the overhang to a multiple of the cube root of the number of bricks, and uses techniques from the area of random walks. A follow-up paper shows that the upper limit is six times the cube root of the number of bricks.
--- Mike Breen
"Darwin, the Reluctant Mathematician," by Julie Rehmeyer. Science News, 11 February 2009.
On the occasion of Charles Darwin's 200th birthday, Rehmeyer and other journalists are examining Darwin's inspirations and influence. Rehmeyer notes that in his autobiography Darwin recalled he didn't like studying mathematics, yet, as she goes on to explain, "history played a joke on the great biologist: it made him a contributor to the development of statistics. It was the wildflower common toadflax that got the whole thing started." Darwin observed while experimenting with cross- and self-fertilizing these that the hybrids, on average, were taller. He consulted his cousin, Francis Galton, who was "a leader in the emerging field of statistics" and inventor of the concept of standard deviation. But Galton reported that there was not a large enough sample to determine if the height differences were random. Forty years later, in an effort to unite Darwin's theory with Mendelian genetics, Sir Ronald Aylmer Fisher "noticed something Galton had missed: Galton had ignored Darwin's clever method of pairing the plants. He had calculated the standard deviation of the plants as a single, large group." Fisher further refined the calculations to determine that indeed, the hybrids really did grow taller than the purebred plants. "Fisher's analysis was only possible because Darwin had designed his experiment so well." Two current-day statisticians, Susan Holmes of Stanford University, and David Brillinger of UC Berkeley, confirm Fisher's analysis of Darwin's experiments. Rehmeyer concludes, "Darwin himself came around eventually in his attitude toward mathematics. While he wrote in his autobiography of his youthful distaste for math, he also wrote that he wished he had learned the basic principles of math, 'for men thus endowed seem to have an extra sense.'"
--- Annette Emerson
"A New Puzzle Challenges Math Skills," by Will Shortz. New York Times, 9 February 2009.
"And the Oscar Goes to...Not Its Voting System," by Carl Bialik. Wall Street Journal, 6 February 2009.
Millions of Americans tuned in to witness the pageantry of the Oscars, but few may know that the methods used to choose the nominees and winners are far from flawless. The nominees were chosen by instant runoff, whereby voters rank the candidates, and the ones with the fewest first-place votes are eliminated. The eliminated candidates' votes are then redistributed based on their supporters' next favorite choices. The name "instant runoff" implies that this system simulates a series of runoff elections. Once the nominees are chosen, a plurality vote determines the final winner.
A system that truly reflects the wishes of the voters is ideal, and many mathematicians agree that instant runoff is not ideal. However, there is no consensus among voting theorists on the best voting system. We can imagine a scenario in which a movie is liked second best by everyone but is immediately eliminated by instant runoff since it received no first place votes. The Gore-Bush-Nader race demonstrates the flaws of the plurality system since few of Nader's supporters would have ranked Bush over Gore. Given that so many seemingly unfair outcomes can be cooked up with a given system, one might question whether any voting system can truly reflect the will of the voters. It is suggested in this article that the best choice of system is contextual, and that the debate over which is the best system overall is getting in the way of useful reform. The article also provides one carefully crafted data set which when seen through the lenses of eight different systems (including the plurality and instant run-off mentioned above) elects eight different candidates. Meanwhile, there are no plans to change the nomination process for the Oscars, and the records of past votes are not being revealed to the public, providing no data for mathematicians to analyze. So if you don't approve of the winners, you can lay some of the blame on the voting system and not the members of the Academy.
--- Brie Finegold
"Playing with Numbers: Teaching Parents to Teach Math to Pre-School Children," by Supawadee Inthawong. Bangkok Post, 5 February 2009.
In his book Kindergarten is Too Late, Sony's founder Masaru Ibuka makes the claim that children around the age of two or three have a "natural ability to learn algebra," according to Bangkok Post writer Supawadee Inthawong. She reports that, upon reading Ibuka's book, a stock investor who uses the alias "Pho Thee" decided to test this idea by seeing if his soon-to-be-born twin daughters could solve the following problem before entering kindergarten: "If there are eight animals and 20 legs, and the animals are comprised of only birds and turtles, how many birds and how many turtles are there?" Upon finding a method to successfully do this, Pho Thee "posted his story on an online web board to show that 'math is easy and can be taught to pre-kindergarten children.'" That post was eventually edited and published as a book. Four of Pho Thee's activities are included at the end of this article.
When using his method, Pho Thee stresses that parents should not "pressure [their children] with their own expectations" of success. Such expectations can lead these children to value winning competitions above all else, and to eventually find math "dull and soulless." Rather, parents should use his method to help their children realize their own potential, to have fun, and "to build a good relationship with the kids, while developing their mathematical skills, which will allow them to become self-learners."
--- Claudia Clark
"What Are the Odds a Handy, Quotable Statistic Is Lying? Better Than Even": Review of The Numbers Game: The Commonsense Guide to Understanding Numbers in the News, in Politics, and in Life, by Michael Blastland and Andrew Dilnot. Reviewed by Barry Gewen. New York Times, 3 February 2009.
The Numbers Game, a new book by Michael Blastland and Andrew Dilnot, aims to show readers the value of questioning how a statistic is computed and the conclusions it seems to imply. For example, a book reported a large discrepancy in the number of hours low and middle-income parents read to their children, as a possible explanation of disparate achievement levels, but the author used only her own family for the middle-income value. People often search for patterns and order in the world around them and place great trust in numbers to describe them. The authors assert, however, that this trust is misplaced; in reality, the concepts we try to measure, such as unemployment or the number of fish in the sea, are too nebulous or vast to be accurately captured. They stress a balance between being skeptical of numbers used without proper caveats and valuing statistics for the descriptive assistance---but not definitive conclusions---they can provide.
The Week magazine regularly includes a small excerpt of a book or longer article in each issue. The excerpt from The Numbers Game includes this passage: "One useful trick is to imagine [large] numbers in seconds. A million seconds is about 11.5 days. A billion seconds is nearly 32 years." Examples of how numbers are used or reported include mammogram results, federal budget numbers, the U.S. Census, estimates of the risk of getting colorectal cancer from eating bacon, the geography of cancer, numbers used in the reporting of birth rates, immigration, etc. The authors point out that "for grown-ups hopeful of putting counting to practical use, it has to lose that [childhood] innocence. The difference is between counting, which is easy, and counting something, which is anything but."
--- Lisa DeKeukelaere and Annette Emerson
If while preparing to go through airport security you have ever wondered about the effectiveness of the screening procedures used to identify terrorists, you are not alone. In this article, writer Sandra Blakeslee describes square root bias sampling, a method proposed by computational biologist and computer scientist William H. Press as a more mathematically optimal way to identify individuals who should undergo more thorough airport screenings.
For example, suppose persons from a profiled group are considered to be 16 times more likely to be terrorists than persons from an average group. Strong profiling—performing additional screening on profiled individuals at least 16 times more often than on the average group—would cause too many screenings of innocent people. On the other hand, not using any profiling "is bad because an opportunity is missed to take a closer look at high-profile passengers," writes Blakeslee. Press's suggestion to apply the square root bias sampling method would lead to persons from the profiled group being subjected to additional screenings four times as frequently as others.
You can read Press's open access article "Strong Profiling is Not Mathematically Optimal for Discovering Rare Malfeasors" online in the Proceedings of the National Academy of Sciences.
--- Claudia Clark
"Chaos and the Catch of the Day," by Paul Raeburn. Scientific American, February 2009, pages 76-78.
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