This month's topics:
"How mathematicians rule the markets"is the subtitle of "Quant trading," a piece by Richard Anderson posted on BBC News, September 24, 2011. "Trading floors were once the preserve of adrenalinfuelled dealers aggressively executing the orders of brokers who relied on research, experience and gut instinct to decide where best to invest" is how it starts, with a photo of an obviously adrenalinfuelled fellow in coat and tie, juxtaposed with one of a serenelooking young man wearing a plaid shirt and large, darkrimmed eyeglasses. Yes, "Investment decisions are no longer being made by financiers, but increasingly by PhD mathematicians and the immensely complex computer programs they devise." This is quant trading, and it is "taking over the world's financial capitals." For good or for evil? Anderson refers to "The Future of Computer Trading in Financial Markets," a working paper put out by the U. K. Government Office for Science (downloadable as a PDF), which "found that quant trading helped to reduce dealing costs and improve liquidity, and did not harm overall market efficiency." But which did call attention to the danger of "selfreinforcing feedback loops." This is the phenomenon that led to the Flash Crash on May 6, 2010, where the Dow "plummeted 700 points in less than five minutes, wiping out about $800 billion." (Fortunately, "the market bounced back within half an hour"). The fluctuation of the Dow Jones Industrial Average between 2 and 4PM on May 6, 2010. This anomaly, known as the Flash Crash, has been attributed to feedback in mathematically computerprogramdriven trading. The market went from 10703 at 2PM to 9869 at 2:42; then closed at 10520. Image adapted from the Wall Street Journal. Some critics are less sanguine about the involvement of mathematicians in trading. "... the problem is more fundamental. Mathematicians, they say, do not understand markets. They deal in absolutes, not the irrational human behaviour that drives so many investment decisions." In the words of Paul Wilmott (identified by Anderson as "a prominent lecturer in quantitative finance"), "Do they appreciate the human side of finance, the herding behaviour of people, the unintended consequences?" "How to fix Math Educationin High School and College" was posted on Tori Bosch's "future tense" blog ("The Citizen's Guide to the Future"), on the Slate website, November 10, 2011. Bosch was just back from the Cuidad de las Ideas in Puebla, Mexico, where she heard experts hold forth on mathematics education. Arthur Benjamin (Harvey Mudd College), professor and mathemagician (does huge calculations in his head) "hopes that mathematical education will be less about computationwe've got calculators for that!and more conceptual, like "understanding when you need to do integrals, [and] when you need to do a square root.'" He also believes that the primary sequence of high school and college courses leads students in the wrong direction. Bosch quotes the professor himself: "For the last 200 years, the mathematics that we've learned starts with arithmetic and algebra, and everything we do after that is taking us toward one subject, calculus. I think that is the wrong mathematical goal for 90 percent of our students. We're now living in an age of information and data, and the mathematics that will be most relevant to our daily lives is probability and statistics." Fixing Math Education (cont.)Sol Garfunkel and David Mumford's New York Times OpEd (reported here) elicited such a warm response that the authors have opened a forum on the Math is More website to continue the dialogue. Registration. Why Science Majors change their minds, and why notIt starts with an "Education Life" piece in the November 4 2011 New York Times: "Why Science Majors Change Their Minds (It's Just So Darn Hard)" by Christopher Drew. As Drew notes, "Studies have found that roughly 40 percent of students planning engineering and science majors end up switching to other subjects or failing to get any degree." (60% if you count premeds). As Mitchell Chang (Education, UCLA) puts it, "We're losing an alarming proportion of our nation's science talent once the students get to college" (Chang notes that paradoxically the attrition rate can be higher at more selective schools). The reasons seem to be:
Monte Carlo methods in mammalian evolution"Multiple routes to mammalian diversity," by Chris Venditti, Andrew Meade and Mark Pagel, appearing in Nature for November 17, 2001, proposes a largescale reassessment of the 190 millionyear history of our class. The authors dispute the conventional view "that the radiation of extant mammals underwent a burst of bodysize evolution that occurred early in its history and coincided with the appearance of the mammalian orders, and that this was followed by a gradual slowdown towards the present." Instead, they propose a model in which "rates of evolution were low and stable for about the first 60 million years, only starting to increase around 90 million years ago and then showing only about a twofold increase over the previous 'baseline" rate." Their analysis detects "shortterm explosive increases of between 10 and 52fold in the rate of evolution, distributed widely among clades." [A clade is the complete set of species sharing a branch on the evolutionary tree.] "It is these that shape the diversity of mammals, in striking ways. ... An important subset of the shortterm bursts occurs in the single evolutionary lineage that leads to the common ancestor of a large monophyletic group. These 'singlelineage ancestral bursts' seem to correspond to drastic changes in an animal from some ancestral state to a size that seems to form the basis for a new radiation, and might be akin to the concept of quantum evolution." The research is based on a new application of Monte Carlo analysis to the study of evolution. In quantum field theory MC analysis allows integration over a hugedimensional space by an intelligent choice of where to sample (the Metropolis algorithm). Here the space is the set of all possible ancestral trees, together with the set of all possible branch lengths on each tree. The new element is "Reversible Jump Markov Chain Monte Carlo Computation," devised in 1995 by Peter J. Green (Biometrika 82 711732); it allows a Metropolis process to operate on a space with many disjoint components of possibly different dimensions (in this application, the components correspond to different tree structures).
Tony Phillips 
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