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March 2013
Brie Finegold summarizes blogs on networks and MOOCs: "The Mathematics of Averting the Next Big Network Failure," by Natalie Wolchover (Simons Science News). Wired Science blog, 19 March 2013. In the case of large personalities, we often say "the bigger they are, the harder they fall". But is the same true for networks of networks? Interdependence has both advantages and disadvantages during emergency. When networks such as those for electricity, gas, and telecommunications become linked at many points, the failure of a small percentage of nodes in the network may lead to a cascade of failures. Datadriven mathematical models measure the fragility of a network in terms of its connectivity. Such models may help us determine strategies for preserving the daily conveniences afforded by linking networks while preventing epidemic failure of our infrastructure in an emergency. For example, research suggests that Chile's success in handling the 2010 earthquake was partly due the decoupling of one network (power) from another (telecommunications). This allowed power to be controlled locally until the larger network was repaired. This is just one of several blog posts having a mathematical theme by Ms. Wolchover that have been picked up by Wired. You can read more about the author.
An Overview: Mathematicians (and other academics) Opine on Online Courses  The Latest on MOOCS: As Massively Open Online Courses begin to lose their novelty, we may get a better picture of their effectiveness, feasibility, and relationship to traditional classroom environments. While some may view MOOCS as a means of bringing college students up to speed on their algebra, others are using the format for upper level and graduate level material such as the "Graph Partitioning and Expanders" course listed below. Detractors point to the lack of interaction between the expert and the student and the fact that many professors teaching MOOCS would not, given the option, want to give those completing the MOOC the same credit as those students completing it's facetoface counterpart. Enthusiasts point to the benefits of having a large online community of classmates and the ability to bring high quality lectures and materials to a wide audience. Professionals who simply need training in certain areas may find this format especially appealing even without the college credit that some MOOCS offer. For example, for those Mathematics Phd's considering careers in industry, the statistics, computer science, and machine learning courses offered could be especially beneficial. Despite the fact that one could pick up a text on these materials, having an email reminder and some multimedia display of the subject matter can make taking the time to work on the ideas more appealing. Will this replace the university system over time? Mathematician Peter Cameron writes "I believe that this is not unthinkable, and therefore the time to think about it is now rather than later." Available Courses: Intro to Stats from University of Toronto (through Coursera), Graph Partitioning and Expanders (through Stanford), Mathematical Biostatistics Bootcamp (through Johns Hopkins on Coursera), Probabilistic Graphical Models (through Stanford on Coursera), and Machine Learning> (through Stanford on Coursera).  Brie Finegold "It's a science for crash detectives," by Gregory Smith. The Providence Journal, 26 March 2013, pages 1 and 6. The front page of this issue of The Providence Journal featured momentum equations and references to Newton's Three Laws of Motion. Police officers, especially "certified traffic reconstructionists," are now frequently using math and physics to determine what happened at an accident. They use their own equipment and data from carsmotor vehicle data event recorders are in over half of the cars and light trucks on the roadto answer questions such as How fast were the vehicles going? and Were traffic signals obeyed? University of Rhode Island police officer Paul Hanrahan related one law that Newton missed: "No one ever tells the truth" at the scene of an accident.  Mike Breen "The Minds Behind the MOOCs," by Steve Kolowich. The Chronicle of Higher Education, 22 March 2013, pages A20A23. This article profiles some of the developers of MOOCs (massive open online courses), includes survey results from people who have taught such courses, as well as their experiences with the courses. Near the end of the article, Robert Ghrist talks about his semester teaching Calculus: Single Variable at the University of Pennsylvania. His course is one of five Coursera courses that the American Council on Education has endorsed for credit. Ghrist designed his course to mimic the traditional classroom course that he's taught for eight years and notes that he did not water it down in an attempt to make it more appealing. As for the future, he hopes that the number of creditworthy MOOCs will increase and expects they will be an important part of his children's college education.  Mike Breen Media coverage of the Abel prize: "Abel Prize for Belgian Pierre Deligne", by R. Ramachandran. The Hindu, 20 March 2013; "Belgian mathematician rewarded for shaping algebra", by Philip Ball. Nature, 20 March 2013;"Shapely algebra breakthrough wins milliondollar prize", by Jacob Aron. New Scientist, 20 March 2013. The above articles represent a sampling of the coverage of the 2013 Abel Prize, which was awarded to Pierre Deligne of the Institute for Advanced Study in Princeton. The Abel Prize, totaling about US$ 1 million, is presented by the Norwegian Academy of Science and Letters and is comparable in prestige to the Nobel Prize (there is no Nobel in mathematics). In addition to profound work in algebraic geometry that helped to revolutionize that subject, Delige has made important contributions to representation theory, number theory, and automorphic forms. In a news release, Deligne's IAS colleague, mathematician Peter Sarnak, said: "Deligne's focus extends beyond establishing fundamental mathematical truths; he seeks to understand why they are inevitable. In his work, this is often achieved by brilliant abstract reasoning, after which the result becomes clear and conceptual. Deligne is responsible for many of the standard tools in modern algebraic geometry, and a range of striking theorems, theories, mathematical objects and constructions bear his name." Deligne is the recipient of many major awards in mathematics, including the Fields Medal (1978) and the Wolf Prize (2008, jointly with Phillip Griffiths and David Mumford). Deligne received another major Scandinavian prize, the Crafoord Prize of the Royal Swedish Academy of Sciences. The US$400,000 Crafoord Prize was awarded jointly to Deligne and his collaborator Alexander Grothendieck in 1988 (Grothendieck declined the prize). An interview with Deligne was created for the Science Lives project of the Simons Foundation.  Allyn Jackson "Should business be allowed to patent mathematics?", by Stephen Ornes. New Scientist, 18 March 2013. Is mathematics discovered or invented? This question has been discussed many times and from many different points of view. "The latest incarnation [of the question] concerns something very down to earth: money," Ornes writes. "[M]athematics powers the algorithms that drive computer software, and software is big business, worth over US$300 billion a year to the global economy." He cites an article in the April 2013 issue of the Notices of the AMS, "Platonism is the Law of the Land," by David A. Edwards, who contends that mathematical ideas should be patentable. "[Edwards] argues patents should be granted for every new formula and algorithm, including those that power computer software," Ornes writes. "His position is extreme, but proponents of software patentability similarly argue that the system fuels growth and rewards people for their work." The article goes on to discuss some of the difficulties surrounding efforts to patent software. Some believe software is mathematics and therefore should not be patentable. "The odds are stacked against them, though," Ornes writes, "there's too much money at stake."  Allyn Jackson "Math geeks everywhere are enjoying a slice of pi today," by Reid Champagne (The News Journal, Wilmington, DE), USA Today, 14 March 2013. This timely article begins by providing a very brief introduction to pi, an irrational number equal to the ratio of a circle’s circumference to its diameter. (Unfortunately, the author mixes up the definition of rational and irrational numbers, calling decimal numbers such as 0.333… irrational because they repeat and do not end, as opposed to decimal numbers, such as 0.5 or 0.125, which end and do not repeat.) The writer also interviews a few math faculty from area universities, as well as the founder of Pi Day Princeton, Mimi Omiecinski, who started this community celebration 5 years ago to commemorate the March 14th birthday of one of Princeton's most famous residents, Albert Einstein. The article also lists some interesting and entertaining information about pi at the end of the article, including the fact that, "in the Star Trek episode 'Wolf in the Fold,' Spock foils the evil computer by commanding it to 'compute to the last digit the value of pi.'" The article also mentions the San Francisco Exploratorium, which just hosted its 25th annual Pi Day event at the museum and online. Don't forget to mark your calendar for next year's Pi Day celebrations!  Claudia Clark "Missing Math Experts," by Carl Straumsheim. Inside Higher Ed, 13 March 2013. Where have all the jobs gone? Well, a few at least have gone into math education, and they seem to have left the jobseekers behind. According to a new study to be published in the Notices of the AMS this April, about a quarter of the positions open to doctorates in mathematics education during the 20112012 academic year went unfilled. While this small sector of the job market shrunk in the recession following 2008's stock market crash  data from 2006 show twice as many positions going unfilledthere is still an unmet need for mathematical educators. Nor are the unfilled positions any too shabbyof the 94 positions offered in 20112012, 90 percent of them were at the assistant professor level, and about half of them offered salaries in the $6070 thousand range, about $10 thousand more than most positions for doctorates in mathematics. According to Robert Reys, a coauthor of the study and professor emeritus of math education at the University of Missouri, the actual number of unfilled positionswhen regional and junior colleges are countedmay be higher still. Reys posits that schools are increasingly seeking candidates with prior teaching experience, something many math educators possess. However, many doctorates in math education never enter the U.S. academic job market, instead returning to prior jobs or their countries of origin. Indeed, only about 50 doctorates per year have entered the job market over the past 15 years. Says Reys, "We just need to be able to recruit. The job opportunities are there." Of course, the number of those opportunities is not enormouswe are talking on the order of 25 unfilled positions last year, some of which may have remained unfilled due to budget constraints. So let's not all rush out and get our doctorates in math education at oncelet's start with 20 or so of us who really want tenuretrack jobs. The full report is "An Update on Jobs for Doctorates in Mathematics Education at Institutions of Higher Education in the United States," by Robert Reys, Barbara Reys, and Anne Estapa, Notices of the American Mathematical Society, April 2013.)  Ben Polletta "Pixar's Senior Scientist explains how math makes the movies and games we love," by Tim Carmody. The Verge, 7 March 2013. Mathematician Tony DeRose has helped Pixar depict bouncy, lifelike red hair, the precise shape of an old man’s nose, and countless other features in animated films using his mathematical skills, in particular by pioneering the use of parabolas rather than polygons to make images more realistic. In a lecture at the Museum of Mathematics in Manhattan, DeRose emphasized that future animators and video game designers will need a strong background in math in order to develop and improve the algorithms for modeling the physics behind the movement of objects like hair, clouds, smoke, and fire. He explained that such algorithms are so complex that they can require new spatial data structures, and while compression of data is tempting, a high level of detail must be maintained for the quality of the film. Prior to the late 1990s, animation studios used polygons to build these models, but DeRose applied his mathematical expertise with wavelets to transform Pixar’s depiction characters and objects into smoother surfaces. (Photo of DeRose at the 2013 Joint Mathematics Meetings, by Sandy Huffaker.)  Lisa DeKeukelaere "As Math Grows More Complex, Will Computers Reign?," by Natalie Wolchover. Wired, 4 March 2013. From citing them as perennial coauthors to shunning them completely, this perceptive article runs the gamut of mathematicians' attitudes towards computers in its exploration of the frontiers of experimental and computational mathematics, surveying the ways in which computers are already beginning to shape and our arcane and erudite field. Testimony from algebraists Constantin Teleman and Minhyong Kim illuminates the reasons many mathematicians shy away from computers: the primacy of insight, creativity, and understanding in mathematical culture, and the paucity of these virtues in computations and computer proofs. “Pure mathematics is not just about knowing the answer; it’s about understanding,” says Teleman. “If all you have come up with is ‘the computer checked a million cases,’ then that’s a failure of understanding.” But things may be more complicated than that  and by "things" here, we mean math itself. “The time when someone can do real, publishable mathematics completely without the aid of a computer is coming to a close,” says mathematician and computer scientist David Bailey. As mathematical questions get more and more complicated, those going without computer assistance may find themselves "increasingly restricted into some very specialized realms." Opinions aside, just as computation is playing a larger role in life in general, it is creeping inexorably into mathematics. Even Teleman has been a party to the computer creep, assigning problems he suspects require computation to students who know how to program. But while there are myriad ways computers are currently being used in mathematics  from generating conjectures to checking large finite numbers of cases to constructing proofs with symbolic and formal algorithms  there are few professional standards governing these uses. Code that is sloppy or very long, or even unacknowledged, may be opening up gaps in the formal fabric used to weave mathematical theory. Peter Sarnak, a former editor of the prestigious Annals of Mathematics, acknowledges that while that journal accepts computerassisted proofs, "in cases where the code is very difficult to check by an ordinary single referee, we will make no claim about the code being correct.” Of course, producing flawed proofs is not an ability unique to computers. Indeed, one of the most exciting uses of computing may be in formally verifying arguments and computations. Eventually, computers may generate large parts of mathematical arguments with little human guidance, both eliminating human error and freeing up human mathematicians to do things we, their intellectual ancestors, can only dream of. "If we were to imagine a future in which all the theorems we currently know about could be proven on a computer," says Jordan Ellenberg of the University of Wisconsin, "we would just figure out other things that a computer can’t solve, and that would become ‘mathematics.’ ” (Photo: Doron Zeilberger, Rutgers University, "believes computers are overtaking humans in their ability to discover new mathematics." Photo by Tamar Zeilberger)  Ben Polletta "In Pursuit of the Impossible Equation," by David Wescott. The Chronicle of Higher Education, 4 March 2013, page B17. The Chronicle reviews Lance Fortnow's book The Golden Ticket: P, NP, and the Search for the Impossible. Fortnow, a professor and chair of the School of Computer Science at the Georgia Institute of Technology, says the book's target reader is "the highschool sciencenerd type." The book draws its name from an unproven computer science theorem called "P versus NP", where P stands for problems with easy solutions and NP stands for very very difficult problems. This unsolved problem is considered by many to be the most important open question in computer science and the Clay Mathematics Institute announced in 2000 that it would pay $1million for its solution. Whether the theorem is true is really important. If P were to equal NP, then according to Fortnow we inhabit a world where eventually computers will solve basically every problem there is to solve. Fortnow's main reason for writing the book is to get young people interested in the world changing problems computer science can solve. At times he worries that computers have become too easy to use. "My biggest fear is that the computer has become like the car, everyone can drive a car, but people never try to get under the hood," says Fortnow.  Baldur Hedinsson "A Museum of Math," Mo Rocca. Sunday Morning, CBS, 3 March 2013. At the beginning of this short segment on the CBS Sunday Morning show, correspondent Mo Rocca spoke briefly with mathematician Glen Whitney about the fact that many people in the United States do not like mathematics and feel it is OK to not be good at math. As one of the founders of the recently opened National Museum of Mathematics—the only mathematics museum in North America—Whitney hopes to do something about that. He points out that students are typically given "one road to go through mathematics" based on curriculum designed over 50 years ago to help the United States win the space race against the Soviet Union. However, "math is this extremely varied, beautiful landscape," as well as a part of our daily lives, and the museum gives children and adults an opportunity to explore this landscape. Among other activities, museum goers can ride a squarewheeled triangle around a circular track, explore sculptures that connect music with mathematics, and create fractals with their bodies. Rocca concluded that "math may not be as easy as "pie", but it's not so square either." Learn more about this Manhattanbased museum.  Claudia Clark "Rolla Math Professor Lands Spot on Jeapordy [what is sic?]." KMOX (St. Louis), 2 March 2013; "S&T professor places 2nd on 'Jeopardy!'," Rolla Daily News, 6 March 2013. Ilene Morgan (left, with Alex Trebek), an associate professor of mathematics and statistics at Missouri University of Science and Technology, appeared on Jeopardy! on Tuesday, March 5. She described her experience as "bucketlist awesome" and had a lot of fun during the taping of the show. The university organized a viewing party on campus so people could watch with her. She answered/questioned Final Jeopardy correctly, but didn't win. Later she said, "The most frustrating thing was being outbuzzed on the math question...Still, I had a wonderful time and an irreplaceable experience." (Photo courtesy of Ilene Morgan.)  Mike Breen "This Is the Hardest Math Problem in the World," by Ian Stewart. Publishers Weekly, 1 March 2013. Columnist Ian Stewart explains, in layman's language, the quandary he terms as possibly "the most elusive mathematical problem ever"—the P/NP problem. First, Stewart describes "class P" algorithms as sets of instructions that a computer can perform quickly enough to be useful, and "class P" problems as those for which an efficient algorithm exists. He then defines "class NP" problems as those for which finding the answer essentially requires guessing, but checking your guess is simple. The milliondollar question, Stewart explains, is whether problems exist that are NP but not P. While most mathematicians believe that such NPbutnotP problems exist, the difficulty lies in the proof. Stewart notes that even if a potential NP problem appears to lack an efficient algorithm for a solution, in reality such an algorithm may exist but simply be yetundiscovered.  Lisa DeKeukelaere The MarchApril issue of American Scientist had three nice articles about mathematics, which are summarized below. "The Music of Math Games," by Keith Devlin. American Scientist, MarchApril 2013, pages 8791. Despite the multitude of "math education" video games available, Keith Devlin asserts that few games are actually built soundly upon the principles wellknown in the education world for helping children learn math. Instead, most games focus on rote memorization of basic skills, such as multiplication tables, reinforcing the idea that mathematics is about manipulating symbols rather than about using logic to solve realworld problems. Devlin cites a study indicating that while many people have difficulty with symbolbased problem solving, leading to discouragement and disinterest, most people can master the mathematical problems they encounter in everyday situations. According to Devlin, welldeveloped video gameswith input from both mathematical education and game design expertscan provide a valuable tool for enhancing mathematical skill by providing engaging enticement to tackle mathematical puzzles free from the constraints of symbols. Learning math through this method is analogous to music in that "learning by doing the real thing," such as playing increasingly difficult pieces on the piano, rather than spending hours memorizing sheet music before first touching the keys, provides more enjoyment and motivation to improve. Image courtesy of Keith Devlin.  Lisa DeKeukeleare "First Links in the Markov Chain," by Brian Hayes. American Scientist, MarchApril 2013, pages 9297. This article, which recounts the birth of the Markov chain, makes up for its meandering path with wit and lucidity, and serves as an excellent introduction to the subject. Andrei Andreevich Markov senior (his son, also a mathematician, had the same name and also published under A.A. Markov) lived through the Russian revolution, and was a leftist and a supporter of the revolutionaries Leo Tolstoy and Maxim Gorky. It turns out he was also something of a pugnacious jerk, who looked down on the work of his contemporaries, and was not shy about insulting friends and foes alike. Markov developed his theory of chains to disprove an (admittedly ridiculous) claim made by his most contemptible colleaguethe religious, monarchist mathematician (listen for Markov rolling over in his grave) Pavel Nekrasov. Nekrasov published a paper that claimed to use the law of large numbers to prove that humans have free will. His logic was that if the law of large numbers applies only to independent events, and sociological data follow the law of large numbers, then human actions must be independent events, and thus governed by free will rather than determinism. To foil this proof, Markov showed that dependent events can also follow a law of large numbers. He described a chain of random events, each drawn from a finite set of possible outcomes. In this chain, each event biases the probable outcome of the next event. (Technically, a Markov chain is a stochastic process on a finite state space in discrete time, in which the state at a given time point depends only on the state at the previous time point.) The probabilities of transitions from each outcome to every other outcome can be organized into a matrix; the probabilities of transitions from one outcome to another over n steps are then given by the entries in the nth power of this transition matrix. This fact makes Markov chains amenable to analysis both numerical and mental, and a growth industry in science and mathematics. Furthermore, as n increases, the powers of the transition matrix limit on a matrix which is constant along its columns. The value of each column is the longrun probability of the corresponding outcome. In other words, the probability of occurrence of each event in the chain reaches a limit as the number of observations increasesso Markov chains, although they are chains of dependent events, obey a law of large numbers. Markov thus exposed Nekrasov's error, but he was not content with this theoretical effort. In a farsighted study of Alexander Pushkin's poem "Eugene Onegin," Markov also pioneered their application. Painstakingly calculating the transition probabilities between vowels and consonants in the first 20,000 letters of the Russian children's standard, Markov discovered that vowels and consonants are not independently distributed (surprise, surprise), but have a tendency to alternate. In his explication of Markov's paperan English translation of which was not widely available until 2006Hayes goes above and beyond. He not only retreads Markov's footsteps (not to say walks in Markov's bootsthat would be impossible, as I leave the reader to verify) and performs some of his analyses by hand, he also provides entertaining examples of how higherorder Markov chains can be used to generate gibberish that becomes more and more Pushkinlike. Thank goodness for computers, and for A.A. Markov, may they rest in peace.  Ben Polletta "Adventures in Mathematical Knitting," by sarahmarie belcastro. American Scientist, MarchApril 2013, pages 124133.

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