On Media Coverage of Math
Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Mathematician George Lusztig (left), an MIT professor who recently visited Hong Kong to receive the Shaw Prize for his work weaving together geometry and algebra, explains that he does math for the beauty of it, rather than for the prospect of real-world applications. He notes that understanding the beauty requires years of study, which is why many people never see it. Recalling that he spent hours each day solving math problems during his childhood in Romania, Lusztig says that doing math had the benefits of minimizing his exposure to politics and allowing him to be judged on his merits. He opines that success in mathematics requires a "good mind" and good luck, and he notes that he has a single-minded focus when trying to solve problems. The annual Shaw Prize honors recipients with $1 million for achievements in the categories of astronomy, medicine and life sciences, and mathematics.
See "When problems equate to happiness," by Raquel Carvalho. South China Morning Post, 30 September 2014. (Photo: Massachusetts Institute of Technology.)
--- Lisa DeKeukelaere
Thomas Bayes was a minister's son who became a mathematician and theologian in the first half of the 18th century. He was elected as a Fellow of the Royal Society, probably on the strength of his book defending the foundations of Newton's calculus, and never published the work for which he is most known today - a specific form of Bayes' rule, shepherded to publication by his friend Richard Price, and generalized by Laplace (Wikipedia: "Bayesian probability"). Bayes' rule says, simply, that the probability of a hypothesis given certain evidence, is proportional to both how likely the evidence is given the hypothesis, and the probability of the hypothesis in the first place. In the past 40 years, this idea, fertilized by the widespread availability of tremendous computing power, has flowered into a statistical approach Bayes might not have recognized, yielding fruit for countless scientific disciplines.
The key to so-called Bayesian statistics is that second term--the probability of the hypothesis of interest. This term plays no role in so-called frequentist statistics, the competing approach to statistical testing which developed in the 18th century and has been widely embraced by science. In frequentist statistical tests, the important thing is the likelihood of the evidence given a certain "null" hypothesis. If the evidence is highly unlikely, the null hypothesis is rejected. In Bayesian statistics, not only the likelihood of the evidence, but also the likelihood of the hypothesis in the absence of the evidence--its so-called prior probability, computed from past evidence or first principles--are taken into account. This allows for more robust calculation of the likelihood of the hypothesis given the current evidence, and for the iterative refinement of that calculation as more experiments are conducted. This Bayesian approach fits with our intuitive reasoning about events and their causes. Say, for instance, that you wake one morning from uneasy dreams to find yourself presented with a photo of a human-sized cockroach occupying your bed, next to a digital alarm clock reading the current date and time. Is it likely that you have transformed into a giant insect? The evidence--a time-stamped photo of a huge cockroach in your bed where you should be--is extremely unlikely if you remain human, suggesting that perhaps you are now, in fact, a horrible vermin. But thanks to the very low probability of such overnight transformations, you may decide that the likelihood that you are still human is quite high, unless you are presented with further evidence of a metamorphosis.
Statisticians like Columbia's Andrew Gelman have argued that a Bayesian approach may be the key to ameliorating a crisis of unreproducible results in fields as varied as oncology ("Drug development: raise standards for preclinical cancer research," by C. Glenn Bagley & Lee M. Ellis), neuroscience ("Power failure: why small sample size undermines the reliability of neuroscience," by Katherine S. Button et al.), and psychology ("False-positive psychology: undisclosed flexibility in data collection and analysis allows presenting anything as significant"). Low statistical power (which makes "statistically significant" results less likely to reflect true effects), a high accepted rate of false negatives (typically one in twenty, which pales in comparison to the vast number of scientific papers being published and the vast number of ways most scientists analyze their data prior to publication), and a bias towards publishing unexpected and counterintuitive findings all play a role in the preponderance of spurious results. The calculation of prior probabilities promotes the utilization of prior research and logic when evaluating hypotheses. For instance, Gelman re-analyzed data suggesting that ovulating single women were more likely to vote for Obama than their non-ovulating peers using a Bayesian framework. When data suggesting that people rarely change their voting preference over an election cycle were incorporated, the significance of the results vanished.
While Bayesian statistics can be misused ("Posterior-hacking: selective reporting invalidates Bayesian results also," by Uri Simonsohn), and properly executed frequentist statistics can solve many of the problems that cause unreproducibility, Bayesian methods clearly have an edge when it comes to guided, iterative hypothesis selection. For instance, astronomers have used Bayesian statistics to narrow down the age of the universe, and to deduce the properties of distant planets, even while the relevant evidence depends on many underlying, unknown parameters. Bayesian analysis also plays a prominent role in the Coast Guard's search and rescue operations ("Wikipedia: Search and Rescue Optimal Planning System"), and many researchers suggest that the brain is essentially an engine for Bayesian inference ("Wikipedia: Bayesian approaches to brain function"). In the absence of evidence, my prior suggests that Thomas Bayes, if he understood, would be thrilled.
(Image: Search and Rescue Optimal Planning System (SAROPS) probability grid.)
See "The Odds, Continually Updated," by F.D. Flam, New York Times, 29 September 2014.
--- Ben Pittman-Polletta
There is that moment twice each year when the hearts of all untenured faculty momentary stop: “Student evaluations are now available online.” A recent article in The Chronicle of Higher Education examines what makes these evaluations so heart-stoppingly terrible, and so deeply ineffective.
The dreaded evaluations are typically a combination of Likert scale and free-response questions. The Likert scale responses are tabulated, averaged, compared across the faculty, and then used to decide the high-stakes outcomes of promotion and tenure. Being numbers, the Likert scale responses give a false sense of security, and suggest objectivity, even if they are anything but. It doesn’t take a PhD in statistics to realize that getting a whole bunch of 1’s and whole bunch of 9’s is not the same thing as getting a whole bunch of 5’s. One is not necessarily better or worse, but they are undeniably different, and the current evaluation structure sweeps this difference under the rug.
These sorts of pitfalls are discussed in a research paper by Philip B. Stark, a professor of statistics at Berkeley, and Richard Freishtat, a senior consultant at Berkeley’s Center for Teaching and Learning. The co-authors address the current model of student evaluations as a high-stakes popularity contest from a statistical perspective.
Beyond the fallacy of objectivity in the outcomes, they argue that the questions themselves are often too open-ended. The article notes that one common question asks “was the course valuable?” This is a broad question, that perhaps a freshman student in a large section of calculus doesn’t really have the expertise to answer. A better question, the article argues, would be “Could you hear the instructor during lectures?” or “Did you leave more or less enthusiastic about the subject matter?” These are still subjective questions, but ones that any participating member of the class could answer with authority.
In closing, this particular untenured faculty feels inclined to point out that if the numerical portion is bad, the free-response part is even worse. I have gotten comments ranging from the benign to the bizarre: “Good class,” “Everybody hates this professor,” “This professor is fantastic, she should be given tenure right now,” “I like your clothes, where do you shop?” Point being, they are all over the place and really have nothing to do with the way I plan my syllabus, execute my lectures, evaluate my students, and in short, pass on a knowledge of calculus.
See "Scholars Take Aim at Student Evaluations’ ‘Air of Objectivity’," by Dan Berrett. The Chronicle of Higher Education, 18 September 2014.
--- Anna Haensch
During Teachers' Week on the television show Wheel of Fortune, middle school math teacher Sarah Manchester became the third person in the show's history to win its $106 grand prize. Manchester is also the coach of the math team at her school, Takoma Park Middle School, which is where she herself went to school. After her win, Wheel of Fortune host Pat Sajak asked Manchester if she had figured out the probability of winning a million dollars. She answered, "I assessed that the probability was low, but even unlikely events sometimes happen!"
You can see video of her performance at the link below.
"Silver Spring math teacher, Sarah Manchester, wins $1million on 'Wheel of Fortune,'" Jay Korff. WJLA ABC 7, 17 September 2014.
--- Mike Breen
Yitang "Tom" Zhang (left) and Jacob Lurie (right). Photos courtesy of the MacArthur Foundation.
Each year, the MacArthur Foundation (of John D. and Catherine T. fame, as those who grew up watching public television will know) invites hundreds of nominators, known for their accomplishments in a wide array of fields of human endeavor, to nominate the most creative people they know. Out of this group, a smaller committee selects 20 to 30 MacArthur Fellows, who receive no-strings-attached financial support for the next five years. Although the past accomplishments of candidate Fellows are reviewed during the selection process, the award is given primarily on the basis of future potential for creative work. The Fellowship is intended to free recipients from financial constraints, allowing them to exercise their creativity to its fullest (see MacArthur Fellows: Our Strategy", on the MacArthur Foundation website). Past recipients include not only artists and academics, but labor organizers (including this year's winner Ai-Jen Poo), papermakers, blacksmiths, barbershop owners turned literacy advocates. The Foundation (somewhat ineffectually) discourages the use of the word "genius" to describe the grants and their recipients ("Five Myths About the MacArthur 'Genius Grants'," by Cecilia Conrad for The Washington Post, September 20, 2013).
Of the 918 awards given since the awards' inauguration in 1981, 28 have been given to practitioners of the mathematical arts (see "MacArthur Fellows Program" on Wikipedia). This year, two of the recipients are mathematicians, and two more use mathematics in their work. Yitang "Tom" Zhang, whose gigantic first step towards a proof of the twin primes conjecture rocked the math world last May, is one of the 2014 class. So is Jacob Lurie, a mathematician at Harvard University who uses the theory of infinity-categories to generalize homotopy theory, and other topological aspects of algebraic topology. Lurie's work on quantum field theories links the categorical concept of duality to the topology of manifolds, as well as providing a classification scheme for quantum field theories. "I think of mathematics as a large number of interconnected stories, and I feel like my job as a mathematician is to take one or a few of those stories that I understand well and try and tell them in a way that other mathematicians can appreciate, enjoy, and maybe use in their own work," says Lurie. He is also a teacher, and in his interview for the MacArthur Foundation, he shares his opinion on the quaqmire that is mathematics education. "Mathematics is a giant playground filled with all kinds of toys that the human mind can play with, but many of these toys have very long operating manuals, but some of them don't, and I think that there are a number of mathematical insights that are very interesting that you really could teach to someone in a freshman course," he says. "I would like it to be viewed as just part of the intellectual culture in the same way that taking a class in Plato or taking a class in Shakespeare would be..."
Fellow Craig Gentry, a computer scientist, has proven that it is possible to manipulate encrypted data without ever lifting the encryption, and physicist Danielle Bassett uses graph-theoretic measures to study the dynamic reconfiguration of brain networks over time, with learning, memorization, and disease. Bassett has discovered that those who learn the best have the most flexible brain networks, suggesting the MacArthur Foundation's emphasis on creativity and the popular press' fascination with genius may not be orthogonal after all.
See "Meet the 2014 Winners of the MacArthur 'Genius Grants'," National Public Radio, September 17, 2014, and Meet the Class of 2014 MacArthur Fellows. Also, read about some of their previous awards.
--- Ben Pittman-Polletta (Posted 9/24/14)
The movements of sharks and other predators are like Lévy flight, "a seemingly complex form of random walk comprising clusters of short step lengths with longer movements between them" to locate prey. Andy Reynolds, at Rothamsted Research, published his study in Proceedings of the Royal Society A, and notes Lévy flight "can be advantageous when searching for randomly distributed resources because they reduce 'over sampling' without the need for cognitive maps and sophisticated navigational abilities." In the case of sharks, they use turbulent waters around them as cues to change direction. The natural behavior seems to have worked, as sharks have survived for over 10 million years.
See "Sharks Act Like Math Geniuses," by Jennifer Viegas, Discovery News, 16 September 2014.
--- Annette Emerson
The last few years have seen tremendous growth in the number of online courses being offered by universities and organizations through sites like Coursera. In this article, writer George Anders describes one of Coursera's most popular courses—Calculus 1—and the man behind it: Jim Fowler, an assistant professor of mathematics at The Ohio State University. The "geeky, high-energy" Fowler is on "a one-man crusade to make secants sexy and integrals irresistible." His dynamic presentations, augmented by clever visuals and "a kicky musical soundtrack," have captured the interest of well over 100,000 students, many of whom are older adults. In addition to the 25 hours of lectures and quizzes provided on Coursera, Fowler also provides an online textbook and “highly nuanced problem sets that make the most of online learning’s feedback loops” on a separate site. Between the time he has spent developing the course, making improvements, visiting discussion boards, and providing one-on-one tutoring, Fowler has devoted some 1,350 hours to the online class. As a result of the success of the course, the university has placed Fowler—initially hired for a non-tenure track position—on the tenure track. (Image courtesy of Jim Fowler.)
See: "Forget Cat Photos: This Prof Is Making Calculus Go Viral," by George Anders. Forbes, 10 September 2014.
--- Claudia Clark
The Science Museum in London will open a new mathematics gallery that will "tell the stories that place mathematics at the heart of our lives, exploring how mathematicians, their tools and ideas have helped to shape the world from the turn of the 17th century to the present." The gallery is made possible by a five-million pound gift (about US$8 million) to the museum from David Harding, a hedge fund manager in London who used mathematics in his trading. Harding says, "Mathematics is a fascinating and mysterious but, for some, forbidding subject. The new gallery has been created to convey something of that fascination in a way that will appeal to a wide audience." The new mathematics gallery will open in late 2016. (Image: View from East gallery entrance, Science Museum Mathematics Gallery, Zaha Hadid Architects.)
See "Science Museum unveils £5m plan for 'world's foremost' mathematics gallery," by Alex Bellos. The Guardian, 10 September 2014.
--- Mike Breen
In a paper recent published in the Proceedings of the National Academy of Sciences, four mathematicians and a biologist describe how they used fluid flow modeling and dissection of crayfish nerves to understand how neural pathways of crustaceans direct their small, paddle-like limbs to provide efficient propulsion. The researchers discovered that the neural pathways activate each pair of paddle limbs to perform a front-to-back "power stroke" slightly before the next pair of limbs closer to the head, resulting in a tail-to-head Mexican wave motion. The fluid flow modeling indicated that staggered tail-to-head pair activations were more efficient than pairs paddling in unison or pair activation starting at the head, and the dissection showed that the crayfish nerve circuit reached equilibrium when paddle pairs operated about a quarter power stroke cycle apart—the same result the mathematicians found to be most efficient.
See "Math Explains How Lobsters Swim," by Clare Pain, ABC Science Online for Discovery News, 9 September 2014, and the original research paper "Neural mechanism of optimal limb coordination in crustacean swimming," (abstract) by Calvin Zhang, Robert D. Guy, Brian Mulloney, Qinghai Zhang, and Timothy J. Lewis. Proceedings of the National Academy of Sciences of the U.S., published ahead of print September 8, 2014.
--- Lisa DeKeukelaere
There must be something in the red rock of the New Mexico desert that's good for interdisciplinary science. The state is home to both the Los Alamos National Laboratory and the Santa Fe Institute, a research center dedicated to the study of complex phenomena using tools from physics, mathematics, biology, social science, and the humanities. There's so much science in New Mexico that the Santa Fe New Mexican features a column written by researchers from the Santa Fe Institute. This month, Ben Althouse, an Omidyar Fellow who uses the science and mathematics of complex systems to study viral epidemiology, takes the helm. Althouse points out that, as opposed to non-infectious diseases, such as cancer and heart disease, the study of virally-transmitted infectious diseases is complicated by the fact that an individual's susceptibility to them depends not only on his or her own health behaviors--how much sleep they get and how often they wash their hands--but also on the health behaviors of those they interact with. With his collaborator Sam Scarpino, Althouse has begun to reveal how the existence of asymptomatic carriers of whooping cough have been crucial to the recent resurgence of the disease. His work also focuses on the spread of mosquito-transmitted viruses, such as dengue fever and Chikungunya, common in the Caribbean and Florida. These diseases introduce the extra complication of interactions between species.
Beyond virally transmitted diseases, Althouse has employed new technologies to study public health more generally, using Google search terms to study patterns in health-related behaviors. With his collaborator John Ayers and others, Althouse found that searches related to quitting smoking, and healthy behaviors more generally, are more common early in the week, on Sundays, Mondays, and Tuesdays ("Circaseptan (weekly) rhythms in smoking cessation considerations," by John Ayers et al; "What's the healthiest day?: circaseptan (weekly) rhythms in healthy considerations," by John Ayers et al). Google searches also reveal the health burden of the 2008 recession--after which queries related to the symptoms of headaches, stomach ulcers, heart disease, and joint and tooth pain increased unexpectedly and dramatically (Population health concerns during the United States' great recession," by Ben Althouse et al).
Althouse's most-cited paper, though, reveals how the impact factor--a measure of the scientific influence of a journal based on the number of citations its articles receive--varies over time and across disciplines ("Differences in impact factors across fields and over time," by Ben Althouse et al). It turns out that, as science has grown, so have impact factors; and that differences in impact factors across fields depend more on which citations are counted than on which fields are growing fastest.
--- Ben Pittman-Polletta (Posted 9/15/14)
Edward Burger is the professor in question. A former math professor at Williams College and now the president of Southwestern University in Texas (but as a photo shows, not beneath helping students move in), Burger answers questions about being a president and the difference between his current and former jobs. When asked how he made the transition from professor to university president, he said: "Maybe some of our problems in education today stem from the fact that someone like me is considered an unconventional choice. Maybe academic institutions should be run by academics, the way they used to be."
See "A Professor in the President's Chair: Pushing for a 'Friendly Revolution'." People, The Chronicle of Higher Education, 5 September 2014, page A34.
--- Mike Breen
Can you imagine a time when using computer software to construct and verify your own, as well as other people's, proofs becomes the standard way to do mathematics? According to mathematician Vladimir Voevodsky, this is on the verge of happening. Writer Florian Meyer describes how Voevodsky "has developed an approach [to doing mathematics] that could... revolutionize mathematics and its foundations: He has been able to show in principle that homotopy theory, which deals with the deformation of geometric objects, expresses the same ideas as the theory of programming languages and mathematical logic, only in a different language." The result of his work means that mathematical proofs will be able to be "translated into a programming language for computer proof assistants much more easily than they can be today." (Image: HLFF/Kreutzer).
-- Claudia Clark
Nothing ruins a day at the beach like washed-up garbage. Unsurprisingly, not just our beaches, but also the oceans themselves are piling up with garbage. But where does it all come from? As reported by nbcnews, a group of scientists from the University of New South Wales (Austrailia) may have found a mathematical approach to understanding how our garbage travels through our oceans. (Image courtesy of Flickr, epSos.de.)
The earth’s oceans are partitioned into 5 distinct gyres--or vortices--and these describe the major ocean currents. Scientists previously thought that the gyres should be self contained. In particular, they believed that once a piece of garbage got swept up in the North Pacific gyre, it would get drawn to the center and join its fellow debris in a so-called garbage patch somewhere in the North Pacific. But recent efforts to track and identify garbage has shown that this junk is traveling farther than we had thought. (Image courtesy of Wikimedia Commons.)
To better understand this flow of trash, Gary Froyland, a professor of mathematics at the University of New South Wales, and his colleagues have approached this problem in a totally new way: by modeling it is a dynamical system. They modeled the surface of the oceans using a Markov chain model, which is able to account for the three-dimensional upwelling and downwelling of the ocean. Using this model, they identified the major attracting regions. And although these regions were mostly consistent with the known ocean gyres, they did find some unexpected inter-connectedness between some really distant parts of the oceans. This has also led to their follow-up study: how hard is it for floating garbage to cross the boundaries of a gyre?
What this means is that we may all be more connected than we thought. Even though oceans may separate us, we are all connected by our joint responsibility for our shared oceans and our planet.
See "Math Might Help Nail Oceans' Plastic 'Garbage Patch' Polluters," by Miguel Llanos. nbcnews.com, 2 September 2014.
--- Anna Haensch (Posted 9/16/14)
Long before there were computers, the term "computer" referred to people who were adept at performing arithmetic operations. In this article, Hayes introduces us to "one of the finest computers of the Victorian era," William Shanks, and Shanks's work calculating the value of Pi to 707 decimal digits. You might not be surprised to find that some errors crept into Shanks' calculation--beginning at around digit 530--and it is Hayes's attempt to determine the possible reasons for these errors that makes up the bulk of the article. Hayes begins with the fact that most calculators of Shanks' era used arctan formulas (and their equivalent infinite series) for determining the value of Pi. Shanks worked with a formula discovered by mathematician John Machin in 1706 that required the evaluation of two arctan series: π/4 = 4arctan(1/5) – arctan(1/239). Hayes then explores some of the computational methods that Shanks may have used. Finally, Hayes describes the method he uses to identify, and provide a reasonable hypothesis for, three of Shanks' errors.
--- Claudia Clark
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