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Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers Breakthrough Prize in Mathematics winners. Photo: (LR) Terence Tao, University of California, Los Angeles; Maxim Kontsevich, Institut des Hautes Etudes Scientifiques; Simon Donaldson, Stony Brook University and Imperial College London; Jacob Lurie, Harvard University; and Richard Taylor, Institute for Advanced Study. Photo by Steve Jennings/Getty Images for Breakthrough Prize. "The news should start with mathematics, then poetry, and move down from there," from The Humans, by Matt Haig. Recent Posts:
See also: The AMS Blog on Math Blogs: Two mathematicians tour the mathematical blogosphere. Editors Evelyn Lamb and Brie Finegold, both PhD mathematicians, blog on blogs that have posts related to mathematics research, applied mathematics, mathematicians, math in the news, mathematics education, math and the arts, and more. Recent posts: "Simple words, Complicated Math." From rubber sheets to happy apps, Discover tunes in to the top math stories of 2014, by Anna Haensch The JanuaryFebruary 2015 issue of Discover Magazine features the top 100 science stories of 2014. Included among them are 5 stories about boundarybreaking mathematics and mathematicians that have taken center stage this year. One such boundary breaker was Maryam Mirzakhani, who became the first woman to win the Fields Medal at this year’s ICM (International Congress of Mathematicians, held in Seoul) . Her work is in geometry and hyperbolic surfaces and Etienne Ghys of the Fields Medal committee describes it as "amazing formidable work." Her life and work were beautifully described by Quanta Magazine. Breaking through a layer of mold, a team of researchers found a set of ancient bamboo strips which appear to be the world’s oldest calculator. The set belongs to a larger collection of 2,500 strips which are marked with numbers 1/2 to 90 and could have been used just like multiplications tables. At Brown, a team came up with a mathematical formula to describe how rubber surfaces fold. Their equation can predict precisely how a material will fold given its composition and stiffness, and how much pressure will be need to flatten it out. Obviously this is a breakthrough for materials engineers, but people in the medical field can also use this to analyze what happens to the skull in a traumatic head injury. The original article appeared in the Proceedings of the Royal Society. Another captivating story came out of the University College London, where researchers discovered that the key to happiness is lowered expectations. In small experiments conducted in the lab, and larger ones conducted using a smart phone app, they found that people’s happiness was highest when they just got more than they expected. The original article appeared in PNAS (Proceedings of the National Academy of Sciences of the United States of America). A final story is on number theorist Ken Ono and his collaborators, who made a key discovery about the beautiful and captivating RogersRamanujan identities last summer. According to Ono, he and his collaborators Michael Griffin and Ole Warnaar were studying these identities, and "found a framework that shows why they’re true.…They turned out to be two golden nuggets that suggested the existence of a whole mother lode of identities out there." To pick the top story of the year, Discover is asking readers to vote and the story about the RogersRamanujan identities has made the Final Four. Go here to cast your vote in the semifinals (by December 17), because wouldn’t it be cool if the number one science story of the year was about math! See "100 Top Stories of 2014," Discover, JanuaryFebruary 2015. (Access to the full articles online requires a subscription.)  Anna Haensch (Posted 12/8/14) Nailing Down The Primes, by Ben PittmanPolletta No, you aren’t hard of reading, and that isn’t a typo. After refusing to budge for decades, the prime gap started shrinking rapidly last May, and now it appears to be growing. Thankfully, they’re not the same gaps. There are fewer primes as you travel out along the number line, but in order to understand their distribution, one wants to know not only how many primes there are on average, but how close together and how far apart they can get. In May of 2013, Yitang Zhang showed that no matter how far out you get on the number line, there are always pairs of primes close togetherat latest count, no more than 246 apart. But progress on the opposite questionin a given finite stretch of the number line, how big is the largest gap between consecutive primes?has been stuck since 1938. Or it was, until this past August, when two groups of mathematiciansone composed of Terence Tao, Ben Green, Kevin Ford, and Sergei Konyagin, and the other composed of, well, James Maynardwon the largest Erdős prize ever by improving upon a 76yearold lower bound on the size of prime gaps due to Robert Alexander Rankin. Rankin proved that for large enough numbers X, the largest prime gap below X is at least 1/3 the size of log(X)loglog(X)loglogloglog(X)/(logloglog(X))^2. Number theorists suspect that the gaps can in fact be much largeras large as log(X)^2, the size of gaps in collections of random numbers. But for close to 80 years, no one has been able to improve on Rankin’s bound, “a ridiculous formula,” according to Tao, “that you would never expect to show up naturally.” There was a conjecture of Paul Erdős: for any constant, you can choose large enough X so that Rankin’s bound holds with 1/3 replaced by that constant. Erdős offered a whopping $10,000 to anyone who could prove his claim, which longtime collaborator Ronald Graham has offered to pay to Tao, Maynard, and their collaborators. Proving lower bounds on prime gaps comes down to, er, coming up with large sequences of consecutive composite numbers, such as n! + 2, n! + 3, …, n! + n. All these numbers are composite, since n! is divisible by each number from 2 through n. However, this sequence occurs very far out in the number line. To improve on Rankin’s conjecture, the five mathematicians had to find numbers much smaller than n! to which 2, 3, …, n could be added to obtain composite numbers. They exploited new results on the structure of prime numbers, including work Maynard did, ironically enough, to understand small prime gaps. Erica Klarreich’s lucid explanation of both the mathematics and the history behind this other prime gap conjecture is well worth reading, even if only to hear Terence Tao’s favorite number theory joke. See "Prime Gap Grows After DecadesLong Lull," by Erica Klarreich, Quanta Magazine, 10 December 2014.  Ben PittmanPolletta (Posted 12/18/14) "Show and Tell": Google turns pictures into words, by Anna Haensch If you’ve used Google translate recently, you know it’s a totally different beast than the wordforword online translators of the days of old. Google’s algorithm doesn’t just translate words, it turns each word into a vector built out of the words that commonly appear around it, sentences become collections of vectors, and the whole thing becomes this gigantic linear algebra problem. And now, as reported in the MIT Technology Review, Google is trying to use this same type of algorithm for a new type of translation: pictures into words. This system, called Neural Image Caption (NIC), uses a data set of 100,000 images and generates captions using the same linear algebra technique. It generates a set of words associated to the image, and then places those words in a vector based on relationship between the words. From here, it works pretty much like a Google language translator. To check how well their algorithm fared, the team from Google set up an Amazon Mechanical Turk experiment, meaning two humans reviewed each caption, rating them on a scale from 1 to 4. They evaluated the captions based on how effective they were at describing the image. You can see a few results from that ranking in the image below. They also measured effectiveness on something called the BLEU scale, and NIC scored a 59 compared to human performance which has a score of 69. Prior to NIC, the best machine technology only scored a 25.
Take a closer look at the image in the first column, second row. It’s pretty impressive that a computer is able to understand that a tiny little white speck is a frisbee. Computer vision really has come a long way, although there is something undeniable hilarious and cute about the fact that facial detection software still struggles to tell the difference between humans and cats. See "How Google 'Translates' Pictures Into Words Using Vector Space Mathematics." MIT Technology Review, 1 December 2014.  Anna Haensch Making A Big Production Out of Optimizing Fluid Flows, by Ben PittmanPolletta What do raspberries, polyester, and blood pressure drugs have in common? All make for great performance art, sure, but I'll bet you didn't know that all three have also benefitted from the attention of applied mathematician Eleanor "Lea" Jenkins of Clemson University. Jenkins and her collaborators employ a multifaceted approach involving fluid dynamics, simulation, and optimization to tackle problems in agriculture and industry having to do with fluid flow through porous mediabringing them right to the border of engineering and mathematics. One of Jenkins' projects involves the mathematics of purifying biopharmaceuticals, or biologics ("The Chemistry of Math"). These complex moleculesincluding blood pressure medications, hormones such as insulin, vaccines, antibodies, antiviral agents, cancer drugs, and viral gene therapiesare "manufactured" inside living cells or tissues (Biopharmaceutical" Wikipedia). Complicated manufacturing processes have been one contributor to recent drug shortages. Jenkins and her collaborators hope to increase the yield of biopharmaceuticals and lower their pricesby optimizing the filtering process used to separate them from the cells in which they're manufactured. With longtime collaborator Kathleen Fowler and others, Jenkins has also worked on optimizing the lifetimes of filters used purify melted polymers woven into plastic fibers ("A simulation filter approach to polymer extrusion filter design," by Kathleen R. Fowler et al., Proceedings from American Filtration & Separation Society Annual Conference, 2008). In both problems, there are multiple competing objectivesfor example, maximizing filter lifetime while still filtering out a sufficient quantity of debris (a filter with large enough pores, though ineffective, would last forever). Furthermore, the functions relating filter parameters to the optimized quantities (such as filter effectiveness and lifetime) are obtained from computational fluid simulations, so calculusbased methods such as gradient descent can't be applied. The way forward is to reformulate these multiobjective optimizations as single objective optimizations with boundary conditions, and then solve them with the implicit filtering algorithm (no pun intended)a steepest descent algorithm for noisy optimization problems with bounded constraints (Implicit Filtering). As for berries, Fowler, Jenkins, and others began working with the berry cooperative Driscoll's at a 2011 Workshop on Sustainability Problems at the American Institute of Mathematics. To help farmers maximize their production and profits in the face of reduced water use limits during the state's ongoing drought ("A Role for Modeling, Simulation, and Optimization in an Agricultural Water Crisis," by Kathleen Fowler and Eleanor Jenkins, SIAM News, December 1, 2014), an interdisciplinary team of mathematicians and engineers formulated the a number of choices available to farmerssuch as which crops, crop rotation schemes, and irrigation and fertilization strategies to employas stochastic optimization problems coupled to a multiscale farm and groundwater simulation. The mathematicians came to a number of conclusions, including the suggestion that farmers plant more raspberrries. Currently, they're deploying their simulations to help design aquifer recharge networks to replenish depleted water sources. To address issues of sustainability, they say, "A common theme is the critical need for interdisciplinary teams from science, engineering, and computational applied mathematics ... in partnerships with industry, policymakers, and practitioners." It takes a village to raise a (sustainable) raspberry. See "The Chemistry of Math: Making and modeling better membranes could help unclog pharmaceutical pipelines," by Anna Simon, Lab Manager, November 29, 2014.  Ben PittmanPolletta (Posted 12/10/14) On making mathematicians glamorous, by Lisa DeKeukelaere (Photo on left: (LR) Terence Tao, University of California, Los Angeles; Maxim Kontsevich, Institut des Hautes Etudes Scientifiques; Simon Donaldson, Stony Brook University and Imperial College London; Jacob Lurie, Harvard University; and Richard Taylor, Institute for Advanced Study. Photo by Steve Jennings/Getty Images for Breakthrough Prize. Photo on right: (LR) Terence Tao, Jacob Lurie, Richard Taylor, Maxim Kontsevich and Simon Donaldson onstage during the Breakthrough Prize Awards. Photo by Steve Jennings/Getty Images for Breakthrough Prize.) Math professor Michael Harris takes an acerbic look at attempts to elevate mathematicians to rock star status at the 2014 Breakthrough Awards, an event created by tech investor Yuri Milner to bestow Oscarstyle treatment on scientific achievement. Harris describes the celebritystudded event, musically accompanied by Christina Aguilera, broadcast on BBC world News (and in the U.S. on the Discovery Channel ad Science Channel), and for the first time honoring a bevy of mathematicians. Their work largely devoid of applications easily explainable to the public, however, the mathematical rock stars appeared to be sidelined as "spare stage props," according to Harris, with their segment of the show confined to a single video clip. Harris notes that the mathematicians themselves were reticent to reach for rock stardom, with Breakthrough Prize winner Terence Tao having attempted to decline the award. At the start, Harris questions how much Hollywood glamour would rub off on mathematics, and the answer is, not much. He notes that mass media attention focused more on famous presenters than award recipients, and the mathematicians "flew home rocking neither more nor less than when they arrived." See "Dispatch From the Oscars of Science," by Michael Harris. Slate, 19 November 2014; See media coverage of the awards event "Scientists Mingle with the Stars at the 2015 Breakthrough Prize," by Lauren Goodman, Vanity Fair, 10 November 2014. Read about the Breakthrough Prize in Mathematics and the winners: Five Winners Receive Inaugural Breakthrough Prize In Mathematics, 23 June 2014; AMS news item, 23 June 2014; "Winners Announced For The World's Richest Science Award: The $3 Million Breakthrough Prize," F.D. Flam, Forbes, 9 November 2014. Below is a video clip from the award ceremony.  Lisa DeKeukelaere Free Your Eyes, by Ben PittmanPolletta It's the perfect cinematic setup: A lone scientist is working late at night in his lab, with a computer program which finds threedimensional objects corresponding to twodimensional polygonal line drawings. He stretches, ready to take a break; as a lark, he inputs a drawing of the Penrose staircasethe constantly descending steps discovered by Roger Penrose and his father Lionel, and made famous by M. C. Escherand, chuckling to himself, leaves the lab to get a cup of coffee. Upon his return, his coffee cup crashes to the floor as he gapes at the computer. The program has finished its work, and the cursor blinks at the end of the phrase: OBJECT FOUND. This goosebumpinducing scenario isn't from a horror movie; it's from the real life magic show that is mathematician Kokichi Sugihara's career. As Sugihara discovered during his early research into computer vision, many drawings of seemingly impossible objects, such as the Penrose staircase and the Penrose triangle, really do depict views of three dimensional polygons, just not the ones your mind assumes they depict. Sugihara doubted his program's results until he began creating paper models of the objects it had found. Soon, he'd moved away from computer vision, and into what's been his main line of work for the last thirty years: creating threedimensional visual illusions, such as the "supermagnet" illusionin which balls of wood and glass roll up ramps to a platform as if attracted by a powerful magnetwhich won World's Best Illusion in 2010: As Erica Klarreich discusses in her delightful article for Nautilus, Sugihara's illusions not only win awards, they also illuminate some of the shortcuts the human visual system uses when interpreting twodimensional images as threedimensional objects. According to Sugihara, one of these is a tendency to perceive sets of perpendicular lines as meeting at right angles. Many of Sugihara's illusions depend on this biasand when he instructs his software to pick threedimensional structures with the maximal number of right angles, it usually arrives at the same interpretations of line drawings as human viewers. Another quirk of the visual system, says Stephen Macknik, a neuroscientist who runs the World's Best Illusion contest with his colleague Susana MartinezConde, is that it does much of its work locally. For example, each small piece of the Penrose staircase is coherent; it's only the global picture that seems nonsensical. This local processing happens mostly automatically, and as a result illusions like the threedimensional Penrose staircase and "supermagnet" video persist even after the brain has learned the underlying trick. But if the human visual system is fallible in predictable (and exploitable) ways, it is certainly rapid and efficientand yet unmatched by any vision software. Indeed, part of the visual system's success comes from the fact that it has evolved (or learned) to be right, as quickly as possible, most of the time. Intriguingly, as Klarreich reports, robots that learned by trial and error to distinguish different shades of gray in natural scenes were also susceptible to "White's illusion" (no pun intended), in which a single shade of gray seems darker or lighter depending on surrounding shades. So, while our robot descendants may be aware of the unusual structures giving rise to Sugihara's illusions, their artificiallyevolved visual systems might be just as susceptible to these illusions as ours. That's just fine with Sugihara: “It’s my pleasure," he says, "to make other people [and sentient robots of the future] surprised." (Image: The gray rectangles under A are the same shade as the gray rectangles under B.) Many more of Sugihara's static illusions are viewable on his webpage, as well as a video of several of his "impossible motion" illusions, and even files allowing you to construct some of his illusions for yourself. See "The Illusion Machine That Teaches Us How We See," by Erica Klarreich, Nautilus, 13 November 2014.  Ben PittmanPolletta (Posted 12/1/14) Terence Tao appears on The Colbert Report, by Claudia Clark In this 6minute segment, comedian Stephen Colbert interviewed UCLA professor of mathematics Terence Tao about his interest in mathematics, specifically his interest in prime numbers. After defining prime numbers for the audience, and noting that there are an infinite number of them, Tao talked about twin primes, explaining that we don't know whether there are a finite or infinite number of pairs. He also talked about cousin primes (prime numbers that differ from each other by 4), and sexy primes (prime numbers that differ from each other by 6), all the while gamely responding to Colbert’s witty comments with good humor. He told the audience that, just this year, it was proven that at least one of these three classes of primes is infinite since the sum of the formulas used to calculate the number of twin primes, the number of cousin primes, and the number of sexy primes has been shown to be infinite. See the segment Terence Tao, guest, The Colbert Report, Comedy Central, 12 November 2014.  Claudia Clark On how math explains why hipsters look alike, by Lisa DeKeukelaere Drawing parallels to the behavior of neurons, mathematician and neuroscientist Jonathan Touboul uses mathematical modeling to explain why nonconformist "hipsters" end up conforming to each other. Touboul lays out his work in a recently published paper, first illustrating how the preferences of a population will shift –seemingly chaotically—as conformists and nonconformists each change their own preferences based on their observations of those around them. The key to Touboul's work, however, is applying to his model the idea of a delay in such observations. Increasing the delay to a certain point elicits a pattern out of the chaos and demonstrates how hipsters seeking to be nonconformist actually fall into sync with each other as they react to an outdated perception of the norm. Touboul concedes that his model is an unrealistic simplification of the world—hipsters have numerous ways to exhibit nonconformity, for example—but he notes that simplification is the point of mathematics, to facilitate understanding of complex ideas. Images courtesy of Jonathan Touboul: Simulations of the discrete system for n = 5,000, inverse temperature β = 2, and q = 1 (fully anticonformist system) and different delays (left: 0.5, center: 0.7, and right: 1.5). Top row: time evolution of all particles as a function of time; bottom row: empirical (blue) and theoretical (red) total trend. See "The mathematician who proved why hipsters all look alike," by Jeff Guo, Washington Post, 11 November 2014; "Math Finally Sleuths Why So Many Hipsters Look Alike," by John Hendrickson. Esquire, November 14, 2014; and "The hipster effect: When anticonformists all look the same," by Jonathan Touboul on arxiv.org.  Lisa DeKeukelaere Combining Food and Formulas, by Claudia Clark Mercedes Siles Molina, a mathematician at the Universidad de Málaga, in Spain, is passionate about both mathematics and cooking. Molina collaborated with Chef José Carlos Garcia and photographer Pedro Reyes Dueñas to create a collection of photographs of beautifully presented culinary creations based on mathematical formulas. These photographs are part of an exhibit called "The Taste of Mathematics" (El sabor de las Matemáticas), which has appeared in several cities in Spain and Panama. Molina recently spoke about her love for both math and cooking at an event entitled "Cooking Up Math" on November 5th at the Museum of Mathematics in New York. "The motivation for the work "The Taste of Mathematics" was the Centennial of the Royal Spanish Mathematical Society (RSME) in 2011. I'm member of the governing board of the RSME and wanted to do something special on that occasion. That was the starting point for my work on divulgation of Mathematics," wrote Molina about the "El sabor de las Matemáticas"("The taste of Mathematics"). (Photos: "Strawberries with tiles of coral and cocoa" and "Cylinder," works by Mercedes Siles Molina and Chef José Carlos Garcia, photographs by Pedro Reyes Dueñas. Courtesy Mercedes Siles Molina.) See "Mouthwatering Math: Culinary Creations Combine Food and Formulas," by Tanya Lewis. LiveScience, 10 November 2014, and photographs from the exhibit.  Claudia Clark He ought to be in papers, by Ben PittmanPolletta The Imitation Game is a new biopic about scientist and mathematician Alan Turing's life and work. With the recent premiere (see coverage of the premiere by Roger Friedman on his blog Showbiz 411) and a cover story in Time magazine, there's no shortage of media coverage on the film, but Dan Rockmore's review in The New Yorker is a welcome deeper look into the intellectual life of the film's protagonist. Turing is most famous for his work in computer science and artifical intelligence, exemplified by the eponymous concepts of the Turing machine and the Turing test. Turing was also a homosexual, and his conviction for "gross indecency" and the subsequent sentence of chemical castration led to his suicide 16 days before his 42nd birthday. These facts provide the "narrative hooks" for the new movie, which, according to Rockmore, focuses on Turing's work cracking the Enigma code during World War II. But Turing was also a pioneer in logic, number theory, and mathematical biology, modeling the brain and development. Rockmore, chair of the Department of Mathematics and professor of computer science at Dartmouth College, uses his review of The Imitation Game as an opportunity to highlight one of Turing's most prescient and impressive intellectual workshis essay "Intelligent Machinery." In this paper, Turing sets down his ideas about how a thinking machine might be built. He puts forward a simple model of a network of neurons, and ideas for how it might be educated from "experience"and especially from rewarding and aversive stimuliwhich are foundational to the influential connectionist paradigm in mind and brain research ("Connectionism," Wikipedia). This paper, as well as others collected in editor B. Jack Copeland's The Essential Turing (reviewed by Andrew Hodges in the AMS Notices), such as "The Chemical Basis of Morphogenesis," illuminate critical issues in science with a clarity rarely seen in today's literature. For instance, in "Intelligent Machinery, a Heretical Idea," Turing describes how "indexes" might be used to organize information in memory. "New forms of index," he suggests, "might be introduced on account of special features observed in the indexes already used." Here and elsewhere, Turing exemplifies his own rationale for modeling human thought. "The whole thinking process is still rather mysterious to us," he says in "Can Machines Think?", "but I believe that the attempt to make a thinking machine will help us greatly in finding out how we think ourselves." Revisiting Turing's work reminds us how much we have to learn from historyhow not only the emotional, but also the intellectual struggles of our forebears, mirror and inform our own. Indeed, this is an idea Turing himself discusses in "Intelligent Machinery." At the end of the essay, he proposes that essentially all problems are search problems, and discusses both evolutionary and intellectual searches in this light. "The remaining form of search is what I should like to call the 'cultural search'," he writes, "... the isolated man does not develop any intellectual power ... the search for new techniques must be carried out by the human community as a whole." See "What's Missing From 'The Imitation Game'" by Dan Rockmore. The New Yorker, 6 November 2014.  Ben PittmanPolletta (posted 11/20/14) Musicians and Birds Converge on the Harmonic Series, by Anna Haensch
These are exactly the terms in the harmonic series. Doolittle and her team were surprised to find that the songs of Hermit Thrush contain this same harmonic series. What is interesting about this, is that tones in the harmonic series, although so familiar to our human ears, are not really a human aesthetic construct. It's just math! So while the results from Doolittle and her lab are surprising, they don't suggest that birds have some understanding of music theory. But as Doolittle points out to The Smithsonian, "If an aspect of music is found not only in humans, but also in a variety of nonhuman species, this would suggest that there may be something in our shared biology that predisposes us to find that aspect interesting, or attractive, or easy to sing." See "This Bird's Songs Share the Same Mathematical Qualities as Human Music," by Allison Eck, NOVA, 4 November 2014.  Anna Haensch (Posted 11/21/14) Matt Parker turns math into standup comedy, by Anna Haensch Rarely does mathematics inspire laughter, and it would seem downright impossible to find a mathematician who could sell out a 3,500 seat comedy show. Meet Matt Parker, a most unusual mathematician. Parker, who grew up in Australia, says he "never got the memo about maths being boring." Since he was a kid, Parker has always been fascinated by math, and now as an adult it’s his mission to spread that fascination around. And as The Independent explains in its recent article, Parker has found a most unconventional way to spread his good newsthrough standup comedy! Parker maintains a Youtube channel full of videos of his standup and various other mathematical entertainment, along with other not so mathy things, like how to make ice cream using a fire extinguisher. In addition to his own channel, he is also a frequent contributor to the very popular Numberphile, where he gives his totally watchable take on some of our most pressing math questions, like why the heck can’t we divide by 0? His most recent effort at connecting people to math is his new book, Things to Make and Do in the Fourth Dimension, full of games, puzzles, and explanations of all the exciting parts of math that we may have missed in school. Parker laments the fact that math is "the one thing you’re still allowed to be proud of being bad at…people are almost proud of being ignorant of this huge achievement of humankind." Parker is out to change that, and he sees that now is the time. "There has been a huge resurgence of maths as entertainment," Parker says, which we can see playing out all over social media. For Parker’s latest, follow him on Twitter @standupmaths. See "'Standup mathematician' Matt Parker is using comedy nights to preach maths to big audiences," by Simon Usborne. The Independent, 30 October 2014.  Anna Haensch (Posted 11/10/14)

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