Math Digest

Edited by Mike Breen and Annette Emerson, AMS Public Awareness Officers
Contributors: Mike Breen (AMS), Claudia Clark (writer and editor), Lisa DeKeukelaere (2004 AMS Media Fellow), Annette Emerson (AMS), Brie Finegold (University of Arizona), Baldur Hedinsson (2009 AMS Media Fellow), Allyn Jackson (Deputy Editor, Notices of the AMS), and Ben Polletta (Harvard Medical School)

See a recent article accompanied by a video on the mathematics of juggling--good preparation for International Juggling Day on June 16.
(Image: Rod Kimball juggling three balls using the 4-4-1 pattern, courtesy of George Hart.)

This Month's Math Digest Summaries Posted here June 2013: breakthrough result on pairs of prime numbers, mathematics of juggling and mathematicians doing cancer research ...

See also: Blog on Math Blogs: Two mathematicians tour the mathematical blogosphere. Editors Brie Finegold and Evelyn Lamb, both Ph. D. mathematicians, blog on blogs --on topics related to mathematics research, applied mathematics, mathematicians, math in the news, mathematics education, math and the arts and more. Recent posts: "On Pregnancy and Probability," "This Week in Number Theory," "Building the World Digital Mathematical Library," "Binary Bonsai and Other Mathematical Plants," "Win at Math! ".

Earlier this year, the Norwegian Academy of Science and Letters announced that it would award the 2013 Abel Prize to Belgian mathematician Pierre Deligne of the Institute for Advanced Study in Princeton, New Jersey, "for his contributions to algebraic geometry and for their transformative impact on number theory, representation theory and related fields." Deligne is the 13th winner of the Abel Prize, established "to recognize outstanding mathematicians and encourage math developments." As the winner, he will also receive a monetary prize of 6 million kroner (almost 1 million US dollars). This article describes the events of May 21st, the day of the ceremony when the award was presented to Deligne, as well as the festivities that will round out the celebration of "Abel Week" in Norway. These include several lectures—including Deligne's prize lecture, "Hidden symmetries of algebraic varieties"—a reception, a symposium, and a meeting with schoolchildren who will be working on mathematical problems. While Deligne is "happy and proud" to receive this award, he has "insisted" that it is not just for him: "The most important thing will be if it can inspire and lead to young people getting interested in mathematics." Read all about this event and hear the Abel lectures. (Photo: Pierre Deligne receives The Abel Prize. Heiko Junge/Scanpix.)

Category theory, a system for describing any mathematical field, is highly abstract--typically a hallmark difficulty in explaining practical utility--yet it has proven useful in a diverse range of disciplines such as biology, music, and philosophy. First developed in the 1940s to examine the similarities between abstract algebra and topology, category theory breaks each field down into objects and the arrows that link them. For example, one category could contain theorems as objects connected by arrows signifying that one is used to derive another. Scientists can then evaluate maps between categories, and maps between those maps. Using such a system not only has helped reveal similarities between mathematical fields, but also has proven useful for modeling complex biological systems and for distinguishing the meaning of words in a sentence based on language theories.

The Business Standard broke the news that 'Slumdog Millionaire' star Dev Patel will play Indian mathematics genius Srinivasa Ramanujan in an upcoming biopic titled "The Man Who Knew Infinity: A Life of the Genius Ramanujan." Director and screen writer Matthew Brown is tasked with adapting the incredible story of Ramanujan's life to the wide screen. Ramanujan was an unlikely math pioneer--a self-taught mathematician, college dropout and isolated from the larger mathematical community. Despite the odds being stacked against him, he made extraordinary contributions to mathematical analysis, number theory and infinite series. In 1912 at the age of 25 Ramanujan sent some of his mathematical discoveries to three academics at the University of Cambridge in England. G H Hardy recognized his brilliance and invited him to visit and work with him at Cambridge. Sadly, Ramanujan died due to illness at the young age of 32.

"Don’t bristle at blunders," by Mario Livio. Nature, 16 May 2013, page 309.

Astrophysicist Mario Livio argues that blunders--incorrect scientific results and theories--are not simply wastes of time and money, but an important and valuable part of the scientific process. As an example, he highlights Lord Kelvin's theory that atoms are "knotted vortex tubes in the ether," rather than point-like objects. Though Kelvin's idea was wrong, it increased the scientific community's interest in knots, which eventually led to development of knot theory in the 1980s and links to quantum field theory and string theory. While the scientific community does have numerous experts in any given field to identify and correct a blunder, the complexity and cost of research today often deter scientists from checking each other's work. Citing the previous allocation of 10% of research time on the Hubble Space Telescope to projects with a low probability of success but a potentially high return, however, Livio asserts that scientists must continue to undertake novel, risky research.

In this article, the writer describes a study published in the journal Current Biology by a team of University of Oxford researchers. These researchers wanted to determine the impact, if any, on a person’s ability to perform calculations with numbers while having transcranial random noise stimulation (TRNS) applied to a certain part of the skull. Researchers found that the 25 student volunteers who underwent the stimulation while performing math calculations over a period of 5 days were "significantly faster at doing the calculations than the control group," Walton writes. A six-month follow-up with a subset of these participants showed that those who'd undergone TRNS were still 28% faster than those who had not. "With just five days of cognitive training and non-invasive, painless brain stimulation, we were able to bring about long-lasting improvements in cognitive and brain functions," concludes Oxford researcher Roi Cohen Kadosh.

Critics have raised some concerns in their responses to this and other articles that have reported on this study. Some have questioned the researchers' conclusions, particularly of a long-term impact, given the small number of participants. Others have pointed out that the tasks measured don't really reflect the process by which math skills are learned. Further research is needed to determine how helpful this technique could be for children who struggle with math.

The research finding was picked up extensively in news media around the world, including Forbes, Chicago Tribune, US News & World Report, The Guardian, BBC News, New Zealand Herald, Nature, HealthCentral, Latinos Post and French Tribune.

The distribution of the primes--those quarks and gluons of the natural numbers--is a source of endless fascination for number theorists and math fans everywhere. How many are there? What are they divisible by? While we may never get the answers to these timeless questions, this month brings a welcome advance on the question of how far primes get from each other as one wanders "into the deserts of the truly gargantuan prime numbers" (as described in "Unheralded Mathematician Bridges the Prime Gap," by Erica Klarreich, Simons Foundation, 19 May 2013)--from an unexpected source. On April 17th, Yitang "Tom" Zhang, described as "a popular math professor at the University of New Hampshire" ("The Beauty of Bounded Gaps," by Jordan Ellenberg, Slate, 22 May 2013), sent a paper to the Annals of Mathematics detailing a new proof of the bounded prime conjecture, which it would be a whopping understatement to call an important step on the road to the proof of the twin prime conjecture.

The prime number theorem tells us that primes become less common as numbers get larger, and as one might expect, the average distances between them get larger at the same time. For large numbers, the expected size of the gap between prime numbers is about 2.3 times the number of digits--so between the average pair of 100-digit primes, there are 230 non-primes, as explained by Klarreich. But the actual gaps between primes may be a great deal larger or smaller. It is not hard to show that the gaps between consecutive primes can be made arbitrarily large ("Progress on the Twin Primes Conjecture," by jrosenhouse on ScienceBlogs, 24 May 2013). The twin primes conjecture makes the audacious (but widely believed) complementary claim that consecutive primes separated by a single non-prime number--so-called twin primes--recur infinitely often. While some say this conjecture goes as far back as Euclid, until Tom Zhang, no one had proven that any finite gap recurs infinitely often--leaving open the possibility that the gaps between consecutive primes grow without bound. According to a lightning-fast referee report from the Annals of Mathematics, Zhang has eliminated this possibility, proving that there are infinitely many pairs of primes separated by no more than 63,374,610 non-primes (see the comments on the ScienceBlog piece).

To explain how surprising Zhang's achievement is, consider that the 51-year-old's last published paper came out in 2001; that after graduating from Purdue in 1991 Zhang had a hard time finding an academic job and spent brief stints working as an accountant and in a Subway sandwich shop; and that number theory was not the subject of Zhang's dissertation! Compound this with the fact that his new paper--reportedly a model of clarity--builds on an approach taken by number theorists Daniel Goldston, Janos Pintz, and Cem Yildrim in a landmark paper from 1995. Klarreich's piece includes these quotes from Granville and Zhang: “The big experts in the area tried and failed,” says number theorist Andrew Granville. “I personally didn’t think anyone was going to be able to do it any time soon.” As for the man who did it, he's taking his success in stride. “My mind is very peaceful," he says, "I don’t care so much about the money, or the honor. I like to be very quiet and keep working by myself." His advice to the rest of us? "There are a lot of chances in your career, but the important thing is to keep thinking."

All parents want their children to do well in life and many parents believe being good at math is one of the essential skills that their kids need have. The social news site reddit has a number of discussion threads on how to get your kids interested in math at an early age. Walter Hickey went through these threads and summarized the most popular submissions from real life math professionals about what made them interested in math early on. The submissions range from letting your kids play with legos to teaching them oragami, from taking your kids to music lessons to teaching them how to play games like chess and Go. All the entries on the list are worth trying out and the list in general is a recommended read, however as Walter Hickey puts it, "results are not guaranteed. As your future mathematician might tell you, correlation does not imply causation."

Scientific American links to a fun article at Simon Science News about the mathematics of juggling. The article is accompanied by a great video by George Hart that describes the many ways juggling involves math. The video also illustrates how to approach juggling with a mathematical mindset, which could be a big help if you want to improve your juggling skills. Juggling has been recorded in many early cultures and the activity dates back to ancient Egypt. However, juggling has advanced enormously in recent decades, thanks in part to the mathematical study of possible juggling patterns. The late computer scientist Claude Shannon, who was also an avid unicyclist, juggler and tinkerer, was a pioneer in applying rigorous mathematics to juggling. Shannon, who is often called the father of information theory, published the first formal mathematical theorem of juggling in the early 1980s. In the paper Shannon correlated the length of time balls are in the air with how long each ball stays in the juggler's hand, demonstrating that hand speed is essential for successful juggling. Mathematicians have been fascinated by juggling ever since. "I think it's a matter of making sense of the order that's in the juggling patterns," said Jonathan Stadler, a math professor at Capital University in Ohio who started juggling as a teenager. "It has to do with understanding how things fit together." (Image: Rod Kimball juggling three balls using the 4-4-1 pattern, courtesy of George Hart.)

In this article, Bruce Newman reports on the 2013 Julia Robinson Mathematics Festival hosted by Stanford University on May 11. During this half-day event, almost 250 students from grades 6 to 12 worked collaboratively in small groups on math problems and puzzles with names like "Instant Insanity" or "Space Chips." The purpose of the festival, according to event founder Nancy Blachman, is to "expose kids to ideas that we think will be fun and engaging, and show them that math can be delightful." In addition, "we thought that if we named it after a woman, girls would look into her and see that women can do math problems." That tactic may have worked: Newman estimates that there were more girls than boys in attendance at the event. Associate director of the San Francisco Math Circle Brandy Wiegers also sees this event as an opportunity to reduce the "dweeb" stigma surrounding the enjoyment of math: "As a culture, we have become more math illiterate. It's completely acceptable in a public setting to say, 'Oh, I'm horrible at math'…" Based on the fun these kids appeared to have, it seems unlikely they'll ever utter these words. Learn more about Julia Robinson Mathematics Festivals around the country. (Photo: Julia Robinson Mathematics Festival, a program of the Mathematical Sciences Research Institute. Photo by Katelyn Weingart.)

"A Fresh Start for Foam Physics," by Denis Weaire. Science, 10 May 2013, pages 693-694.

A foam is a structure formed by trapping pockets of gas in a liquid or a solid--both bath sponges and the lather they carry are examples--and anyone familiar with minimal surfaces could guess that foams bring plenty of interesting mathematics to the bathroom. The geometry of soapy foams is largely dictated by the minimization of surface energy, and conforms to laws first written down by the 19th century Belgian physicist Joseph Plateau. Plateau's model--which certainly approaches the optimal name for an equilibrium theory--has received great attention from the mathematical and physical community. Highlights include Jean Taylor's 1976 geometric measure theory proof that minimal surfaces satisfy Plateau's laws, and Kenneth Brakke's 1992 creation of the software Surface Evolver, which simulates the mean curvature flow under quite general conditions to find the configurations of complex minimal surfaces. But time waits for no foam, and for no science. While equilibrium foams have occupied mathematicians and physicists for more than a century, the more practical pressures of industrial applications--such as controlling the formation of lightweight metal and plastic foams--have increasingly called scientific attention to the problem of foam dynamics.

Like many of the hardest problems in science, this one is multiscale. In equilibrium soap foams, liquid drains from fluid boundaries due to the combined effects of forces like surface tension, surfactant concentration, and gravity. This drainage results in instabilities which cause bubbles to rupture, and these ruptures in turn cause rapid rearrangement of the foam configuration. In the May 10th issue of Nature, mathematicians Robert Saye and James Sethian of Lawrence Berkeley National Labs introduce a multiscale computational model of liquid foam dynamics. "In such 'multiscale problems'," the authors write, "the unfolding of small-scale processes, depending on physics, chemistry, and biology, combine to produce large scale effects, and these macroscopic dynamics subsequently affect the interplay of microscopic forces. ... Fortunately, the details at one space or time scale are not necessarily important at another scale. By devising different models and equations at different scales, we can 'separate scales' and compute physics at different resolutions, allowing these different models to communicate across the scales." Saye and Sethian's model divides foam dynamics into three phases - rapid evolution towards an equilibrium macroscopic structure (on the order of fractions of a second), followed by the slow drainage of fluid at microscopic scales from the foam's liquid boundaries (on the order of tens or hundreds of seconds), followed by the fast process of bubble rupture, leading to a nonequilibrium macroscopic structure and the repetition of the first phase. "These efforts arrive just in time," writes Dennis Weaire in his Perspectives overview, "to confront the mass of new data soon to be provided by x-ray tomography". Multiscale computational models and big data? Who knew lathering up could be so intellectually simulating--er, stimulating. (Image: James Sethian and Robert Saye, UC Berkeley)

Mathematics and Music: Composition, Perception and Performance, a new textbook by two University of Wisconsin-Eau Claire (UW-EC) professors, James Walker and Gary Don, highlights the links between the two disciplines by explaining such topics as the math behind musical scales and the graphical depiction of pitch. The authors explain that UW-EC was an ideal setting for development of the book because of its strong focus on music and its student-faculty research program, which led to support for a music/math course. As a result, Walker and Don were able to draw useful examples for the book from their students, one of whom was responsible for bringing the two professors together when she undertook a double major. Walker and Don also note that adding the angle of music helps make the study of mathematics more interesting, which increases student performance.

Read about the research that led to the book in the article Walker and Don co-authored with Karyn K. Muir and Gordon B. Volk: "Music: Broken Symmetry, Geometry, and Complexity," in the January 2010 issue of Notices of the AMS. Photo of James Walker (left) and Gary Don by Andrea Paulseth, Volume One.

Like the deadly fog in Stephen King's The Mist, mathematicians--with their coffee, blackboards, and a conspicuous lack of test tubes and microscopes--continue to ooze their way around every door in biology, harboring powerful ideas whose inhuman tentacles threaten to revolutionize modern medicine. This article takes a peek at the (not really) unholy hybrid that is Moffitt Cancer Center's Integrated Mathematical Oncology department - the first, but surely not the last, of its kind. Run by the (probably not actually) diabolical Dr. Robert Gatenby, the department - at whose heart lurks the "collaboratorium", a room filled only with couches, "a blizzard of numbers, lines and diagrams", and the bone-chilling smell of coffee - has as its goal the building of mathematical models which will allow doctors to better predict cancer growth, and to better design effective tumor treatments. One of the department's methods is the use of game theory to improve cancer treatment. Predicting how tumors will respond to medical interventions has resulted in what Dr. Gattenby terms "adaptive treatment". An example is reduced chemotherapy doses, currently being tested by the group in the treatment of prostate and breast cancers, which keep the population of drug-resistant cancer cells in check by allowing some drug-sensitive cancer cells to survive. Another is the combination of chemotherapy and immunotherapy; in adapting to the latter immune-boosting therapy, cancer cells may take on traits that make them more vulnerable to chemotherapy, resulting in an "uneasy stalemate" keeping the tumor in check.

Alexander "Sandy" Anderson, one of the first members of the (surely not) unholy research alliance, turned to cancer research after studying stochastic PDEs describing chemotaxis. After developing a series of equations to describe how nematodes use the concentrations of soil chemicals to find potatoes, Dr. Anderson made a connection with cancer growth: tumor cells release growth factors that attract sustaining blood vessels. When Anderson proposed to use his model to study cancer growth, his medical colleagues were less than enthused, asking him, " 'Why do I need a mathematician? Why do I need a model?' " But their resistance was futile. Dr. Anderson continued his research with collaborators at Vanderbilt University, and in 2008, he was recruited by Dr. Gatenby, who had agreed to head Moffitt's radiology department on the condition that he be allowed to start the nation's first free-standing mathematical oncology department. Anderson and his colleagues "want to be taking a patient, get a biopsy from them, get their imaging, get their blood work, get as much information from them as we can" and plug that information into a model used to predict and treat their tumor. Their proof of concept is a model used to predict the growth of gliomas. These brain cancers are fatal but highly idiosyncratic: while some remain dormant for decades, others grow into high grade glioblastomas, killing their hosts within months. Predicting how a glioma will grow is thus a matter of great importance, but existing methods are far from accurate. The Moffitt group's model includes new variables, and matches real tumor growth in simulations. Anderson's group recently received a $3 million grant from the NIH to try their approach on prostate cancer. "I think it's a really exciting time for the field," says Anderson. "The bottom line is," (I like to imagine) Dr. Gatenby cackles maniacally as lightning and thunder split the sky behind him, "how can we make clinical care better." (Photo: Alexander Anderson. Image courtesy of Nicholas J. Gould.)

Summaries of Media Coverage of Math

This Month's Math Digest Summaries Posted here June 2013: breakthrough result on pairs of prime numbers, mathematics of juggling and mathematicians doing cancer research ...

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