
Mike Breen and Annette Emerson
Public Awareness Officers
paoffice at ams.org
Tel: 4014554000
Fax: 4013313842 

The abc conjecture in Nature
There doesn't seem to be any connection between the prime factors of two integers and those of their sum. After all, $3+ 17 = 4\cdot 5$. But in fact there is a subtle link. That's the point of the "$abc$ conjecture," and the subject of an unusually prominent look at mathematics in Nature. "The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof," by Davide Castelvecchi (October 7, 2015) ran with the subhead: "A Japanese mathematician claims to have solved one of the most important problems in his field. The trouble is, hardly anyone can work out whether he's right." The conjecture stems from the observation that if three relatively prime integers are related by $a+b=c$, then $c$ is usually smaller than the prime content ${\rm rad}(abc)$, the product of all the distinct primes occurring in the three factors. For example in $3+ 17 = 4\cdot 5$, ${\rm rad}(abc)= 3\cdot 17\cdot 2\cdot 5 = 510$, certainly larger than $20$. But this is not always the case. For example, (from Wikipedia), $3+125=128$. Here ${\rm rad}(abc)=3\cdot 5\cdot 2 = 30 < 128$. There are in fact infinitely many such triples. The $abc$ conjecture quantifies how "bad" they can be:

For every $\epsilon >0$ there are only finitely many triples $a+b=c$ with ${\rm rad}(abc)^{1+\epsilon} < c$.
Castelvecchi explains how, after the conjecture was formulated (19851988) by Joseph Oesterlé and David Masser, it was soon recognized to "have profound implications for the study of equations concerning whole numbers  also known as Diophantine equations ... ." Noam Elkies "found that a proof of the $abc$ conjecture would solve a huge collection of famous and unsolved Diophantine equations in one stroke." So the appearance of Mochizuki's proof, in 2012, was a big event. The article does not go into mathematical details of the proof, and for good reason: as stated in the subtitle, even after three years almost nobody in the profession understands it. And worse: "... the few who have understood the work have struggled to explain it to anyone else. 'Everybody who I'm aware of who's come close to this stuff is quite reasonable, but afterwards they become incapable of communicating it,' says one mathematician who did not want his name to be mentioned. The situation, he says, reminds him of the Monty Python skit about a writer who jots down the world's funniest joke. Anyone who reads it dies from laughing and can never relate it to anyone else."
Chaos theory without equations
"A Twisted Path to EquationFree Prediction," by Gabriel Popkin, appears in the October 13, 2015 Quanta. The subhead: "Complex natural systems defy standard mathematical analysis, so one ecologist is throwing out the equations." Quanta runs a lovely animated GIF of the Lorenz attractor, but Popkin's story is that in general, the chaotic dynamical systems occurring in nature are too complex to be usefully modeled by equations. An equation implies that some relation between present and future, which can be expected to persist, has been explicitly understood. But even when this understanding is lacking, the data themselves can do the job. Popkin gives the example of George Sugihara (Scripps), an ecologist whose team analyzed flutuations in the salmon population off British Columbia by developing "an approach based on chaos theory that they call 'empirical dynamic modeling,' which makes no assumptions about salmon biology and uses only raw data as input." Sugihara: "It's allowing the data to say what the relationships are." The work is reported in a PNAS article published in March, 2015.
(Image: The Lorenz attractor. Image courtesy of davidope.)
As Popkin tells us, the underlying theory goes back to the Dutch mathematician Floris Takens, whose Embedding theorem (1981) showed how "the full state of a chaotic system can, in theory at least, be embedded in a time series of a single variable." One concentrates on that variable. "The essence of the method involves identifying points in a system's attractor graph that are close to the point representing the system's present state. For one or two time steps, one can then predict that the system will evolve similarly to how it did in the past. ... Takens' theorem works best when there are enough data points to make a dense attractor, making it easier to find times when a system's present state is close to a past one."
Popkin spoke with Don DeAngelis, an ecologist with the U.S. Geological Survey in Miami, who called Sugihara's work "a huge theoretical breakthrough," and who wrote, with Simeon Yurek, in a commentary in the same issue of PNAS,"This approach also suggests that, with highperformance computation, the study of dynamic systems is moving away from formulation and parameterization of equations and toward letting data directly determine the model. Because of the central role equations hold in science, it also raises questions: How will these changes affect the way scientists communicate, the way they understand, and the way they think?"
Bedtime math boosts kids' scores
Last month, we learned how parents can transmit their math anxiety to their children by "helping" with homework. Now comes more positive news, in the form of "Math at home adds up to achievement in school," a research report in Science (October 9, 2015) that describes how having children engage in math story time with their parents can bring about significant improvement in the children's mathematical progress, even when it happens as seldom as once a week and especially when the parents have math anxiety themselves. As the authors, Talia Berkowitz, Marjorie Schaeffer and five collaborators (Psychology, Chicago), state in their abstract, "Brief, highquality parentchild interactions about math at home help break the intergenerational cycle of low math achievement."
The study was large (587 families with firstgraders, from 22 Chicagoarea schools) and meticulously organized. Basically there were two groups, a "math" group and a "reading" group that served as a control. "Children and their parents were asked to read topical math (or reading) passages and answer corresponding math (or reading) questions, delivered by an iPad app called Bedtime Learning Together (BLT), several times per week over the course of the school year. ... Participating families were given an iPad Mini to access the story problems." The experimenters were able to track how often parents used the app with their children, and investigate the relation between that frequency and the children's mathematical progress during the year, comparing the results with those from the reading group. They were also able to test the parents' math anxiety levels before the start of the experiment, and analyze the high and lowlevelanxiety groups separately. Among the reported results:

"The more times children and their parents used the app, the higher the children's math achievement at school year's end (controlling for beginningofyear math achievement), but only for children in the math group."

"For children of highmathanxious parents, we found a significant effect of group, with children in the math group outperforming childen in the reading group by almost 3 months in math achievement by school year's end. We did not find this same pattern for children of lowmathanxious parents."

"When families used the app on average once a week or more, children with highmathanxious parents made gains in math achievement by the end of the school year that did not significantly differ from those made by children with lowmathanxious parents."
The Bedtime Math app can be downloaded free. Instructions are here.
"Maths whizz solves a master's riddle"
The "maths whizz" is Terence Tao, the master is Paul Erdős, the story is told by Chris Cesare in Nature, October 1, 2015. The riddle is the Erdős discrepancy problem, posed by the master in the 1930s. "Like many puzzles in number theory, the Erdős discrepancy problem is simple to state but devilishly difficult to prove. Erdős ... speculated that any infinite string of the numbers 1 and $1$ could add up to an arbitrarily large (positive or negative) value by counting only the numbers at a fixed interval for a finite number of steps."
Paul Erdős and Terence Tao in 1985. Highresolution image. Photograph courtesy of Terence Tao.
Cesare sketches out the recent history of the problem, including the involvement of Tim Gowers and Polymath where the contrapositive (and to my mind more intelligible) version of the problem can be found: "Is it possible to find a $\pm 1$valued sequence $x_1,x_2,\dots$ and a constant $C$ such that $x_d+x_{2d}+\cdots+x_{nd} \leq C$ for every $n$ and every $d$?" (For a nice presentation of this take on the problem, as well as an idea of how intractable it seemed only 18 months ago, I recommend James Grime's YouTube clip). "The [Polymath] effort fizzled out in 2012, but participants did manage to show that proving the conjecture for a certain family of sequences was good enough to prove it in general." Tao, who was part of the project, had been working on another problem early this September when "a timely comment on his blog suggested that the problem might be related to the Erdős conjecture." Tao "quickly realized that combining the commenter's fresh insight with previous results could lead to a solution. He submitted his paper less than two weeks later" with an acknowledgement thanking the commenter. A more mathematically detailed account of Tao's achievement, "A Magical Answer to an 80YearOld Puzzle" by Erica Klarreich, appeared in Quanta, October 1, 2015.
Tony Phillips
Stony Brook University
tony at math.sunysb.edu

Math Digest includes posts throughout each month, with summaries of math stories and unique insights (and occasionally videos, interviews and podcasts) on mathrelated topics recently covered by the media.
Recently posted:
On an interview with Talithia Williams on increasing opportunities for minorities in STEM, by Annette Emerson
Maria Klawe, president of Harvey Mudd College, talks with Harvey Mudd mathematics Professor Talithia Williams about an annual math and science conference for African American middle and high school girls to get them interested in and excited about STEM careers. Williams started the conference, now in its fourth year, in partnership with Sacred SISTAHS (Sisters In Solidarity Teaching And Healing our Spirit), a local nonprofit organization.
"We engage the girls in fun, handson math and science lessons. For example, one of our math professors led a session in which the girls related math concepts to handson experiments in fluid dynamics. I led a session on working with data and how to use statistics to better understand ourselves and our world. A local graphic artist taught a session on how she uses computer graphing techniques and software in her work. In both the talks and the handson activities, we showcase African American women who are successful and excited about their careers in STEM. We try to get the girls excited about studying math, science and engineering." Williams says the girls will struggle, but she tells them it's important for them to work hard and seek help. Williams herself is a role model, and says, "I know from my own experience that seeing a role model who looks like you, who is successful in a STEM career, can be a pivotal moment."
In sessions for parents the conference covers "The top 10 questions you should ask your daughter's guidance counselor," and gives parents a list of courses their daughters should take to prepare for college.
See "Increasing Education Opportunities For Minorities In STEM," an interview by Maria Klawe with Talithia Williams, Forbes, 7 October 2015. See also posts by Talithia Williams on the eMentoring Network in the Mathematical Sciences Blog.
 Annette Emerson
Also now on Math Digest: Alan Turing's work, devising encryption methods that will foil quantum computers, avoiding traffic,...
Citations for reviews of books, plays, movies and television shows that are related to mathematics (but are not aimed solely at the professional mathematician). The alphabetical list includes links to the sources of reviews posted online, and covers reviews published in magazines, science journals and newspapers since 1996.
