Mirzakhani Makes Nature's Top Science Stories of 2014, by Annette Emerson
Various journals are listing the top science stories of 2014 (see above) and we are watching for the mathematical sciences stories. The editors of Nature have featured Maryam Mirzakhani as one of the ten newsmakers of the year, as the first woman to receive the Fields Medal. She is on "The Year in Science: Wins and Losses" timeline (page 303) and in a piece by Erica Klarreich, who notes that the attention Mirzakhani received "threw a spotlight on the vast under-representation of women in mathematics." Klarreich cites, "according to a 2012 survey of U.S. universities by the American Mathematical Society, women make up only 30% of PhD students—a number that has not budged for years—and only 12% of tenured faculty members at PhD-granting institutions. Those who do become tenured mathematics professors receive disproportionately small number of scholarly awards." Klarreich also summarizes Mirzakhani's decade-long work on surfaces, "tying together disparate mathematical fields such as geometry, topology and dynamical systems" and earning her a 2014 Fields Medal .
See "Surface Explorer," by Erica Klarreich (part of "One Year, Ten Stories. Newsmakers of the Year"). Nature, 18/25 December 2014, page 316.
--- Annette Emerson
On fun (calendar) dates, by Claudia Clark
In this article, Victoria Jaggard speaks with Neil J. A. Sloane, founder of the Online Encyclopedia of Integer Sequences, about some of the dates containing interesting sequences that “numberphiles” can look forward to, now that 12/13/14 has passed. The first date they discuss involves a sequence of prime numbers: 11/13/17. The second date contains a sequence of Mersenne primes, prime numbers that equal a power of 2 minus 1: 3/7/31. If you look at the Fibonacci numbers, you get a third date 8/13/21. Two more dates—7/13/20 and 8/25/43—come from Recamán’s sequence, which is defined as follows: a_{1} = 1. For n > 1, a_{n} = a_{n-1} - n if the result is a positive number that is not already in the sequence. Otherwise, a_{n} = a_{n}_{-1} + n. (Can you guess where 8, 25, and 43 fall in the sequence?) Another date—1/11/21—is composed of the first three terms of the look-and-say sequence: 1, 11, 21, 1211, 111221… (Each term is a description of what is seen in the previous term: 1 is “one one,” 11 is “two ones,” 21 is “one two and one one,” and so on.) So, mark your calendars!
See "After 12/13/14, What Are the Next Fun Dates for Math Lovers?" by Victoria Jaggard, Smithsonian Magazine, 11 December 2014. (The article's web page also has a musical interpretation of Recamán’s sequence.)
--- Claudia Clark
Nailing Down The Primes, by Ben Pittman-Polletta
No, you aren’t hard of reading, and the title of the article ("Prime Gap Grows ...") 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 together--at latest count, no more than 246 apart.
But progress on the opposite question--in 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 mathematicians--one composed of Terence Tao, Ben Green, Kevin Ford, and Sergei Konyagin, and the other composed of, well, James Maynard--won the largest Erdős prize ever by improving upon a 76-year-old 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 larger--as 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 Decades-Long Lull," by Erica Klarreich, Quanta Magazine, 10 December 2014.
--- Ben Pittman-Polletta (Posted 12/18/14)
On surveys of mathematically gifted teens, 30 years later, by Lisa DeKeukelaere
Starting with a list of 1650 individuals identified as “mathematically gifted” as 13-year-olds in the 1970s, researchers from Vanderbilt University crunched survey data to track and interpret how the prodigies had fared, 40 years later, in education, career, income, and happiness. The paper notes that the study is the first to follow mathematically talented individuals during a time in which women have high-level career options. The study showed that the mathematically inclined cohorts far exceeded the average U.S. participation rate in higher education, and the rate for different types of terminal degrees was roughly similar between men and women. The data indicates, however, that the female participants earned on average $60,000 less than their male counterparts and tended to be married to men who earned significantly more than they did, whereas the men tended to be married to women who earned significantly less. The study also included several measures to track the value placed on career advancement, family, and having an impact on society, and noted a significant dichotomy between the sexes.
See “Unequal fates for maths superstars.” Research Highlights, Nature, 4 December 2014, page 11, which is based on the article, “Life Paths and Accomplishments of Mathematically Precocious Males and Females Four Decades Later,” by David Lubinski, Camilla P. Benbow, Harrison J. Kell. Psychological Science, 10 November 2014.
--- Lisa DeKeukelaere
"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 word-for-word 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 (Posted 12/8/14)
On modeling the immune system's response to the flu, by Claudia Clark
Many virus-causing diseases like measles and chicken pox change very little over time, therefore conferring lifelong immunity upon the infected or vaccinated person. Flu (or influenza) viruses, however, change enough over time that immunity to last year’s flu virus does not guarantee immunity to this year’s flu virus. In this article, Adam Kucharski, a research fellow at the London School of Hygiene and Tropical Medicine, describes his work building a mathematical model of influenza based on observed data from the 2009 flu pandemic. (Studies of that pandemic “have shown that immunity against regular seasonal flu viruses tends to peak in young children, drop in middle-aged people and then rise again in the elderly.”) Along the way, Kucharski’s “work provides new support for a quirky hypothesis--first proposed more than half a century ago and known as original antigenic sin--about why the body’s response to this illness is biased toward viruses seen in children.”
See "Immunity's Illusion," by Adam J. Kurcharski. Scientific American, December 2014, pages 80-85.
--- Claudia Clark
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