On the sphere packing problem in dimensions 8 and 24, by Rachel Crowell
Suppose you have a set of oranges. According to this article, in 1611 Johannes Kepler conjected that in three-dimensional space, a pyramid formation is the densest way to pack these and other spherical objects. In 1998, Thomas Hales, who is now Andrew Mellon Professor of Mathematics at the University of Pittsburgh, proved Kepler's conjecture, which solves the "sphere packing" problem in the third dimension. Mathematicians have also studied the "sphere packing" problem in higher dimensions. A high-dimensional sphere is the set of points in the space that are a fixed distance away from a given point.
In higher dimensions, the sphere-packing problem is more complicated to solve than in the third dimension, with two exceptions: dimensions 8 and 24. In dimension eight, the densest sphere packing is called E_{8}. The densest sphere packing in dimension 24 is called the Leech lattice. E_{8} and the Leech lattice are symmetrical sphere packings. Until last month, mathematicians were confident that E_{8} and the Leech lattice were the densest packings for spheres in their respective dimensions, but lacked proofs to support these conjectures. In March, Maryna Viazovska, a postdoctoral researcher at the Berlin Mathematical School and Humboldt University of Berlin, published a proof that E_{8} is the densest packing for spherical objects in dimension 8. She developed this proof by using the theory of modular forms to find an "auxiliary" function for dimension 8. Auxiliary functions enable mathematicians to calculate the largest sphere density allowed in a given dimension.
Viazovska, along with Henry Cohn (Microsoft Research New England) and three other mathematicians, extended her method for finding an auxiliary function in dimension eight to the quest for an auxiliary function in dimension 24. After finding the appropriate auxiliary function, they were able to show that the Leech lattice is the densest packing for spherical objects in the 24th dimension. Prior to Viazovska's publication, "The sphere packing problem in dimension 8," mathematicians, including Thomas Hales, believed that if the right auxiliary functions were found, they could show that E_{8} and the Leech lattice were the densest spherical packings for the 8th and 24th dimensions, respectively. However, they did not know where to look for these functions. Hale told Quanta Magazine that Viazovska, "pulled a Ramanujan," when she published her work detailing the auxiliary functions for E_{8} and the Leech lattice. Hale's comment refers to Srinivasa Ramanujan, a mathematician in the early 20th century who was known for his innovative mathematical thought. Ramanujan also studied modular forms.
See "Sphere Packing Solved in Higher Dimensions," by Erica Klarreich. Quanta Magazine, 30 March 2016. (Image: A visual representation of the E_{8} root system, courtesy of Henry Cohn.)
--- Rachel Crowell (Posted 4/11/16)
On designing a marijuana-license lottery, by Samantha Faria
When the state of Washington decided to issue licenses to marijuana retailers they wanted to make sure it would be fair and random. Not entirely sure how to go about doing this the state turned to Washington State University Ph.D. math student, Sharif Ibrahim. In his presentation, "Joint Mathematics: Lessons from a Marijuana License Lottery," at the 2016 Joint Mathematics Meeting in Seattle, he explained his methodology. Using the Fisher-Yates shuffle in software he developed, Ibrahim was able to affectively 'scramble the applicants in random order' similar to shuffling a deck of cards. First, though, random numbers needed to be assigned to each card, or applicant. Ibrahim employed two different and independent sources for securing random numbers. Once those numbers were fed in to his card-shuffling algorithm, the lottery was ready to go. "It really turns out to be somewhat anticlimactic... you press a button on a computer and it says, Okay, here are the results. It doesn't have a drumroll or bouncing ball," explains Ibrahim. Unfortunately, a small glitch popped up when the state had mistakenly disqualified some candidates from participating in the lottery. Ibrahim was able to devise a "supplemental lottery," to include these new candidates while retaining the original positions of the initial applicants.
See "How to Design a Marijuana-License Lottery," by Roberta Kwok, The New Yorker, 22 March 2016.
--- Samantha Faria (Posted 4/1/16)
On preferences of prime numbers, by Rachel Crowell
All primes except 2 and 5 end in 1, 3, 7, or 9. Mathematicians Kannan Soundararajan and Robert Lemke Oliver of Stanford University recently published research that says primes prefer to be immediately followed by primes ending in different digits than their own. For example, if one prime ends in a 3, it is more likely that the prime immediately following it will have a final digit of 1, 7, or 9 than 3. The finding doesn't end there. Soundararajan and Lemke Oliver found that, for example, primes ending in 3 prefer to be followed by primes ending in 9 than those ending in 1 or 7.
Number theory tells us to assume that primes behave similarly to random numbers, with the addition of one rule: There is an inverse relationship between the approximate density of primes near a number and amount of digits the number has. Does Soundararajan and Lemke Oliver's result violate this assumption? Not according to Soundararajan, who asked Quanta Magazine, "Can we redefine what 'random' means in this context so that once again, [this phenomenon] looks like it might be random?" He added, "That's what we think we’ve done."
The preference of primes that Soundararajan and Lemke Oliver published can be explained using the prime k-tuples conjecture that G.H. Hardy and J.E. Littlewood originally stated in 1923. The conjecture estimates the frequency at which different combinations of primes with certain spacing patterns will appear. The biases Soundararajan and Lemke Oliver's study of consecutive primes found are similar to the distribution of different constellations of primes predicted by the prime k-tuples conjecture.
See "Mathematicians Discover Prime Conspiracy," by Erica Klarreich, Quanta Magazine, 13 March 2016 and "Mathematicians shocked to find patterns in 'random' prime numbers," by Jacob Aron. New Scientist, 19 March 2016 (Print version: "'Random' primes pair up on the sly," page 12).
--- Rachel Crowell (Posted 3/30/16)
On coverage of Pi Day 2016, by Samantha Faria
Pi, is it really that meaningful? Depends on who you ask. Mathematician Carlos Castillo-Chavez, who studies epidemics at Arizona State University, explained that pi is used to study everything that cycles, like his own research in to the cyclical reoccurrence of the flu. Vi Hart, well known for her fun and engaging math videos claims that pi just isn’t special. She would rather see tau (2 pi) celebrated as it would make math less confusing to students and simplify equations. In Rhode Island, the AMS held its annual Pi Day Who Wants to Be a Mathematician competition. The state’s top math students competed for cash and prizes by answering difficult math problems. Classmates and fans of the big winner, Jonny Zhang, congratulated him with a piece of pie (left). The mathematical giant, John Conway, recently stated, "Pi may be irrational, but free pizza is anything but." He teamed up with Pizza Hut by writing three math problems of varying difficulty to offer a unique challenge to "consumers and mathematic wizards." The prize for the first person to correctly solve and submit the correct answer was 3.14 years of free pizza. Find more stories about Pi Day 2016 by searching for hashtag #PiDay on Twitter on 3.14 and see more Pi Day information.
See "On Pi Day, Let's Gawk At The Beauty And Controversy Of The Math Constant," by Eyder Peralta. National Public Radio, 14 March 2016; "Freshman Wins Annual RI Pi-Day Mathematics Competition," by John Bender. RI Public Radio, 14 March 2016 "You Could Win 3.14 Years of Free Pizza Hut in Honor of National Pi Day," by John Kell. Fortune, 11 March 2016; "Happy Pi Day 2016: Incredible facts about the enigmatic number that has fascinated mathematicians for generations," by Sam Webb. Mirror, 14 March 2016.
--- Samantha Faria (Posted 4/1/16)
Stephen Hawking's tribute to his math teacher, by Mike Breen
Above is a video tribute from Stephen Hawking to Dikran Tahta, one of his math teachers at St. Albans School. Hawking says that Tahta's classes "were lively and exciting. Everything could be debated." Near the end he says: "Today we need teachers more than ever." The video was posted to help promote the Global Teacher Prize.
For more on Hawking's memories of Tahta, see "Stephen Hawking remembers best teacher," by Sean Coughlan. BBC News, 8 March 2016.
--- Mike Breen (Posted 3/9/16)
On math and music, by Claudia Clark
In this article, writer Lee Phillips writes about the "Mathematics and Music" session at the 2016 annual American Association for the Advancement of Science conference in Washington, D.C. Three researchers presented their work, beginning with Harvard University mathematician Noam Elkies speaking about "The Entropy of Music: How Many Possible Pieces of Music Are There?" Phillips writes that Elkies' "basic idea was to apply concepts similar to those used in statistical mechanics and information theory to approach the question posed in his title. Elkies addressed how much a piece of music needs to change before it is a different piece, rather than a variation on the original." Elkies ended his presentation with an "impressive performance, from memory, of a piece made from a baroque-style repeating arpeggiation where the root of the chord changed from measure to measure based…on the digits of π." See and more about Elkies' piece.
The second presentation, entitled "Geometry of Music," was given by Princeton University professor of music Dmitri Tymoczko. "[Tymoczko's research] focuses on the harmonic structure and voice leading (the passage of voices from chord to chord, which largely results in counterpoint) of the Western common-practice period," Phillips explains. "He argues that this is most naturally represented through the language of differential geometry."
"Overtones and Tuning" was the topic of the third presenter, David Wright, chair of the mathematics department at Washington University. Wright discussed and gave examples of tuning that preserves "just intonation" compared with tuning that uses "equal temperament." According to Phillips, "the highlight here was a recording of a barbershop quartet in which Wright discovered a subtle use of microtonics during a chord change."
See "Mathematics meets music," by Lee Phillips. Ars Technica, 6 March 2016.
--- Claudia Clark
On infinity in the real world, by Allyn Jackson
This short video appears on the New Scientist web site and on its YouTube channel, Explanimator. The video discusses mind-bending aspects of infinity in a nontechnical and appealing way. The visuals convey some of the ideas very well, like the infinite bag of potato chips (labeled Cantor's Chips) in which every 10th chip is a bit burnt. Matching up the burnt chips with the set of all chips, the video demonstrates the seemingly paradoxical fact that the set of all multiples of 10 has the same size as the set of all counting numbers. A bit less effective is the Bigfoot figure who rides around on a unicorn, to suggest that infinity is an imaginary concept. The video also discusses Stephen Hawking's result that the amount of information contained in any part of space is finite and asks what this result might imply about whether the universe is finite or infinite.
See "Explanimator: Does infinity exist in the real world?," by MacGregor Campbell. New Scientist, 3 March 2016. [Any explanations about why the 10,000,000,000th chip gets mapped to the 1,111,111,111,111th chip are welcome. Write us at paoffice at ams dot org.]
--- Allyn Jackson (Posted 3/7/16)
On Michael Atiyah, by Claudia Clark.
Among physicists and mathematicians, Michael Atiyah hardly needs any introduction. He has been the recipient of many honors, has held several prestigious positions, and has won both the Fields Medal (in 1966) and the Abel Prize (in 2004, along with colleague Isador Singer). And while he may be best known for the index theorem and K-theory, the author of this article proposes that "[Atiyah] is nonetheless perhaps most aptly described as a matchmaker. He has an intuition for arranging just the right intellectual liaisons, oftentimes involving himself and his own ideas, and over the course of his half-century-plus career he has bridged the gap between apparently disparate ideas within the field of mathematics, and between mathematics and physics." Indeed, at 86, the author notes, Atiyah is "still tackling the big questions, still trying to orchestrate a union between the quantum and the gravitational forces."
Following an introduction, the remainder of the article is an interview with Atiyah in which he talks about his lifelong interest in beauty and science, how he gets his ideas, K-theory, his collaboration with string theorist Edward Witten (who won the Fields Medal in 1990), and his current work.
See "Michael Atiyah’s Imaginative State of Mind," by Siobhan Roberts. Quanta Magazine, 3 March 2016.
--- Claudia Clark
On American students doing more complex mathematics, by Claudia Clark
The "math revolution" to which Tyre refers is the growth over the past several years in the number of American students, starting in the elementary grades, who are doing more complex mathematics than what they are learning in a typical classroom. The author explores why this is happening, starting with the growth of outside-of-school math programs that now exist. These include summer math camps, math circles, special schools, math competitions, and online math forums, where students can learn and practice problem solving. For example, Inessa Rifkin, at one point a mechanical engineer in the Soviet Union, co-founded the Russian School of Mathematics after observing that her children "were being taught to solve problems by memorizing rules and then following them like steps in a recipe, without understanding the bigger picture." At Rifkin's school, Tyre continues, "at every level, and with increasing intensity as they get older, students are required to think their way through logic problems that can be resolved only with creative use of the math they've learned." Another program that Tyre discusses in detail is Bridge to Enter Advanced Mathematics (beam), a nonprofit organization based in New York City. Founder Daniel Zaharopol started this program to serve a population that has seen very little benefit from this revolution. "We know that math ability is universal and interest in math is spread pretty much equally through the population," Zaharopol says, "and we see there are almost no low-income, high-performing math students. So we know that there are many, many students who have the potential for high achievement in math but who have not had opportunity to develop their math minds, simply because they were born to the wrong parents or in the wrong zip code. We want to find them."
See "The Math Revolution," by Peg Tyre, The Atlantic, March 2016.
Math Digest Archives
|| 2010 || 2009 ||
2008 || 2007 || 2006 || 2005 || 2004 || 2003 || 2002 || 2001 || 2000 || 1999 || 1998 || 1997 || 1996 || 1995
Click here for a list of links to web pages of publications covered in the Digest. |