Zhang and the Twin Primes...

If you've never seen this 2015 documentary (uploaded to YouTube about a month ago) on Yitang Zhang and his Twin Prime Conjecture work it's worth a watch:



It's almost an hour long (starting off with fairly basic information) and very enjoyable, both for the remarkable storyline and for the appearances of lots of key people.

Unlikely Movie Star

The George Csicsery film, "Counting From Infinity," recounting Yitang Zhang's surprising, groundbreaking work on the Twin Prime Conjecture was released earlier this year. If you missed it, the latest trailer for it has been going around recently:



also, an older, similar trailer here:  https://vimeo.com/116199555
and an earlier review of the film (calling it a "Math-erpiece") here:  http://tinyurl.com/q8npanb

Csicsery, by the way, has previously done film work on Paul Erdös, Julia Robinson, and Paul Halmos.

Thursday Potpourri

1) I'm delighted to learn that Noson Yanofsky's "The Outer Limits of Reason" has won a PROSE Award in the "Popular Science and Popular Mathematics" category for 2013.

"The PROSE Awards annually recognize the very best in professional and scholarly publishing by bringing attention to distinguished books, journals, and electronic content in over 40 categories. Judged by peer publishers, librarians, and medical professionals since 1976, the PROSE Awards are extraordinary for their breadth and depth."

I LOVED this volume, calling it a "phenomenal book" in my review last November (indeed it is my FAVORITE book of the last couple decades, and I'm glad to see it get further acknowledgement!):

http://mathtango.blogspot.com/2013/11/phenomenal-book.html

Congratulations to Dr. Yanofsky!

2) And in the 'suddenly-it-came-to-me' category, another nice story on the continuing saga of Yitang Zhang and his work on the twin-prime saga conjecture:

http://tinyurl.com/mbukg36

excerpt:

" 'There's nothing wrong with working at a Subway, but normally these proofs, these breakthroughs, are achieved by those that are working at Princeton, Harvard, these kind of really elite places,' Tony Padilla, a physics professor at the UK's University of Nottingham, says... 'And now we've got somebody who's literally come out of nowhere, that no one expected to produce this kind of results, and has done something impressive that many great minds were unable to do'...
"Zhang himself, a self-described 'shy person,' said in a UNH statement that the proof came to him during a vacation in Colorado, when he was feeling particularly relaxed. 'I didn't bring any notes, any books, any paper,' he said. 'And suddenly it came to me.' "

3) Last week I was writing about the age-old topic of math and beauty over at MathTango and now a new study points to a neuroscience substrate linking the two:

http://www.bbc.co.uk/news/science-environment-26151062 

4) Finally, h/t to Derek Smith of AMS blogs for pointing out a nice listing of interesting math-related documentaries available from this MathOverflow page:

http://mathoverflow.net/questions/100033/interesting-mathematical-documentaries

JMM and MoMath

Brian Hayes of "bit-player" offers a wonderful account of his experience at the recent Joint Math Meetings in Baltimore:

http://bit-player.org/2014/notes-from-the-jmm

… really, a must-read; especially interesting stuff on Yitang Zhang, of twin-prime fame (and apparently now tackling the Goldbach conjecture), including this quote from one of his former Purdue colleagues T.T. Moh:

"When I looked into his [Zhang's] eyes, I found a disturbing soul, a burning bush, an explorer who wanted to reach the north pole, a mountaineer who determined to scale Mt. Everest, and a traveler who would brave thunders and lightnings to reach his destination."

Meanwhile, Peter Woit reports on a "cagematch" between Ed Frenkel (Platonist) and Jim Holt (non-Platonist) held at the Museum of Mathematics in New York recently; he calls it "a no-holds-barred discussion of Platonism and mathematics in front of a standing-room-only crowd"... sounds like it was quite entertaining:

http://www.math.columbia.edu/~woit/wordpress/?p=6609

Don't know if MoMath will eventually upload the session to their YouTube channel, but if you wish to watch for it, or explore some of their other events, the channel is here:

http://www.youtube.com/museumofmathematics

Yin and Yang… No, Zipf and Zhang

A couple of pieces to catch up on if you've missed them:

Yitang Zhang, the largely unknown, humble New Hampshire mathematician who sort of broke open the twin prime conjecture earlier this year, is now deservedly being awarded the Frank Nelson Cole Prize in Number Theory come January (he has already received the Ostrowski Prize). Read another lovely news story about this unforeseen math champion here:

http://www.unionleader.com/article/20131215/NEWS04/131219488

And the below article notes that Zipf's Law is finding more applications to human society than its original linguistic roots with which I was familiar (however, note that several commenters take issue with the article's author):

http://tinyurl.com/lakytzh

Of Primes and Probability

Two bits for a Wednesday...:

1) In a fascinating long-read for Quanta Magazine, Erica Klarreich covers the astounding progress so far made in the "prime number gap" of the Twin Primes Conjecture, in just six months since Yitang Zhang postulated a limit to the gap of 70 million!:

http://tinyurl.com/pw6hyyz

James Maynard, a post-doc, has wrestled the gap down to no more than 600, well below the result that even Terry Tao's Polymath group had yet achieved.

an excerpt to whet your appetite:

"Zhang’s work and, to a lesser degree, Maynard’s fits the archetype of the solitary mathematical genius, working for years in the proverbial garret until he is ready to dazzle the world with a great discovery. The Polymath project couldn’t be more different — fast and furious, massively collaborative, fueled by the instant gratification of setting a new world record.
"For Zhang, working alone and nearly obsessively on a single hard problem brought a huge payoff. Would he recommend that approach to other mathematicians? 'It’s hard to say,' he said. 'I choose my own way, but it’s only my way.'
"Tao actively discourages young mathematicians from heading down such a path, which he has called  'a particularly dangerous occupational hazard' that has seldom worked well, except for established mathematicians with a secure career and a proven track record. However, he said in an interview, the solitary and collaborative approaches each have something to offer mathematics.
“ 'It’s important to have people who are willing to work in isolation and buck the conventional wisdom,' Tao said. Polymath, by contrast, is 'entirely groupthink.' Not every math problem would lend itself to such collaboration, but this one did."

2) I mentioned a couple of posts back that in the Preface to his new book ("Will You Be Alive 10 Years From Now?") Paul Nahin gives an example of a Marilyn vos Savant column where the famous Mensa-ite gives the WRONG answer to a math question and sometime later corrects herself. The question, and her initial ill-fated answer ran as follows:

Q.: "I manage a drug-testing program for an organization with 400 employees. Every three months, a random-number generator selects 100 names for testing. Afterward, these names go back into the selection pool. Obviously, the probability of an employee being chosen in one quarter is 25 percent. But what’s the likelihood of being chosen over the course of a year?"

A.: "The probability remains 25 percent, despite the repeated testing. One might think that as the number of tests grows, the likelihood of being chosen increases, but as long as the size of the pool remains the same, so does the probability. Goes against your intuition, doesn’t it?"

Nahin points out that the actual probability of being chosen at some point during the four quarters of testing works out to 0.6836, considerably greater than Marilyn's 0.25.

In a later column Marilyn 'fessed up, "My neurons must have been napping" and corrected herself:

http://www.parade.com/45916/marilynvossavant/22-sunday-column-2/

You can also see the math involved at this Forum site where the problem was discussed:

http://www.aiqus.com/questions/34413/ask-marilyn-new-probability-question-controversy

Weekend Reads

Just a couple of good reads for the weekend….

For any who missed it, this somewhat low-key, yet fascinating (and non-technical) interview with Yitang Zhang ('rags-to-riches/fame' worker on the twin-prime conjecture) last week in Nautilus:

http://nautil.us/issue/5/fame/the-twin-prime-hero

Secondly, a Scientific American excerpt from Edward Frenkel's new book, "Love and Math: The Heart of Hidden Reality" (now showing up in bookstores):

http://tinyurl.com/mrpsxuk

70,000,000 to 7000? Impressive!

Articles just keep coming along about the twin-primes work started by Yitang Zhang… this time a good piece from the always layman-accessible Amir Aczel (who it turns out actually worked with Zhang in a university math department a few years back):

http://tinyurl.com/ml6ula2

Mathematicians on the Web taking a collaborative approach to improve Zhang's initial 70,000,000 figure for an upper boundary twin-primes gap, have, in just a couple of months, reduced the figure to barely over 12,000; possibly even 7000 (awaiting some technical verifications)… and at this rate, only a short matter of time likely before even those figures may be improved upon... an amazing and fascinating accomplishment over such a brief time period, on a conjecture that had evaded great progress for decades.
Terry Tao's "progress report" on all of this (for the technically-inclined) was posted less than two weeks ago:

http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/

Hive Mind and Collaboration… Driving the Future

(via Waugsberg/Wikimedia)

Yitang Zhang's recent proof regarding twin primes keeps generating press pieces… including some good recent ones for lay readers, first from the Wall Street Journal:

http://www.livemint.com/Opinion/fodS89bFVdFCaO17Sh9fLN/From-70-million-to-two.html

and then from popular science-writer Amir Aczel over at HuffPost Science:

http://www.huffingtonpost.com/amir-aczel/the-mystery-of-the-primes_b_3391666.html

And finally, this update from New Scientist:

http://www.newscientist.com/article/dn23644-game-of-proofs-boosts-prime-pair-result-by-millions.html

(I think all 3 of these are worth a gander as they touch on different aspects of the twin prime/Zhang story.)

Things are moving fast… Zhang's work was quickly taken up by a bevy of individual mathematicians, and now also by the Polymath Project group (a Web collaboration of many mathematicians). Zhang's computed upper limit of ~70 million has, in just a few brief weeks, been reduced to under 400,000! (the latest figure I've seen, but it will keep changing). Phenomenal!

The Polymath Project (largely inspired by math icons Tim Gowers and Terry Tao) is, to my mind, an example of the premier use of the internet, to utilize the 'hive mind,' as never before in history, to solve or work on significant problems (in this case math, but it can be applied to other subject areas). In the past, dozens or perhaps 100s of individuals could be brought together to work somewhat collaboratively on a problem, but today 100s of 1000s can be easily drawn from. Books have already been written about the power of the "hive mind" (some quite critical). Solutions to many problems will come at a warp-speed previously unattainable. Can the 'hive mind' be cluttered with junk science… pseudoscience… quackery… tomfoolery… Yes, of course, GOBS of it;  But the beauty of the hive is how rapidly and efficiently it can sort through masses of information and ignorance to arrive at the productive core. "Open access," "crowdsourcing," collaboration, and the hive, are the wave of the future.

My current post over at MathTango is a doff-of-the-cap to those leading the way in the current evolution (revolution???) of math education, with a focus, by the end, again on how rapid/broad Web collaboration is currently leading the way to shape the 'flipped classrooms' and MOOCs of the near future… and, obviously, so much more.

The Exhilaration of Math... (Zhang/primes)

"Math is a beautiful thang!" (I can't say that often enough)… or perhaps I should say, math is a beautiful Zhang!!

The words I've seen describing the newly-announced work of mathematician Yitang Zhang include not just "beautiful," but also "breakthrough," "incredible," "astounding," "stunning."

If you somehow missed the math buzz of the last 48 hours, Zhang claims to have proven that there are an infinite number of paired primes that are some given finite distance apart… and that distance apart is less than 70,000,000… that's right, I said 70 million digits apart!!  And, mathematicians are THRILLED! Thrilled because this is the first time someone has shown there to be an upper limit or boundary to what the gap between adjacent primes can be. This may be a first, sort of baby step, toward trying to prove the long-held (but unproven) 'twin-prime' conjecture (that there are an infinite number of paired primes with a gap of 2).
Yeah, sure, working down from 70,000,000 to 2 might require some work yet, but ya gotta start somewhere (and you have to understand that prior to Zhang the only boundary was infinity, so stepping us down from infinity to 70,000,000 is no minor accomplishment -- though it can be difficult for lay folks to comprehend how teeny a number like 70,000,000 is within mathematics).

The Zhang story began on the Web (so far as I'm aware) with Peter Woit's rumor report Monday on his "Not Even Wrong" blog, that a special seminar by Zhang, was about to take place at Harvard, related to “bounded gaps between primes.”
Following the seminar, and its positive reception, the story got picked up quickly, first by Nature, here:

http://www.nature.com/news/first-proof-that-infinitely-many-prime-numbers-come-in-pairs-1.12989

…and then by others, including these two pieces, also very accessible to the lay public:

http://io9.com/mathematician-makes-astounding-prime-numbers-breakthrou-505577996

http://www.newscientist.com/article/dn23535-proof-that-an-infinite-number-of-primes-are-paired.html

If you haven't followed the story, read some of the above links to catch up.
I won't try to re-cap what those articles tell you, but will simply comment on how fun it is to see mathematicians get excited over a piece of news (and also watch the press pick up on it). One suspects most of the public will merely scratch their heads in bemusement that mathematicians can be so enamored of a number like 70,000,000 when they are really aiming for 2! I wish, alas, we could translate the excitement and wonderment into some form that the general public might better fathom! I can only hope that those who don't grasp what all the fuss is about, might at least be fascinated that human minds exist which are subject to exhilaration by such findings!

(Having said all this, I should add that while the experts seem optimistic, Zhang's proof still must pass vigorous peer review.)

ADDENDUM (5/19/13): a very good follow-up to the story from the Simons Foundation here:

http://tinyurl.com/am5fvjp