A Name to Know; Work to Be Aware Of

From Erica Klarreich at Quanta, a fascinating piece about a fascinating young mathematician, his fascinating work in number theory, making fascinating, groundbreaking connections between disparate areas of math:

https://www.quantamagazine.org/20160628-peter-scholze-arithmetic-geometry-profile/

Did I mention this is fascinating stuff....

Peter Scholze is a 28-year-old German wunderkind, probable Fields Medal candidate, and by several accounts, "one of the most influential mathematicians in the world," who works at the intersection of number theory and geometry. That might sound simple, but it is cutting edge, and for most, unexplored territory. Yet Peter seems to possess a strong intuitive sense for it (the article is aptly titled, "The Oracle of Arithmetic"). 

A couple of quick sentences from the piece:
"'I’m interested in arithmetic, in the end,' he [Scholze] said. He’s happiest, he said, when his abstract constructions lead him back around to small discoveries about ordinary whole numbers."

Almost makes it sound as if we rookies could understand what he does ;-); but that'll be the day. He's delving into deep, rich, abstract areas of mathematics, that most of us will never encounter, but the article makes clear he is also open, generous, and patient in his willingness to explain it to those who are able to take the leap.

Klarreich writes that Scholze "avoids getting tangled in the jungle vines by forcing himself to fly above them," which reminded me so much of Keith Devlin's early metaphor of reaching the top of a mathematical woodland canopy where he could look down and suddenly see that the whole forest was inter-connected.
Part of Scholze' work deals with what is called "reciprocity" and its linkage to hyperbolic geometry, including "perfectoid spaces," all of which leads to the Langlands Program and "frontiers of knowledge" which may eventually unify the field of mathematics (slightly akin to the so-called "Theory of Everything" searched for in physics).

But I can't do Scholze or Klarreich's writing justice here, so go read her article NOW!

Don Knuth Explains the Surreals

Wonderful new video from Numberphile, of Donald Knuth describing where "surreal numbers" came from:



If you missed it, less than a year ago Jim Propp ran this great post on the surreals at his MathEnchantments blog:

https://mathenchant.wordpress.com/2015/08/12/the-life-of-games/

40 Years Ago, But So Timely...

Another no-math Sunday reflection; just a hackneyed piece of movie nostalgia (1976) with "Howard Beale," that now seems all too prescient:



(movie: "Network" screenplay: Paddy Chayefsky)

Political Navel-gazing

So Stephen Hawking, Terence Tao, Scott Aaronson, and Leonard Susskind have all now weighed in very publicly to essentially denounce Donald Trump (and many others have voiced shorter, but similar, sentiments through Twitter, Facebook, Google+, and the like). Four folks I'd never feel too bad being in agreement with ;-)

I'm still not convinced Trump will get his party's nomination (is the Republican Party establishment reeeeally that craaaazy? -- granted, in the near-term they may wreck their party either way: nominating Trump or denying him the nomination; their best hope is for him to become disabled, forcing another choice)... but if he does get nominated, the lengthy parade of brilliant minds springing forth to castigate him may be unlike anything ever seen in American politics... not merely because he is obviously narcissistic, misogynistic, bigoted, incompetent, hedonistic, naive, dishonest, inconsistent, simplistic, egocentric, and ignorant... but, because many surmise he is mentally-ill in some psycho-or-socio-pathic sense, and dangerous (...but hey, nobody's perfect!). 

STEM people often leave politics outside their public personae, viewing it as a private matter held separate from their profession. This year could be very different. Albert Einstein  famously spoke out on various political and philosophical fronts. Perhaps the time is long overdue for today's prominent scientists to also be more actively/loudly involved. Maybe they'd just be preaching to the choir... but just maybe enough Americans will listen when their most brilliant, creative, productive, insightful minds speak up in concert, out of genuine concern for their nation's future.

Muppet Beaker (via Wikipedia) refusing to release his tax returns

ADDENDUM:  I wrote the above yesterday for posting this morning... and now wake up to find 'Brexit' approved in Britain. Incredible! World stock markets are expected (at least temporarily) to plummet. David Cameron is resigning. Uncertainties about other EU members now become quickly more real. The anger of the 95% against the 5% is worldwide, and reminiscent of 1930s German mentality, looking for scapegoats as targets for that anger. 
Some even speculate this all helps Trump's campaign... shake things up, just for the sake of shaking things up, because hey, trying to do things rationally hasn't exactly been a booming success... so goes the visceral logic of the masses.  ANY change is better than the status quo.
When the 2008 crash happened I told friends I believed we were in for essentially a 20+ year recession; it seemed clear it would require a generation to correct the long baked-in maladies of our banking and corporate system. But perhaps I underestimated the degree and scope of the problems. In fact, maybe what we call "recession" is simply the new "normal."
We live in bizarre times. Rocky days ahead. But at least have a good weekend! ;-)

  

Putting Lipstick on a Pig

"Any sufficiently crappy research is indistinguishable from fraud"... that's the gist of a recent post from Andrew Gelman taking off on Arthur C. Clarke's 3rd Law, in the realm once again, of research papers displaying poor statistical analysis (be it incompetency or deliberate deception): 

http://andrewgelman.com/2016/06/20/clarkes-law-of-research/

The post gets quite a bit of commentary in follow-up (mostly backing Gelman up): 

And in a funny bit of timing, I came to Gelman's post very shortly after seeing a political cartoon on the Web showing Paul Ryan putting lipstick on a pig drawn as Donald Trump. Just struck me as an odd juxtaposition... how often politicians put lipstick on pigs, and, so too, researchers.


ADDENDUM:  just this morning "Retraction Watch" tweets out this abstract from a John Ioannidis group indicating that the majority of randomly-controlled studies evaluating "efficacy and safety" are sponsored by industry, and, lo-and-behold, 95+% of published results favor the sponsor:

http://www.jclinepi.com/article/S0895-4356(15)00058-X/abstract

Hard and Easy Problems...

For today's Sunday reflection, Steven Pinker (from "The Language Instinct"):

“The main lesson of thirty-five years of AI research is that the hard problems are easy and the easy problems are hard. The mental abilities of a four-year-old that we take for granted – recognizing a face, lifting a pencil, walking across a room, answering a question – in fact solve some of the hardest engineering problems ever conceived…. As the new generation of intelligent devices appears, it will be the stock analysts and petrochemical engineers and parole board members who are in danger of being replaced by machines. The gardeners, receptionists, and cooks are secure in their jobs for decades to come.”   

James Propp Explores Ramanujan

"What can you say about a thirty-two-year-old mathematician who died?  That he loved numbers and equations.  That he had a mysteriously intimate understanding of infinite numerical processes..." 
so begins a brand new post from James Propp on Indian savant mathematician Srinivasa Ramanujan.

Ramanujan, of course, has been much in the news of late, due to a major motion picture on his life, and also a Ken Ono bio of him; now comes along the best single post I've ever seen on him from Propp, that provider of once-a-month thought-provoking, "enchanting" posts:

https://mathenchant.wordpress.com/2016/06/16/sri-ramanujan-and-the-secrets-of-lakshmi/

This is just a great, succinct compendium of Ramanujan's life and work (if I were you, I'd print it out and keep on hand, just for inspiration!). Not only is there a bit of the wonderful life story in brief, but lots of the math wonderment for which Ramanujan was famous. Jim too goes a little into the mysterious connection between Ramanujan and his family Hindu "Goddess" Namagiri Thayar (who supposedly provided him his math insights), as I did in an earlier post here.

This is a not-to-be-missed post! (with lots of good links and endnotes as well). And next month Jim will be doing a followup specifically on the current film biopic of Ramanujan's life ("The Man Who Knew Infinity"). His take should be very interesting.
Anyway, a great post to take you into the weekend.

Inspired By Escher

For a different Sunday reflection today re-running this lovely 3-yr.-old Cristobal Vila film-short in tribute to M.C. Escher (always worth seeing again):

A Puzzle to End the Week

                     9 7 3 1 4 8 2 5 6

Britain's Matt Parker has fi-i-i-i-i-inally hit the big time! He's in "Futility Closet" ;-):

http://www.futilitycloset.com/2016/06/09/divide-and-conquer-3/


The puzzle is to put the digits 1-9 in an order whereby the first two digits represent a number evenly divisible by 2, the first three digits evenly divisible by 3, first four divisible by 4... the full number is divisible by 9. The simple order 1234356789 won’t work: 12 and 123 are ok, but 1234 is not evenly divisible by 4.
Visit the site to find the answer.

Brian Hayes in Pursuit of Prime Numbers

Ulam Spiral via WikimediaCommons

There are many questions, notably in theology, but in science as well, that may NEVER give way to human reason; their formulation is so far beyond the limitations of measly, squishy brains and hard-wired computers. One of the beauties of math, however, is that such a high percentage of its questions ARE amenable to comprehension with mere human logic and persistence.
Just maybe understanding prime numbers is one such subject (...though maybe it is not!).

Brian Hayes' latest post on the non-randomness of primes is a beautiful read (pretty typical for Brian actually). I can't pretend to comprehend 70% of it :-(  but that doesn't prevent me from appreciating and sensing the work he has put in to it and the direction it takes.
With its visual power (reminiscent of the Ulam Spiral above), Brian's post yields, even without a full understanding, that ineffable sense that SOMETHING significant is going on here... something tantalizingly, almost tauntingly just within, or, just beyond human grasp? And with a little more time, or effort, or computer power, perhaps we can tap into it. The primes toy with us, tease us, and Brian falls under their siren spell.

Hayes writes that he's been working for a couple of months to get to the point of what he presents in the post, a continuation of previously-discussed recent findings about non-randomness in the order of primes. There is of course the far-more famous case of Andrew Wiles wiling away secretly for 6+ years to prove Fermat's Last Theorem -- I admire the dogged, focused persistence and willingness of humans to secrete themselves away with their own brains as lone company to wrestle with such abstract knowledge, not even knowing if anything useful may result from it... the passion for knowledge/pure-math for its own sake.
What does it mean that primes, the building blocks of our number system, seem to have order/pattern, even if we can barely discern it; and yet any such order/pattern seems to change/evolve as one goes farther and farther out in the run of primes toward infinity? Maybe by now Erdös has devoured 'God's Book' and knows all these answers, but we're still scratching our heads in confused wonder.

When the original Lemke Oliver/Soundararajan work was reported to much fanfare, I wrote that it looked like the sort of thing that would swing open the door (floodgates?) to much further study. Brian's work is likely just one of the many paths one might go down. It is the sort of thing even amateurs, with some computer skills and interest, can play with almost endlessly... and, just maybe, strike gold.

Do prime numbers exist only in our heads, amenable to full self-discovery, or do they lie in some more mystical Platonic realm forever just beyond our reach? I wish I knew. In the end, both Brian's hope and frustration is palpable:

"The complexity of the mathematical treatment leaves me feeling frustrated, but it's hardly unusual for an easily stated problem to require a deep and difficult solution. I hang onto the hope that some of the technicalities will be brushed aside and the main ideas will emerge more clearly with further work. In the meantime, it's still possible to explore a fascinating and long-hidden corner of number theory with the simplest of computational tools and a bit of graphics."

Anyway, read what all Brian has done. It's 23 pages (if printed out) of deliciousness, even though the last 1/3 of it may be especially tough going for general readers:

http://bit-player.org/2016/prime-after-prime

Float Like a Butterfly....

Foregoing the usual Sunday math reflection this morning in favor of some music:



"I am leaving, I am leaving," but the fighter still remains...."

Mathematician Done Good

from Wikipedia

I love this piece about billionaire hedge fund manager/philanthropist/mathematician Jim Simons (h/t to Egan Chernoff for pointing to it)... so many things to like about the article. Read it yourself to see if you don't agree:

http://tinyurl.com/zdzcy4y

One little bit that I enjoyed:

"One of the reasons that Renaissance Technologies has been so successful, is because Simons has taken an unusual approach. When he hires an employee, Wall Street experience is generally looked down upon. Instead, he hires mathematicians, astronomers, physicists, cryptographers, computational linguists, programmers, and scientists."

Jim is a founder of the Simons Foundation, whose work so many of us relish (including bringing us Quanta Magazine), and has devoted much time/money to math/science education.
Just about a year ago, Numberphile did this great (and somewhat rare) 1-hour interview with the then 77-year-old Simons (try to find time for it, if you've never seen it):

Recreational Math Suggestions

I noticed today on Twitter someone asked for a listing of recreational math books suitable for students, so thought I'd post a quick list of some favorites I have on hand, but these aren't necessarily the best or most recent volumes. Especially if the individual is seeking interesting/fun problems I'm guessing either Mike Lawler or Tanya Khovanova (among others) may be able to provide a better list. Also, Quora-math did a great posting some while back with many fun puzzle/problem book suggestions:
http://tinyurl.com/z2dsvep

Still, here's my own list of some recreational-related volumes that I've enjoyed:

Classics from Martin Gardner:

The Colossal Book of Mathematics
Aha!
Gotcha 
(...and of course many others from Martin)

three from Ian Stewart:

Professor Stewart's Casebook of Mathematical Mysteries
Professor Stewart's Cabinet of Mathematical Curiosities
Professor Stewart's Hoard of Mathematical Treasures

Clifford Pickover:

A Passion For Mathematics
Wonders of Numbers

Alfred Posamentier:

Math Charmers
Mathematical Curiosities
Mathematical Amazements and Surprises

David Wells: Games and Mathematics 
Matt Parker: Things to Make and Do in the Fourth Dimension 

any of Raymond Smullyan's books

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Finally, I love paradoxes and believe they offer an excellent gateway for introducing young people to 'critical/mathematical thinking.' They aren't always very mathematical themselves, but rather, often nicely straddle boundaries between math, philosophy, and science. A few of my favorite volumes, suitable for young minds are:

Labyrinths of Reason by William Poundstone
Paradoxes in Mathematics by Stanley Farlow
Paradoxes from A to Z by Michael Clark
Paradoxes: Adventures In the Impossible by Gary Hayden and Michael Picard
This Sentence Is False by Peter Cave
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Hope this listing helps someone out, and of course feel free to add your own faves in the comments below.