hi all, not feeling all that inspired to write anything in particular based on current zeitgeist [which doesnt seem to be popping anywhere in particular at the moment], but wanted to write something, so am dipping into the backlog (ie some retro zeitgeist instead).
around early may Zhang recently announced an amazing breakthrough on the twin prime conjecture that decreased the gap to a finite width for a large interval, 7 x 107. this was decreased to 5414 in a recent polymath project. very impressive! another huge feather in the cap of Team Polymath. (have they come up with a cool mascot yet?) 😀
the twin prime conjecture is very ancient, at least about two millenia old. the current wikipedia history on it is poorly written, but according to britannica, euclid conjectured it. its neat that there remains an unbroken line/thread between ancient greek thought and very modern theory/math.
mckee of Nature states, “It’s a result only a mathematician could love.” ouch! to me, the appeal is broader than that, but I must admit that number theory is fairly regarded by the masses as esoteric, it takes quite a bit of imagination to grasp the beauty that hardy wrote of:
The mathematician’s patterns, like the painter’s or the poet’s must be beautiful; the ideas like the colours or the words, must fit together in a harmonious way. Beauty is the first test: there is no permanent place in the world for ugly mathematics.
the proof is apparently not extremely difficult because the math community seemed to quickly verify it and endorse it. this is in striking contrast to the recent claims about the abc conjecture by Mochizuki. a nice writeup on that basic quandary [verging on imbroglio] appeared by Chen below.
this is all covered very well in many refs, and its really neat to see such great media coverage (dont have a huge amt to add, but did have at least one different angle). true breakthroughs are rare. the primes seem to be yielding a few secrets. I am reminded of the AKS primality testing breakthrough. lets not forget the 1-½ century old Riemann conjecture is worth $1M. have always wanted to play with that one too.
there are two lines of thought with these very difficult problems. one is that you have to learn everything you can about the problem to make any kind of serious contribution. and it can be fun and appealing to survey known ideas. but another contrarian view might be that all known methods have been tried and failed, and therefore one has a chance at attacking them and finding new angles/properties using novel ideas from new theory as long as there is some kind of unusual novelty involved (its a little bit like the Big Orthodox Corporation vs Tiny Scrappy Startup dichotomy).
ok so heres my unique angle/take on the Zhang breakthrough to offer, more on the behind-the-scenes significance. what struck me in the great wired article below is the emphasis on how “unknown” Zhang is to the math community, where there is a relatively small group of people who work on the problem and most know each other, and there was some real shock that someone who hadnt published anything prior on the subject was the source of a huge breakthrough. below was this passage:
Rumors swept through the mathematics community that a great advance had been made by a researcher no one seemed to know — someone whose talents had been so overlooked after he earned his doctorate in 1991 that he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop.
in some ways this is both inspiring and depressing. (as the japanese say, the reverse side also has a reverse side.) so Zhang is certainly not an amateur, and given his Phd in math, even calling him an “outsider” is a stretch. but this is not an uncommon story as far as accomplished Phds not fitting into academia. I am sure there are many very talented Phds who cant get jobs in academia, there seem to be many anecdotes circulating on this and even many solid studies.
of course many other media accounts of Zhang omit some of this fascinating story which I personally find quite riveting and relevant, in the kind of typical airbrushing that scientific discoveries/breakthroughs undergo. it would be interesting to hear more of the details. did he really find it hard to get a job in academia? or could he have been burned out? maybe he hadnt published much? or was there real visibility of promise that was not recognized? which makes one question the publication system [so called publish-or-perish] as academia’s main measurement of worth/viability/efficacy/promise/potential etc.
on the other hand we should also try to avoid looking at everything through the [“discriminatory”] academic lens/pov/perspective/bias/bubble, by which Zhangs employment path seems like sort of a failure. working in academia is obviously not the only outcome of getting a Phd but its at times the only conceivable option in the academic culture. were his ambitions frustrated/thwarted or did he not really want to engage with academia yet work on an academic subject? which reminds me of the so-called “eccentric” Perelman.
I have struggled with this quandary somewhat myself, deciding not to go into academia/theory, one of my early loves/passions, because of the apparently unimpressive pay (and sometimes, prestige) in general, and work on theory more as a hobby.
recently Suresh was blogging on this with some excellent refs about the joyless grind of academic life (reputedly sometimes verging on soul-crushing), and I snarfed a few choice links below, and am including some others. I didnt really go searching for these links, just kind of ran into them as maybe a current (anti-?)zeitgeist (disspirits of the times?). its both reality and yeah, at times depressing (just reading the titles of those articles below would make grown men flinch—eg “why are so many college graduates driving taxis” from Bloomberg, and not the Onion!). there were stories like this when I went through college two decades ago, alas it looks to me like the problem may have gotten worse in my lifetime. but as they say in self help groups, “admitting theres a problem is the 1st step to a solution”.
anyway sincere congratulations to Zhang and hope there are more breakthroughs to come from him, aka rabbits out of the magicians hat. hey Zhang why dont you start a blog! cyberspace can be a new kind of support group/network outside of academia (or industry). seriously! its gradually evolving and being woven together.
footnote: recently I tried to play up the deep/cool 😎 connection between number theory and undecidability (and therefore TCS theory) on that really cool/popular/recent Reyzin question on TCS stackexchange, and I was not alone, in eg the twin prime conjecture, odd perfect number conjecture [as old as the greeks also!], goldbach conjecture, and collatz conjecture, but the response was cricket-chirping silence [like in rock band when you miss the Big Rock Ending notes]—or controversy, and there was some strong pushback. (Collatz got ~17 upvotes as a comment but only ~1 as an answer. ?!?) to me the link/bridge is quite unmistakable and impressive in eg Hilberts 10th problem (the still-open rational variant coincidentally a very high-voted answer on the Reyzin post), but the TCS crowd can be very ivory tower insular narrowminded selective at times.
guess sometimes ya gotta just be satisfied with a footnote or the “fine print”. guess we gotta come up with a few more breakthroughs before anyone takes it really seriously. hey, how about a breakthrough result that uses TCS theory to vanquish some difficult number theory problem! hey, Im working on it! (sometimes feeling like its as singlehanded, outfield, and longshot as Zhang, except of course without the breakthough yet.) 😛
a. twinprime breakthru
- 1. Unknown Mathematician Proves Elusive Property of Prime Numbers | Wired Science | Wired.com
- 2. Solving a Riddle of Primes – NYTimes.com
- 3. First proof that infinitely many prime numbers come in pairs : Nature News & Comment
- 4. Philosophy behind Yitang Zhang’s work on the Twin Primes Conjecture – MathOverflow
- 5. Yitang Zhang, twin primes conjecture: A huge discovery about prime numbers—and what it means for the future of math. – Slate Magazine
- 6. Polymath proposal: bounded gaps between primes | The polymath blog
- 7. Online reading seminar for Zhang’s “bounded gaps between primes” | What’s new
- 8. Yitang Zhang Proves ‘Landmark’ Theorem in Distribution of Prime Numbers | Simons Foundation
- 9. Polymath 8 – a Success! | Combinatorics and more
- 10. Twin prime – Wikipedia, the free encyclopedia
- 11. Twin Primes Conjecture: ‘Weak’ Version Of Famed Math Problem Possibly Proven
- 12. twin prime conjecture (number theory) — Encyclopedia Britannica
- 13. The Paradox of the Proof | Project Wordsworth
- 14. computability – A simple problem whose decidability is not known – Theoretical Computer Science Stack Exchange
- 15. The Twin Prime Hero Nautilus ⭐ (Zhang interview)
- 16. twin prime conjecture, britannica
- 17. Sudden Progress on Prime Number Problem Has Mathematicians Buzzing Wired/Klarreich
- 18. Twin primes are useful RJLipton blog
- 19. Game of proofs boost prime pair result by millions New Scientist
- 20. New equidistribution estimates of Zhang type, and bounded gaps between primes “DHJ Polymath” (Polymath8)
b. edu/academia grind
- 1. The Explosive Growth of Postdocs in Computer Science | February 2013 | Communications of the ACM
- 2. Academia’s Indentured Servants | Alternet
- 3. Why Are Associate Professors Some of the Unhappiest People in Academe? – Faculty – The Chronicle of Higher Education
- 4. I’ve Got Tenure. How Depressing. – Manage Your Career – The Chronicle of Higher Education
- 5. Why Are So Many College Graduates Driving Taxis? – Bloomberg
- 6. 10 truths a PhD supervisor will never tell you | Features | Times Higher Education