ABC... A Baez Commentary

ICYMI, the more hardcore among you may want to see John Baez's recent commentary (and the comments that follow) on Mochizuki's "proof" of the ABC conjecture:
https://plus.google.com/+johncbaez999/posts/P7AN48F9pC7

Mathematician Go Yamashita has written a 294-page "summary" of Mochizuki's 500-page inscrutable(?) proof... if that's any encouragement to you ;)

Here's a few lines of the summary as quoted by Baez:

"By combining a relative anabelian result (relative Grothendieck Conjecture over sub-p-adic felds (Theorem B.1)) and "hidden endomorphism" diagram (EllCusp) (resp. "hidden endomorphism" diagram (BelyiCusp)), we show absolute anabelian results: the elliptic cuspidalisation (Theorem 3.7) (resp. Belyi cuspidalisation (Theorem 3.8)). By using Belyi cuspidalisations, we obtain an absolute mono-anabelian reconstruction of the NF-portion of the base field and the function field (resp. the base field) of hyperbolic curves of strictly Belyi type over sub-p-adic fields (Theorem 3.17) (resp. over mixed characteristic local fields (Corollary 3.19))."

...Have at it!

Making the Incomprehensible a Little More Comprehensible

Wow! Seems like everyone has been writing for awhile now about how incomprehensible Shinichi Mochizuki's "proof" of the ABC conjecture is... leave it to Mathbabe to find someone, Brian Conrad, willing to take a stab at making it a little MORE comprehensible! Long, informative (but still technical) post (certainly the best effort I've seen to address the topic... IF you can set some time aside):

http://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/

A Big Serving of Monday Potpourri

Catching up on a few things:

1) First, Keith Devlin continues his fascinating series on his MOOC experience (now 6 recent posts to catch up on in his latest series, if you haven't been following along):

http://mooctalk.org/

2) And if you missed Keith delightfully talking on NPR this weekend about the excitement generated by Yitang Zhang's attack on the Twin-prime conjecture earlier this year, give that a listen here:

http://www.npr.org/2013/12/21/256003488/and-the-number-of-the-year-is-the-lowly-2?ft=1&f=1007

3) Meanwhile, Peter Woit writes about the incredible difficulty involved in verifying last year's proof from Shinichi Mochizuki of the ABC conjecture (perhaps unparalleled in the history of math):

http://www.math.columbia.edu/~woit/wordpress/?p=6514

4) And if anyone has missed it, my own latest interview with popular mathematician James Tanton is currently up at MathTango:

http://mathtango.blogspot.com/2013/12/james-tanton-making-math-accessible.html

5) Perhaps mathematicians are too shy or uncomfortable talking about their (or their colleagues') politics, but I'm still interested to hear from you, if you aren't:

http://math-frolic.blogspot.com/2013/12/are-mathematicians-liberals.html

6) And three for your mathematical entertainment:

a) Hat tip to The Aperiodical for pointing out this fun 30-min. BBC podcast on math and magic:

http://www.bbc.co.uk/programmes/b03ls7y2

b) Just this morning, Presh Talwalkar posted a nice triangle geometry problem with both a traditional algebraic solution and a 'quickie' simpler solution available:

http://tinyurl.com/mtovhpx

c) And finally, a link that got some play on Twitter last week is this old "urban legends" of math discussion from mathoverflow.com:

http://tinyurl.com/l6gapx6
(some interesting, fun, and quite technical 'urban legends' included...)

Mathematicians Having Fun

A couple of disparate items, each of which I thought were quite instructional in their own ways…:

1) One of the lovely things about mathematics is that simple-sounding problems/puzzles may nonetheless lead to sharp disagreements over the correct answer (the Monty Hall puzzle being a classic example of this), but with little more than pencil and pad, the disagree-ers can talk through the differences, reaching agreement and an AHA! moment… no ill feelings had, just the enjoyment of learning and thinking! (…now if only cosmology was that simple!).
Anyway, if you follow mathy things on Twitter you may have seen the recent back-and-forth-and-back-again dispute over a recent Paul Krugman blog post in the NY Times in which Krugman nonchalantly gave the answer to a conundrum dealing with traffic flow.

http://krugman.blogs.nytimes.com/2013/07/26/friday-night-music-sprawl-again/?_r=0

A certain mathematician of some repute, who we shan't name, but will simply call Stefanovich Stroganoff ;-) took issue with Krugman's answer on Twitter (and was quickly warned by some that questioning Krugman on math matters was playing with fire -- there were also many replies/comments directly to the post itself in the Times ). What followed were many more tweets debating the proper approach to the problem (fascinating that a mathematical debate can even be carried out in 140 characters or less!).
Anyway, in the end (given certain assumptions) Krugman's answer was vindicated after-all (the last tweet I recall from Stefanovich was hashtagged #EggOnFace), and Krugman himself responded with another short NY Times piece about 48 hrs. later:

http://krugman.blogs.nytimes.com/2013/07/28/life-in-the-slow-lane-trivial/

A good time was had by all, I do believe… and now further, a Cornell engineering student has put up a longer piece (pdf) recounting the whole affair with fuller explanation of the mathematics involved:

http://ruina.tam.cornell.edu/research/topics/miscellaneous/KrugmanTraffic.pdf

2) Meanwhile, and a bit heavier than the above, over at Huffington Post, of all places, there is a very good (...as far as I can tell) elucidation of the ABC conjecture for anyone wishing to read more on that subject that was brought into the limelight about a year ago by Shinichi Mochizuki:

http://www.huffingtonpost.com/shunsuke-katayama/abc-conjecture_b_3352657.html

"the dark unknown of mathematics"

 Is Shinichi Mochizuki "travelling alone"...?:

Hat tip to The Aperiodical for pointing to this fabulous, longread on Shinichi Mochizuki’s claimed 2012 proof of the ABC conjecture:

http://projectwordsworth.com/the-paradox-of-the-proof/

It's not a technical piece, but an overview of the issues/controversy generated by the proof (including its incomprehensibility!), and the need to resolve its accuracy.
I'd pick out several choice quotes, but it's all so good I just recommend setting aside some time to read the whole article, which does end as follows:

" [Mochizuki] may have found the key that would redefine number theory as we know it. He has, perhaps, charted a new path into the dark unknown of mathematics. But for now, his footsteps are untraceable. Wherever he is going, he seems to be travelling alone."

Mochizuki, Gowers, Alan November

First, a wonderful article from The Boston Globe emphasizing how difficult it will be to check Shinichi Mochizuki's lengthy claimed proof of the ABC conjecture, because of the complexity and newness of the math involved:

http://tinyurl.com/a8srze4

The piece ends thusly:

"And this, [Minhyong] Kim thinks, might pose the greatest challenge of all. 'When you’ve been wrapped up in your own research program for a long time sometimes you lose a sense of what it is that other people don’t understand,' he says. 'Other people feel quite mystified as to what he’s [Mochizuki] doing and part of him, I suspect, doesn’t quite understand why.'"

On a simpler note, do you wish to engage your own math students… well, so does Tim Gowers...
British Field Medalist Gowers (who deserves to be read/discussed whenever possible) has a piece in The Spectator regarding math education:

http://www.spectator.co.uk/features/8744071/should-alice-marry-bob/

from it: "...rather than explaining mathematical ideas (about statistics, say) and then discussing how they can be applied to the real world, a teacher should instead start with a question that is interesting for non-mathematical reasons and keep a completely open mind about what mathematics has to contribute to the discussion."

Gowers' take is that teachers need to be utilizing real 'real-life' examples in the classroom, not the 'if 2 painters can paint 2 houses in 5 days how many houses can 4 painters paint in 15 days' sort of story problem that often gets passed off as an application.

This is more-or-less a followup to a fantastic earlier piece he did on same subject:

http://gowers.wordpress.com/2012/06/08/how-should-mathematics-be-taught-to-non-mathematicians/

[On a side note, just yesterday, Gowers put up another interesting post about probabilities and surgery he is undertaking for an atrial fibrillation condition… hopefully, we'll soon hear that all went well!]:
http://gowers.wordpress.com/2012/11/05/mathematics-meets-real-life/

Finally, hat tip to John Golden for leading me to this TED video (~16 mins.) by Alan November that piggy-backs nicely onto Tim Gowers' views, in describing student engagement in learning:

Huge News... That Few Can Fathom

Shinichi Mochizuki's claim of proving the 'abc conjecture' is receiving widespread notice in the scientific press, although it may take years to confirm the highly-complex, 500-page-long finding, which according to some is currently only comprehensible to Mochizuki himself (who invented new math in the process of reaching his results)!
The conjecture is viewed as being central to number theory and the underlying relationships of prime numbers; described by some "as a sort of grand unified theory of whole numbers."
For those who wish to follow along a bit, a few more of the many Web entries that have been reporting on the claim:

http://michaelnielsen.org/polymath1/index.php?title=ABC_conjecture

http://www.scientificamerican.com/article.cfm?id=proof-claimed-for-deep-connection-between-prime-numbers

http://bit-player.org/2012/the-abc-game

I suspect Amir Aczel ;-) or somebody may already be at work writing a book for the rest of us to understand!

Will Another Conjecture Bite-the-dust?

Peter Woit reports the claim that the "abc conjecture" has been proven:

http://www.math.columbia.edu/~woit/wordpress/?p=5104

The abc conjecture is a number theory conjecture from 1985 which has been called "the most important unsolved problem in Diophantine analysis."

Peter links to this more detailed report and discussion of the claim:

http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/

and here's a more layman-friendly interpretation of it from a few years back:

http://bit-player.org/2007/easy-as-abc