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/

"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."

Four For Tuesday

A few of the things that crossed my screen yesterday…:

1) Another interesting article from the Simons Foundation, this time on the nature and fragility of networks, or as they say, "extreme fragility of interdependency" in the modern world. Be it urban infrastructure, the stock market, even the human body, or other networked systems the risks of small failures leading to chaotic cascading effects seems ever-present, as scientists try to figure out ways to evade catastrophes. From the article:

"While scientists remain cautious about using the results of simplified mathematical models to reengineer real-world systems, some recommendations are beginning to emerge. Based on data-driven refinements, new models suggest interconnected networks should have backups, mechanisms for severing their connections in times of crisis, and stricter regulations to forestall widespread failure."

Read further here: http://tinyurl.com/bom9eym

2) Hat tip to The Aperiodical for bringing this to my attention… a 40-page Shinichi Mochizuki paper intended to assist those trying to understand his "proof" of the ABC conjecture. I certainly don't grasp the paper (although there are a few words I can understand: "the," "of," "with," "for"… ;-) and 99% of folks won't comprehend it, but what's absolutely amazing is that there exists human brain wiring that IS capable of such production/comprehension!:

http://tinyurl.com/d6pje5l

3) And a hat tip to Patrick Honner for pointing me to this NY Times piece about "online proctoring" for MOOC course testing, a topic that is sure to draw more discussion as time goes on:

http://tinyurl.com/cstz6xa

4) Finally, even if you're tired of the question, "Is mathematics discovered or invented?" you may find the below New Scientist article interesting… it moves that question into the realm of business, software, and the debate over patentability, before concluding: "The rallying cry of a good many critics remains 'software is mathematics', meaning that software shouldn't be patentable. The odds are stacked against them, though – there's too much money at stake":

http://www.newscientist.com/article/mg21729086.300-should-business-be-allowed-to-patent-mathematics.html

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!