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Tony PhillipsTony Phillips' Take on Math in the Media
A monthly survey of math news

This month's topics:

Logic and Lewis Carroll

That was the title of a "Books & Arts" Comment piece in Nature, November 19, 2015: "As Alice's Adventures in Wonderland reaches 150, Francine Abeles surveys its creator's wide-ranging legacy." As Abeles tells us, besides Alice and Through the Looking-glass, Charles Dodgson (Lewis Carroll was his pen name) "produced many pamphlets and ten books on mathematical topics." Many of them involved puzzles, or shortcuts to computation like an 1897 piece in Nature itself, "Brief Method of Dividing a Given Number by 9 or 11" (Nature 56 565-566). "Carroll did not influence his contemporary colleagues in the devopment of mathematical ideas. However, posthumously ..." it was recognized that he had anticipated several significant developments.

A nice example of one of his "knots," his name for mathematical puzzles, is "Knot II, Mad Mathesis: 'I waited for the train'" on page 76 of the 1880 Monthly Packet (of Evening Readings for Members of the English Church).

Brian Greene in The New Yorker

The "Talk of the Town" on November 30, 2015 had a "Brave New World" department with a piece by Rebecca Mead entitled "Mathphilic." The occasion was a performance of Brian Greene's multimedia work Light Falls at the opening of a two-day Institute for Advanced Study conference celebrating the 100th anniversary of Einstein's completion of the theory of General Relativity. "Greene ... narrates, telling the story of Einstein's discovery in language accessible even to audience members without a Ph.D." Mead spoke backstage with Greene about his personal history with physics and mathematics, and quotes him: "Math is hard for just about everybody. It is not what our brains evolved to do. ... You didn't have to take exponentials or use imaginary numbers in order to avoid that lion or that tiger or to catch that bison for dinner. So the brain is not wired, literally, to do the kinds of things we now want it to do." He goes on to give us a picture of how a physicist uses math. "He starts with a blank page, and he thinks of some kind of mathematical relation that may be able to describe the physics that he's thinking about, and then he starts with that equation. He thinks, O.K., with that as a starting point, let me manipulate it. And then you come to the act of inspiration, where, after manipulating the equations, you say, Wow, look at that--the pattern right there in the equations aligns with the pattern of gravity, or that pattern of motion."

Math in The New York Times

A busy week:

Update on "abc"

News of the abc conjecture was covered in this column two months ago. Meanwhile a top-tier conclave assembled in Oxford, December 7-11, to try and bring some light on the problem. "Biggest mystery in mathematics in limbo after cryptic meeting" was Davide Castelvecchi's report on the proceedings for Nature, posted on December 16, 2015. "A collective effort to scrutinize one of the biggest mysteries in mathematics has ended with a few clues but no firm answers." What is the mystery? Shinichi Mochizuki, a very well respected number theorist, posted a series of 4 papers in August, 2012 which purport to prove the "abc conjecture." A correct proof would have enormous repercussions in the field. The mystery is, how does Mochizuki's proof work? So far, nobody understands it well enough to communicate it, even to the most battle-tested experts. As Castelvecchi explains, "Mochizuki's papers, which totalled more than 500 pages, were exceedingly abstract and cryptic even by the standards of pure mathematics. That has made it tough for others to read the proof, let alone verify it. Moreover, the papers built on an equally massive body of work that he had accumulated over the years, but that few were familiar with." One source of frustration, as Mochizuki himself explained (via Skype), is that "over many years he had developed a host of tools that he thought would be useful to prove abc--but that in the end he realized he did not need all of them" and unfortunately no one in the community knows yet which are the right ones to concentrate on.

Castelvecchi reports one ray of hope: "A consensus emerged that the highlight of the workshop was a lecture on 9 December by Kiran Kedlaya, an arithmetic geometer from the University of California, San Diego. He zeroed in on a result from a 2008 paper by Mochizuki that linked the statement of the abc conjecture to another branch of maths called topology. The link was immediately recognised as a crucial step in Mochizuki's grand strategy." But "Even Kedlaya agrees that the insight needs to be followed up by many others, and by an understanding of the strategy that links those key passages to one another."

A comprehensive survey of the Oxford meeting, by Brian Conrad, is posted on the Mathbabe website.

Tony Phillips
Stony Brook University
tony at math.sunysb.edu