This month's topics:

$360^{\circ}$ of hyperbolic reality in the New York Times

Every day since November 1, 2016 the New York Times has posted a short $360^{\circ}$ video on The Daily 360. As they put it, "To understand the world see it from every angle." On August 27, 2017 they brought us more angle than we might have thought possible, with "Bending the Rules of Geometry," a you-are-there tour of two non-Euclidean 3-dimensional manifolds (the hyperbolic plane $\times$ a line, and 3D hyperbolic space) where, as they show us, a regular hexagon can have six right angles.

still from movie
A detail of a still from "Bending the Rules." This space (hyperbolic horizontally and Euclidean vertically) is made up of truncated cubes, with adjacent ones differently colored. Looking up or down, in the Euclidean direction, gives the standard perspective of a chain of cubes. But looking sideways, in the hyperbolic directions, the cubes seem to get narrower very rapidly. Full image. Image courtesy of Henry Segerman.

The movie is credited to Vi Hart, Henry Segerman, Elisabetta Matsumoto, M Eifler, Andrea Hawksley and Samantha Quick, and narrated by Vi. Among other adventures the camera takes us on a guided tour around one of those hexagons. Vi: "Let's start in this olive-green cell and move around one of the columns. First we move straight into a neighboring teal cell. At the center we'll turn left $90^{\circ}$ and go straight into the sky-blue cell. From the center of this cell we'll make another $90^{\circ}$ left turn to move into the light peach cell. From there $90^{\circ}$ brings us not to the original olive-green cell but to this coral-orange cell. Another $90^{\circ}$ left and we're heading into the deep red cell. And finally our last turn will bring us back to the olive-green cell we started in. The path we took around that column was a regular hexagon with six right angles. That's the kind of thing that can happen when your space is curved." Along the tour we are free to look up or down, sideways or back, just as we choose, as if we were tourists on a double-decker bus in non-Euclidean space.

New trove of Turing correspondence

As The Guardian's North of England correspondent John Halliday reported on August 27, 2017, Collection of letters by codebreaker Alan Turing found in filing cabinet. "The correspondence, dating from early 1949 to Turing's death in 1954, was found by chance when an academic cleared out an old filing cabinet in a storeroom at the University of Manchester. Turing was deputy director of the university's computing laboratory from 1948, after his heroic wartime codebreaking at Bletchley Park." The correspondence is almost entirely routine exchanges of academic/administrative information: requests for reprints, invitations to conferences, requests for Turing to review manuscripts, evaluation of research students, etc. Halliday quotes James Peters, archivist at the University: "There is very little in the way of personal correspondence, and no letters from Turing family members. But this still gives us an extremely interesting account and insight into his working practices and academic life whilst he was at the University of Manchester." The letters, itemized and categorized but not reproduced here, make it clear that two weeks before his death he was actively engaged in his profession.

One item picked up by the Guardian and elsewhere in the international press was a note in which he turned down without further explanation an invitation to speak at a conference in the US: "I would not like the journey, and I detest America."

Women mathematicians on "Science Friday"

The November 10, 2017 episode of "Science Friday," broadcast on PRI, was The Infinitely Surprising Career Of A Mathematician. Ira Flatow, the host, starts it off: " ... we know scientists study weird things ... But can you imagine studying something that's not even a real physical thing, something that you can't see or hear or feel? And imagine trying to do that while people tell you that you can't do it because you're a woman. Joining me now to talk about all of these topics are three mathematicians." He introduces Eugenia Cheng (Art Institute of Chicago), Rebecca Goldin (George Mason University), and Emily Riehl (Johns Hopkins), asks them about the math they do.

Flatow brings up the woman question: " ... you must have faced the problem of being one woman mathematician in a field of many men mathematicians."


Voevodsky obituary in Nature

Vladimir Voevodsky, who died September 30, is one of the few mathematicians to have an obituary in Nature (November 6, 2017). This one was written by Daniel Grayson (Illinois), a long-time friend and collaborator. "Vladimir Voevodsky revolutionized algebraic geometry and is best known for developing the new field of 'motivic homotopy theory'. His contributions to computer formalization of proofs and the foundations of mathematics also made an immense impact." Grayson explains how Voevodsky was initially inspired by Alexandre Grothendieck's "dream .. to produce, for any system of polynomial equations, the essential nugget that would remain after everything apart from the shared topological flavour of the system was washed away. Perhaps borrowing the French musical term for a recurring theme, Grothendieck dubbed this the motif of the system." But how he diverged from Grothendieck's vision: "In Voevodsky's motivic homotopy theory, familiar classical geometry was replaced by homotopy theory -- a branch of topology in which a line may shrink all the way down to a point. He abandoned the idea that maps between geometric objects could be defined locally and then glued together, a concept that Grothendieck considered to be fundamental. A colleague commented that if mathematics were music, then Voevodsky would be a musician who invented his own key to play in."

Grayson tells us how Voevodsky had been working since 2002 on the computer representation of mathematical proofs. "Like others before him, Voevodsky dreamed of a global repository of mathematical statements and proofs. This would help mathematicians to accomplish, verify and share their work. ... A mechanism called univalence would allow mathematicians to use each other's work even if they had different approaches to the same underlying concepts." Using type theory (as opposed to set theory) as the formal language for the repository, he "succeeded in developing a library of thousands of pieces of code for his basic definitions and theorems. He called this repository Foundations." Later, "Univalent Foundations," which "provides the basis for a global mathematics repository and offers the first potentially viable alternative to set theory as a foundation for all of mathematics."

"Motivic homotopy theory is blossoming, despite Voevodsky's change of focus about ten years ago. Similarly, Univalent Foundations is destined to remain a vibrant area of research. Formalizing Voevodsky's work on motives in the Univalent Foundations would close the circle in a fitting way and fulfil one of his dreams."

Other obituaries, with different details, appeared in the New York Times (October 6, 2017), the Washington Post (October 7, 2017) and on John Baez's website. In particular, Martin Weil wrote in the Post: "As a pure mathematician, Mr. Voevodsky possessed powers of imagination, visualization and reasoning that could be applied at levels of abstraction almost impossibly remote from the minds and lives of most people. Aware of this, he was also troubled by it. 'I cannot explain--even to a very good student in his final year at university--the details of my work!' he once said in an interview."

Tony Phillips
Stony Brook University
tony at