- $360^{\circ}$ of hyperbolic reality in the
*New York Times* - New trove of Turing correspondence
- Women mathematicians on "Science Friday"
- Voevodsky obituary in
*Nature*

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.

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.

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."

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.

- "Rebecca, ... from a mathematician's point of view, a donut and a coffee cup are the same thing. How is that?" Goldin: "A lot of times, mathematicians like to study the way things can be equivalent if you mush them around or deform them. So if you take a coffee cup, you can imagine that it's got exactly one hole in it, and that's where you put your hand to grab onto your coffee. And if you mush that around, you can kind of morph it into a donut. So people in the field of topology would call those things equivalent."
- "Eugenia, you and Emily are in the same math field. You're both category theorists. What does that mean? Do you have an example to help us envision what that means?" Cheng: "I like to call category theory the mathematics of mathematics. And to understand that, first I have to explain what mathematics is. ... I think math is about understanding how things work very deeply. And we make analogies between different situations and say, what can we understand about these two cookies, for example, that is also going to be true about these two apples or these two people, these two other things? And that analogy turns into the number two. And category theory actually does that for mathematics itself. So it says, what about this branch of math can we understand that is similar to this other branch of math that's also similar to this other branch?"
- "I'm going to ask Emily Riehl to tell us about [infinite dimensional category theory]." Riehl: "And a way to think about what that does is a category provides a template for a mathematical theory. So there are the mathematical objects, which are like the nouns of the theory, and then there are the transformations between those objects, which are the verbs. But as mathematics becomes more complicated, ... we might need pronouns, adjectives, adverbs, prepositions, conjunctions, interjections, and so on. And so that's where the infinite dimensions come in. We have not only the objects and the transformations, but there might be transformations between transformations ..."
- Cheng: "... just like we can consider relationships between people but we can also compare relationships and say, well, whose relationship is better? But then we can say, well, maybe this relationship is better because they have more fun. But this other relationship is better because they don't argue very much. And then we could compare the ways of comparing and say, well, which is better? Is it better to have more fun or is it better to avoid arguing? And this is another way that we get many dimensions."

- Goldin: "I think, at least speaking for myself, being female within mathematics has definitely played a role for how it's come out, whether that's good or bad. If somebody identifies you as a female in the field, of course you don't want that to be the only thing about you that is viable. ... But I don't think it's irrelevant, either. If it were irrelevant, we wouldn't have such a shortage of women in mathematics ... it's a very real issue that has social consequences."
- Cheng: "When I was younger and I was just starting out, I really also didn't want to draw attention to the fact that I was a woman. ... And then as I became more advanced and I became more secure and I got a permanent job and I was respected in the community, I became more active about it. If I complain that the images of [people] in mathematics as presented by the media are largely older white men, then I need to put myself forward to help dispel those images."
- Riehl: "At this point in my career, I am mostly attentive to how under-representation along gender axes --but other axes as well-- affects my students. So when I teach a large multivariable calculus course to 350 incoming students at Johns Hopkins, I start with a comic, an XKCD comic by Randall Munroe entitled "How it works." And I use this comic to explain how it's harder for students who feel like they stand out in the sea of faces in the classroom to ask a question and then to really engage with the mathematics in a way that I think makes it easiest to learn."
- Goldin: "I think there's another feature or bias or division that has to do with young versus old people. So there's this idea in mathematics that you peak when you're young .... So if you're not showing some kind of brilliant sparky shiny something by the time you're 14, you're just not all that good. And no one is going to invest in you. And I think that is something that's really quite pernicious because people often come to mathematics a little bit later ... it's not until they get quite far into the game mathematically that they see something really beautiful, something beyond calculus, something that really kind of gets them to fall in love. ... You've fallen in love and now you're, say, 21. You're basically an old maid. I mean, thats a really incorrect assumption about how mathematics is learned, how it grows on people, and how they become mathematicians."

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 math.sunysb.edu